ANALYTIC HYPOELLIPTICITY IN DIMENSION TWO

MICHAEL CHRIST

Abstract. A simple geometric condition is sufficient for analytic hypoellipticity of sums of squares of two vector fields in ℝ^{2}. This condition is proved to be necessary for generic vector fields and for various special cases, and to be both necessary and sufficient for a closely related family of operators.

Contents

1. Introduction 1 2. Coordinates and canonical forms 7 3. Analytic preliminaries 9 4. Necessity: Reduction to eigenvalue problems 14 5. Nonlinear eigenvalue problems 19 6. Analysis of special cases 24 7. Sufficiency 32 8. Microlocal analytic hypoellipticity for factored operators 37 9. Proof of Lemma 4.2 41 10. Comments 44 References 46

1. Introduction

Consider a linear partial differential operator L, representable as a sum of squares ∑_{j} X_{j}^{2} of finitely many real vector fields with real analytic coefficients. Under what conditions is L analytic hypoelliptic? Many examples and some partial results are known, but the techniques presently available fall far short of resolving the question, and no plausible candidate for a necessary and sufficient condition has been put forward. In this paper a nearly complete treatment will be given of the simplest possible case, that of two vector fields in ℝ^{2}. The main results are:

Date: February 4, 1996.

Research supported by NSF grant DMS-9306833 and at MSRI by NSF grant DMS-9022140.

• Formulation of a conjectured necessary and sufficient condition for analytic hypoellipticity.

• Proof of sufficiency of this condition.

• Complete reduction of the question of necessity to an eigenvalue problem for certain ordinary differential operators.

• Solution of generic eigenvalue problems, and hence proof of the conjecture for generic vector fields.

• Treatment of certain examples.

• Formulation and proof of a necessary and sufficient condition for microlocal analytic hypoellipticity of (X_{1} +iX_{2})∘(X_{1} −iX_{2}), under a natural pseudoconvexity hypothesis.

• Introduction of a new geometric invariant, a rational number q, associated to a

pair of vector fields in ℝ^{2}. One upshot of this analysis is that analytic hypoellipticity is very rare.

Definition. Let {X_{1}, X_{2}}, {Y_{1}, Y_{2}} be two pairs of real vector fields with real analytic coefficients, defined in a neighborhood of a point p. We say that

span{X_{1}, X_{2}} ≡ span{Y_{1}, Y_{2}}

in a neighborhood U of p if each X_{j} may be expressed as a linear combination, with analytic coefficients, of the Y_{i}, and conversely each Y_{i} may be so expressed in terms of the X_{j}, in U.

Let X_{1}, X_{2} be real vector fields with C^{ω} coefficients, defined in a neighborhood of p ∈ ℝ^{2}. Suppose that the Lie algebra generated by them spans the tangent space at p.

Conjecture 1.1. L is analytic hypoelliptic in some neighborhood of p if and only if there exists a system of coordinates (x, t) with origin at p, in which

span{X_{1}, X_{2}} ≡ span{∂_{x}, x^{m−1}∂_{t}}  (1.1)

in some neighborhood of the origin, for some m ≥ 1.

That (1.1) implies analytic hypoellipticity is very well known in the elliptic case m = 1 and symplectic case m = 2. If a point p_{0} is fixed, then within the class of all pairs of C^{ω} vector fields for which X_{1}, X_{2}, [X_{1}, X_{2}] are linearly dependent at p_{0} but which satisfy the bracket hypothesis at p_{0} to some fixed order, the condition of Conjecture 1.1 is violated generically. On the other hand, for any pair of vector fields satisfying the bracket hypothesis, the set of points p ∈ ℝ^{2} at which the condition of Conjecture 1.1 fails to hold is discrete.

Theorem 1.2. Suppose that for some m ≥ 1 there exist coordinates (x, t) in a neighborhood U of 0 in which span{X_{1}, X_{2}} ≡ span{∂_{x}, x^{m−1}∂_{t}}. Then X_{1}^{2}+X_{2}^{2} is analytic hypoelliptic in U.

This is a special case of a relatively elementary theorem of Grušin [12]. We include a proof in Section 7 in order to show how it follows from the same point of view, and from a subset of the machinery, that we will develop in demonstrating nonhypoellipticity in other cases. With this result in hand, our aim is to prove that analytic hypoellipticity fails in all nonelementary cases.

A few definitions are required before our main results can be properly formulated. The Lie algebra generated by X_{1}, X_{2} is always assumed to span the tangent space. By a normalized homogeneous polynomial of degree m − 1 we will mean a function Q(x, z) of the form

Q(x, z) = x^{m−1} + ∑_{j=0}^{m−3} c_{j}z^{m−1−j}x^{j},  (1.2)

for some m ≥ 1, where each c_{j} ∈ ℝ.^{1}

Definition. Two normalized homogeneous polynomials Q, P of degree m − 1 are equivalent if there exists a nonzero real constant c such that for all (x, t),

P (x, t) = Q(x, ct).

Define m = m(p) to be the type of the set {X_{1}, X_{2}} at p. This invariant is 1 if the vector fields are linearly independent at p, and otherwise is defined to be the smallest integer such that X_{1}, X_{2} together with all of their iterated Lie brackets having m or fewer factors span the tangent space to ℝ^{2} at p. Denote by ℚ^{+} the set of all positive rational numbers.

The following lemma will be proved in Section 2.

Lemma 1.3. Suppose that X_{1}, X_{2} are real vector fields with C^{ω} coefficients defined near p ∈ ℝ^{2}. Let m ≥ 1 be their type at p. Suppose that there do not exist coordinates (x, t) with origin at p such that span{X_{1}, X_{2}} ≡ span{∂_{x}, x^{m−1}∂_{t}} in a neighborhood of 0. Then there exist q ∈ ℚ^{+}, coordinates (x, t) with origin at p, and an analytic function Θ̃(x, t) of the form

Θ̃(x, t) = x^{m−1} + ∑_{j=0}^{m−3} β_{j}(t)x^{j}

1If m < 3 then the sum over j is taken to be empty.

such that near 0,

span{X_{1}, X_{2}} ≡ span{∂_{x}, Θ̃(x, t)∂_{t}}

with the following properties: Each β_{j} ∈ C^{ω} is real-valued, β_{j}(0) = 0, each β_{j}(t) = c_{j}t^{(m−j−1)q}(1 + O(|t|^{δ})) for some δ > 0, and at least one coefficient c_{j} is nonzero.

The quantity q is independent of the coordinate system used and choices made in its definition. The polynomial Q(x, z) = x^{m−1} + ∑_{j=0}^{m−3} c_{j}z^{m−j−1}x^{j} is likewise invariant, modulo the equivalence relation defined above.

Note that necessarily c_{j} = 0 whenever the exponent (m − j − 1)q is not an integer.

To a pair (L, p), where L = X_{1}^{2} + X_{2}^{2} and p ∈ ℝ^{2}, we associate a family of ordinary differential operators

L_{z} = −∂_{x}^{2} + Q^{2}(x, z), (x, z) ∈ ℝ × ℂ

depending polynomially on the complex parameter z, where Q is a member of the equivalence class of normalized homogeneous polynomials associated to L via Lemma 1.3.

Definition.

NE(L, p) = {z ∈ ℂ : there exists 0 ≠ f ∈ S(ℝ) satisfying L_{z}f ≡ 0}. (1.3)

This definition does not quite make sense, because associated to L is not a single family of ordinary differential operators L_{z}, but rather an equivalence class. Therefore we define two subsets S, S′ of ℂ to be equivalent if there exists a nonzero real number γ such that S′ = γS = {γζ : ζ ∈ S}. NE(L, p) is then defined to be the equivalence class of the set described in (1.3).

When we are discussing a family of ordinary differential operators L_{z} of the form (1.3), not necessarily associated to any partial differential operator, we denote the same set of nonlinear eigenvalues by E(Q).

It is a general principle that, at least for certain classes of partial differential operators, the question of analytic hypoellipticity is linked to global eigenvalue problems. In the present context this link may be formulated rigorously, as follows.

Theorem 1.4. Suppose that L is a sum of squares of two real, C^{ω} vector fields in ℝ^{2} satisfying the bracket hypothesis. If NE(L, p) is nonempty, then L is not analytic hypoelliptic in any neighborhood of p.

Conjecture 1.5. If Q is a normalized homogeneous polynomial of degree m−1 ≥ 1, then E(Q) = ∅ if and only if Q(x, z) ≡ x^{m−1}.

Three pieces of evidence will be offered to support this conjecture. Firstly, in Section 5 the analogous assertion for a family of ordinary differential operators that is formally very similar to L_{z} will be formulated and proved. Secondly, representative special cases, as listed in Proposition 1.7, will be treated. Thirdly, the conjecture will be proved to be valid for generic Q.

The following notion of genericity is natural in this discussion. Recall that a subset E of ℂ^{d} is said to be pluripolar if there exists a nonconstant plurisubharmonic function h : ℂ^{d} ↦ [−∞, ∞], satisfying h ≡ −∞ on E. A subset of ℝ^{d} is said to be pluripolar if it is pluripolar as a subset of ℂ^{d}. We identify the set of all normalized homogeneous polynomials Q of degree m − 1 with ℝ^{m−2} ⊂ ℂ^{m−2} via the correspondence ζ ↦ Q_{ζ}(x, z) = x^{m−1} + ∑_{j=0}^{m−3} ζ_{j+1}z^{m−1−j}x^{j}. In Section 5 it will be shown that Conjecture 1.5 is valid for generic Q, in this sense. This and Theorem 1.4 together have the following consequence.

Theorem 1.6. For each m ≥ 3, there exists a pluripolar set B_{m} ⊂ ℝ^{m−2} such that whenever the pair {X_{1}, X_{2}} is of type m at p and some normalized homogeneous polynomial Q associated to L belongs to ℝ^{m−2}\B_{m}, L fails to be analytic hypoelliptic in any neighborhood of p.

The following is a list of examples for which it is relatively easy to show that E(Q) is indeed nonempty.

Proposition 1.7. Let Q(x, z) = x^{m−1} − z^{k}x^{m−k−1}. If at least one of the following conditions holds, then E(Q) ≠ ∅, and consequently for any positive integer r and for either choice of the ± sign,

L = ∂_{x}^{2} + ([x^{m−1} ± t^{r}x^{m−k−1}]∂_{t})^{2}

is not analytic hypoelliptic in any neighborhood of 0 :

• m/k is not an integer

• m is odd

• k is even

• m is divisible by 4.

We turn now to the analysis of the closely related operators

L = (X_{1} + iX_{2})(X_{1} − iX_{2}) + c_{1}X_{1} + c_{2}X_{2} + c_{3}

where the c_{j} are C^{ω}, complex valued coefficients; the lower order terms c_{1}X_{1} +c_{2}X_{2} + c_{3} play no essential role. These are lower-dimensional analogues of the operators ¯∂_{b} ¯∂_{b}^{∗} which arise on three-dimensional CR manifolds, and it is natural to impose a supplemental hypothesis.

Definition. {X_{1}, X_{2}} is a pseudoconvex pair in a neighborhood of p ∈ ℝ^{2} if either X_{1}, X_{2} span ℝ^{2} at p, or they are dependent at p and there exist a real, smooth vector field T transverse to span{X_{1}, X_{2}} at p, and smooth coefficients h, b_{1}, b_{2} such that [X_{1}, X_{2}] ≡ hT + b_{1}X_{1} + b_{2}X_{2} and h does not change sign in some neighborhood of p.

This condition is clearly independent of T and of the choice of basis for span{X_{1}, X_{2}}. However, choosing T does not uniquely determine the coefficients h, b_{j}.

If X_{1}, X_{2} are linearly independent at p then L is elliptic, hence analytic hypoelliptic, near p. In the linearly dependent case, assuming the bracket hypothesis to be satisfied, let (x, t) be a coordinate system with the properties of Lemma 1.3. Let (ξ, τ) be Fourier variables dual to (x, t). The characteristic variety Σ of L is near 0 ∈ ℝ^{2} a trivial line bundle over the analytic variety V = {(x, t) ∈ ℝ^{2} : Θ̃(x, t) = 0} consisting of all points in the base space at which X_{1}, X_{2} are dependent^{2}. Restricting attention to a neighborhood of 0 in which V is connected, Σ splits in a unique way as a union of two half line bundles Σ^{±}, where each half line has vertex at ξ = τ = 0. It will be shown in Section 8 that exactly one of these two, denoted Σ^{+}, has a conic neighborhood Γ^{+} in which the principal symbols σ_{1} of the vector fields satisfy

σ_{1}(i[X_{1}, X_{2}]) = μ + a_{1}σ_{1}(iX_{1}) + a_{2}σ_{1}(iX_{2})

for some functions μ, a_{1}, a_{2} such that μ ≤ 0. From this it is straightforward to deduce, following the method of Kohn [14] and exploiting the bracket hypothesis and Garding’s inequality, that L is microlocally C^{∞} hypoelliptic in Γ^{+}.

Our final result characterizes microlocal analytic hypoellipticity of L in Γ^{+}. Denote by W F_{a}(u) the analytic wave front set, as defined in [17].

Theorem 1.8. Assume X_{1}, X_{2} are linearly dependent at p. Under the hypotheses of finite type and pseudoconvexity,

W F_{a}(u) ∩ Γ^{+} ⊂ W F_{a}(Lu) ∩ Γ^{+} for all distributions u

if and only if there exist m ≥ 2 and coordinates (x, t) with origin at p such that

span{X_{1}, X_{2}} ≡ span{∂_{x}, x^{m−1}∂_{t}}

in a neighborhood of 0.

Assuming pseudoconvexity, L will not be microlocally analytic hypoelliptic in any conic neighborhood of Σ^{−}, at least in the special case where L = (X_{1}+iX_{2})(X_{1}−iX_{2}). Indeed, the theorem of Trepreau [22] asserts that (X_{1} − iX_{2}) is not microlocally analytic hypoelliptic there, hence the composition cannot be. The proof of Theorem 1.8 will be outlined in Section 8.

2The hypotheses of pseudoconvexity and finite type force V to have real dimension one.

The principal results of this paper were announced in [4]. The invariant q is already ubiquitous in our estimates, but in terms of Gevrey class hypoellipticity has a deeper significance outlined in [4], which will be the subject of a subsequent publication.

The technique used here to reduce analytic (non)hypoellipticity to eigenvalue problems for ordinary differential equations is identical to that used by the author in establishing a counterexample to global analytic regularity [3]; only the details of the formulas must be changed. Furthermore, various lemmas are contained, at least implicitly, in earlier works [5],[6],[8]. Therefore this paper is not entirely self-contained.

2. Coordinates and canonical forms

In this section we prove Lemma 1.3. Assume that X_{1}, X_{2} are linearly dependent at p ∈ ℝ^{2}. The Lie bracket hypothesis ensures that at least one of X_{1}, X_{2} does not vanish at p, so there exists a C^{ω} coordinate system (x, t) with origin at p in which span{X_{1}, X_{2}} ≡ span{∂_{x}, a(x, t)∂_{t}} for some a ∈ C^{ω}. By the Weierstraß preparation theorem, this span is in turn equal to the span of ∂_{x} and Θ̃(x, t)∂_{t}, for some Θ̃ ∈ C^{ω} of the form

Θ̃(x, t) = x^{m−1} + ∑_{j=0}^{m−2} β_{j}(t)x^{j}

for some m ≥ 1, where each β_{j} ∈ C^{ω} is real-valued and β_{j}(0) = 0. Matters may be further reduced to the case where β_{m−2} ≡ 0 by a change of variables (x, t) ↦ (x − (m − 1)^{−1}β_{m−2}(t), t), and we shall always do so. Thus

Θ̃(x, t) = x^{m−1} + ∑_{j=0}^{m−3} β_{j}(t)x^{j}.  (2.1)

From the expression for Θ̃ one finds that the quantity m defined in this way equals the type as defined in Section 1.

Define τ_{j} to be the order of vanishing of β_{j} at t = 0, that is, β_{j}(t) = d_{j}t^{τ_{j}} +O(t^{τ_{j}+1}) for some d_{j} ≠ 0.

Definition.

q = \begin{cases}∞ & if each β_{j} vanishes identically \\ min_{j} τ_{j}/(m − 1 − j) & otherwise. \end{cases}

The lowest order part of Θ̃ will be denoted by Θ.

Definition. If q < ∞,

Θ(x, t) = x^{m−1} + ∑_{j=0}^{m−3} c_{j}t^{(m−1−j)q}x^{j}  (2.2)

where

\begin{cases}c_{j} = 0 & if (m − 1 − j)q > τ_{j} \\ β_{j}(t) = c_{j}t^{(m−j−1)q}(1 + O(t)) & if (m − 1 − j)q = τ_{j}. \end{cases}

If q = ∞ then Θ(x, t) ≡ Θ̃(x, t) ≡ x^{m−1}. In either case, the normalized homogeneous polynomial Q associated to L is

Q(x, z) = x^{m−1} + ∑_{j=0}^{m−3} c_{j}z^{m−1−j}x^{j}.

If m = 2 then necessarily q = ∞, and L is analytic hypoelliptic by the well known theorems on the symplectic case.

To complete the proof of Lemma 1.3, it remains only to show that q, Q are invariant in the required sense. Assume that {X_{1}, X_{2}} is of finite type, but is linearly dependent at the origin. Let (x, t), Θ̃, q and Θ be as defined above. Denote by (y, s) some other system of coordinates, with the same origin as (x, t). We regard y, s both as coordinates, and as functions of (x, t).

The type m at 0 is an invariant, being intrinsically defined as 1 if X_{1}, X_{2} are independent at 0, and otherwise as the minimal length of any Lie bracket of X_{1}, X_{2} that is not in their span at 0. Suppose that the span of {X_{1}, X_{2}} equals the span of {∂_{y}, Φ̃(y, s)∂_{s}} in a neighborhood of 0, and that Φ̃ = y^{m−1} + ∑_{j≤m−3} γ_{j}(s)y^{j} where each γ_{j}(0) = 0. Then Φ̃(y, s) = h(y, s)Θ̃(x, t) for some analytic function h that does not vanish at 0. Our task is to show that the polynomial Φ associated to Φ̃ as in (2.2) is identical to Θ.

Observe that s = ct + O(t^{2}, xt, x^{m}) for some c ≠ 0. Indeed, consider any monomial D = V_{1}V_{2} … V_{n} where each V_{i} has smooth coefficients and belongs to the span of X_{1}, X_{2} at every point of a neighborhood of 0. To avoid confusion of notation define a function f by f(x, t) = s. If n < m then D(f) must vanish at the origin, as is seen by writing each V_{i} as a linear combination of ∂_{y} and of Φ̃(y, s)∂_{s}. Choosing D to equal ∂_{x}^{n} in turn for each n < m forces s = f(x, t) to have the required form for some constant c. Since the differential of f cannot vanish, c must be nonzero.

The situation is symmetric, so we have likewise that t = cs + O(s^{2}, ys, y^{m}) for some nonzero c. A further consequence is that x = ay + O(s, ys, y^{2}) for some a ≠ 0. Consider first the case where x = ay + bs^{r} + O(y^{2}, ys, s^{r+1}) for some r ≥ 1 and b ≠ 0.

Consider the subcase where r < q. To each monomial y^{α}s^{β} assign weight α +r^{−1}β. Expand h · Θ̃ as a formal Taylor series in powers of y, s at 0, and consider the weights of those monomials in this expansion that might have nonzero coefficients. For 0 ≤ j ≤ m − 3,

x^{m−j−1} = (ay + bs^{r} + O(y^{2}, ys, s^{r+1}))^{m−j−1}.

When this is expanded, no terms of weight < m − 1 − j arise, and the sum of all terms of weight equal to m − 1 − j is (ay + bs^{r})^{m−j−1}. Since β_{j}(t) = O(t^{jq}), each monomial in its expansion has weight > jq/r, and hence each monomial in the expansion of β_{j}(t)x^{m−j−1} has weight ≥ (m − 1 − j) + jq/r, which is strictly greater than m − 1. Consequently the same goes for the expansion of h · β_{j}(t)x^{m−j−1}, and similar reasoning gives the same conclusion for (h − h(0)) · x^{m−1}. Finally x^{m−1} = (ay + bs^{r})^{m−1} plus monomials of weights strictly greater than m − 1, and hence h · x^{m−1} = h(0)(ay + bs^{r})^{m−1} plus monomials of weights > m − 1. Thus in the Taylor expansion of Φ̃ = hΘ̃, no terms of weight < m − 1 occur, and the sum of all monomials of weight m − 1 is exactly h(0)(ay + bs^{r})^{m−1}. Therefore if we expand Φ̃ instead in powers of y with coefficients depending on s, the coefficient of y^{m−2} must equal (m − 1)a^{m−2}bh(0)s^{r} + O(s^{r+1}). The numerical factor (m − 1)a^{m−2}bh(0) is nonzero, and therefore Φ̃ cannot be a normalized polynomial of degree m − 1 in (y, s), a contradiction. Thus r cannot be less than q.

If r = q, the same reasoning leads to the conclusion that again, no monomials of weights < m−1 occur, and that the sum of all monomials of weight m−1 is h(0)Θ(ay+ bs^{q}, cs). The coefficient of y^{m−2} in this polynomial is still (m − 1)a^{m−2}bh(0)s^{q}, so the same contradiction is reached.

The only remaining cases are where either r > q, or x = ay + O(y^{2}, ys, s^{N}) for all N. Define the weight of any monomial y^{α}s^{β} now to be α + q^{−1}β. Then the same reasoning as above leads to the conclusion that no monomials of weight < m − 1 occur with nonzero coefficients in the Taylor expansion of Φ̃, and moreover that the sum of all terms of weight m − 1 equals h(0)Θ(ay, cs).

Writing Φ̃(y, s) = y^{m−1} + ∑_{j≤m−3} γ_{j}(s)y^{j}, it follows that γ_{j}(s) = O(s^{(m−1−j)q}) for all j. Therefore if q̃ denotes the quantity obtained by applying the definition of q in the coordinates (y, s), we find that q̃ ≥ q. The situation is symmetric, so likewise q ≤ q̃.

3. Analytic preliminaries

Let L be given, and adopt coordinates (x, t) as described in Lemma 1.3. Let m be the type at p = 0. Assume that L is not elliptic at 0, and moreover that q < ∞, which implies that m ≥ 3. To simplify notation let E = E(Q) = NE(L, 0). Let E^{1/q} be the set of all q-th roots of elements of E(Q); recall that q is rational.

Write

L = P (x, t, ∂_{x}, Θ̃(x, t)∂_{t})

where P is a C^{ω} function of (x, t) and is an elliptic polynomial of degree 2 with respect to ∂_{x}, Θ̃∂_{t}. More concretely,

L = (a_{1,1}∂_{x} + a_{1,2}Θ̃∂_{t})^{2} + (a_{2,1}∂_{x} + a_{2,2}Θ̃∂_{t})^{2}

for certain real valued, C^{ω} coefficients a_{i,j}, where the two by two matrix with entries a_{i,j}(x, t) is nonsingular for every (x, t) sufficiently close to 0.

Fix an entire holomorphic function Ψ of one complex variable, not identically zero, satisfying Ψ(z) = O(exp(C|z|^{m})) for all z ∈ ℂ and Ψ(z) = O(exp(−|z|^{m})) for all z in some conic neighborhood of ℝ; when m is even, it suffices to set Ψ(z) = exp(−Cz^{m}). Denote by a, b ∈ ℝ constants to be chosen in the course of the proof of Theorem 1.4, and set ψ(x) = e^{iax}Ψ(x − b). For each large λ ∈ ℝ^{+} set

F_{λ}(x, t) = e^{iλ^{m}t}ψ(λx)

Lemma 3.1. Let L be a locally solvable, C^{∞} hypoelliptic linear partial differential operator. Suppose that L is analytic hypoelliptic in some neighborhood of 0. Then there exist open sets U ⊂ ℝ^{2} and Ũ ⊂ ℂ^{2} containing 0 such that for any N there exist C, M < ∞ such that for every λ ≥ 1, there exists G_{λ} ∈ C^{∞}(U) satisfying

LG_{λ} = F_{λ}
∥G_{λ}∥_{C^{N} (U)} ≤ Cλ^{M}

G_{λ} extends to a holomorphic function in Ũ,

and

|G_{λ}(x, t)| ≤ Cλ^{M}e^{Cλ^{m}| Im(x,t)|} for all (x, t) ∈ Ũ.

This type of result originates in work of Oleĭnik and Radkevič [16]; see the proof of Lemma 4.1 of [3] or that of Theorem 1 of [16].

Set

v_{λ}(x, t) = e^{−iλ^{m}t}G_{λ}(x, t),

and

y = λx,  s = λ^{1/q}t.  (3.1)

The final inequality in Lemma 3.1 is also satisfied by v_{λ} and by each of its partial derivatives. The equation satisfied by v_{λ} is

(e^{−iλ^{m}t} ∘ L ∘ e^{iλ^{m}t})v_{λ}(x, t) = ψ(λx) = ψ(y).

Lemma 3.2.

λ^{m−1}Θ̃(λ^{−1}y, λ^{−1/q}s) = Θ(y, s) + R_{λ}(y, s)

where for each ε > 0 there exist δ > 0 and C < ∞ such that for all sufficiently large λ ∈ ℝ^{+}, for all |y| ≤ λ^{1−ε} and s in any fixed complex neighborhood of 0, for any α, β ≥ 0,

|∂_{y}^{α}∂_{s}^{β}R_{λ}(y, s)| ≤ \begin{cases}Cλ^{−δ}(1 + |y|)^{m−1−α} & if m − 1 − α ≥ 0 \\ Cλ^{−δ} & if m − 1 − α < 0. \end{cases}

This follows directly from the fact that Θ̃ = Θ plus terms of higher weight. Define

B_{λ} = λ^{−2}e^{−iλ^{m}t} ∘ L ∘ e^{iλ^{m}t}.

Thus

B_{λ} = λ^{−2}P (λ^{−1}y, λ^{−1/q}s, λ∂_{y}, Θ̃(λ^{−1}y, λ^{−1/q}s)(iλ^{m} + λ^{1/q}∂_{s}))
= P (λ^{−1}y, λ^{−1/q}s, ∂_{y}, λ^{−1}Θ̃(λ^{−1}y, λ^{−1/q}s)(iλ^{m} + λ^{1/q}∂_{s}))
= P (λ^{−1}y, λ^{−1/q}s, ∂_{y}, λ^{−m}(Θ + R_{λ})(y, s)(iλ^{m} + λ^{1/q}∂_{s}))
= P (λ^{−1}y, λ^{−1/q}s, ∂_{y}, i(Θ + R_{λ})(1 − iλ^{−p}∂_{s}))

where

p = m − q^{−1}.  (3.2)

Since q^{−1} equals (m − 1 − j)/τ_{j} for some 0 ≤ j ≤ m − 3, and since each τ_{j} ≥ 1, we have q^{−1} ≤ m − 1 − j ≤ m − 1 and hence^{3} p ≥ 1. Setting

u_{λ}(y, s) = λ^{2}v_{λ}(x, t)  (3.3)

we arrive at the equation

B_{λ}u_{λ}(y, s) = ψ(y).  (3.4)

Define

A_{s} = P (0, 0, ∂_{y}, iΘ(y, s)).  (3.5)

3In the particular case treated in [3], p was equal to 1. The main difference between the proof of Theorem 1.4 and the analysis of [3] is merely that λ is systematically replaced by λ^{p}.

Our plan is essentially to analyze the equation B_{λ}u_{λ} = ψ by a Neumann series argument, writing formally B_{λ}^{−1} = ∑_{0}^{∞}(−1)^{j}[A_{s}^{−1}(B_{λ} − A_{s})]^{j}A_{s}^{−1}. Thus we require bounds on both A_{s}^{−1} and B_{λ} − A_{s}.

The remainder of this section summarizes properties of A_{s}^{−1}. The operator A_{s} may be regarded in either of two ways, as an ordinary differential operator, acting on functions of y and depending on the parameter s ∈ ℂ, or as acting on functions of both variables (y, s). For the present we view it in the former light. E^{1/q} = NE(L, 0)^{1/q} is thus the usual set of nonlinear eigenvalues for the family of ordinary differential operators ζ ↦ A_{ζ}.

Set ⟨x⟩ = (1 + |x|^{2})^{1/2} for any x ∈ ℂ. For ρ ∈ ℝ and k ∈ {0, 1, 2} consider the Sobolev space H_{ρ}^{k} = H_{ρ}^{k}(ℝ) of (equivalence classes of) measurable functions f : ℝ ↦ ℂ for which the following norms are finite:

∥f∥_{H_{ρ}^{0}}^{2} = ∫_{ℝ} |f(x)|^{2} ⟨x⟩^{−2(m−1)} e^{ρ|x|^{m}} dx
∥f∥_{H_{ρ}^{1}}^{2} = ∫_{ℝ} (⟨x⟩^{−2(m−1)}|∂_{x}f|^{2} + |f|^{2}) e^{ρ|x|^{m}} dx
∥f∥_{H_{ρ}^{2}}^{2} = ∫_{ℝ} (⟨x⟩^{−2(m−1)}|∂_{x}^{2}f|^{2} + |∂_{x}f|^{2} + ⟨x⟩^{2(m−1)}|f|^{2}) e^{ρ|x|^{m}} dx.

Likewise for any open set D ⊂ ℂ define the spaces H_{ρ}^{k}(ℝ × D) to consist of all (equivalence classes of) functions of (x, z) ∈ ℝ×D that are holomorphic with respect to z, for which the following norms are finite:

∥f∥_{H_{ρ}^{0}}^{2} = ^{∫∫} _{ℝ×D} |f(x)|^{2} ⟨x⟩^{−2(m−1)} e^{ρ|x|^{m}} dx dzdz̄
∥f∥_{H_{ρ}^{1}}^{2} = ^{∫∫} _{ℝ×D} (⟨x⟩^{−2(m−1)}|∂_{x}f|^{2} + |f|^{2}) e^{ρ|x|^{m}} dx dzdz̄
∥f∥_{H_{ρ}^{2}}^{2} = ^{∫∫} _{ℝ×D} (⟨x⟩^{−2(m−1)}|∂_{x}^{2}f|^{2} + |∂_{x}f|^{2} + ⟨x⟩^{2(m−1)}|f|^{2}) e^{ρ|x|^{m}} dx dzdz̄.

Lemma 3.3. For any normalized homogeneous polynomial Q having real coefficients, the set E(Q) is discrete, and E(Q) ∩ ℝ = ∅ = (E(Q)^{1/q}) ∩ ℝ.

Proof. That E(Q)∩ℝ = ∅ follows from positivity of the ordinary differential operator, for ζ ∈ ℝ. Indeed, for any f in the Schwartz class,

∫_{ℝ}(−∂_{x}^{2}f + Q^{2}(x, ζ)f) · f̄ = ∫_{ℝ}[|f′|^{2} + Q(x, ζ)^{2}|f|^{2}] dx ≥ c_{ζ}∥f∥_{L^{2}(ℝ)}^{2},

because Q has real coefficients and does not vanish identically as a function of x for any ζ. Hence E ∩ ℝ = ∅. The same reasoning applies to the set E^{1/q} of nonlinear eigenvalues associated to the family of operators A_{z}, because Θ(x, z) is likewise real valued whenever x, z are real.

As was proved in [5] for similar families of operators, E(Q) is equal to the set of all zeros of an entire holomorphic function; see also Section 6 below. Since E(Q) ≠ ℂ, E(Q) must be discrete.

Lemma 3.4. For each compact set K ⊂ ℂ\E^{1/q} there exists r > 0 such that for every ρ ∈ [−r, r] and ζ ∈ K, A_{ζ} : H_{ρ}^{2} ↦ H_{ρ}^{0} is an isomorphism, whose inverse is bounded uniformly in ζ ∈ K, |ρ| ≤ r.

Proof. This follows from the method of [3], once it is shown that the (two dimensional) nullspace of A_{ζ} contains no functions that are O(exp(−δ|x|^{m})) for some δ > 0, since any function in the nullspace of A_{ζ} either decays at such a rate as |x| → ∞, or tends to ∞ in modulus as x → +∞ or as x → −∞. The hypothesis that K does not intersect E^{1/q} means that ∂_{x}^{2} − Θ(x, ζ)^{2} has no such solutions, for any ζ ∈ K.

To relate A_{ζ} to this operator, write A_{ζ} = α∂_{x}^{2} + iβ(∂_{x} ∘ Θ(x, ζ) + Θ(x, ζ)∂_{x}) − γΘ(x, ζ)^{2} where α, β, γ ∈ ℝ, α, γ are positive, β^{2} < αγ, and Θ denotes both a function and the operator defined by multiplication by that function. Fix a polynomial R(x, z), with real coefficients, satisfying ∂R/∂x = Q(x, z), and consider

e^{iβR(x,ζ)/α} ∘ A_{ζ} ∘ e^{−iβR(x,ζ)/α} = α∂_{x}^{2} + (α^{−1}β^{2} − γ)Θ(x, ζ)^{2}.

The coefficient α^{−1}β^{2} − γ is negative. Dilating both variables x, ζ by appropriate factors and multiplying the operator by a constant reduces it to ∂_{x}^{2} − Θ(x, ζ)^{2}.

Now the factor exp(iβR(x, ζ)/α) may not be a bounded function of x when ζ is not real, but since R has real coefficients and β/α is real, this factor and its inverse are O(exp(C|x|^{m−2})) for ζ in any compact set and x ∈ ℝ. Therefore since ∂_{x}^{2} − Θ(x, ζ)^{2} has no solutions bounded by exp(−δ|x|^{m}) as x → ±∞, no matter how small the exponent δ, neither has A_{ζ}.

The proofs of the remaining lemmas of this section are essentially identical to those in Section 3 of [3], and will not be repeated here. Define Ψ_{a,b}(x) = exp(iax)Ψ(x − b).

Lemma 3.5. Suppose that Ψ is a Schwartz function that does not vanish identically. For each ζ_{0} ∈ E^{1/q}, for any circle Γ centered at z_{0} whose closure contains no other points of E^{1/q}, there exist σ ∈ {0, 1, 2, … }, a, b ∈ ℝ and φ ∈ C_{0}^{∞}(ℝ) such that

∫_{ℝ} ∮_{Γ} φ(x)A_{ζ}^{−1}Ψ_{a,b}(x) ζ^{σ}dζ dx ≠ 0.

The proof relies on the fact that the span of all functions Ψ_{a,b} is dense in L^{2}(ℝ). Otherwise it is essentially the same as the proof of Lemma 3.2 of [3].

For ζ, τ ∈ ℂ define

A_{ζ,τ} = P (0, 0, ∂_{x}, iτΘ(x, ζ))

Then for 0 ≠ τ in a small conic neighborhood of ℝ,

A_{ζ,τ} = P (0, 0, ∂_{x}, τ^{1/m}iΘ(τ^{1/m}x, τ^{1/mq}ζ))
= τ^{2/m}P (0, 0, ∂_{w}, iΘ(w, τ^{1/mq}ζ))

where w = τ^{1/m}x.

Lemma 3.6. If τ belongs to a sufficiently small conic neighborhood of ℝ then for any ζ ∈ ℂ, A_{ζ,τ} annihilates some nonzero Schwartz class function if and only if τ^{1/mq}ζ ∈ E^{1/q}. If τ^{1/mq}ζ ∉ E^{1/q} then A_{ζ,τ} : H_{ρ}^{2} ↦ H_{ρ}^{0} is invertible, for all sufficiently small |ρ|. This holds uniformly for any compact set of such (ζ, τ).

Lemma 3.7. There exists a conic neighborhood of ℝ that is disjoint from E^{1/q}.

4. Necessity: Reduction to eigenvalue problems

This section contains the core of the proof of Theorem 1.4, which asserts that analytic hypoellipticity fails to hold whenever a set of eigenvalues is nonempty. Recall that B_{λ}u_{λ} = ψ, and

|u_{λ}| + |∇u_{λ}| + |∇^{2}u_{λ}| ≤ λ^{M} exp(Cλ^{m}| Im(t)|) ≤ exp(C_{0}λ^{p})  (4.1)

for all |y| ≤ cλ and |s| ≤ C_{1}, for some C_{0} < ∞, where (y, s) ∈ ℝ × ℂ and C_{1} may be taken to be as large as we wish provided that λ and C_{0} are sufficiently large. Here ∇ denotes the gradient in both variables y, s.

Assume that E^{1/q} is nonempty, as hypothesized in Theorem 1.4, and fix any ζ_{0} ∈ E^{1/q} having strictly negative imaginary part. E = Ē because the coefficients of Q are real, and E^{1/q} ∩ ℝ = ∅, so such a point exists. Define Γ = {|s − ζ_{0}| = r} to be a circle centered at ζ_{0}, contained in the open lower half plane, such that the intersection of E^{1/q} with the closed disk bounded by Γ contains only the point ζ_{0}.

Consider (y, s) ∈ ℝ × Γ, and write s = ζ_{0} + re^{iθ}. When acting on restrictions to ℝ × Γ of functions holomorphic with respect to s, B_{λ} takes the form

B_{λ} = P (λ^{−1}y, λ^{−1/q}(ζ_{0} + re^{iθ}), ∂_{y}, i[Θ + R_{λ}](1 − iλ^{−p}(−ir^{−1}e^{−iθ}∂_{θ})))

where Θ + R_{λ} is evaluated at (y, ζ_{0} + re^{iθ}) = (y, s). Denote by B_{λ}^{∗} the transpose of B_{λ} in L^{2}(ℝ × Γ, dydθ), where functions of s ∈ Γ are identified with periodic functions of θ, |θ| ≤ π. Thus

B_{λ}^{∗} = P^{∗}(λ^{−1}y, λ^{−1/q}s, −∂_{y}, i(1 − iλ^{−p}(ir^{−1}∂_{θ} ∘ e^{−iθ})) ∘ [Θ + R_{λ}])

where

P^{∗}(x, t, −∂_{x}, −∂_{t} ∘ Θ̃) = (−∂_{x} ∘ a_{1,1} − ∂_{t} ∘ Θ̃a_{1,2})^{2} + (−∂_{x} ∘ a_{2,1} − ∂_{t} ∘ Θ̃a_{2,2})^{2}, and where a_{i,j} denotes also the operator defined by multiplication by the function a_{i,j}.

Let σ, φ be as in Lemma 3.5. The proofs of the next two lemmas will be discussed at the end of this section.

Lemma 4.1. There exist ρ, δ, ε, C ∈ ℝ^{+} such that for each sufficiently large λ ∈ ℝ^{+} there exists f : ℝ × Γ ↦ ℂ, supported where |y| ≤ λ^{1−ε}, satisfying

∫∫ _{ℝ×Γ} |B_{λ}^{∗}f − s^{σ}(∂s/∂θ)φ(y)|^{2}e^{ρ|y|^{m}} dy dθ  ≤ e^{−δλ^{p}}  (4.2)
∫∫ _{ℝ×Γ}(|f|^{2} + |∇f|^{2})e^{ρ|y|^{m}} dy dθ  ≤ C  (4.3)
∫∫ _{ℝ×Γ} |f(y, θ) − s^{σ}(∂s/∂θ)A_{s}^{−1}φ(y)|^{2}e^{ρ|y|^{m}} dy dθ ≤ Cλ^{−δ}  (4.4)

where s = ζ_{0} + re^{iθ}.

The next lemma is conditional in nature; we will soon see that u_{λ} cannot satisfy its conclusion, hence must not satisfy its hypotheses.

Lemma 4.2. Suppose that B_{λ}u_{λ} = ψ and that u_{λ} satisfies the bound (4.1). Then for any C_{1} < ∞ and μ, ε, ϱ > 0 there exists C < ∞ such that for all sufficiently large λ and all |s| ≤ C_{1},

∫_{|y|≤λ^{1−ε}}[|u_{λ}(y, s)|^{2} + |∇u_{λ}(y, s)|^{2}]e^{−ϱ|y|^{m}} dy ≤ e^{μλ^{p}}.  (4.5)

Theorem 1.3 may now be proved, as follows. Suppose that L were analytic hypoelliptic in some neighborhood of the origin. Define F_{λ} as in Section 3, solve LG_{λ} = F_{λ} as in Lemma 3.1, and define first u_{λ} and then v_{λ} as was done following Lemmas 3.1 and 3.2, respectively. Then Lemma 4.2 yields a strong bound for u_{λ}. Let f be an approximate solution for the adjoint operator B_{λ}^{∗}, satisfying the conclusions of Lemma 4.1. Define

ω = ∫_{ℝ} ∮_{Γ} φ(y)u_{λ}(y, s)s^{σ} ds dy.

Then ω = 0 since u_{λ} is holomorphic with respect to s. We will prove that

ω = c_{0} + O(λ^{−δ}) as λ → +∞  (4.6)

for some c_{0} ≠ 0 and δ > 0, thus arriving at a contradiction.

Choosing μ in Lemma 4.2 to be less than δ in Lemma 4.1, and ϱ in Lemma 4.2 to be less than ρ in Lemma 4.1,

ω = ^{∫∫} _{ℝ×Γ}[φ(y)s^{σ}(∂s/∂θ)]u_{λ}(y, s) dθ dy
= ^{∫∫} _{ℝ×Γ}(B_{λ}^{∗}f)(y, s)u_{λ}(y, s) dθ dy + O(e^{−ελ^{p}})

for some ε > 0, by (4.2) and (4.5). Because f has compact support with respect to y and Γ has no boundary, it is permissible to integrate by parts to rewrite the last line as

ω = ^{∫∫} _{ℝ×Γ} f(y, s)B_{λ}u_{λ}(y, s) dθ dy + O(e^{−ελ^{p}}).

Note that for each s ∈ ℂ, A_{s} is its own transpose. Therefore

ω = ^{∫∫} _{ℝ×Γ} f(y, s)ψ(y) dθ dy + O(e^{−ελ^{p}})
= ^{∫∫} _{ℝ×Γ}[s^{σ}(∂s/∂θ)A_{s}^{−1}φ(y)]ψ(y) dθ dy + O(λ^{−δ})
= ∮_{Γ}(∫_{ℝ} A_{s}^{−1}φ(y) · (A_{s}A_{s}^{−1}ψ)(y) dy)s^{σ} ds + O(λ^{−δ})
= ∮_{Γ} ∫_{ℝ} φ(y)A_{s}^{−1}ψ(y) dys^{σ} ds + O(λ^{−δ})
= c_{0} + O(λ^{−δ})

where c_{0} ≠ 0 by Lemma 3.5.

We next prove Lemma 4.1. It is convenient for B_{λ}^{∗} to be globally defined, on the Cartesian product of ℝ with a complex neighborhood of Γ. This can be accomplished as in the first paragraph of the proof of Lemma 5.1 of [3]: Fix a constant ε_{0} > 0 satisfying

(1 − ε_{0})m > p = m − q^{−1}.  (4.7)

Extend the coefficients a_{i,j}(λ^{−1}y, λ^{−1/q}s) to ã_{i,j}(λ^{−1}y, λ^{−1/q}s) so that ã_{i,j}(λ^{−1}y, λ^{−1/q}s) is identically equal to a_{i,j}(λ^{−1}y, λ^{−1/q}s) when |y| ≤ λ^{1−ε_{0}} and is equal to a_{i,j}(0, λ^{−1/q}s) when |y| ≥ 2λ^{1−ε_{0}}, and so that for λ^{1−ε_{0}} ≤ |y| ≤ 2λ^{1−ε_{0}},

ã_{i,j}(λ^{−1}y, λ^{−1/q}s) − a_{i,j}(λ^{−1}y, λ^{−1/q}s) = O(λ^{−1}y) = O(λ^{−ε_{0}})

and

∂_{s}^{β}∂_{y}^{α}ã_{i,j} = O(λ^{−α−β/q}) for all α, β ≥ 0.

Similarly, R_{λ} may be extended so that

|∂_{y}^{α}∂_{s}^{β}R_{λ}| ≤ Cλ^{−ε_{0}}(1 + |y|)^{max(0,m−1−α)}

for all |y| ≤ λ^{1−ε_{0}}, |s| ≤ C.

Let Λ ∈ ℝ^{+} be a large constant to be chosen below, and given any large λ, construct concentric open annuli Ω_{0} ⊃ Ω_{1} ⊃ ⋯ ⊃ Ω_{N+1} = Ω ⊃ Γ so that

1. E^{1/q} ∩ \overline{Ω_{0}} = ∅,

2. The three sets Ω_{0}, Ω_{1}, Ω are all independent of λ,

3. N ≥ cΛ^{−1}λ^{p}, and

4. distance (∂Ω_{j}, Ω_{j+1}) ≥ Λλ^{−p} for all j. Set ϕ(y, s) = s^{σ}(∂s/∂θ)φ(y) = is^{σ}(s − ζ_{0})φ(y).

Decompose

B_{λ}^{∗} = A_{s} + K + E

where

A_{s} = P (0, 0, ∂_{y}, iΘ(y, s)),
A_{s} + K = P^{∗}(λ^{−1}y, λ^{−1/q}s, −∂_{y}, iΘ + iR_{λ}).

Thus the remainder E represents all contributions of ∂_{s}. A_{s} and K will be regarded both as ordinary differential operators depending holomorphically on the parameter s, and as operators acting on functions of (y, s) that are holomorphic with respect to s.

For all sufficiently small ρ > 0, A_{s} : H_{ρ}^{2}(ℝ) ↦ H_{ρ}^{0}(ℝ) is an isomorphism, uniformly for all s ∈ \overline{Ω_{0}}, by Lemma 3.4. Since A_{s}^{−1} depends holomorphically on s ∈ ℂ\E^{1/q}, it follows that A_{s} : H_{ρ}^{2}(ℝ × U) ↦ H_{ρ}^{0}(ℝ × U) is also an isomorphism for every open set U ⊂ Ω_{0}, uniformly in ρ, U provided that ρ is sufficiently small.

By Cauchy’s inequality, ∂_{s} : H_{ρ}^{k}(ℝ × Ω_{j}) ↦ H_{ρ}^{k}(ℝ × Ω_{j+1}) has norm O(Λ^{−1}λ^{p}), uniformly in all parameters, because the distance between the boundaries is at least Λλ^{−p}, so

λ^{−p}∂_{s} : H_{ρ}^{k}(ℝ × Ω_{j}) ↦ H_{ρ}^{k}(ℝ × Ω_{j+1}) has norm ≤ CΛ^{−1} for all j, λ.

Moreover, the norm is O(λ^{−p}) for j = 0, since the distance from ∂Ω_{0} to ∂Ω_{1} is independent of λ. From this, from the definitions of the norms, and from straightforward computation there follows the first conclusion of the following lemma.

Lemma 4.3. The remainder terms K, E satisfy the following bounds.

E : H_{ρ}^{2}(ℝ × Ω_{j}) ↦ H_{ρ}^{0}(ℝ × Ω_{j+1}) is O(Λ^{−1} + λ^{−ε})

for some ε > 0, as λ → +∞. For j = 0, it is O(λ^{−ε}). Secondly,

K : H_{ρ}^{2}(ℝ × Ω_{j}) ↦ H_{ρ}^{0}(ℝ × Ω_{j}) is O(λ^{−ε})

for some ε > 0, as λ → +∞. These bounds hold for all |ρ| ≤ ρ_{0} ≪ 1, uniformly in λ, j.

The second conclusion follows from the bounds of Lemma 3.2 on R_{λ} and its derivatives, and the presence of negative powers of λ in the expressions λ^{−1}y, λ^{−1/q}s.

Fix ρ > 0 sufficiently small that Lemma 3.4 applies for all ζ ∈ \overline{Ω_{0}}. As an approximate solution to B_{λ}^{∗}f ≈ ϕ define

f = ∑_{j=0}^{N}(−1)^{j}[A_{s}^{−1}(K + E)]^{j}A_{s}^{−1}ϕ.

Then

B_{λ}^{∗}f − ϕ = ±[(K + E)A_{s}^{−1}]^{N+1}ϕ.

Lemma 4.3 implies

∥[(K + E)A_{s}^{−1}]^{j}ϕ∥_{H_{ρ}^{0}(ℝ×Ω_{j})} ≤ C^{j}λ^{−ε}Λ^{1−j}  (4.8)

for all j ≥ 1, and

∥[A_{s}^{−1}(K + E)]^{j}A_{s}^{−1}ϕ∥_{H_{ρ}^{2}(ℝ×Ω_{j})} ≤ C^{j}λ^{−ε}Λ^{1−j}.  (4.9)

Thus if Λ is chosen to be greater than 2C for the constants C in (4.8) and (4.9), the fact that N ≥ cΛ^{−1}λ^{p} may be used to conclude that

∥B_{λ}^{∗}f − ϕ∥_{H_{ρ}^{0}(ℝ×Ω)} ≤ C(C/Λ)^{N} ≤ Ce^{−δλ^{p}}

for some δ > 0. (4.3) follows by summing over j, if Λ is chosen to be sufficiently large. And

f − A_{s}^{−1}ϕ = ∑_{j=1}^{N}(−1)^{j}[A_{s}^{−1}(K + E)]^{j}A_{s}^{−1}ϕ = O(λ^{−δ})

in H_{ρ}^{2}(ℝ × Ω) norm. Specializing to s ∈ Γ and invoking Cauchy’s inequality to majorize the L^{2}(Γ) norm by that of L^{2}(Ω), this is (4.4).

This f cannot be expected to have compact support. So fix a cutoff function η ∈ C_{0}^{∞}[−1, 1] that is ≡ 1 for |y| ≤ 1/2, and define f̃(y, s) = η(λ^{−1+ε_{0}}y)f(y, s). Then f̃ has all the required properties if ρ is replaced by ρ/2 and ε_{0} is sufficiently small. In particular, f̃ is supported where |y| ≤ λ^{1−ε_{0}}, in which region the coefficients of the modified operator B_{λ}^{∗} agree with those of the original operator B_{λ}. In each of the three conclusions of Lemma 4.1, the contribution of f̃− f, in the L^{2} norm with respect to the weight exp(ρ|y|^{m}/2), is O(exp(−cλ^{(1−ε_{0})m})) by the bound (4.4) for f, ∇f with respect to the weight exp(ρ|y|^{m}). Since (1 − ε_{0})m > p, this last bound is ≪ exp(−λ^{p}) for large λ.

The proof of Theorem 1.4 is now complete modulo the proof of Lemma 4.2, which is outlined in Section 9.

5. Nonlinear eigenvalue problems

Assume that Q(x, z) = x^{m−1} + ∑_{j=0}^{m−3} c_{j}z^{m−1−j}x^{j} where each c_{j} ∈ ℝ, x ∈ ℝ, z ∈ ℂ, and m ≥ 2. Let L_{z} = −∂_{x}^{2} + Q(x, z)^{2}, and recall that E(Q) denotes the set of all nonlinear eigenvalues of the family of operators L_{z}. This section contains various results connected with Conjecture 1.5, which says that E(Q) = ∅ if and only if Q(x, z) ≡ x^{m−1}.

Lemma 5.1. If Q(x, z) ≡ x^{m−1} then E(Q) = ∅.

This is implied by other results, for L = ∂_{x}^{2} + x^{2(m−1)}∂_{t}^{2} is analytic hypoelliptic by Theorem 1.2, which by Theorem 1.4 precludes the existence of eigenvalues.

Proof. L_{z} does not depend on z, and we have already remarked that E(Q) ∩ ℝ is always empty.

Before discussing the proofs of Theorem 1.6 and Proposition 1.7, we introduce a closely related class of nonlinear eigenvalue problems for which we are able to prove the analogue of Conjecture 1.5. Let Q be any normalized homogeneous polynomial, with real coefficients, of odd degree m − 1. Define

𝕃_{z} = (∂_{x} + Q(x, z)) ∘ (−∂_{x} + Q(x, z)),
E′(Q) = {z ∈ ℂ : there exists 0 ≠ f ∈ S such that 𝕃_{z}f = 0}.

Theorem 5.2. Let Q be a normalized homogeneous polynomial of odd degree m − 1, with real coefficients. Then E′(Q) = ∅ if and only if Q(x, z) ≡ x^{m−1}.

Proof. Introduce

P (x, z) = ∫_{0}^{x} Q(y, z) dy.

The function

ψ_{z}^{−}(x) = e^{P (x,z)} ∫_{−∞}^{x} e^{−2P (y,z)} dy

satisfies 𝕃_{z}ψ_{z}^{−} ≡ 0, and ψ_{z}^{−} tends rapidly to 0 as x → −∞. On the other hand, exp(P (x, z)) is a solution of 𝕃_{z} whose modulus tends rapidly to +∞ as x → −∞. Therefore since 𝕃_{z} has only a two-dimensional nullspace, each solution of 𝕃_{z} either tends rapidly to ∞ in modulus as x → −∞, or is a scalar multiple of ψ_{z}^{−}. Thus z ∈ E′(Q) if and only if ψ_{z}^{−}(x) behaves as a Schwartz function as x → +∞. Defining

W (z) = ∫_{ℝ} exp(−2P (x, z)) dx,

one has

E′(Q) = {z ∈ ℂ : W (z) = 0}.  (5.1)

Indeed, if W (z) ≠ 0, then ψ_{z}^{−}(x) ∼ W (z) exp(P (x, z)) tends to ∞ in modulus as x → +∞. If W (z) = 0 then ψ_{z}^{−}(x) = − exp(P (x, z)) ∫_{x}^{∞} exp(−2P (y, z))dy, which tends to 0 as x → +∞. If Q(x, z) = (x − cz)^{m−1} then a change of the contour of integration reveals that W (z) = c_{1} exp(c_{2}z^{m}) for some constants c_{j}, so W has no zeros.

Since P is a polynomial of even degree with positive leading coefficient independent of z, W is clearly an entire holomorphic function. For any z ∈ ℂ,

|W (z)| ≤ ∫_{ℝ} e^{−2x^{m}/m}e^{C(|z|^{m−1}|x|+|z||x|^{m−1})} dx
≤ ∫_{ℝ} e^{−x^{m}/m}e^{C|z|^{m}} dx
= C′e^{C|z|^{m}},

so W is of finite order, and its order does not exceed m. Any entire function of finite order with no zeros must be of the form exp(R(z)) for some polynomial R, so in order to prove that W must have zeros, it suffices merely to obtain sufficient information on the asymptotic behavior of W as ℝ ∋ z → +∞ to rule out W = e^{R}.

Restrict attention for the remainder of the proof to the case where z ∈ ℝ^{+}. Substituting x = zy yields

W (z) = z ∫_{ℝ} e^{−2z^{m}P (y,1)} dy.

Define

η = max_{x:Q(x,1)=0} −P (x, 1) = max_{y∈ℝ}−P (y, 1).

Let x_{1}, … x_{N} be those real points at which Q(x_{j}, 1) = 0 and −P (x_{j}, 1) = η. Since P has even degree, real coefficients, and negative leading coefficient, −P has at least one global maximum, so there exists at least one such point x_{j}.

These points determine the asymptotic behavior of W (z) as z → +∞. To compute the asymptotics choose 0 < δ < \frac{1}{2} min_{i,j} |x_{i} − x_{j}|. Then for some ε > 0,

∫_{ℝ\∪_{j}[x_{j}−δ,x_{j}+δ]} e^{−2z^{m}P (y,1)} dy = O(e^{2(η−ε)z^{m}})

by straightforward majorization.

For each j, let k_{j} be the smallest integer for which ∂^{k_{j}}P (y, 1)/∂y^{k_{j}}, evaluated at y = x_{j}, is nonzero. Since −P (y, 1) has a local maximum at x_{j}, each k_{j} must be even, and the derivative P^{(k_{j})}(x_{j}, 1) is strictly positive. By the method of real stationary phase,

∫_{x_{j}−δ}^{x_{j}+δ} e^{−2z^{m}P (y,1)} dy = e^{−2ηz^{m}}(c(k_{j})(z^{m}P^{(k_{j})}(x_{j}, 1))^{−1/k_{j}} + O(z^{−2m/k_{j}})),

where each c(k_{j}) is strictly positive. Defining k = max k_{j}, we find that as ℝ^{+} ∋ z → +∞,

W (z) = ce^{−2ηz^{m}}z^{1−m/k}(1 + O(z^{−ε}))  (5.2)

for some c, ε > 0. Thus the logarithm of W, restricted to ℝ^{+}, cannot be a polynomial unless 1 − m/k = 0. In that event k = m, that is, y ↦ P (y, 1) has a zero of order m at some x_{j}. Since P is a polynomial of degree m, this is only possible if P (y, 1) takes the form c(y − x_{j})^{m}. Thus because of the normalizations already imposed on Q, Q(y, 1) ≡ y^{m−1}.

Our next result concerns the ordinary differential operators L_{z} = −∂_{x}^{2}+Q(x, z)^{2}, for generic Q. Generalize the setting by permitting the coefficients of Q to be complex, and the coefficient of x^{m−2} to be nonzero.^{4} The definitions of L_{z} and of E(Q) still make sense.

Proposition 5.3. Suppose that for some n ≥ 1, for each 0 ≤ j ≤ m − 2 there is given an entire holomorphic function g_{j} : ℂ^{n} ↦ ℂ. Set

Q_{ζ}(x, z) = x^{m−1} + ∑_{j=0}^{m−2} g_{j}(ζ)z^{m−1−j}x^{j},

L_{z}^{ζ} = −∂_{x}^{2} + Q_{ζ}(x, z)^{2},

and

E_{ζ} = {z ∈ ℂ : there exists 0 ≠ f ∈ S(ℝ) such that L_{z}^{ζ}f = 0}.

Suppose that there exists ζ_{0} ∈ ℂ^{n} for which E_{ζ_{0}} ≠ ∅. Then

E_{ζ} ≠ ∅ for generic ζ,

4A normalization was carried out in Section 2 to reduce the study of analytic hypoellipticity to the case c_{m−2} = 0. A term c_{m−2}zx^{m−2} can likewise be eliminated in the eigenvalue theory, as follows: Suppose that Q(x, z) = x^{m−1} + ∑_{j=0}^{m−2} c_{j}z^{m−1−j}x^{j} where each c_{j} ∈ ℝ, x ∈ ℝ, z ∈ ℂ, and m ≥ 2. Define Q̃(x, z) = Q(x − (m − 1)^{−1}c_{m−2}z, z), and let L̃_{z} be the associated family of Schrödinger operators. If L_{ζ}f_{ζ} = 0 then f_{ζ} extends to an entire holomorphic function of x ∈ ℂ, and if moreover f_{ζ}(x) → 0 as ℝ ∋ x → ±∞, then ℝ ∋ x ↦ f(x + b) defines a Schwartz function, for any b ∈ ℂ [10]. Thus f̃(x) = f_{ζ}(x − (m − 1)^{−1}c_{m−2}ζ) is a Schwartz function annihilated by L̃_{ζ}.

Therefore there is a one to one correspondence between E(Q) and E(Q̃).

in the sense that the set of ζ ∈ ℂ^{n} for which E_{ζ} = ∅ is pluripolar.^{5}

For the normalized homogeneous polynomial Q(x, z) = x^{m−1} − z^{m−1}, E(Q) is already known [8] to be nonempty. Therefore, taking n = m−2 and defining g_{m−2} ≡ 0 and g_{j}(ζ) = ζ_{j} for all 0 ≤ j ≤ m − 3 where ζ = (ζ_{0}, … ζ_{m−3}) ∈ ℂ^{m−2}, we obtain the following corollary.

Proposition 5.4. The set of all c = (c_{0}, … c_{m−3}) ∈ ℝ^{m−2} for which E(Q) = ∅ for Q = x^{m−1} + ∑_{j≤m−3} c_{j}z^{m−1−j}x^{j} is pluripolar.

Combining Proposition 5.4 with Theorem 1.4, we deduce Theorem 1.6 as well.

We turn to the detailed analysis of the nonlinear eigenvalue problem for L_{z} = −∂_{x}^{2} + Q(x, z)^{2}, assuming always that Q is normalized, homogeneous, and has real coefficients. Define P (x, z) = ∫_{0}^{x} Q(y, z) dy.

Lemma 5.5. Suppose that m is even. For each z ∈ ℂ there exist unique entire holomorphic functions x ↦ ψ_{z}^{±}(x), depending holomorphically on z and on the coefficients c_{j} of Q for fixed m, satisfying

L_{z}ψ_{z}^{±} = 0

and

\begin{cases}ψ_{z}^{+}(x) & = e^{−P (x,z)}( x)^{−(m−1)/2}(1 + O(|x|^{−1})) as ℝ ∋ x → +∞ \\ ψ_{z}^{−}(x) & = e^{−P (x,z)}(−x)^{−(m−1)/2}(1 + O(|x|^{−1})) as ℝ ∋ x → −∞. \end{cases}  (5.3)

Any solution f of L_{z}f = 0 on the real axis either tends to ∞ in modulus as x → +∞, or is a scalar multiple of ψ_{z}^{+}. Likewise any solution either tends to ∞ in modulus as x → −∞, or is a scalar multiple of ψ_{z}^{−}. For any Q, the asymptotics (5.3) remain valid for all x in some conic neighborhood of ℝ.

If m is odd then all assertions remain valid if the asymptotics for ψ_{z}^{−} are changed to

ψ_{z}^{−}(x) = e^{+P (x,z)}(−x)^{−(m−1)/2}(1 + O(|x|^{−1})) as ℝ ∋ x → −∞.

For each Q there exists C < ∞ such that

|ψ_{z}^{+}(x)| + |∂_{x}ψ_{z}^{+}(x)| ≤ C exp(C|z|^{m}) for all x ≥ 0

and

|ψ_{z}^{−}(x)| + |∂_{x}ψ_{z}^{−}(x)| ≤ C exp(C|z|^{m}) for all x ≤ 0.

5I am indebted for this device to a paper of D. Barrett [2]; it is a general idea in spectral theory

Lemma 5.6. Let Q be fixed. There exists C < ∞ such that for all z ∈ ℂ and all x ≥ C + C|z|, for k = 0, 1,

|∂_{x}^{k}ψ_{z}^{+}(x) − (−1)^{k}e^{−P (x,z)}x^{−(m−1)/2}Q(x, z)^{k}| ≤ C|e^{−P (x,z)}|x|^{−(m−3)/2}Q(x, z)^{k}|.

The corresponding bound holds for ψ_{z}^{−}(x) for all x ≤ −C − C|z|.

These two lemmas are proved as in [5] and as in the proofs of Lemmas 6.1 and 6.4 below, by rewriting the ordinary differential equation as a first order system, diagonalizing the system modulo a coefficient matrix that is suitably small for large |x|, and solving an integral equation. Holomorphic dependence on the coefficients c_{j} is not addressed in Lemmas 6.1 and 6.4, but follows directly from their proofs.

Central to our analysis is the Wronskian

W (z) = Det (\begin{matrix}ψ_{z}^{+} & ψ_{z}^{−} \\ ∂_{x}ψ_{z}^{+} & ∂_{x}ψ_{z}^{−} \end{matrix}) (0).  (5.4)

The function W in the proof of Theorem 5.2 may be interpreted as such a Wronskian.

Corollary 5.7. W has the following properties.

1. W depends holomorphically on z and on the coefficients c_{j} of Q.

2. z ∈ E(Q) if and only if W (z) = 0.

3. For each Q there exists a constant C such that |W (z)| ≤ C exp(C|z|^{m}) for all z ∈ ℂ.

Proof. The first two conclusions follow directly from Lemma 5.5. The third requires a small additional argument in order to pass from upper bounds on ψ_{z}^{±} at ±C|z|, respectively, to bounds at 0. Assume |z| to be large, Lemma 5.5 guarantees that ψ_{z}^{+} and its first derivative are O(exp(C|z|^{m})) at x = C|z|, if C is chosen to be sufficiently large. On the interval [−C|z|, C|z|], the potential Q^{2}(x, z) is O(|z|^{2(m−1)}). Therefore |∂_{x}^{2}ψ_{z}^{+}| ≤ C|z|^{2(m−1)}|ψ_{z}^{+}| on this interval. A simple comparison argument then gives ψ_{z}^{+} = O(exp(C|z|^{m})) at x = 0, and likewise for its first derivative. The same reasoning applies to ψ_{z}^{−}, yielding the desired bound on the Wronskian.

Proof of Proposition 5.3. By the first two conclusions of the preceding lemma, there exists an entire holomorphic function W (z, ζ) such that for each ζ ∈ ℂ^{n}, E_{ζ} = {z ∈ ℂ : W (z, ζ) = 0}. Thus the exceptional set of all ζ for which there exist no nonlinear eigenvalues z equals the set of all ζ for which z ↦ W (z, ζ) has no zeros. By hypothesis, there exists at least one ζ_{0} for which W (z, ζ_{0}) has at least one zero. Therefore the set of all ζ ∈ ℂ^{n} for which there are no zeros is pluripolar [1].

6. Analysis of special cases

This section is devoted to the proof of Proposition 1.7. Let Q(x, z) = x^{m−1} − z^{k}x^{m−k−1}. W (z) is an entire holomorphic function of z^{k}, so

W(ζ) = W (ζ^{1/k})

is independent of the choice of k-th root, and hence is also entire holomorphic. By Corollary 5.7, |W(z)| ≤ C exp(C|z|^{m/k}) for all z ∈ ℂ.

There are now two difficulties. Firstly, a superficial examination of [5],[8] might lead one to expect a lower bound |W (z)| ≥ c exp(c|z|^{m}) for all z ∈ ℝ, but this is false^{6} for Q = x^{2} + z^{2}. Secondly, whereas the strategy in [5] and other closely related works was to show that W is an entire function of nonintegral order and hence must have zeros, it turns out that for the class of examples contemplated in Proposition 1.7, W is indeed an entire function of the expected order m/k, but this order may be integral. In that case a more refined analysis is required; this same point arose in the proof of Theorem 5.2 above.

Factor

Q(x, z) = x^{m−1−k}(x^{k} − z^{k}).

As in the proof of Theorem 5.2, the zeros of ℝ ∋ x ↦ Q(x, z) are the key to the asymptotic behavior of W (z), for z^{k} ∈ ℝ. Assume always that z^{k} ∈ ℝ. Then x ↦ Q(x, z) has two zeros when k is odd, and has either one zero or three, depending on the sign of z^{k}, when k is even.

The differential equation is ∂_{x}^{2}ψ_{z}^{±} = Q(x, z)^{2}ψ_{z}^{±}. Its solutions ψ_{z}^{±} are real valued for real x, z^{k}. The asymptotics (5.3) imply that ψ_{z}^{+}(x) is positive and ∂_{x}ψ_{z}^{+}(x) negative for all sufficiently large x ∈ ℝ^{+}, and the differential equation then implies that ψ_{z}^{+} is a positive, convex, decreasing function of x on the whole real axis. Likewise ψ_{z}^{−} is positive, convex, and increasing. Consequently

W (z) = ψ_{z}^{+}(0)∂_{x}ψ_{z}^{−}(0) − ψ_{z}^{−}(0)∂_{x}ψ_{z}^{+}(0) > ψ_{z}^{+}(0)∂_{x}ψ_{z}^{−}(0).  (6.1)

Thus in order to derive a lower bound for W (z), it suffices to derive lower bounds for ψ_{z}^{+} and for ∂_{x}ψ_{z}^{−} at 0. We have P (x, z) = m^{−1}x^{m} − (m − k)^{−1}z^{k}x^{m−k}, so

P (z, z) = (m^{−1} − (m − k)^{−1})z^{m} = −c_{0}z^{m}

where c_{0} > 0.

6For Q = x^{2} + z^{2}, W is a polynomially bounded function on the real axis, as follows from the proof of Lemma 6.4 below, but satisfies the lower bound c exp(c|z|^{3}) on the imaginary axis (by the proof of Lemma 6.1).

Lemma 6.1. There exists c > 0 such that

ψ_{z}^{+}(z) ≥ exp(cz^{m})

for all sufficiently large z ∈ ℝ^{+}. If k is even, so that Q(−z, z) = 0, then likewise

∂_{x}ψ_{z}^{−}(−z) ≥ exp(cz^{m})

for all sufficiently large z ∈ ℝ^{+}.

Since ψ_{z}^{+}(z) < ψ_{z}^{+}(0) and ∂_{x}ψ_{z}^{−}(−z) < ∂_{x}ψ_{z}^{−}(0) by monotonicity, these same bounds hold at x = 0, as well. In particular, the lemma implies that W (z) ≥ exp(cz^{m}) as z → +∞, when k is even. More generally, it implies such a lower bound for W whenever we can show that for some finite exponent N, ∂_{x}ψ_{z}^{−}(0) ≥ z^{−N} as z → +∞.

Proof of Lemma 6.1. The analysis of ψ_{z}^{−}(−z) for even k is completely parallel to the analysis for ψ_{z}^{+}(z), so we treat only the latter. Set

σ = (m − 2)/2.

We will prove that for all sufficiently large ρ ∈ ℝ^{+}, for all x ≥ z + ρz^{−σ}, for k = 0, 1,

∂_{x}^{k}ψ_{z}^{+}(x) = (−1)^{k}Q(x, z)^{k− \frac{1}{2}} e^{−P (x,z)}(1 + o(1)) as ρ → +∞,  (6.2)

uniformly in x, z for all sufficiently large z ∈ ℝ^{+}. Since |P (z+ρz^{−σ}, z)−P (z, z)| ≤ C_{ρ} uniformly for z ≥ 1, since Q(z + ρz^{−σ}, z) ≥ cz^{−N} for some finite N, and since ψ_{z}^{+} is a decreasing function of x, this would imply that

ψ_{z}^{+}(z) ≥ ψ_{z}^{+}(z + ρz^{−σ}) ≥ exp(cz^{m})

for some c > 0 as z → +∞, as desired.

To simplify notation in the proof, write ψ = ψ_{z}^{+}, Q(x), P (x) for Q(x, z), P (x, z), and write f′ for ∂f/∂x. To prove (6.2) set

u = (\begin{matrix}ψ \\ ψ′ \end{matrix}) ,  A = (\begin{matrix}0 & 1 \\ Q^{2} & 0 \end{matrix})  (6.3)

so that u′ = Au. Set

S = (\begin{matrix}1 & 1 \\ Q − \frac{1}{2}Q′/Q & −Q − \frac{1}{2}Q′/Q \end{matrix}) for x > z,

in which region Q is always nonzero.

If ρ is chosen to be sufficiently large, then for all sufficiently large z and all x ≥ z + ρz^{−σ}, 0 < Q′(x, z)/Q^{2}(x, z) ≤ ε(ρ) where ε(ρ) → 0 as ρ → +∞; this calculation is contained in the proof of Lemma 6.2 below. We assume always that x ≥ z + ρz^{−σ}, where ρ is sufficiently large. S is then clearly invertible, and

S^{−1} = −\frac{1}{2}Q^{−1} (\begin{matrix}−Q − \frac{1}{2}Q′/Q & −1 \\ −Q + \frac{1}{2}Q′/Q & 1 \end{matrix}) .

This formal matrix calculation and others below may be found in [6], and the details will not be reproduced here.

The column vector v satisfies the first order system of equations

v′ = (B + E)v,  (6.4)

where

B = (\begin{matrix}Q − \frac{1}{2}Q′/Q & 0 \\ 0 & −Q − \frac{1}{2}Q′/Q \end{matrix})

and E is a continuous matrix valued function satisfying

|E| ≤ C(Q′)^{2}/|Q|^{3} + C|Q′′|/Q^{2}.  (6.5)

That is, each entry of E satisfies this upper bound, uniformly in x, z for x ≥ z +ρz^{−σ} for any fixed ρ, assuming always that z ≥ 1.

Set

w = (\begin{matrix}0 \\ Q^{−1/2}e^{−P} \end{matrix} ) ,

and

Λ(x) = (\begin{matrix}Q^{−1/2}e^{P} & 0 \\ 0 & Q^{−1/2}e^{−P} \end{matrix} )

so that Λ′ = BΛ. Define the integral operator

T f(x) = ∫_{x}^{∞} Λ(x)Λ^{−1}(y)(Ef)(y) dy,  (6.6)

acting on functions f defined on [z + ρz^{−σ}, ∞) and taking values in the space of column vectors with two (real) entries.

If v satisfies the integral equation

v(x) = w(x) − T v(x)  (6.7)

then it satisfies the differential equation v′ = (B + E)v. Indeed,

v′(x) = (w − T v)′(x) = w′(x) − ∫_{x}^{∞} Λ′(x)Λ^{−1}(y)(Ev)(y) dy + (Ev)(x)

= (Bw)(x) − ∫_{x}^{∞} B(x)Λ(x)Λ^{−1}(y)(Ev)(y) dy + (Ev)(x)

= Bw(x) − B(x)(T v)(x) + (Ev)(x)
= Bv(x) + Ev(x).

Denote by ε(ρ, z) any quantity that tends to zero as min(ρ, z) → ∞; this quantity is permitted to change from one line to the next.

Lemma 6.2. The remainder coefficient matrix E satisfies

∫_{z+z^{−σ}}^{∞} |E(y)| dy ≤ C < ∞,

uniformly as z → +∞. Moreover

∫_{z+ρz^{−σ}}^{∞} |E(y)| dy ≤ ε(ρ, z).

Proof. Write A ∼ B to mean that A, B are positive quantities whose ratio is bounded above and below by positive constants independent of z, provided that z is sufficiently large. Assume always that x ≥ z ≫ 1. Then

|Q(x, z)| ∼ x^{m−2}(x − z)
|Q′(x, z)| ≤ Cx^{m−2}
|Q′′(x, z)| ≤ Cx^{m−3},

where all derivatives indicated are taken with respect to x. Recalling the pointwise bound (6.5) for E, we obtain

∫_{2z}^{∞} |E(x, z)| dx ≤ C ∫_{2z}^{∞}[x^{2(m−2)−3(m−1)} + x^{m−3−2(m−1)}] dx ≤ Cz^{−m}.

Likewise

∫_{z+ρz^{−σ}}^{2z} |E(x, z)| dx ≤ C ∫_{z+ρz^{−σ}}^{2z}[x^{2(m−2)−3(m−2)}(x − z)^{−3} + x^{m−3−2(m−2)}(x − z)^{−2}] dx
≤ Cz^{2−m} ∫_{z+ρz^{−σ}}^{2z}(x − z)^{−3} dx + Cz^{1−m} ∫_{z+ρz^{−σ}}^{2z}(x − z)^{−2} dx
≤ Cz^{2−m}(ρz^{−σ})^{−2} + Cz^{1−m}(ρz^{−σ})^{−1}
= Cρ^{−2}z^{0} + Cρ^{−1}z^{1−m+ \frac{m}{2} −1}
≤ ε(ρ, z).

For each x ≥ z + z^{−σ} introduce the norm

∥f∥_{∗,x} = sup_{y≥x} Q^{1/2}(y)e^{P (y)}|f(y)|  (6.8)

on the space of all continuous 2 dimensional column vector valued functions on [x, ∞).

Lemma 6.3. For each ρ ≥ 1, for all x ≥ z + ρz^{−σ},

∥T f∥_{∗,x} ≤ C_{ρ}∥f∥_{∗,x}

for all f for which the right hand side is finite, uniformly for all z ≥ 1 and uniformly in x. Moreover C_{ρ} → 0 as ρ → +∞.

Proof. Suppose that ∥f∥_{∗,x} ≤ 1. Then

|T f(x)|

≤ ∫_{x}^{∞}|(\begin{matrix}Q^{−1/2}(x)e^{P (x)}Q^{1/2}(y)e^{−P (y)} & 0 \\ 0 & Q^{1/2}(x)e^{−P (x)}Q^{−1/2}(y)e^{P (y)} \end{matrix})|

· |E(y)| Q^{−1/2}(y)e^{−P (y)} dy
≤ Q^{−1/2}(x)e^{−P (x)} ∫_{x}^{∞} |E(y)|dy
≤ CQ^{−1/2}(x)e^{−P (x)}

by Lemma 6.2, because e^{−P} and Q^{−1/2} are both decreasing functions of x for x ≥ z.

If ρ is chosen to be sufficiently large, then we find that the map f ↦ w − T f is a strict contraction on the space of all continuous functions on [z + ρz^{−σ}, ∞) for which ∥f∥_{∗,[z+ρz^{−σ}]} is finite. Therefore by the contraction mapping principle, there exists a unique solution of the integral equation (6.7) for which this norm is finite. We define v to be that solution.

It follows from the same reasoning together with the second conclusion of Lemma 6.2 that

|(v − w)(y)| ≤ ε(ρ)Q^{−1/2}(y)e^{−P (y)}  (6.9)

for all y ≥ z + ρz^{−σ}, where ε(ρ) → 0 as ρ → ∞.

Defining u = Sv, u satisfies the differential equation u′ = Au. Thus

u = Sv = (\begin{matrix}Q^{−1/2}e^{−P} (1 + ε(ρ)) \\ Q^{+1/2}e^{−P} (1 + ε(ρ)) \end{matrix})

for all x ≥ z + ρz^{−σ}, since |Q′|/Q^{2} → 0 as ρ → ∞ in that region, uniformly for z ≥ 1. ψ_{z}^{+} is defined (for large z ∈ ℝ^{+}) in terms of u by equation (6.3). In particular,

ψ_{z}^{+}(x) = Q^{−1/2}(x, z)e^{−P (x,z)}(1 + ε(ρ))

for x ≥ z + ρz^{−σ}. At x = z + ρz^{−σ}, this is bounded below by e^{cz^{m}} for some c > 0, as z → +∞. This concludes the proof of Lemma 6.1.

Lemma 6.4. If k is odd then there exist C, N ∈ ℝ^{+} such that for all z ≥ 1,

C^{−1}z^{−N} ≤ ψ_{z}^{−}(0) ≤ Cz^{N},

and the same holds for ∂_{x}ψ_{z}^{−}(0).

Proof. Consider first the case where m is even. Let B, E, W, S, Λ be as defined in the proof of Lemma 6.1. Then v′ = (B + E)v. Redefine

T f(x) = ∫_{−∞}^{x} Λ(x)Λ^{−1}(y)(Ev)(y) dy

and denote by v a putative solution of the integral equation

v = w + T v.

Redefine

σ = k/(m − k).

We again have

∫_{−∞}^{−ρz^{−σ}} |E(x, z)| dx ≤ Cε(ρ, z).

Indeed, for z ≥ 1 and x ≤ 0,

|Q(x, z)| ∼ |x|^{m−k−1}(|x|^{k} + z^{k})
|Q′(x, z)| ≤ C|x|^{m−k−2}(|x|^{k} + z^{k})
|Q′′(x, z)| ≤ C|x|^{m−k−3}(|x|^{k} + z^{k}).

Thus

∫_{−∞}^{−z} |E(x, z)| dx ≤ C ∫_{−∞}^{−z}[|x|^{2(m−2)−3(m−1)} + |x|^{m−3−2(m−1)}] dx ≤ Cz^{−m},

while the integral of |E(x, z)| over [−z, −ρz^{−σ}] is majorized by

C ∫_{−z}^{−ρz^{−σ}}[z^{2k−3k}|x|^{2(m−k−2)−3(m−k−1)} + z^{k−2k}|x|^{m−k−3−2(m−k−1)}] dx
≤ Cz^{−k} ∫_{−z}^{−ρz^{−σ}} |x|^{−m+k−1} dx
≤ Cz^{−k}(ρz^{−σ})^{k−m}
= Cρ^{k−m}.

Redefine

∥f∥_{∗,x} = sup_{y≤x} |Q|^{1/2}(y)e^{P (y)}|f(y)|.

Because k is odd, |Q|^{−1/2}e^{−P} is a monotone increasing function on (−∞, 0]. Therefore for x ≤ 0, for any f with finite ∗, x norm,

|T f(x)| ≤ ∥f∥_{∗,x} ∫_{−∞}^{x}[|Q|^{−1/2}(y)e^{−P (y)} + |Q|^{1/2}(x)e^{P (x)}|Q|^{−1}(y)e^{−2P (y)}]|E(y)| dy
≤ ∥f∥_{∗,x}|Q|^{−1/2}(x)e^{−P (x)} ∫_{−∞}^{x} |E(y)| dy.

Thus for x ≤ −ρz^{−σ},

∥T f∥_{∗,x} ≤ ε(ρ, z)∥f∥_{∗,x}.

Consequently, when ρ, z are both sufficiently large, T is a strict contraction on the space of all continuous two dimensional column vector valued functions f having finite ∗, −ρz^{−σ} norms, and so there exists a unique solution with finite norm, v, of the integral equation v = w + T v. Moreover ∥v − w∥_{∗,−ρz^{−σ}} ≤ ε(ρ, z).

Defining u = Sv, we have

u = (\begin{matrix}ψ_{z}^{−} \\ ∂_{x}ψ_{z}^{−} \end{matrix}) .

The equation v = w +T v and bound on T v = v −w bound obtained in the preceding paragraph imply that

u = (\begin{matrix}|Q|^{−1/2}e^{−P} · (1 + ε(ρ, z)) \\ |Q|^{+1/2}e^{−P} · (1 + ε(ρ, z)) \end{matrix})

for x ≤ −ρz^{−σ}. At x = −ρz^{−σ}, |Q| ∼ z^{k}|x|^{m−k−1} ∼ z^{k−[(m−k−1)k/(m−k)]} = z^{k/(m−k)} = z^{σ}. For −ρz^{−σ} ≤ x ≤ 0 we have Q(x) = O(z^{σ}), so for all such x, |P (x, z)| = | ∫_{0}^{x} Q(y, z)dy| = O(1). Thus

C^{−1}z^{−σ/2} ≤ ψ_{z}^{−}(−ρz^{−σ}) ≤ Cz^{−σ/2},
C^{−1}z^{+σ/2} ≤ ∂_{x}ψ_{z}^{−}(−ρz^{−σ}) ≤ Cz^{+σ/2}.

For x ∈ [−ρz^{−σ}, 0] we have a differential inequality

|(ψ_{z}^{−})′′(x)| ≤ Cz^{2σ}ψ_{z}^{−}(x).

From a simple comparison argument and the upper bounds for ψ_{z}^{−} and its first derivative at −ρz^{−σ}, it follows that ψ_{z}^{−}(0) = O(z^{−σ/2}) and ∂_{x}ψ_{z}^{−}(0) = O(z^{σ/2}). On the other hand, convexity of ψ_{z}^{−} implies that

∂_{x}ψ_{z}^{−}(0) > ∂_{x}ψ_{z}^{−}(−ρz^{−σ}) ≥ C^{−1}z^{σ/2}.

The case of odd m is treated in the same way, except that the formulas are changed as in Lemma 5.5, with P replaced by −P throughout.

Corollary 6.5. For every m ≥ 2 and k ≥ 1, W is an entire holomorphic function of order exactly m.

Proof. If k is even then

ψ_{z}^{+}(0) ≥ ψ_{z}^{+}(z) ≥ e^{cz^{m}},  (6.10)

while

∂_{x}ψ_{z}^{−}(0) ≥ ∂_{x}ψ_{z}^{−}(−z) ≥ e^{cz^{m}}

and hence

W (z) > e^{cz^{m}}

for all large z ∈ ℝ^{+}. Since |W (z)| ≤ C exp(C|z|^{m}) for all z ∈ ℂ, W has order exactly m.

If k is odd then (6.10) remains valid, while the lower bound of Lemma 6.4 implies that as ℝ ∋ z → +∞,

W (z) > ψ_{z}^{+}(0)∂_{x}ψ_{z}^{−}(0) ≥ C^{−1}z^{−N}e^{cz^{m}} > e^{δz^{m}}

for some δ > 0.

Lemma 6.6. If k is even then there exist C, N ∈ ℝ^{+} such that whenever z^{k} ∈ ℝ^{−} and |z| sufficiently large,

C^{−1}|z|^{−N} ≤ |W (z)| ≤ C|z|^{N}.

Proof. For x ∈ ℝ and z^{k} ∈ ℝ^{−}, for k even, |Q(x, z)| ∼ |x|^{m−k−1}(|x|^{k} + |z|^{k}). Both ψ_{z}^{±} may therefore be treated as was ψ_{z}^{−} in Lemma 6.4, for x ≥ 0 in the case of the plus sign, and for x ≤ 0 in the case of the minus sign.

Lemma 6.7. For any m ≥ 2 and k ≥ 1,

|W (z)| ≤ C|z|^{N}

for all |z| ≥ 1 such that z^{k} ∈ iℝ.

Proof. When z^{k} ∈ iℝ and x ∈ ℝ, |Q(x, z)| ∼ |x|^{m−k−1}(|x|^{k} + |z|^{k}). Now |e^{−P (x)}| ≡ e^{−x^{m}/m}, so the ∗, x norms are slightly different though they are defined by the same formal expressions. |e^{−P} | and |Q| are still monotone. The proof is thus nearly identical to that of Lemma 6.4.

With these lemmas in hand we now prove Proposition 1.7. Any entire holomorphic function F of finite, strictly positive order either has infinitely many zeros, or takes the form exp(R) for some polynomial R whose degree equals the order of F. From Corollary 6.5 we know that in all cases of Proposition 1.7, W is an entire function of order exactly m/k. Hence if m/k is not an integer, W must have zeros.

Note that whenever k is odd, W is an even function, since ∂_{x}^{2} and (x^{m−1}−ζx^{m−k−1})^{2} are invariant under the substitution (x, ζ) ↦ (−x, −ζ). Consequently if W had no zeros and hence equalled exp(R) for some polynomial, R would also have to be even, and hence R would have to have even degree, so m/k would be forced to be even.

Suppose m is odd. If k is even, then m/k is not integral, so W has zeros. If k is odd, then m/k is odd, and hence by the preceding paragraph, W must have zeros.

Suppose k is even. By Lemma 6.6, |W(ζ)| and its reciprocal have at most polynomial growth as ℝ^{−} ∋ ζ → −∞, while |W(ζ)| ≥ exp(cζ^{m/k}) as ℝ^{+} ∋ ζ → +∞. There exists no polynomial R of degree m/k such that exp(R(ζ)) behaves in this way, so W cannot have a polynomial logarithm, hence must have zeros.

Suppose m is divisible by 4. By the preceding paragraph it suffices to examine the case where k is odd. Then the order m/k of W is also divisible by 4. Suppose W = exp(R) where R is a polynomial. Then since |W(ζ)| ≥ exp(c|ζ|^{m/k}) for large ζ ∈ ℝ^{+}, we have Re(R(ζ)) ≥ c|ζ|^{m/k} for large ζ ∈ ℝ^{+}, for some c > 0. Since the degree m/k of R is divisible by 4, this forces Re(R(ζ)) ≥ |ζ|^{m/k}/2 whenever ζ ∈ iℝ has sufficiently large modulus. This contradicts Lemma 6.7. The proof of Proposition 1.7 is therefore complete.

7. Sufficiency

In this section we outline a proof of Theorem 1.2. The method is a straightforward combination of the FBI transform with a subset of the machinery used in Section 4 to prove the negative results.

Define an FBI transform of any compactly supported function by

Fu(x, t; ξ, τ) = ∫_{ℝ^{2}} e^{i[(x−y)ξ+(t−s)τ]−|τ|(x−y)^{2}−|τ|(t−s)^{2}} u(y, s) dy ds.  (7.1)

It will always be assumed that |ξ| ≤ |τ|. For any point p = (x_{0}, t_{0}; ξ_{0}, τ_{0}) ∈ T^{∗}ℝ^{2} with |ξ_{0}| < |τ_{0}|, p belongs to the complement of W F_{a}(u), the analytic wave front set of u, if and only if there exist ε > 0 and a conic neighborhood Γ of p such that

|Fu(x, t; ξ, τ)| ≤ C exp(−ε|τ|)  (7.2)

for all (x, t; ξ, τ) ∈ Γ [17]. Fix a coordinate system in which the span of {X_{1}, X_{2}} coincides with the span of {∂_{x}, x^{m−1}∂_{t}}. Suppose that Lu ∈ C^{ω} in some neighborhood U of 0; we wish to prove that u ∈ C^{ω} in the intersection of U with a fixed neighborhood U_{0} of 0. Analyticity at other points follows by ellipticity where x ≠ 0, and by a change of coordinates to reduce the case x = 0 to x = t = 0. Since L is C^{∞} hypoelliptic, u ∈ C^{∞} near 0.

The analytic wave front set of u is contained in the union of the characteristic variety Σ of L with the analytic wave front set of Lu [17],[13]. In our coordinates, Σ is simply {(x, t, ξ, τ) : x = ξ = 0}. Thus in order to prove that u ∈ C^{ω}(U ∩ U_{0}), it suffices to prove that (7.2) holds for all (x′, t′) in any fixed compact subset of U ∩ U_{0} and for all (ξ, τ) satisfying |ξ| ≤ |τ|.

Fix any (ξ, τ) satisfying |ξ| ≤ |τ| and a point (x′, t′) near 0. Define a differential operator L_{τ}, acting with respect to the variables (x, t) and depending on the parameters (t′, τ), by

L_{τ} = (e_{(t′,τ)})^{−1} ∘ L^{∗} ∘ e_{(t′,τ)}

where e_{(t′,τ)}(x, t) = exp(i(t′ − t)τ − |τ|(t′ − t)^{2}) and L^{∗} denotes the transpose of L. The notation L_{τ} is misleading because the operator depends on t′ as well, but the dependence on τ is of greater importance.

The following lemma is analogous to Lemma 4.1. Denote by D_{δ} the open disk in ℂ of radius δ, centered at the origin.

Lemma 7.1. For any sufficiently small δ > 0 there exist C, ε ∈ ℝ^{+} such that for each (x′, t′) ∈ ℝ^{2} satisfying |(x′, t′)| ≤ δ/8 and for each (ξ, τ) ∈ ℝ^{2} satisfying |ξ| ≤ |τ|, there exists a function g = g_{(x′,t′,ξ,τ)}(x, t) defined for (x, t) ∈ (−δ, δ) × D_{δ} and holomorphic with respect to t, satisfying the equation

L_{τ}g(x, t) = e^{i(x′−x)ξ−|τ|(x′−x)^{2}} + O(e^{−ε|τ|})  (7.3)

with the bounds

|g(x, t)| ≤ Ce^{−ε|τ|}  for all |x| ≥ δ/4 and t ∈ D_{δ}  (7.4)
|g(x, t)| ≤ C  for all (x, t) ∈ [−δ, δ] × D_{δ}.  (7.5)

Proof of Theorem 1.2. Granting the lemma, and supposing that Lu ∈ C^{ω} near 0, there exist δ and a C^{∞} function v supported in (−δ, δ)×(−2δ, 2δ), such that Lv ≡ Lu for all (x, t) ∈ ℝ^{2} satisfying |x| ≤ δ/2 and |t| ≤ δ. Consider any (ξ, τ) satisfying |ξ| ≤ |τ|, and suppose for simplicity of notation that τ > 0. Consider likewise any (x′, t′) ∈ ℝ^{2} sufficiently close to the origin. Then writing g = g_{(x′,t′,ξ,τ)},

Fv(x′, t′; ξ, τ) = ^{∫∫} v(x, t)e^{i(x′−x)ξ−τ(x′−x)^{2}}e^{i(t′−t)τ−τ(t′−t)^{2}} dx dt
= ^{∫∫} ve^{i(t′−t)τ−τ(t′−t)^{2}}L_{τ}g + O(e^{−ετ})
= ^{∫∫} v · L^{∗}(e^{i(t′−t)τ−τ(t′−t)^{2}}g) + O(e^{−ετ})
= ^{∫∫} Lv · e^{i(t′−t)τ−τ(t′−t)^{2}}g + O(e^{−ετ}).

Replace the two dimensional contour of integration by the contour in ℝ × ℂ parametrized by (x, t) ↦ (x, t + ih(x, t)) where h is smooth, takes values in (−1/2, 0], is identically equal to zero where |(x, t)| ≥ δ/2, and is strictly negative but small where |(x, t)| < δ/2. Since Lv is a holomorphic function of t where |(x, t)| < δ/2, we obtain

Fv(x′, t′; ξ, τ) = O(e^{−ετ})

+^{∫∫} Lv(x, t + ih(x, t))e^{i(t′−t−ih(x,t))τ−τ(t′−t−ih(x,t))^{2}}g(x, t + ih(x, t)) J(x, t) dx dt

where J(x, t) = 1 + i∂h/∂t.

Where |(x, t)| ≤ δ/2, both Lv(x, t_{i}h) and g(x, t + ih) are bounded uniformly in (x′, t′; ξ, τ), while the real part of i(t′ − t − ih(x, t))τ − τ(t′ − t − ih(x, t))^{2} equals [h(x, t) − h(x, t)^{2}]τ, which is ≤ −ετ for some ε > 0, provided that (x′, t′) lies in a sufficiently small neighborhood of 0 that is independent of ξ, τ. Thus the exponential factor in the integral is O(exp(−ετ)), hence so is the integrand.

Where |x| ≥ δ/4, both Lv and the exponential factor are still O(1), and g is O(exp(−ετ)). Lastly, where |t| ≥ δ/4, the exponential factor is O(exp(−ετ)) and Lv, g are O(1). Thus Fv(x′, t′; ξ, τ) = O(exp(−ετ)), as desired.

The proof of Lemma 7.1 is parallel to that of Lemma 4.1, with one essential difference: the real part of Θ̃(x, t)^{2} is nonnegative for all (x, t) in some neighborhood of 0 in ℝ × ℂ if (and only if) Θ(x, t) ≡ x^{m−1}. Consequently the proof of Lemma 4.1 may be executed in a fixed neighborhood of 0 with respect to the t coordinate, rather than in a neighborhood that shrinks to {0} as τ → ∞. This in turn allows us to carry out the Neumann series to cτ terms, rather than to cτ^{p/m} terms, resulting in an error that is O(exp(−ετ)), rather than merely O(exp(−ετ^{p/m})) where p/m = 1 − (mq)^{−1}.

Denote by D any open disk in ℂ centered at the origin. The Sobolev spaces appropriate for the Neumann series argument in the present context are as follows:

∥f∥_{H_{ρ,τ}^{0} (ℝ×D)}^{2} = ^{∫∫} _{ℝ×D} |f(x)|^{2} (1 + τ^{2/m}x^{2})^{−(m−1)} e^{τρη(x)} dx dtdt̄
∥f∥_{H_{ρ,τ}^{1} (ℝ×D)}^{2} = ^{∫∫} _{ℝ×D} ((1 + τ^{2/m}x^{2})^{−(m−1)}|∂_{x}f|^{2} + τ^{2/m}|f|^{2}) e^{τρη(x)} dx dtdt̄
∥f∥_{H_{ρ,τ}^{2} (ℝ×D)}^{2} = ^{∫∫} _{ℝ×D} ((1 + τ^{2/m}x^{2})^{−(m−1)}|∂_{x}^{2}f|^{2} + τ^{2/m}|∂_{x}f|^{2}
+ τ^{4/m}(1 + τ^{2/m}x^{2})^{(m−1)}|f|^{2}) e^{τρη(x)} dx dtdt̄,

where η ∈ C_{0}^{∞} is nonnegative, η(x) ≡ 0 for all |x| ≤ δ/8 and η(x) > 0 for all |x| ≥ δ/4. The parameter ρ is to be chosen to be positive but sufficiently small, and is independent of τ.

In order to work in these spaces we must modify L so that its coefficients are defined for all (x, t) in the Cartesian product of ℝ with a complex neighborhood of 0, and are holomorphic with respect to t. Θ̃(x, t) is identically equal to x^{m−1} near 0, so we extend it to equal x^{m−1} everywhere. L takes the form (a_{1,1}∂_{x} +a_{1,2}Θ̃∂_{t})^{2} +(a_{2,1}∂_{x} +a_{2,2}Θ̃∂_{t})^{2} near 0, so it suffices to extend the coefficients a_{i,j} to ℝ×ℂ so that they are independent of x outside a small neighborhood of the origin, and so that the coefficient matrix (a_{i,j}) is real and invertible everywhere on the product of ℝ with a small real neighborhood of 0, and depends holomorphically on t in ℝ × D_{δ} for some δ > 0.

After conjugation, ∂_{t} takes the form

e_{(t′,τ)}^{−1} ∘ ∂_{t} ∘ e_{(t′,τ)} = ∂_{t} − iτ + 2(t′ − t)τ = −iτ(1 + 2i(t′ − t)) + ∂_{t}.

Write

L^{∗} = P (x, t, ∂_{x}, Θ̃∂_{t}) = (a_{1,1}∂_{x} + a_{1,2}Θ̃∂_{t})^{2} + (a_{2,1}∂_{x} + a_{2,2}Θ̃∂_{t})^{2} + b_{1}∂_{x} + b_{2}Θ̃∂_{t} + b_{3}

where the coefficients a_{i,j} and b_{i} are real and analytic. Then the conjugated operator L_{τ} takes the form

L_{τ} = P (x, t, ∂_{x}, [−iτ(1 + 2i(t′ − t)) + ∂_{t}]Θ̃).

Define the ordinary differential operators

A_{t} = P (x, t, ∂_{x}, [−iτ(1 + 2i(t′ − t))]Θ̃).

which depend on the parameters t, t′, τ. Define an operator A, acting on functions of (x, t) ∈ ℝ × D that are holomorphic with respect to t in a small disk D ⊂ ℂ centered at 0, by letting A_{t} act with respect to the variable x, for each t.

Lemma 7.2. For all t, t′ ∈ ℂ and ρ ∈ ℝ sufficiently close to 0 and all sufficiently large τ ∈ ℝ^{+}, the map

A_{t} : H_{ρ,τ}^{2}(ℝ) ↦ H_{ρ,τ}^{0}(ℝ)

is invertible, uniformly in t, t′, τ. For all sufficiently small t′, δ, ρ and sufficiently large τ, the map

A : H_{ρ,τ}^{2}(ℝ × D_{δ}) ↦ H_{ρ,τ}^{0}(ℝ × D_{δ})

is invertible, uniformly in t′, δ, τ.

Proof. The main point is the inequality

∥ϕ∥_{H_{0,τ}^{2} (ℝ)}^{2} ≤ C∥A_{t}ϕ∥_{H_{0,τ}^{0} (ℝ)}^{2} for all ϕ ∈ C_{0}^{2}(ℝ).  (7.6)

To prove this, consider

− Re ∫_{ℝ} A_{t}ϕ · ϕ̄ ≥ c ∫_{ℝ} |∂_{x}ϕ|^{2} + τ^{2}x^{2(m−1)}|ϕ|^{2} − C ∫_{ℝ} |ϕ| · (|∂_{x}ϕ| + τ|x^{m−1}ϕ| + |ϕ|)

for some c, C ∈ ℝ^{+}, provided that |t − t′| is sufficiently small. Since

τ^{2/m} ∫_{ℝ} |ϕ|^{2} ≤ C ∫_{ℝ} |∂_{x}ϕ|^{2} + τ^{2}x^{2(m−1)}|ϕ|^{2},

the last inequality plus Cauchy-Schwarz lead to

− Re ∫_{ℝ} A_{t}ϕ · ϕ̄ ≥ c′ ∫_{ℝ} |∂_{x}ϕ|^{2} + τ^{2/m}(1 + τ^{1/m}|x|)^{2(m−1)}|ϕ|^{2}

for some c′ > 0, provided that τ is sufficiently large. Applying Cauchy-Schwarz to the left hand side yields

\begin{aligned}∫_{ℝ} |∂_{x}ϕ|^{2} + τ^{2/m}(1 + τ^{1/m}|x|)^{2(m−1)}|ϕ|^{2} & ≤ C ∫_{ℝ} |A_{t}ϕ|^{2}τ^{−2/m}(1 + τ^{1/m}|x|)^{−2(m−1)} \\ & ≤ Cτ^{2/m}∥A_{t}ϕ∥_{H_{0,τ}^{0}}^{2} . \end{aligned} (7.7)

In order to prove (7.6), it remains only to bound the H_{0,τ}^{0} norm of ∂_{x}^{2}ϕ by the H_{0,τ}^{0} norm of A_{t}ϕ. The operator A_{t} is a quadratic polynomial, with C^{ω} coefficients, in ∂_{x} and τΘ̃. The result of applying any monomial of degree two or less, excepting only ∂_{x}^{2}, to ϕ has already been estimated in the H_{0,τ}^{0} norm in (7.7). The missing term ∂_{x}^{2} may be expressed as a linear combination, with C^{ω} coefficients, of A_{t} and all the other monomials. Therefore the H_{0,τ}^{0} norm of ∂_{x}^{2}ϕ is majorized by

∥∂_{x}^{2}ϕ∥_{H_{0,τ}^{0}}^{2} ∼ ∫_{ℝ}(1 + τ^{1/m}|x|)^{−2(m−1)}|∂_{x}^{2}ϕ|^{2} dx
≤ C∥A_{t}ϕ∥_{H_{0,τ}^{0}}^{2} + C ∫_{ℝ} τ^{2/m}|∂_{x}ϕ|^{2} + Cτ^{4/m}(1 + τ^{1/m}|x|)^{2(m−1)}|ϕ|^{2}
≤ C∥A_{t}ϕ∥_{H_{0,τ}^{0}}^{2} .

Thus we obtain the inequality (7.6) for ρ = 0. The case of small |ρ| follows from the case ρ = 0 by conjugating with e^{ρτη(x)}, as in the final paragraph of the proof of Lemma 3.1 of [3].

The spaces H_{ρ,τ}^{k} are defined so that A_{t} is automatically a bounded operator from H^{2} to H^{0}. Inequality (7.6) implies that A_{t} has closed range; the same analysis applies to its transpose and consequently invertibility of A_{t} follows from a duality argument as in the proof of Lemma 3.1 of [3]. Since A_{t} depends holomorphically on t and is invertible uniformly in t, A_{t}^{−1} also depends holomorphically on t. Therefore as an operator from H_{ρ,τ}^{2}(ℝ × D_{δ}) to H_{ρ,τ}^{0}(ℝ × D_{δ}), A may be inverted by applying the inverse of A_{t} in the x variable for each value of t.

With Lemma 7.2 in hand, imitating the proof of Lemma 4.1 leads directly to the conclusions of Lemma 7.1, hence to Theorem 1.2.

8. Microlocal analytic hypoellipticity for factored operators

Let X_{1}, X_{2} be vector fields defined in a neighborhood of p = 0, satisfying the hypotheses of Theorem 1.8. Assume that X_{1}, X_{2} are linearly dependent at the point 0. Suppose that L = (X_{1} + iX_{2})(X_{1} − iX_{2}) + c_{1}X_{1} + c_{2}X_{2} + c_{3} where c_{j} ∈ C^{ω} are complex valued.

Let (x, t) be a system of coordinates with the properties of Lemma 1.3. Writing X_{1} = a_{1,1}∂_{x} + a_{1,2}Θ̃∂_{t} and X_{2} = a_{2,1}∂_{x} + a_{2,2}Θ̃∂_{t},

[X_{1}, X_{2}] = Det(a)(∂_{x}Θ̃)∂_{t} + O(X_{1}, X_{2})

where Det(a) is the determinant of the matrix a = (a_{i,j}). This determinant is real valued and nonvanishing near 0 in ℝ^{2}; by replacing t by −t if necessary, we may require it to be negative.

Let (ξ, τ) be coordinates dual to (x, t). Near 0, decompose the characteristic variety Σ of L as the disjoint union Σ^{+} ∪ Σ^{−}, where Σ^{+} = Σ ∩ {τ > 0}. Note that the fiber of Σ over 0 is {(ξ, τ) : ξ = 0}.

Lemma 8.1. Assume that {X_{1}, X_{2}} is a pseudoconvex pair near 0. Then the function ∂_{x}Θ is everywhere nonnegative, in some neighborhood of 0, and the type m at 0 is even.

Proof. The pseudoconvexity hypothesis is

∂_{x}Θ̃ = h + bΘ̃,  (8.1)

where h, b are smooth functions and h does not change sign. Consider first the case where h ≥ 0.

Expand ∂_{x}Θ̃ and Θ̃ in Taylor series, and assign weights 1, q^{−1} respectively to x, t, as in Section 2. Only monomials with weights ≥ m − 2 arise in the expansion of ∂_{x}Θ̃, and the sum of all monomials having weight m − 2 is equal to ∂_{x}Θ. Any smooth multiple of Θ̃ itself involves only terms of weights m − 1 and greater. Thus h = ∂_{x}Θ modulo terms of weight > (m − 2). Consequently for any (y, s),

∂_{x}Θ(y, s) = lim_{ε→0} ε^{−(m−2)}∂_{x}Θ̃(εy, ε^{1/q}s)

= lim_{ε→0} ε^{−(m−2)}∂_{x}Θ̃(εy, ε^{1/q}s) − ε^{−(m−2)}b(εy, ε^{1/q}s)θ̃(εy, ε^{1/q}s)

= lim_{ε→0} ε^{−(m−2)}h(εy, ε^{1/q}s)

is a limit of nonnegative quantities, hence must be nonnegative.

In the case h ≤ 0, the same reasoning leads to the conclusion that ∂_{x}Θ(x, t) ≤ 0 for all (x, t) ∈ ℝ^{2}. This is impossible, since ∂_{x}Θ(x, 0) ≡ (m − 1)x^{m−2}. Likewise nonnegativity of ∂_{x}Θ forces m to be even, for otherwise (m − 1)x^{m−2} would change sign.

The principal symbol of the commutator satisfies

σ_{1}(i[X_{1}, X_{2}]) = − Det(a)(∂_{x}Θ̃)τ + O(σ_{1}(iX_{1}), σ_{1}(iX_{2})).

Thus modulo the span of the symbols of X_{1}, X_{2}, this is nonnegative when τ ≥ 0. The function Θ̃ cannot vanish identically along any line segment where the t coordinate is constant, by the bracket hypothesis. Therefore arbitrarily close to any point where Θ̃(x, t) = 0 there exist points where Θ̃ ≠ 0 but the ratio |∂_{x}Θ̃/Θ̃| is arbitrarily large. For (x, t, ξ, τ) ∈ Σ^{−} one has τ < 0, so by the pseudoconvexity hypothesis, −τ∂_{x}Θ̃(x, t) ≥ 0 modulo a bounded multiple of |τΘ̃(x, t)|, so τ∂_{x}Θ̃(x, t) must be negative, even modulo any bounded multiple of |τΘ̃(x, t)|, wherever the ratio |∂_{x}Θ̃/Θ̃| is sufficiently large. Consequently Σ^{−} has no conic neighborhood in which τ∂_{x}Θ̃ is nonpositive modulo the span of the symbols of X_{1}, X_{2}, so the half line bundle Σ^{+} is uniquely specified by the existence of such a neighborhood.

Consider first the case where span{X_{1}, X_{2}} ≡ span{∂_{x}, x^{m−1}∂_{t}}, in which microlocal analytic hypoellipticity is to be proved by the same method as in Section 7. Then Θ̃(x, t) ≡ x^{m−1}. Let L_{τ} be the operator defined by conjugating the transpose of L as in Section 7. The ordinary differential operators A_{t} of that section should now be replaced by

A_{t} = [(a_{1,1} − ia_{2,1})∂_{x} + (a_{1,2} − ia_{2,2})iτ(−1 − 2i(t′ − t))Θ̃]

∘ [(a_{1,1} + ia_{2,1})∂_{x} + (a_{1,2} + ia_{2,2})iτ(−1 − 2i(t′ − t))Θ̃] + O(∂_{x}, τΘ̃, 1) where the coefficients in the term O(∂_{x}, τΘ̃, 1) are C^{ω} and are bounded uniformly in t, x, t′, τ. Here the coefficients a_{i,j} are still functions of (x, t); we do not freeze coefficients as was done in Section 4. The principal part of A_{t} is −Y Y^{∗}, where

Y = (a_{1,1} − ia_{2,1})∂_{x} − (a_{1,2} − ia_{2,2})iτΘ̃

and for any operator T, we now denote by T^{∗} the formal adjoint of T, with respect to the usual Hilbert space structure of L^{2}(ℝ), rather than the transpose. The coefficients a_{i,j} are real valued for (x, t) ∈ ℝ^{2}. Therefore for (x, t) ∈ ℝ × ℂ,

−A_{t} = Y ∘ Y^{∗} + O(∂_{x}, τΘ̃, 1) + O(| Im(t)|)O(∂_{x}, τΘ̃)^{2},

that is, −A_{t} − Y Y^{∗} is a quadratic polynomial, with real analytic coefficients, in ∂_{x}, τΘ̃, 1 where 1 denotes the identity operator, and the coefficients of all monomials of degree 2 in ∂_{x}, τΘ̃ are O(| Im(t)|).

For any ϕ ∈ C_{0}^{2}(ℝ),

∫_{ℝ} Y Y^{∗}ϕ · ϕ̄ = ∥Y^{∗}ϕ∥_{L^{2}(ℝ)}^{2} = ∥Y ϕ∥_{L^{2}(ℝ)}^{2} + ∫_{ℝ}[Y, Y^{∗}]ϕ · ϕ̄,

and

[Y, Y^{∗}] = −2τ Det(a)(∂_{x}Θ̃) + O(∂_{x}, τΘ̃, 1),

where norms without subscripts are L^{2}(ℝ, dx) norms. By the pseudoconvexity hypothesis and the sign conventions Det(a) < 0 and ∂_{x}Θ̃ ≥ 0, the right hand side equals a nonnegative function, modulo an operator that is O(∂_{x}, τΘ̃, 1). Therefore

2 ∫_{ℝ} Y Y^{∗}ϕ · ϕ̄ ≥ ∥Y ϕ∥^{2} + ∥Y^{∗}ϕ∥^{2} − C ∫_{ℝ} |ϕ|(|∂_{x}ϕ| + τ|x^{m−1}ϕ| + |ϕ|),

which for large τ is

≥ c ∫_{ℝ} |∂_{x}ϕ|^{2} + τ^{2/m}(1 + (τ^{1/m}|x|))^{2(m−1)}|ϕ|^{2} − C ∫_{ℝ} |ϕ|(|∂_{x}ϕ| + τ|x^{m−1}ϕ| + |ϕ|),

by Cauchy-Schwarz. Replacing Y Y^{∗} by −A_{t} introduces additional terms, but all these are majorized by

ε∥ϕ∥_{H_{0,τ}^{2}}^{2} + ε ∫_{ℝ} |∂_{x}ϕ|^{2} + τ^{2/m}(1 + (τ^{1/m}|x|))^{2(m−1)}|ϕ|^{2}

+ C ∫_{ℝ} |ϕ|(|∂_{x}ϕ| + τ|x^{m−1}ϕ| + |ϕ|)

where ε may be made as small as desired by taking t, t′ to be sufficiently small. For τ > 0 sufficiently large and t, t′ sufficiently small we thus obtain

− Re ∫_{ℝ} A_{t}ϕ · ϕ̄ ≥ c ∫_{ℝ} |∂_{x}ϕ|^{2} + τ^{2}x^{2(m−1)}|ϕ|^{2} − C ∫_{ℝ} |ϕ|(|∂_{x}ϕ| + τ|x^{m−1}ϕ| + |ϕ|).

As in the proof of Lemma 7.2, this leads to the conclusion that for all sufficiently large τ and for all complex t, t′ and real ρ sufficiently close to 0, A_{t} maps H_{ρ,τ}^{2}(ℝ) invertibly to H_{ρ,τ}^{0}(ℝ), uniformly in all parameters. Reasoning just as in the proofs of Lemma 7.1 and Theorem 1.2 establishes the microlocal analytic hypoellipticity of L in Γ^{+}, when Θ̃ is divisible by x^{m−1}.

Consider next the case where Θ̃(0) = 0 and Θ̃ is not divisible by x^{m−1}, where it is to be shown that L cannot be microlocally analytic hypoelliptic. The first obstacle to imitating the proof of Theorem 1.4 is that under our hypotheses, (X_{1} + iX_{2}) is not locally solvable near 0, whence neither is (X_{1} + iX_{2})(X_{1} − iX_{2}), so we cannot hope to solve the equation LG_{λ} = F_{λ} for arbitrary F_{λ} as was possible in Lemma 3.1. As a substitute, fix a bounded C^{∞} auxiliary function n(ξ, τ) which (i) is supported in the conic neighborhood {τ < 0, |xi| < −τ} of the ray {ξ = 0, τ < 0}, (ii) is identically equal to 1 in a smaller conic neighborhood minus some compact set, and whose inverse Fourier transform (iii) is C^{ω} except at the origin, and (iv) agrees with a Schwartz function away from the origin.^{7} Define the Fourier multiplier operator (Bf)ˆ (ξ, τ) = n(ξ, τ)f̂(ξ, τ).

Lemma 8.2. Let {X_{1}, X_{2}} satisfy the bracket and pseudoconvexity hypotheses. Then there exists a neighborhood U of 0 such that for all f ∈ L^{2}(U) there exist v, g ∈ L^{2}(U) satisfying Lg + Bv = f in U, such that the L^{2}(U) norms of g, v are bounded by a fixed constant times the norm of f.

Proof (sketch). L is elliptic outside Σ and is subelliptic in a conic neighborhood of Σ^{+}, because of the bracket and pseudoconvexity hypotheses, as follows from the reasoning of Kohn [14]. On the other hand, the pseudodifferential operator B is elliptic in a conic neighborhood of Σ^{−}. This is all that is needed for the proof of the nearly identical Lemma 7.2 of [9] to apply.

Defining F_{λ} as in Lemma 3.1, solve LG_{λ} = F_{λ} + Bv_{λ} with v_{λ} = O(1) in L^{2} norm. Proceeding as in Sections 3 and 4 with the notations and definitions introduced there, e^{−iλ^{m}t}G_{λ} satisfies a conjugated equation, with right hand side e^{−iλ^{m}t}F_{λ} + e^{−iλ^{m}t}Bv_{λ}. We are interested in this equation in the region where Im(t) < 0 and | Im(t)| = λ^{−1/q}| Im(s)| ∼ λ^{−1/q}, and hence | exp(−iλ^{m}t)| = O(exp(−cλ^{m−q^{−1}})) for some c > 0. Because Bv_{λ} ∈ L^{2}(U) uniformly in λ and because the Fourier transform of Bv_{λ} is supported where τ < 0 and |ξ| < |τ|, Bv_{λ} extends to a holomorphic function of t near 0 in the half space {Im(t) < 0}, and each partial derivative of Bv_{λ}, with respect to (x, t), is O(λ^{N}) there for some finite N; this follows by writing Bv_{λ} as the inverse Fourier transform of its Fourier transform, and noting that the resulting integral converges absolutely for Im(t) < 0. Therefore

7Such functions n may be constructed using the cutoff functions of Andersson and Hörmander, which are discussed in [13].

e^{−iλ^{m}t}Bv_{λ} = O(exp(−cλ^{m−q^{−1}})) = O(exp(−cλ^{p}))

in this region, for some c > 0, as λ → +∞.

The situation is therefore the same as in Sections 3 and 4, except for the presence of an extra term of magnitude O(exp(−cλ^{p})) on the right hand side of the conjugated equation for exp(−iλ^{m}t)G_{λ}(x, t). Such a term has no effect in the argument immediately following Lemma 4.2, Following the nor in the proof of Lemma 4.2 outlined in Section 9. In other respects the analysis of L = (X_{1}+iX_{2})(X_{1}−iX_{2})+c_{1}X_{1}+c_{2}X_{2}+c_{3} is identical to that for X_{1}^{2} + X_{2}^{2}, modulo minor changes in formulas. In this case the existence of eigenvalues was established for all Q in Theorem 5.2, so it follows in full generality that L cannot be analytic hypoelliptic if Θ̃ is not divisible by x^{m−1}.

9. Proof of Lemma 4.2

In this section we outline the proof of the remaining step in the reduction of analytic hypoellipticity, for sums of squares of two vector fields in ℝ^{2}, to eigenvalue problems for ordinary differential operators. The proof is a straightforward adaptation of that given for the corresponding Lemma 4.3 of [3], with certain systematic changes in formulas. We will therefore give the definitions, notations, and statements of sublemmas, armed with which the determined reader will be able to construct a full proof by following the exposition of [3] line by line.

Recall the conjugated operator

B_{λ} = P (λ^{−1}y, λ^{−1/q}(ζ_{0} + re^{iθ}), ∂_{y}, i[Θ + R_{λ}](1 − iλ^{−p}∂_{s})),

where p = m − q^{−1} and q < ∞. Assume that

B_{λ}u_{λ} = ψ + O(e^{−δλ^{p}})

for all large λ ∈ ℝ^{+}. The extra term O(exp(−δλ^{p})) on the right is absent in Lemma 4.2, but occurs in the analogue needed for the proof of Theorem 1.8. This equation holds for all (y, s) ∈ ℝ × ℂ satisfying |y| ≤ cλ and C^{−1} ≤ − Im(s) ≤ C, for some c > 0 and for any C ∈ ℝ^{+}, where δ depends on C. By (4.1) we know that u_{λ} and all its partial derivatives of order ≤ 2 are bounded by exp(Bλ^{p}), for some large constant B, for all (y, s) in this same region. What is desired is essentially a bound u_{λ} = O(exp(μλ^{p})), for arbitrarily small μ > 0, in the smaller region where |y| ≤ λ^{1−ε_{0}} and s belongs to a neighborhood of ζ_{0}.

The proof requires the selection of auxiliary constants A_{0}, ν, N, σ_{0}, … σ_{N}, A, γ satisfying various constraints. These depend on B, μ, ζ_{0}, but not on λ. Let A_{0} be a large constant to be chosen later, and set C_{0} = 2|ζ_{0}|.

Lemma 9.1. There exist 0 < ν < μ, N < ∞, and 3C_{0} > σ_{0} > σ_{1} > ⋯ > σ_{N} > 2C_{0} such that

\frac{1}{2}μ < A_{0} − (N + 1)ν < μ

and

(s − iσ_{j})(1 + r)^{1/qm} ∉ E^{1/q}  (9.1)

for all s ∈ ℝ, 0 ≤ j ≤ N and r ∈ ℝ^{+} satisfying

(A_{0} − jν) ≥ r ≥ (A_{0} − (j + 1)ν).

A small twist needed to adapt the proof of the corresponding lemma in [3] may be found in [21].

The constants γ, A will be required to satisfy finitely many constraints, to be encountered in the course of the proof, all of the two forms

γ and γ · A are sufficiently small  (9.2)
γ · A^{2} is sufficiently large  (9.3)

relative to various quantities depending on B, μ, ζ_{0}, ν, N, σ_{0}, … σ_{N}. Any finite collection of such constraints is satisfied by some pair γ, A ∈ ℝ^{+}; we assume henceforth that γ, A do satisfy them.

The first constraint on γ, A is that

(s − iσ_{j})[1 + r − i2γ(s − iσ_{j})]^{1/mq} ∉ E^{1/q}  (9.4)

for all s ∈ ℝ satisfying |s| ≤ 4A, all 0 ≤ j ≤ N, and all r ∈ I_{j} where

I_{j} = [A_{0} − (j + 1)ν, A_{0} − jν].

This follows from (9.1), using Lemma 3.7, as shown in [3].

Extend the coefficients of B_{λ} so as to be globally defined with respect to y, as was done in the proof of Lemma 4.1. Define

f_{σ}(y, s) = e^{−γλ^{p}(s−iσ)^{2}}u_{λ}(y, s − iσ)η(A^{−1}s)

where η ∈ C_{0}^{∞}(ℝ) is supported in (−4, 4) and is identically equal to 1 on [−2, 2]. Set

𝕃_{σ} = e^{−γλ^{p}(s−iσ)^{2}} ∘ B_{λ} ∘ e^{γλ^{p}(s−iσ)^{2}}
= P (λ^{−1}y, λ^{−1/q}(ζ_{0} + re^{iθ}), ∂_{y}, i[Θ + R_{λ}](1 − 2iγ(s − iσ) − iλ^{−p}∂_{s})),

Then

𝕃_{σ}f_{σ} = Ψ_{σ} where Ψ_{σ}(y, s) = e^{−γλ^{p}(s−iσ)^{2}}ψ(y) + O(e^{−δλ^{p}})e^{−γλ^{p}(s−iσ)^{2}}

for all s ∈ ℝ satisfying |s| ≤ 2A, provided that σ ≥ C^{−1}. For |s| > 2A the cutoff function γ comes into play, but the factor exp(−γλ^{p}(s−iσ)^{2}) is O(exp(λ^{p}[C −γA^{2}])), with C dependent only on quantities fixed before γ, A are chosen. Thus both Ψ_{σ} and f_{σ} are O(exp(−C_{2}λ^{p})) where |s| ≥ A/2, for any desired C_{2} < ∞, provided that γA^{2} is chosen to be sufficiently large. The same holds for their partial derivatives of first and second orders with respect to y, s.

Define the partial Fourier transform

f̂_{σ}(y, ξ) = ∫_{ℝ} f_{σ}(y, s)e^{−isξ} ds

for all ξ ∈ ℝ.

Lemma 9.2. For any constant A_{0} < ∞, for all sufficiently large λ, for all |ξ| ≤ A_{0}λ^{p}, |y| ≤ λ^{1−ε_{0}} and 0 ≤ σ, σ′ ≤ 3C_{0},

f̂_{σ}(y, ξ) = e^{(σ−σ′)ξ}f̂_{σ′}(y, ξ) + O(e^{−λ^{p}}).

This is proved by shifting the contour of integration with respect to s in the definition of the partial Fourier transform. For |s| ≥ A the integrand is not holomorphic, resulting in an error term that is O(exp(−δγA^{2}λ^{p} + Cλ^{p})) for some δ > 0, which is negligible if γA^{2} is sufficiently large.

Lemma 9.3. There exists a large constant A_{0} such that for all sufficiently large λ and all |y| ≤ λ^{1−ε_{0}}, for all | Im(ζ_{0})|/2 ≤ σ ≤ 3C_{0},

∫_{|ξ|≥A_{0}λ^{p}} (|f̂_{σ}(y, ξ)|^{2} + |∂_{y}f̂_{σ}(y, ξ)|^{2}) dξ ≤ e^{−2λ^{p}}.

This is proved by combining the preceding lemma with the weak bound O(exp(Bλ^{p})) for u_{λ}, choosing A_{0} sufficiently large to overwhelm the factor exp(Bλ^{p}).

For functions φ of y ∈ [−λ^{1−ε_{0}}, λ^{1−ε_{0}}] define the dual norms

∥φ∥_{H_{ρ}}^{2} = ∫_{|y|≤λ^{1−ε_{0}}} |φ(y)|^{2}⟨y⟩^{−2(m−1)}e^{+ρ|y|^{m}} dy,
∥φ∥_{H_{ρ}^{∗}}^{2} = ∫_{|y|≤λ^{1−ε_{0}}} |φ(y)|^{2}⟨y⟩^{+2(m−1)}e^{−ρ|y|^{m}} dy

where ρ > 0 is fixed and small. Denote by f̂(ξ) the function y ↦ f̂(y, ξ).

Lemma 9.4. There exists δ > 0 such that for all sufficiently large λ, for all 0 ≤ j ≤ N, whenever λ^{−p}ξ ∈ I_{j},

∥f̂_{σ_{j}}(ξ)∥_{H_{ρ}^{∗}}^{2} ≤ e^{−δλ^{p}} ∫_{|η|≤A_{0}λ^{p}} ∥f̂_{σ_{j}}(η)∥_{H_{ρ}^{∗}}^{2} dη + e^{−δλ^{p}}.

Lemma 4.2 follows from this result and Lemmas 9.3 and 9.2 by an induction on j as in [3], so it remains only to prove Lemma 9.4. This requires a final sublemma.

Denote by 𝕃_{σ}^{∗} the formal transpose of 𝕃_{σ} with respect to the pairing

⟨f, g⟩ = ∫∫_{\begin{matrix}|y|≤λ^{1−ε_{0}} \\ |s|≤A \end{matrix}} f(y, s)g(y, s) dy ds,

ignoring all boundary terms arising from integration by parts in the formal identity ⟨𝕃_{σ}f, g⟩ = ⟨f, 𝕃_{σ}^{∗}g⟩.

Lemma 9.5. There exist δ > 0 and an open set ℂ ⊃ Ω ⊃ {s ∈ ℝ : |s| ≤ A} such that for all sufficiently large λ and sufficiently small ρ > 0, for all 0 ≤ j ≤ N, ξ ∈ λ^{−p}I_{j} and φ ∈ L^{2}(ℝ) supported in [−λ^{1−ε_{0}}, λ^{1−ε_{0}}], there exist g, E defined on [−λ^{1−ε_{0}}, λ^{1−ε_{0}}] × Ω satisfying

e^{isξ}𝕃_{σ_{j}}^{∗}(e^{−isξ}g)(y, s) = φ(y) + E(y, s) in [−λ^{1−ε_{0}}, λ^{1−ε_{0}}] × [−A, A],
∥g(·, s)∥_{H_{ρ}^{2}} ≤ C∥φ∥_{H_{ρ}^{0}} for all s ∈ Ω
∥E(·, s)∥_{H_{ρ}^{0}} ≤ Ce^{−δλ^{p}}∥φ∥_{H_{ρ}^{0}} for all s ∈ Ω.

The norms H_{ρ}^{k} are those of H_{ρ}^{k}([−λ^{1−ε_{0}}, λ^{1−ε_{0}}]), defined as in Section 3 except that the integration with respect to y extends only over [−λ^{1−ε_{0}}, λ^{1−ε_{0}}], rather than over the entire real line. The proof of this lemma is very similar to that of Lemma 4.1; the relation (9.4) together with Lemma 3.6 guarantee that those ordinary differential operators arising in the analysis are indeed invertible. Lemma 9.4 is then deduced by a duality argument like that used in Section 4 to deduce Theorem 1.4 from Lemma 4.1, shifting the contour of integration with respect to s to majorize the principal term resulting after integration by parts, as in the proof of Lemma 6.4 of [3].

10. Comments

1. Denote by Σ the characteristic variety of L, that is, the set of all points in the cotangent bundle where the principal symbol of L vanishes. In our analysis there is no essential distinction between the four coefficients Θ̃:

• x(x^{2} + t^{2}), for which Σ is a (smooth) symplectic manifold,

• x(x^{2} − t^{2}), for which Σ is a singular variety with normal crossings, • x(x^{2} − t^{3}), for which Σ is more singular, and

• x^{2} + t^{2}, for which Σ consists of the single fiber {x = t = ξ = 0}, and no distinction whatsoever between the first two examples. Yet in terms of symplectic geometry, the four characteristic varieties have little in common.^{8}

2. One of the principal hypotheses in the work of Tartakoff [18],[19] and of Treves [20] giving sufficient conditions for analytic hypoellipticity is that the characteristic variety be symplectic. Treves had conjectured that analytic hypoellipticity should fail to hold (for, say, any X_{1}^{2} + X_{2}^{2}) whenever the characteristic variety Σ contains a smooth nonconstant curve γ that is orthogonal to the tangent space of Σ (at each point of γ). For the case of two linearly independent vector fields in ℝ^{3}, any null bicharacteristic for some nonvanishing vector field in the span of X_{1}, X_{2} that lies above the set of points in ℝ^{3} where X_{1}, X_{2}, [X_{1}, X_{2}] are linearly dependent is such a curve γ, and the conjecture has been established [7] in this case. More recently, the conjecture has also been confirmed for a three dimensional example in which the only such curve γ is a fiber of the cotangent bundle [21].

Himonas and Treves asked for examples of operators not analytic hypoelliptic, yet not possessing any such curves γ. The history of this question is a bit obscure; Oleĭnik and Radkevič had already published by 1973 [16] examples of analytic nonhypoelliptic operators with characteristic varieties Σ that are symplectic and hence possess no such curves γ, but had not explicitly discussed the symplectic geometric nature of Σ. Much later it was observed that our examples ∂_{x}^{2} + (x^{k}(x^{2} + y^{2})∂_{t})^{2} likewise have symplectic characteristic varieties {x = ξ = 0}, and by Proposition 1.7 are not analytic hypoelliptic for k = 1 or k even. Special cases of the examples of Oleĭnik and Radkevič were subsequently rediscovered by Hanges and Himonas.^{9}

3. Our results should generalize to sums of squares of any finite number of vector fields in ℝ^{2}. The necessary and sufficient condition for analytic hypoellipticity should be the existence of a coordinate system in which span{X_{j}} ≡ span{∂_{x}, x^{m−1}∂_{t}}, that is, both ∂_{x} and x^{m−1}∂_{t} may be expressed as linear combinations, with C^{ω} coefficients, of the X_{j}, and conversely.

4. In Proposition 1.7 we have singled out certain examples for which Conjecture 1.5 can be proved relatively easily. With more labor, similar reasoning should apply to other special cases, as well, but the conjecture can be formulated more generally by allowing the coefficients of Q to be arbitrary complex numbers, and then neither the ad hoc proof of Proposition 1.7 nor the method of proof of Theorem 5.2 appear likely to lead to a proof in full generality.

8This accords with a remark of Métivier [15]: “Il ne faut sans doute pas vouloir relier trop rigidement le condition “Σ symplectique” à l’hypoellipticité analytique.”

9A higher order example is L = L_{1} ∘ L_{2}, where the L_{j} are chosen to have characteristic varieties Σ_{j} such that Σ_{2} ⊂ Σ_{1}, Σ_{1} is symplectic, and L_{2} is not analytic hypoelliptic. Then L is not analytic hypoelliptic, and its characteristic variety is Σ_{1}. An explicit example is (∂_{x}^{2} + x^{2k}∂_{t}^{2}) ∘ (∂_{x}^{2} + (x^{2} + t^{2})∂_{t}^{2}).

A more promising approach is related to the phenomenon of Stokes lines. A solution ψ_{z}^{+} having the required asymptotics as x → +∞ has this behavior in a conic neighborhood of ℝ^{+}, but not for x in all of ℂ; the Stokes lines separate disjoint sectors in the complex plane in which the solution has distinct asymptotics. Whenever a (normalized, homogeneous) polynomial Q(x, z) is not identically equal to x^{m−1}, then x ↦ Q(x, z) has for each z at least two distinct zeros in the complex plane. As in the proof of Theorem 5.2, these zeros should govern the asymptotic behavior of W (z) as |z| → ∞. One could attempt to show that there exist zeros w_{i}(z) and disjoint sectors Γ_{i}, for i = 1, 2, such that the asymptotic behavior of the Wronskian W (z) as z tends to infinity through Γ_{i} is governed by w_{i}(z); more precisely, that for z ∈ Γ_{i}, W (z) ∼ exp(c_{i}z^{m})z^{γ_{i}}(1 + O(|z|^{−δ}) for some exponents γ_{i}. If one could show further that c_{1} ≠ c_{2}, then W could not have a polynomial logarithm and hence would have zeros. One advantage of this approach is that it requires no analysis of the exponents γ_{i} (let alone of higher order terms in the asymptotic expansions). Some of the ideas of Yu [23] should be useful here.

5. The fact that ordinary differential operators play such a prominent role in our analysis is a feature of the particular class of low-dimensional partial differential operators under consideration. Globally elliptic operators in more than one variable play the corresponding role in other situations.

6. In its present form, the method employed here is too primitive to apply to the general case of two independent vector fields in ℝ^{3}.

References

[1] B. Aupetit, A primer on spectral theory, Springer-Verlag, 1991.

[2] D. Barrett, Duality between A^{∞} and A^{−∞} on domains with nondegenerate corners, Cont. Math. 185 (1995), 77-87.

[3] M. Christ, The Szegö projection need not preserve global analyticity, Annals of Math., to appear.

[4] M. Christ, On local and global analytic and Gevrey hypoellipticity, Journees “Equations aux derivees partielles”, Saint-Jean-de-Monts, Mai-Juin 1995.

[5] M. Christ, Analytic hypoellipticity, representations of nilpotent groups, and a nonlinear eigenvalue problem, Duke Math. J. 72 (1993), 595-639.

[6] M. Christ, A family of degenerate differential operators, J. Geometric Analysis 3 (1993), 579-597.

[7] M. Christ, A necessary condition for analytic hypoellipticity, Math. Research Letters 1 (1994), 241-248.

[8] M. Christ, Certain sums of squares of vector fields fail to be analytic hypoelliptic, Comm. Partial Differential Equations 16 (1991), 1695-1707.

[9] M. Christ, Examples of analytic non-hypoellipticity of ¯∂_{b}, Comm. Partial Differential Equations 19 (1994), 911-941.

[10] E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

[11] A. Grigis and J. Sjöstrand, Front d’onde analytique et sommes de carrés de champs de vecteurs, Duke Math. J. 52 (1985), 35-51.

[12] V. V. Grušin, A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold, Mat. Sb. 84 (1971), 163-195 = Math. USSR Sb. 13 (1971), 155-185.

[13] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin Heidelberg 1983.

[14] J. J. Kohn, Estimates for ¯∂_{b} on pseudoconvex CR manifolds, Proc. Symp. Pure Math. 43 (1985), 207-217.

[15] G. Métivier, Non-hypoellipticité analytique pour D_{x}^{2} + (x^{2} + y^{2})D_{y}^{2}, Comptes Rendus Acad. Sci. Paris 292 (1981), 401-404.

[16] O. A. Oleĭnik, On the analyticity of solutions of partial differential equations and systems, Astérisque 2,3 (1973), 272-285.

[17] J. Sjöstrand, Singularités analytiques microlocales, Astérisque 95 (1992).

[18] D. Tartakoff, Local analytic hypoellipticity for □_{b} on non-degenerate Cauchy-Riemann manifolds, Proc. Nat. Acad. Sci. USA 75 (1978), 3027-3028.

[19] D. Tartakoff, On the local real analyticity of solutions to □_{b} and the ¯∂–Neumann problem, Acta Math. 145 (1980), 117-204.

[20] F. Treves, Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the ¯∂–Neumann problem, Comm. Partial Differential Equations 3 (1978), 475-642.

[21] F. Tolli, Analytic hypoellipticity on a convex bounded domain, UCLA Ph.D. dissertation, March 1996.

[22] J.-M. Trepreau, Sur l’hypoellipticité analytique microlocale des opérateurs de type principal, Comm. Partial Differential Equations 9(1984), 1119–1146.

[23] C.-C. Yu, Nonlinear eigenvalues and analytic-hypoellipticity, UCLA Ph.D. dissertation, June 1995.

Michael Christ, Department of Mathematics, University of California, Los Angeles, CA 90095-1555

  E-mail address: christ@math.ucla.edu

Current address: Mathematical Sciences Research Institute, 1000 Centennial Road, Berkeley, CA 94720