INTERMEDIATE OPTIMAL GEVREY EXPONENTS OCCUR

MICHAEL CHRIST

Abstract. Hypoellipticity in Gevrey classes G^{s} is characterized for a simple family of sums of squares of vector fields satisfying the bracket hypothesis, with analytic coefficients. It is shown that hypoellipticity holds if and only if s is greater than or equal to an optimal exponent that may take on any rational value.

1. Introduction

Let L be a linear partial differential operator, with C^{∞} coefficients. Denote by G^{s} the Gevrey class of order s, for each 1 ≤ s < ∞. The operator L is said to be G^{s} hypoelliptic at a point x if, whenever u is a distribution defined near x for which Lu ∈ G^{s} in some neighborhood of x, necessarily u ∈ G^{s} in some neighborhood, possibly smaller, of x. It is G^{s} hypoelliptic in an open set U if it is so at every point of U.

Definition. The optimal exponent for Gevrey regularity for L at x is defined to be the infimum of all s ≥ 1 such that L is G^{s} hypoelliptic at x. The optimal Gevrey exponent for L in an open set U is defined to be the supremum, over all x ∈ U, of the optimal exponent for L at x.

Suppose that L is locally solvable in L^{2} near x, that is, that there exists a neighborhood V of x such that ∥φ∥_{L^{2}} ≤ C∥L^{∗}φ∥_{L^{2}} for all φ ∈ C_{0}^{∞}(V), where L^{∗} denotes the transpose of L. Under this mild hypothesis, if L is G^{s} hypoelliptic in V for some exponent s, then it is also G^{t} hypoelliptic in V for every t ≥ s; see Theorem 3.1 and Remark 3.2 of Métivier [9]. The word “optimal” is thus appropriate under this hypothesis.

Suppose that L is a sum of squares of real vector fields having real analytic coefficients. Fix a point x, and denote by m ≥ 1 the least integer for which the vector fields satisfy the usual bracket hypothesis to order m at x. Then in some neighborhood of x, L is G^{s} hypoelliptic for every s ≥ m [5]. But s = m need not be optimal; for instance, if L has symplectic characteristic variety and loses one derivative then L is analytic hypoelliptic [11],[13], so G^{s} hypoelliptic for all s ≥ 1. Various examples and special classes that are not analytic hypoelliptic have been studied, and an analysis of the proofs that these fail to be analytic hypoelliptic near x has generally shown that they also fail to be G^{s} hypoelliptic whenever s < m. To this author’s knowledge, it has only recently been pointed out [3] that there exist sums of squares operators for which the optimal Gevrey exponent satisfies 1 < s_{0} < m, and moreover, s_{0} need not be an integer but can take on (certain) rational values.

Date: March 8, 1996.

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

In this note, examples will be given for which s_{0} takes on intermediate values in (1, m).^{1} The analysis of these examples is simpler than that of the operators of [2],[3]. In addition, some general results concerning the connection between Gevrey regularity, variants of the FBI transform, and holomorphic extendibility will be established, generalizing well known results for the real analytic case s = 1.

The operators that we study have previously been considered by Grušin [6], who proved certain of them to be analytic hypoelliptic, and by Oleĭnik and Radkevič [10], who proved all others not to be analytic hypoelliptic. More recently they have been rediscovered by Hanges and Himonas, who observed that their principal symbols have symplectic characteristic varieties, even in the analytic nonhypoelliptic case.

Let 1 ≤ p ≤ q ∈ ℕ, let (x, t) be coordinates in ℝ × ℝ^{2}, and define

L = ∂_{x}^{2} + x^{2(p−1)}∂_{t_{1}}^{2} + x^{2(q−1)}∂_{t_{2}}^{2}.  (1.1)

These are analytic hypoelliptic if [6] and only if [10] p = q. The bracket condition is satisfied to order m = q at 0.

Theorem 1.1. L is G^{s} hypoelliptic in some neighborhood of 0 if and only if s ≥ q/p.

Further generalizations are possible. For instance,

L = ∂_{x}^{2} + a_{1}(x, t)x^{2(p−1)}∂_{t_{1}}^{2} + a_{2}(x, t)x^{2(q−1)}∂_{t_{2}}^{2}

satisfies the same conclusions provided that a_{j} are real-valued, are strictly positive and are analytic.^{2} The variable (x, t_{1}, t_{2}) can be permitted to take values in ℝ^{m_{1}} × ℝ^{m_{2}} × ℝ^{m_{3}} for any m_{j} ≥ 1, with ∂_{x}^{2}, ∂_{t_{j}}^{2} replaced by constant-coefficient elliptic operators having nonpositive principal symbols and x^{2} replaced by |x|^{2}.

1For examples (1.1), s_{0}^{−1} is always a rational number with denominator m, but other rational numbers occur for the examples in [3].

2The proof in Section 3 that L is G^{s} hypoelliptic for all s ≥ q/p applies equally well to such generalizations. The very simple analysis given in Section 4 for s < q/p no longer applies when the coefficients of L depend on t, however. The techniques of [4],[1],[2] can be adapted to this situation, but additional labor would be necessary.

Examples exhibiting a rather different behavior in the Gevrey scale have been analyzed by Yu [14]: In ℝ^{5} with coordinates (x_{1}, y_{1}, x_{2}, y_{2}, t) set X_{j} = ∂_{x_{j}}, Y_{1} = ∂_{y_{1}} + x_{1}^{m−1}∂_{t}, and Y_{2} = ∂_{y_{2}} + x_{2}∂_{t} for m ≥ 2. Then the optimal exponent for Gevrey regularity of L = X_{1}^{2} + X_{2}^{2} + Y_{1}^{2} + Y_{2}^{2} in any neighborhood of the origin is s = 1 for m = 2, and is s = 2 for all even m > 2.^{3} Thus in this family, the optimal Gevrey exponent is independent of the degree of degeneracy of the operator.

Throughout the paper, C, c, ε, ε′, δ denote strictly positive constants whose values are permitted to vary freely from one line to the next. Generally C is a large constant, while the others are small.

2. Characterizations of Gevrey Classes

For y = (y_{1}, … y_{n}) ∈ ℂ^{n} define ⟨y⟩ = (1 + Σy_{j}^{2})^{1/2}, which is well defined and holomorphic in a conic neighborhood Γ of ℝ^{n}. Let (x, ξ) be coordinates on ℂ^{n} × ℂ^{n}. For each γ ∈ [0, 1] define a differential form ω on a conic neighborhood of ℝ^{n} in ℂ^{n} by

ω = dx_{1} ∧ ⋯ ∧ dx_{n} ∧ d(ξ_{1} + ix_{1}⟨ξ⟩^{γ}) ∧ ⋯ ∧ d(ξ_{n} + ix_{n}⟨ξ⟩^{γ})

and define a function α_{γ} by

ω = α_{γ}(x, ξ) dx_{1} ∧ ⋯ ∧ dx_{n} ∧ dξ_{1} ∧ ⋯ ∧ dξ_{n}.

For u ∈ ℰ′(ℝ^{n}), for (x, ξ) ∈ ℂ^{n} × Γ, and for 0 ≤ γ ≤ 1 define

F_{γ}u(x, ξ) = ⟨u, e^{i(x−x′)·ξ−⟨ξ⟩^{γ}(x−x′)^{2}}α_{γ}(x − x′, ξ)⟩ ,

where the pairing is that of distributions with test functions, with respect to the variable x′. The exponent in this expression may be written more explicitly as

i ∑_{j=l}^{n}(x_{j} − x′_{j})ξ_{j} − ⟨ξ⟩^{γ} ∑_{j=1}^{n}(x_{j} − x′_{j})^{2}.

The coefficient α_{γ} is holomorphic with respect to x, and equals 1 + O(⟨ξ⟩^{γ−1}) for x in any bounded subset of ℂ^{n}. Moreover α_{γ}^{−1}(x, ξ) is uniformly bounded, and equals 1 + O(⟨ξ⟩^{γ−1}) for large |ξ|, provided that x is restricted to a sufficiently small but fixed neighborhood of the origin.

3Gevrey hypoellipticity for the indicated ranges of exponents follows from theorems of Treves and of Tartakoff for m = 2 and of Derridj and Zuily for m > 2; that hypoellipticity does not hold for smaller exponents was proved by Yu.

Lemma 2.1. Let 0 ≤ γ ≤ 1. For any u ∈ C^{n+1}(ℝ^{n}) having compact support, for any x ∈ ℝ^{n},

u(x) = (2π)^{−n} ∫_{ℝ^{n}} F_{γ}u(x, ξ) dξ.

Proof. Write the Fourier inversion formula as

u(x) = (2π)^{−n} lim_{ε→0} ∫∫ _{ℝ^{n}×ℝ^{n}} u(x′)e^{i(x−x′)·ξ}e^{−εξ^{2}} dx′ ∧ dξ

where ξ^{2} = ∑_{j} ξ_{j}^{2} for ξ ∈ ℂ^{n}. Consider this integral with some small value of ε, and shift the contour of integration from ℝ^{n} × ℝ^{n} ⊂ ℝ^{n} × ℂ^{n} to the contour

Γ(y, η) = (y, η + i⟨η⟩^{γ}(x − y)) for (y, η) ∈ ℝ^{n} × ℝ^{n}.

The integrand is holomorphic with respect to ξ. It also decays rapidly, for (y, η) ∈ ℝ^{2n}, as

|(x′, ξ)| = |Γ(y, η, t)| → ∞,

while x′ = y remains bounded, where Γ(y, η, t) = (1−t)(y, η)+tΓ(y, η) for 0 ≤ t ≤ 1, provided that either γ < 1, or |x − y| is sufficiently small for all y in the support of u. Indeed,

Re(−εξ^{2}) = −εη^{2} + εt^{2}⟨η⟩^{2γ}(x − y)^{2}

is bounded above by −εη^{2}/2 for all sufficiently large η, in either of these two cases. Since dx′ ∧ dξ = α_{γ}(y, η) dy ∧ dη,

u(x) = (2π)^{−n} lim_{ε→0} ∫∫ _{ℝ^{n}×ℝ^{n}} u(y)e^{i(x−y)·η−⟨η⟩^{γ}(x−y)^{2}}α_{γ}(x − y, η)e^{−ε(η+i⟨η⟩^{γ}(x−y))^{2}} dy dη.

In either of the two favorable cases, fixing any sufficiently largte η and integrating by parts n + 1 times with respect to y yields

∫ u(y)e^{i(x−y)·η−⟨η⟩^{γ}(x−y)^{2}}α_{γ}(x − y, η)e^{−ε(η+i⟨η⟩^{γ}(x−y))^{2}} dy = O(|η|^{−n−1}),

uniformly in ε and uniformly for x in any bounded set. This integral converges, as ε → 0, to F_{γ}u(x, η), which is therefore O(|η|^{−n−1}). Thus (2π)^{−n} ∫ F_{γ}u(x, ξ) dξ converges absolutely, and uniformly on compact sets, to u(x).

In the case γ = 1 an additional argument is needed. By introducing a partition of unity we may assume u to be supported in a ball B(x_{0}, ε) of center x_{0} and radius ε, with ε chosen to be sufficiently small that the inversion formula is valid for all x ∈ B(x_{0}, 2ε). Setting v(x) = (2π)^{−n} ∫ F_{γ}u(x, ξ) dξ, we thus have v ≡ 0 in B(x_{0}, 2ε)\ B(x_{0}, ε). But when γ = 1, a simple calculation shows that v is real analytic outside the support of u, so v ≡ 0 on the complement of B(x_{0}, ε). Thus v ≡ u.

Lemma 2.2. Let γ ∈ [0, 1] and u ∈ ℰ′(ℝ^{n}). Define

u_{ε}(x) = (2π)^{−n} ∫_{|ξ|≤ε^{−1}} F_{γ}u(x, ξ) dξ.

Then u_{ε} → u in ℰ′ as ε → 0.

Proof. This follows from Lemma 2.1 by duality.

Definition. Let U ⊂ ℝ^{n} be an open neighborhood of a point x_{0}, and s ∈ (0, ∞). The distribution u is said to belong to the Gevrey class G^{s} at x_{0} if there exist a neighborhood V ⊂ U of x_{0} such that u ∈ C^{∞}(V), and a constant C ∈ ℝ^{+}, such that for every x ∈ V and every multi-index α,

|∂^{α}u(x)| ≤ C^{|α|+1}|α|^{s|α|}.

Moreover u ∈ G^{s}(U) if u ∈ G^{s} at every point of U.

It is clear, by the Cauchy-Kowalevska theorem, that G^{s} hypoellipticity cannot hold for s < 1, so we restrict attention henceforth to the case s ≥ 1. The principal result of this section relates three characterizations of the classes G^{s}; two of these characterizations are formulated each in two distinct ways.

Theorem 2.3. Let x_{0} ∈ ℝ^{n}, u ∈ 𝒟′(ℝ^{n}), and s ∈ [1, ∞). Then the following five conditions are mutually equivalent:

(1) u ∈ G^{s} at x_{0}.

(2) For every γ ≥ s^{−1} and every distribution v ∈ ℰ′(ℝ^{n}) satisfying v ≡ u in some neighborhood of x_{0}, there exist C, δ ∈ ℝ^{+} and a neighborhood V of x_{0} such that

|F_{γ}v(x, ξ)| ≤ Ce^{−δ⟨ξ⟩^{1/s}} for all (x, ξ) ∈ V × ℝ^{n}.

(3) There exist v ∈ ℰ′ agreeing with u in some neighborhood of x_{0}, γ ∈ [s^{−1}, 1], and a neighborhood V of x_{0} such that

|F_{γ}v(x, ξ)| ≤ Ce^{−δ⟨ξ⟩^{1/s}} for all (x, ξ) ∈ V × ℝ^{n}.

(4) There exist an open neighborhood U ⊂ ℂ^{n} of x_{0} and C, δ ∈ ℝ^{+} such that, for each λ ≥ 1, there exists a decomposition

u = g_{λ} + h_{λ} in U ∩ ℝ^{n}

such that g_{λ} is holomorphic in U,

|g_{λ}(z)| ≤ Ce^{Cλ|Im(z)|} for all z ∈ U.

and

|h_{λ}(x)| ≤ Ce^{−δλ^{1/s}} for all x ∈ U ∩ ℝ^{n}.

(5) There exist an open neighborhood U ⊂ ℂ^{n} of x_{0} and C, δ ∈ ℝ^{+} such that, for each λ ≥ 1, there exists a decomposition

u = g_{λ} + h_{λ} in U ∩ ℝ^{n}

such that g_{λ} is holomorphic in {z ∈ U : |Im(z)| ≤ λ^{−(s−1)/s}} = U_{λ},

|g_{λ}(z)| ≤ C for all z ∈ U_{λ},

and

|h_{λ}(x)| ≤ Ce^{−δλ^{1/s}} for all x ∈ U ∩ ℝ^{n}.

The implication (2) ⇒ (3) is tautologous, and (4) ⇒ (5) will be shown to be rather superficial as well.

Proof that (1) ⇒ (2). Assume that u ∈ G^{s} at x_{0}, γ ≥ s^{−1}, and v ∈ ℰ′(ℝ^{n}) agrees with u in some open neighborhood U of x_{0}. Fix a cube Q centered at x_{0}, whose closure is contained in U, and fix any relatively compact open subset V of the interior of Q. Then uniformly for (x, ξ) ∈ V × ℝ^{n},

F_{γ}v(x, ξ) = ∫_{Q} e^{i(x−y)·ξ} · (v(y)e^{−⟨ξ⟩^{γ}(x−y)^{2}}α_{γ}(x − y, ξ)) dy + O(e^{−δ⟨ξ⟩^{γ}}). (2.1)

Assume without loss of generality that |ξ_{1}| ≥ n^{−1}|ξ| ≥ 1, and let N be a large parameter to be chosen below. Integrating by parts N times with respect to y_{1} in (2.1) yields

±i^{N}|ξ_{1}|^{−N} ∫_{Q} e^{i(x−y)·ξ} ∂_{y_{1}}^{N}(v(y)e^{−⟨ξ⟩^{γ}(x−y)^{2}}α_{γ}(x − y, ξ)) dy,

modulo boundary terms which will be discussed below. To simplify notation set λ = ⟨ξ⟩. For (x, y) in any compact set,

|∂_{y_{1}}^{k}e^{−⟨ξ⟩^{γ}(x−y)^{2}}| ≤ C^{k+1}λ^{γk/2}k^{k/2}

for any k ≥ 0. Since v ≡ u ∈ G^{s} on a neighborhood of the closure of Q, Leibniz’s rule yields

|ξ_{1}|^{−N} · |∂_{y_{1}}^{N}(v(y)e^{−⟨ξ⟩^{γ}(x−y)^{2}}α_{γ}(x − y, ξ))| ≤ λ^{−N}C^{N+1} max_{a+b=N} N^{sa}λ^{γb/2}N^{b/2}.

where λ = ⟨ξ⟩. Fix a small constant ε > 0, and choose N so that |N − ελ^{1/s}| ≤ 1. Then the right-hand side of the preceding display is bounded by

C^{ελ^{1/s}+1} max_{a+b=N}(ελ^{1/s})^{sa}λ^{γb/2}(ελ^{1/s})^{b/2}λ^{−N}

≤ C^{ελ^{1/s}+1} max_{0≤b≤N} ε^{sa+b/2}λ^{( \frac{bγ}{2} + \frac{b}{2s} −b)}.

Now γ/2 + 1/(2s) − 1 ≤ \frac{1}{2} + \frac{1}{2} − 1 = 0, and sa + \frac{1}{2}b ≥ \frac{N}{2} , so this last quantity is bounded above by

C^{ελ^{1/s}+1}ε^{N/2} ≤ C · (Cε^{1/2})^{ελ^{1/s}} ≤ Ce^{−δ⟨ξ⟩^{1/s}}

for some C, δ ∈ ℝ^{+} if ε is chosen to be sufficiently small.

It remains to handle the boundary terms arising from integration by parts in (2.1), which necessitates an improved bound for the derivatives of exp(−⟨ξ⟩^{γ}(x − y)^{2}) away from the diagonal. Setting f(x) = exp(−⟨ξ⟩^{γ}x^{2}), one has, for x in any compact set disjoint from the origin,

|∂_{x_{1}}^{k}f(x)| ≤ C^{k+1}k^{k}e^{−δ⟨ξ⟩^{γ}},

as is seen by applying Cauchy’s integral formula to the holomorphic function f on disks disjoint from the origin. Thus for x ∈ V and y ∈ ∂Q, for any M ≤ N,

|ξ_{1}|^{−M} · |∂_{y_{1}}^{M}(v(y)e^{−⟨ξ⟩^{γ}(x−y)^{2}}α_{γ}(x − y, ξ))| ≤ λ^{−M}C^{M+1}e^{−δ⟨ξ⟩^{γ}} max_{a+b=M} M^{sa}M^{b}
≤ C^{M+1}M^{sM}λ^{−M}e^{−δλ^{1/s}}
≤ C^{M+1}(N^{s}/λ)^{M}e^{−δλ^{1/s}}
≤ C(Cε^{s})^{M}e^{−δλ^{1/s}}
≤ Ce^{−δλ^{1/s}}.

Thus each boundary term is O(exp(−δλ^{1/s})), and there are 2N ≤ Cλ^{1/s} of them.

The transform F_{γ}v(z, ξ) extends, for each ξ ∈ ℝ^{n}, to an entire holomorphic function of z ∈ ℂ^{n}. Observe that the same reasoning as above gives

|F_{γ}v(z, ξ)| ≤ Ce^{−δ⟨ξ⟩^{1/s}}e^{C⟨ξ⟩|Im(z)|}  (2.2)

for z in a sufficiently small neighborhood V ⊂ ℂ^{n} of x_{0}, provided that v ∈ ℰ′ belongs to G^{s} at x_{0} and that 1 ≥ γ ≥ s^{−1}.

Proof that (1) ⇒ (4). Suppose that u ∈ G^{s} at x_{0}. Fix v ∈ ℰ′ that agrees with u near x_{0}, and set γ = s^{−1}, so that (2.2) holds. Thus it suffices to show that (2.2) implies (4). For each large λ ∈ ℝ^{+} define

g_{λ}(z) = (2π)^{−n} ∫_{|ξ|≤λ} F_{γ}v(z, ξ) dξ.

Since |F_{γ}v(z, ξ)| ≤ C exp(−δ⟨ξ⟩^{γ} + C⟨ξ⟩|Im(z)|) in a complex neighborhood U ⊂ ℂ^{n} independent of ξ, and since z → F_{γ}v(z, ξ) is holomorphic in U, each g_{λ} is holomorphic on U and satisfies

|g_{λ}(z)| ≤ C ∫_{r=0}^{λ} e^{−δr^{γ}}e^{Cr|Im(z)|} r^{n−1}dr ≤ Ce^{Cλ|Im(z)|}.

Moreover, for x ∈ U ∩ ℝ^{n},

\begin{aligned}|(u − g_{λ})(x)| & = |v − g_{λ}(x)| = (2π)^{−n} |∫_{|ξ|≥λ} F_{γ}v(x, ξ) dξ| \\ & ≤ C ∫_{r=λ}^{∞} e^{−δr^{γ}} r^{n−1}dr ≤ Ce^{−ελ^{γ}} \end{aligned}

for some ε > 0.

Proof that (4) ⇒ (5). For each λ ∈ ℝ^{+} fix a decomposition u = g_{λ} + h_{λ} having the stated properties. Then there exist a neighborhood V ⊂ ℂ^{n} of x_{0}, relatively compact in U, and a constant ε > 0 such that for all λ ≥ 1 and all z ∈ V satisfying |Im(z)| ≤ ελ^{−(s−1)/s},

|g_{λ}(z) − g_{λ/2}(z)| ≤ e^{−ελ^{1/s}}.  (2.3)

Indeed, the function log |g_{λ} − g_{λ/2}| is subharmonic, is O(Cλ|Im(z)|) for z ∈ U, and is ≤ −δλ^{1/s} for z ∈ U ∩ℝ^{n}, because there it equals log |h_{λ} −h_{λ/2}|. A simple comparison argument, applied in complex one dimensional rectangles each having one boundary segment in ℝ^{n}, then yields (2.3).

Given any large λ ∈ ℝ^{+}, choose k so that 2^{k} ≤ λ < 2^{k+1} and write

g_{λ} = g_{λ/2^{k}} + ∑_{j=1}^{k} (g_{λ/2^{j−1}} − g_{λ/2^{j}}) .

By (2.3), g_{λ}(z) is O(1) for all z ∈ V satisfying |Im(z)| ≤ ε′λ^{−(s−1)/s}. Thus g_{λ} has all the properties required in (5) of g_{μ}, where μ^{−(s−1)/s} = ε′λ^{−(s−1)/s}.

Proof that (5) ⇒ (2). Suppose that v ≡ u in U_{0}, that v is supported in U_{1} and that U_{0} is relatively compact in U_{1}. Assume that γ ≥ s^{−1}. To estimate F_{γ}v(x, ξ) for x ∈ U_{0}, assume without loss of generality that |ξ_{1}| ≥ |ξ|/n ≥ 1 and that ξ_{1} > 0. Fix η ∈ C_{0}^{∞}(U_{1}) satisfying η ≡ 1 on a neighborhood of the closure of U_{0}. Then for x ∈ U_{0},

F_{γ}v(x, ξ) = F_{γ}(g_{λ}η)(x, ξ) + O(e^{−δλ^{1/s}})

where λ is defined to be |ξ|, and u = g_{λ} + h_{λ} in U_{1} with g_{λ}, h_{λ} as in (5). Translate the coordinate system to have origin at x_{0}, and choose U_{0} of the form

U_{0} = {x : |x_{j}| < r for all j},

where r is small enough that x ∈ U_{1} whenever each |x_{j}| < 3r. Choose η ∈ C_{0}^{∞}(U_{1}) so η(x) = 1 whenever each |x_{j}| ≤ 2r. Fix φ ∈ C^{1}(ℝ) such that 0 ≤ φ ≤ \frac{1}{2}, φ(t) > 0 for |t| ≤ 2r, and φ(t) = 0 for all |t| ≥ 3r. Letting ε > 0 be small, shift the contour of integration in the definition of F_{γ}(g_{λ}η)(x, ξ) to

ϕ(y) = (y_{1} + iελ^{−(s−1)/s}φ(y_{1}), y_{2}, …, y_{n})

to obtain

F_{γ}(g_{λ}η)(x, ξ)

= c ∫_{ℝ^{n}} e^{i(x−ϕ(y))·ξ−⟨ξ⟩^{γ}(x−ϕ(y))^{2}}α_{γ} (x−ϕ(y), ξ) (1+iελ^{−(s−1)/s}φ′(y_{1}))·g_{λ}η (ϕ(y)) dy.

Since η(y) ≡ 1 for all y ∈ ℝ^{n} such that ϕ(y) ≠ y, we interpret η(ϕ(y)) to be η(y). The function g_{λ} is holomorphic in a region containing ϕ(y) for all y in the support of η, so g_{λ}(ϕ(y)) is defined and the change of contour is justified.

Now

Re (i(x − ϕ(y)) · ξ − ⟨ξ⟩^{γ}(x − ϕ(y))^{2})
= −⟨ξ⟩^{γ}(x − y)^{2} − ξ_{1}λ^{−(s−1)/s}εφ(y_{1}) + ⟨ξ⟩^{γ}λ^{−2(s−1)/s}ε^{2}φ^{2}(y_{1})
≤ −δλ^{γ}(x − y)^{2} − δλ^{1/s}φ(y_{1})

for some δ > 0, for all sufficiently large λ = |ξ|, if s > 1 or if ε is chosen to be sufficiently small. The integrand is then O(exp(−δλ^{1/s}φ(y_{1}))) = O(exp(−δ′|ξ|^{1/s})) for |y_{1}| ≤ 2r, since g_{λ}(y) = O(1) where |Im(y)| ≤ C|ξ|^{−(s−1)/s}, and for |y_{1}| ≥ 2r is

O (exp(−δλ^{γ}|x_{1} − y_{1}|^{2})) = O (exp(−c|ξ|^{1/s}))

if x ∈ U_{0}.

Proof that (3) ⇒ (1). Suppose that v ≡ u near x_{0}, v ∈ ℰ′, γ ≥ s^{−1}, and

F_{γ}v(x, ξ) = O(exp(−c⟨ξ⟩^{1/s})) for all x ∈ ℝ^{n}

in a fixed neighborhood of x_{0}. There is also the trivial bound

F_{γ}v(z, ξ) = O(exp(C⟨ξ⟩)),

valid for all z in a fixed complex neighborhood of x_{0}. Subharmonicity of log |F_{γ}v(·, ξ)| and a simple comparison argument based on these two bounds yields

|F_{γ}v(z, ξ)| ≤ Ce^{−δ⟨ξ⟩^{1/s}+C|Im(z)|⟨ξ⟩},

valid for all ξ ∈ ℝ^{n} and all z ∈ ℂ^{n} in a smaller neighborhood of x_{0}. In particular, there exist C, δ ∈ ℝ^{+} such that

|F_{γ}v(z, ξ)| ≤ Ce^{−δ⟨ξ⟩^{1/s}}

whenever |z| < δ and |Im(z)| < δ⟨ξ⟩^{−(s−1)/s}. Therefore for all x ∈ ℝ^{n} satisfying |x| < δ/2, we may apply Cauchy’s inequalities to F_{γ}v(·, ξ) in the polydisk of radii δ′⟨ξ⟩^{−(s−1)/s} centered at x to obtain

|∂_{x}^{α}F_{γ}v(x, ξ)| ≤ C^{|α|+1}|α|^{|α|}⟨ξ⟩^{|α|(s−1)/s}e^{−δ⟨ξ⟩^{1/s}}

for all multi-indices α and all ξ ∈ ℝ^{n}. The inversion formula for F_{γ} and these bounds together imply that u = v ∈ C^{∞} in the region where |x| < δ/2, and

|∂_{x}^{α}u(x)| ≤ C^{|α|+1}|α|^{|α|} ∫_{ℝ^{n}}⟨ξ⟩^{|α|(s−1)/s}e^{−δ⟨ξ⟩^{1/s}} dξ
≤ C^{|α|+1}|α|^{|α|} ∫_{0}^{∞} t^{|α|(s−1)/s}e^{−δt^{1/s}}t^{n−1} dt
≤ C^{|α|+1}|α|^{|α|} ∫_{0}^{∞} r^{|α|(s−1)}e^{−δr}r^{ns−1} dr
= C^{|α|+1}|α|^{|α|} Γ(|α|(s − 1) + ns)
≤ C^{|α|+1}|α|^{s|α|}.

Thus u ∈ G^{s} at x_{0}.

We have proved the chain of implications (1) ⇒ (2.2) ⇒ (4) ⇒ (5) ⇒ (2) ⇒ (3) ⇒ (1), so the proof of the theorem is complete.

3. Gevrey Regularity

Let 1 ≤ p ≤ q be integers and set

L = ∂_{x}^{2} + (x^{p−1}∂_{t_{1}})^{2} + (x^{q−1}∂_{t_{2}})^{2},  (3.1)

with coordinates (x, t) ∈ ℝ × ℝ^{2}. Let η = (ξ, τ) be dual coordinates. Our aim is to prove L to be G^{s} hypoelliptic for all s ≥ q/p. The proof is based on characterizations (2) and (5) of G^{s}, together with the following lemma. To simplify notation, set

E = E(t′) = e^{i(t−t′)·τ−⟨η⟩^{p/q}(t−t′)^{2}}.

Lemma 3.1. Let L take the form (3.1), for some 1 ≤ p ≤ q. There exist c, c′, δ ∈ ℝ^{+} and an open neighborhood U ⊂ ℝ×ℂ^{2} of the origin such that for each η = (ξ, τ) ∈ ℝ^{3} satisfying |ξ| ≤ |τ| and each (x, t) ∈ U there exists g ∈ C^{0}(U), holomorphic with respect to t, satisfying

L(Eg)(x′, t′) = α_{p/q}(x − x′, t − t′, η)e^{i(x−x′)·ξ−⟨η⟩^{p/q}(x−x′)^{2}}E(x′, t′) + O(e^{−δ⟨η⟩^{p/q}})

in U ∩ ℝ^{3},

g = O(1) in U,

and for (x′, t′) ∈ U ∩ ℝ^{3},

g(x′, t′) = O(e^{−δ|τ|^{p/q}}) if |x′| > c and |x| < c′,

uniformly in (x, t, η).

The method of proof is the same as that of Lemmas 5.1 of [1] and 4.1 of [2]. Before presenting the details, we indicate how it implies Gevrey hypoellipticity. Set γ = p/q.

Suppose that W ⊂ ℝ^{3} is open, u ∈ 𝒟′(W), and Lu ∈ G^{s}(W) for some s ≥ γ^{−1} = q/p. Because L is elliptic wherever x ≠ 0, u ∈ G^{s} on W \ {(x, t) : x = 0} [7]. Since L is invariant under translation with respect to t, it suffices to show that if 0 ∈ W, then u ∈ G^{s} at 0. Fix a relatively compact open set V ⊂ U ∩ W and h ∈ C_{0}^{∞}(U ∩ W) satisfying h ≡ 1 in a neighborhood of \overline{V}. Replace u by u·h. Then u, Lu ∈ C^{∞}(U ∩W), because L is C^{∞} hypoelliptic.

Consider any (x, t) ∈ V and η = (ξ, τ) ∈ ℝ × ℝ^{2} with |ξ| ≤ |τ|, and let g = g_{(x,t,η)} satisfy the conclusions of Lemma 3.1. Then

F_{γ}u(x, t, η) = ⟨u, L(Eg)⟩ + O(e^{−δ⟨η⟩^{γ}})
= ⟨Lu, Eg⟩ + O(e^{−δ⟨η⟩^{γ}}).

By the characterization (5) of G^{s}, there exists an open set 0 ∈ V_{1} ⊂ ℂ^{3} such that for each η, Lu may be decomposed in V_{1} ∩ ℝ^{3} as

Lu = G + O(e^{−δ⟨η⟩^{1/s}}),

where G is holomorphic and O(1) in

{(x, t) ∈ V_{1} : |Im(x, t)| < ⟨η⟩^{−(s−1)/s}},

uniformly in η. For all (x, t) in a compact subset of an open ball B ⊂ V ∩V_{1} centered at 0, then,

F_{γ}u(x, t, η) = ∫_{B} Lu(x′, t′)g(x′, t′)e^{i(t−t′)·τ−⟨η⟩^{γ}(t−t′)^{2}} dx′dt′ + O(e^{−δ⟨η⟩^{γ}})
= ∫_{B} G(x′, t′)g(x′, t′)e^{i(t−t′)·τ−⟨η⟩^{γ}(t−t′)^{2}} dx′dt′ + O(e^{−δ⟨η⟩^{1/s}}).

The hypothesis s ≥ q/p = 1/γ has been invoked in passing from the first line to the second, while the fact that g(x′, t′) exp(−⟨η⟩^{γ}(t − t′)^{2}) is O(exp(−δ|τ|)) for (x′, t′) outside B has been used to justify restricting the integral to B in the first line; this bound for g holds, by Lemma 3.1, provided that B is chosen to be sufficiently small and x is restricted to a compact subset of B. This last integral is O(exp(−δ⟨η⟩^{1/s})), as is seen by deforming the contour of integration with respect to t′ as in the proof that (5) ⇒ (2) in Section 2, exploiting the holomorphic extendibility of both G and g. Thus we have shown that F_{γ}u(x, t, η) is O(exp(−δ⟨η⟩^{1/s})), in the region where |ξ| ≤ |τ|.

There remains the elliptic region, where |ξ| ≥ |τ|. Then F_{γ}u(x, t, η) is analyzed in the same way, using the following simpler variant of Lemma 3.1. Let

E = e^{i(x−x′, t−t′)·η−⟨η⟩^{p/q}(x−x′, t−t′)^{2}}.

Lemma 3.2. There exist δ ∈ ℝ^{+} and an open set 0 ∈ U ⊂ ℂ^{3} such that for each (x, t) ∈ U ∩ℝ^{3} and each η = (ξ, τ) ∈ ℝ^{3} satisfying |ξ| ≥ |τ|, there exists g holomorphic in U satisfying

L(Eg)(x′, t′) = α_{p/q}(x − x′, t − t′, η) · E(x′, t′) + O(e^{−δ⟨η⟩}) in ℝ^{3} ∩ U

and g = O(1) in U, uniformly in (x, t, η).

This completes the proof of Gevrey regularity for all s ≥ q/p, modulo the proofs of Lemmas 3.1 and 3.2.

To begin the proof of Lemma 3.1, replace (x, t) and (x′, t′) in the statement of the lemma by (x̃, t̃) and (x, t), respectively, so that L acts with respect to (x, t), and the parameters are (x̃, t̃) and η = (ξ, τ). Define γ = p/q and set

E(t) = e^{i(t̃−t)·τ−⟨η⟩^{γ}(t̃−t)^{2}} and L_{η} = E^{−1} ∘ L ∘ E.

Both E and L_{η} depend on the parameters t̃, η. Put

𝒜_{τ} = ∂_{x}^{2} − x^{2(p−1)}τ_{1}^{2} − x^{2(q−1)}τ_{2}^{2}.

The operator 𝒜_{τ} will be regarded sometimes as an ordinary differential operator acting on functions of x and depending on parameters t, t̃, η, and at other times as acting on functions of (x, t). For x ∈ ℝ and 0 ≠ τ ∈ ℝ^{2} set

w(x, t) = (τ_{1}^{2/p} + τ_{1}^{2}x^{2(p−1)} + τ_{2}^{2/q} + τ_{2}^{2}x^{2(q−1)})^{1/2}.

Fix v ∈ C^{∞}(ℝ), real-valued and nonnegative, satisfying v ≡ 0 in a small neighborhood of 0, and v(x) ≡ 1 for all |x| ≥ 1. Let ρ > 0 be a small parameter, which will eventually depend on v, and define weighted Sobolev spaces ℋ_{τ}^{k}(ℝ) with norms

\begin{aligned}∥f∥_{ℋ_{τ}^{0}(ℝ)}^{2} & = ∫_{ℝ} |f(x)|^{2}w(x, τ)^{−2}e^{ρ|τ|^{p/q}v(x)} dx, \\ ∥f∥_{ℋ_{τ}^{1}(ℝ)}^{2} & = ∫_{ℝ}[w(x, τ)^{−2}|∂_{x}f|^{2} + |f|^{2}]e^{ρ|τ|^{p/q}v(x)} dx, \\ ∥f∥_{ℋ_{τ}^{2}(ℝ)}^{2} & = ∫_{ℝ}[w(x, τ)^{−2}|∂_{x}^{2}f|^{2} + |∂_{x}f|^{2} + w(x, τ)^{2}|f|^{2}] e^{ρ|τ|^{p/q}v(x)} dx. \end{aligned}

Lemma 3.3. For all sufficiently small |ρ|, 𝒜_{τ} : C_{0}^{2}(ℝ) → C_{0}^{0}(ℝ) extends to an invertible operator from ℋ_{τ}^{2}(ℝ) to ℋ_{τ}^{0}(ℝ), uniformly in ρ, τ for all |τ| ≥ 1.

The lemma remains true if the exponent p/q in the definition of the norms is replaced by any exponent belonging to [0, 1], but our application requires p/q.

Proof. The definitions are set up so as to imply directly that 𝒜_{τ} : ℋ_{τ}^{2}(ℝ) → ℋ_{τ}^{0}(ℝ) is bounded, uniformly in ρ, τ. To prove invertibility consider first the case where ρ = 0. For any real valued f ∈ C_{0}^{2}(ℝ),

− ∫ 𝒜_{τ}f · f dx = ∥∂_{x}f∥^{2} + ∫ |f|^{2}(τ_{1}^{2}x^{2(p−1)} + τ_{2}^{2}x^{2(q−1)}) dx.

Recall that for any f ∈ C_{0}^{2} and any λ ∈ ℝ^{+},

λ^{2/m}∥f∥^{2} ≤ C∥∂_{x}f∥^{2} + C ∫_{ℝ} |f|^{2}λ^{2}x^{2(m−1)};

the case λ = 1 is elementary, and the general case then follows by scaling. Combining this with the preceding inequality yields

∥∂_{x}f∥^{2} + ∫ |f|^{2}w(x, τ)^{2} ≤ −C ∫ 𝒜_{τ}f · f dx ≤ C∥𝒜_{τ}f∥_{ℋ_{τ}^{0}} · (∫ |f|^{2}w(x, t)^{2})^{1/2}

so that

∥∂_{x}f∥^{2} + ∫ |f|^{2}w(x, τ)^{2} ≤ C∥𝒜_{τ}^{2}f∥_{ℋ_{τ}^{0}} .

We have thus controlled two of the three terms in the definition of the ℋ_{τ}^{2} norm. From the identity ∂_{x}^{2} = 𝒜_{τ} + τ_{1}^{2}x^{2(p−1)} + τ_{2}^{2}x^{2(q−1)}, the a priori inequality

∥f∥_{ℋ_{τ}^{2}(ℝ)}^{2} ≤ C∥𝒜_{τ}f∥_{ℋ_{τ}^{0}(ℝ)}^{2} for all f ∈ C_{0}^{2}  (3.2)

thus follows, uniformly in τ, for ρ = 0. The corresponding inequality for |ρ| small^{4} and |τ| ≥ 1 follows by conjugation with exp(ρ|τ|^{γ}v(x)/2), using the fact that w(x, τ) ≥ |τ| ≥ |τ|^{γ} on the support of ∇v. Now ℋ_{τ}^{2}(ℝ) embeds compactly in ℋ_{τ}^{0}(ℝ), and 𝒜_{τ} is symmetric in L^{2}(ℝ, dx); the invertibility follows from these two facts and from (3.2) as in the proof of Lemma 3.1 of [1]. Uniformity follows from the uniformity of (3.2).

4When p/q is strictly less than 1, |ρ| need not be chosen to be small.

For any open disk D ⊂ ℂ^{2} centered at 0, define Sobolev spaces ℋ_{τ}^{k}(ℝ × D) of measurable, locally square-integrable functions of (x, t) ∈ ℝ×D that are holomorphic with respect to t ∈ D (for almost every x), for which the norms are

\begin{aligned}∥f∥_{ℋ_{τ}^{0}(ℝ×D)}^{2} & = ∫_{ℝ×D} |f(x, t)|^{2}w(x, τ)^{−2}e^{ρ|τ|^{p/q}v(x)} dx dtdt̄, \\ ∥f∥_{ℋ_{τ}^{1}(ℝ×D)}^{2} & = ∫_{ℝ×D}[w(x, τ)^{−2}|∂_{x}f|^{2} + |f|^{2}]e^{ρ|τ|^{p/q}v(x)} dx dtdt̄, \\ ∥f∥_{ℋ_{τ}^{2}(ℝ×D)}^{2} & = ∫_{ℝ×D}[w(x, τ)^{−2}|∂_{x}^{2}f|^{2} + |∂_{x}f|^{2} + w(x, τ)^{2}|f|^{2}] e^{ρ|τ|^{p/q}v(x)} dx dtdt̄. \end{aligned}

Clearly 𝒜_{τ} : ℋ_{τ}^{2}(ℝ × D) → ℋ_{τ}^{0}(ℝ × D) is invertible, as a consequence of Lemma 3.3, uniformly in ρ, τ, D provided that |τ| ≥ 1 and |ρ| is sufficiently small.

Fix h ∈ C_{0}^{∞}(ℝ) satisfying h ≡ 1 in a neighborhood of 0. Define ℰ_{η} by

𝒜_{τ} + ℰ_{η} = ∂_{x}^{2} + x^{2(p−1)}[−iτ_{1} + 2⟨η⟩^{γ}(t̃_{1} − t_{1})]^{2} + x^{2(q−1)}[−iτ_{2} + 2⟨η⟩^{γ}(t̃_{2} − t_{2})]^{2}.

Define ℛ_{η} by L_{η} = 𝒜_{τ} + ℰ_{η} + ℛ_{η}. The perturbation term ℛ_{η} involves differentiation with respect to t, whereas 𝒜_{τ}, ℰ_{η} do not. Let D ⊃ D′ be open disks in ℂ^{2} centered at the origin, with distance (D′, ∂D) ≥ ε. Let r be the radius of D.

Lemma 3.4. If |ρ| is sufficiently small then the perturbation terms ℰ_{η}, ℛ_{η} satisfy the following bounds, uniformly for all |τ| ≥ 1.

h(x) · ℰ_{η} : ℋ_{τ}^{2}(ℝ × D) → ℋ_{τ}^{0}(ℝ × D) with norm O(r + |t̃|).  (3.3)
h · ℛ_{η} : ℋ_{τ}^{2}(ℝ × D) → ℋ_{τ}^{0}(ℝ × D′) with norm O(ε^{−2}|τ|^{−2γ})  (3.4)

whenever distance (D′, ∂D) ≥ ε.

Proof. (3.3) would follow from the inequality

|τ|^{γ}(|x|^{p−1} + |x|^{q−1}) ≤ Cw(x, τ) ∀ x ∈ support(h), ∀ |τ| ≥ 1.  (3.5)

To prove this note that |x|^{q−1} ≤ C|x|^{p−1} for all x in the support of h. If |τ_{1}| ≥ |τ_{2}| then since p ≤ q,

|τ|^{γ}|x|^{p−1} = |τ|^{p/q}|x|^{p−1} ≤ C|τ_{1}||x|^{p−1} ≤ Cw(x, τ).

If |τ_{2}| ≥ |τ_{1}| then

|τ|^{γ}|x|^{p−1}  ≤ C|τ_{2}|^{p/q}|x|^{p−1}  = C|τ_{2}|^{1/q} · (|τ_{2}|^{1/q}|x|)^{p−1}
≤ C|τ_{2}|^{1/q}(1 + (|τ_{2}|^{1/q}|x|)^{q−1}) ≤ Cw(x, τ) .

Thus (3.3) holds.

Note that by Cauchy’s inequality relating the derivative of a holomorphic function at a point to the integral of the function over a disk, ∂_{t_{j}} maps H^{2}(D) to H^{2}(D′) boundedly, with norm O(ε^{−1}), where H^{2} denotes the space of all holomorphic squareintegrable functions, with respect to Lebesgue measure. Therefore

∂_{t_{j}} : ℋ_{τ}^{k}(ℝ × D) → ℋ_{τ}^{k}(ℝ × D′) is O(ε^{−1}) for each k,

uniformly in τ.

By (3.5), h(x)x^{2(p−1)} and h(x)x^{2(q−1)} map ℋ_{τ}^{2}(ℝ) to ℋ_{τ}^{0}(ℝ) with a bound O(|τ|^{−2γ}). Thus

h(x)x^{2(p−1)}∂_{t_{1}}^{2} : ℋ_{τ}^{2}(ℝ × D) → ℋ_{τ}^{0}(ℝ × D′)

is O(ε^{−2}|τ|^{−2γ}); the same goes for h(x)x^{2(q−1)}∂_{t_{2}}^{2}. This implies (3.4).

We turn to the proof of Lemma 3.1. Set

ψ(x, t) = e^{i(x̃−x)ξ−⟨η⟩^{γ}(x̃−x)^{2}}α_{γ}(x̃ − x, t̃− t, η).

It suffices to solve

(𝒜_{τ} + hℰ_{η} + hℛ_{η})g(x, t) = ψ(x, t) + O(e^{−δ⟨η⟩^{γ}})

globally on ℝ × D_{0} for some polydisk D_{0} ⊂ ℂ^{2} centered at 0, for then the original equation is solved in the region where h ≡ 1.

Fix nested polydisks D_{∞} ⊂ D_{2} ⊂ D_{1} ⊂ ℂ^{2} centered at 0, such that each is relatively compact in the next. Let Λ ∈ ℝ^{+} be a large constant to be chosen below. Assume |τ| to be large, and choose an integer N such that | N − Λ^{−1}|τ|^{γ} | < 1. Construct polydisks D_{2} ⊃ D_{3} ⊃ ⋯ ⊃ D_{N} = D_{∞}, centered at 0, satisfying

distance (D_{j+1}, ∂D_{j}) ≥ cΛ|τ|^{−γ},

where c is a constant independent of Λ, τ.

Define g by the Neumann series

g = ∑_{j=0}^{N}(−1)^{j}[𝒜_{τ}^{−1}(hℰ_{η} + hℛ_{η})]^{j}𝒜_{τ}^{−1}ψ.

If D_{1} is chosen to be sufficiently small, but independent of η, then 𝒜_{τ}^{−1}hℰ_{η} maps ℋ_{τ}^{2}(ℝ × D_{j}) to itself with a bound of at most 1/2, for all j, τ, while

𝒜_{τ}^{−1}hℛ_{η} : ℋ_{τ}^{2}(ℝ × D_{j}) → ℋ_{τ}^{2}(ℝ × D_{j+1}) is O(Λ^{−1}).

Therefore it is possible to choose Λ sufficiently large that

[𝒜_{τ}^{−1}(hℰ_{η} + hℛ_{η})] : ℋ_{τ}^{2}(ℝ × D_{j}) → ℋ_{τ}^{2}(ℝ × D_{j+1})

has norm at most 3/4, for all j, for all sufficiently large |τ|. Then

∥g∥_{ℋ_{τ}^{2}(ℝ×D_{∞})} ≤ C < ∞,

uniformly in η, x̃, t̃, provided that the ℋ_{τ}^{0}(ℝ × D_{1}) norm of ψ is bounded uniformly in η, (x̃, t̃). This is so, provided that x̃ belongs to a relatively compact subset of the region where v ≡ 0 and ρ > 0 is chosen to be sufficiently small (recall the weight exp(ρ|τ|^{γ}v(x)) in the definition of the ℋ_{τ}^{k} norms).

Finally,

(𝒜_{τ} + hℰ_{η} + hℛ_{η})g = ψ ± [(hℰ_{η} + hℛ_{η})𝒜_{τ}^{−1}]^{N+1}ψ,

and

[(hℰ_{η} + hℛ_{η})𝒜_{τ}^{−1}]^{N+1}ψ = O(3/4)^{N+1} = O(e^{−δ⟨η⟩^{γ}})

in the ℋ_{τ}^{0}(ℝ × D_{∞}) norm, where δ depends on the choice of Λ but is positive. This completes the proof of Lemma 3.1.

The proof of Lemma 3.2 is parallel, but is much simpler because the principal symbol of L is nonzero where |ξ| ≥ |τ|. The conjugated operator L_{η} is replaced by E^{−1}LE, where

E(x, t) = e^{i(x̃−x, t̃−t)·η−⟨η⟩^{γ}(x̃−x, t̃−t)^{2}}.

The ordinary differential operator 𝒜_{τ} is now replaced by the operator defined by multiplication by the symbol −ξ^{2} − τ_{1}^{2}x^{2(p−1)} − τ_{2}^{2}x^{2(q−1)}, which for x in any bounded region is comparable to ξ^{2}, hence to ⟨η⟩^{2}, uniformly in all parameters. All derivatives with respect to x or t are now incorporated into ℛ_{η}. One works in the simpler Hilbert spaces H^{2}(D_{j}), for a sequence of polydisks D_{j} ⊂ ℂ^{3} centered at 0, where 1 ≤ j ≤ N ≈ Λ^{−1}⟨η⟩ and distance (D_{j+1}, ∂D_{j}) ≥ cΛ⟨η⟩. The result is a solution g of E^{−1}LEg = ψ + O(exp[−δ⟨η⟩]), in H^{2}(D_{∞}) norm, for a certain polydisk D_{∞} ⊂ ℂ^{3} containing the origin and independent of η.

4. Gevrey Irregularity

Proposition 4.1. If q/p > 1 and s < q/p, then L is not G^{s} hypoelliptic in any neighborhood of 0.

The proof follows a well established method, based upon separation of variables and a reduction to certain eigenvalue problems. For the operators under consideration here this procedure is relatively elementary, first because a complete separation of variables is possible, second because the resulting eigenvalue problems are straightforward.

Consider the ordinary differential operators

ℒ_{z} = ∂_{x}^{2} − x^{2(q−1)} + zx^{2(p−1)}, z ∈ ℂ.

Lemma 4.2. [10] For any 1 ≤ p ≤ q ∈ ℕ there exist z ∈ ℝ and a Schwartz class function f ≠ 0, defined on ℝ, satisfying ℒ_{z}f ≡ 0.

Lemma 4.3. If a linear partial differential operator P is G^{s} hypoelliptic in some open set U containing a point x_{0}, then there exists B < ∞ such that for each f ∈ L^{2}(U) satisfying P f = 0 in U,

|∂^{α}f(x_{0})| ≤ B^{|α|+1}|α|^{s|α|}∥f∥_{L^{2}(U)}

for every multi-index α.

The proofs of Lemma 4.2 and of Lemma 4.3 are omitted. The former is in Oleĭnik [10]. The latter is a routine application of the Baire category theorem using the Banach spaces X_{B} of all functions f ∈ L^{2}(U) such that Lf = 0 and such that there exists C < ∞ such that |∂^{α}f(x_{0})| ≤ CB^{|α|}|α|^{s|α|} for all α, as in the case s = 1 discussed in [10].

Proof of Proposition 4.1. Let z, f be as in Lemma 4.1. Suppose that L were G^{s} hypoelliptic in a bounded neighborhood U of 0, for some s < q/p. Fix a square root w ∈ ℂ of z. For each large λ ∈ ℝ^{+} set

F_{λ}(x, t_{1}, t_{2},) = e^{iλt_{2}}e^{λ^{p/q}wt_{1}}f(λ^{1/q}x).

Then LF_{λ} ≡ 0 in ℝ^{3}. Fix k ∈ {0, 1} such that ∂^{k}f/∂x^{k}(0) ≠ 0. Then

|∂_{x}^{k}∂_{t_{2}}^{N}F_{λ}(0)| = λ^{N}λ^{k/q}|∂_{x}^{k}f(0)|

for each N ∈ ℕ, for each λ. On the other hand,

∥F_{λ}∥_{C_{0}(U)} ≤ Ce^{Cλ^{p/q}}

for some C < ∞, since the factors f and exp(iλt_{2}) are uniformly bounded. Thus by Lemma 4.3, there exists B < ∞ such that

λ^{N} ≤ CB^{N}N^{sN}e^{Cλ^{p/q}} for all λ, N.

For each large N set λ = N^{q/p} to deduce that

N^{Nq/p} ≤ CB^{N}N^{sN}e^{CN} for all N.

But such an inequality is clearly false as N → ∞, if q/p > s.

References

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

[2] , Analytic hypoellipticity in dimension two, preprint.

[3] , On local and global analytic and Gevrey hypoellipticity, Journées “Equations aux derivées partielles,” Saint-Jean-de-Monts, Mai-Juin 1995, Exposé IX.

[4] , A necessary condition for analytic hypoellipticity, Mathematical Research Letters 1 (1994), 241–248.

[5] M. Derridj and C. Zuily, Régularité analytique et Gevrey pour des classes d’opérateurs élliptiques paraboliques dégénérés du second ordre, Astérisque 2,3 (1973), 371–381.

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

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

[8] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1980.

[9] G. Métivier, Une classe d’opérateurs non hypoélliptiques analytiques, Indiana Math. J. 29 (1980), 823–860.

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

[11] 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.

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

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

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

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

University of California, Los Angeles

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