Intrinsic Ultracontractivity, Conditional Lifetimes and Conditional Gauge for Symmetric Stable Processes on

Rough Domains

Zhen-Qing Chen^{∗} Departments of Mathematics and Statistics Cornell University Ithaca, NY 14853 Email: zchen@math.cornell.edu and Renming Song Department of Mathematics University of Illinois Urbana, IL 61801 Email: rsong@math.uiuc.edu

Abstract

For a symmetric α-stable process X on R^{n} with 0 < α < 2, n ≥ 2 and a domain D ⊂ R^{n}, let L^{D} be the infinitesimal generator of the subprocess of X killed upon leaving D. For a Kato class function q, it is shown that L^{D} + q is intrinsic ultracontractive on a Hölder domain D of order 0. This is then used to establish the conditional gauge theorem for X on bounded Lipschitz domains in R^{n}. It is also shown that the conditional lifetimes for symmetric stable process in a Hölder domain of order 0 are uniformly bounded.

Keywords and phrases: Symmetric stable processes, Feynman-Kac semigroup, conditional gauge theorem, logarithmic Sobolev inequality, intrinsic ultracontractivity. Running Title: Conditional Gauge Theorem

∗The research of this author is supported in part by NSA Grant MDA904-98-1-0044

1 Introduction

A symmetric α-stable process X on R^{n} is a Lévy process whose transition density p(t, x − y) relative to Lebesgue measure is uniquely determined by its Fourier transform

∫_{R^{n}} e^{ix·ξ}p(t, x)dx = e^{−t|ξ|^{α}}.

Here α must be in the interval (0, 2]. When α = 2, we get a Brownian motion running with a time clock twice as fast as the standard one. Brownian motion has been intensively studied due to its central role in modern probability theory and its numerous important applications in other scientific areas including many other branches of mathematics. In the sequel, symmetric stable processes are referred to the case when 0 < α < 2. During the last thirty years, there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes. Stable processes are now widely used in physics, operation research, queuing theory, mathematical finance and risk estimation. Recently some fine properties related to symmetric stable processes and Riesz potential theory, as counterparts to Brownian motion and Newtonian potential theory, have been studied, for example in [8]–[13].

Let X be a symmetric α-stable process in R^{n} with n ≥ 2. It is well known that X is transient and has Green function G(x, y) = A(n, α)|x − y|^{α−n} where

A(n, α) = \frac{α2^{α−1}Γ(\frac{α+n}{2})}{π^{n/2}Γ(1 − \frac{α}{2})}.  (1.1)

Definition 1.1 A Borel measurable function q on R^{n} is said to be in the Kato class K_{n,α} if

lim_{r↓0} sup_{x∈R^{n}} ∫_{|x−y|≤r} \frac{|q(y)|}{|x − y|^{n−α}} dy = 0.

For q ∈ K_{n,α} and t > 0, define

e_{q}(t) = exp (∫_{0}^{t} q(X_{s})ds) .

For a domain D ⊂ R^{n}, the gauge function g of (D, q) is defined by g(x) = E^{x}[e_{q}(τ_{D})], x ∈ D where τ_{D} = inf{t > 0 : X_{t} ∉ D}. Chung and Rao [16] showed the following gauge theorem for domain D with finite Lebesgue measure: the function g is either identically infinite or is bounded on D. In the latter case, (D, q) is said to be gaugeable (with respect to X). By Theorem 1 of Chung [14], (D, 0) is gaugeable for any domain D with finite Lebesgue measure. It is proved in [12] that (D, q) is gaugeable if and only if the first eigenvalue of L^{D} + q is negative, where L^{D} is the non-positive definite infinitesimal generator of the part process X^{D} of X killed upon leaving domain D.

Now assume that D is a bounded Lipschitz domain. Recall that the Green function G_{D} and Poisson kernel K_{D} of X in D are determined by the following equations. For x ∈ D,

E^{x} [∫_{0}^{τ_{D}} f(X_{s})ds] = ∫_{D} G_{D}(x, y)f(y)dy for f ≥ 0 on D,
E^{x} [φ(X_{τ_{D}})] = ∫_{D^{c}} K_{D}(x, z)φ(z)dz for φ ≥ 0 on D^{c}.

Fix an x_{0} ∈ D, it is shown in [13] (see also [9]) that for each z ∈ ∂D and x ∈ D,

M_{D}(x, z) = lim_{y→z, y∈D} \frac{G_{D}(x, y)}{G_{D}(x_{0}, y)} exits and is finite.  (1.2)

M_{D} is called the Martin kernel of X in D.

Definition 1.2 Let D be a domain in R^{n}. A locally integrable function f defined on D taking values in (−∞, ∞] and satisfying the condition ∫_{{|x|>1}∩D} |f(x)||x|^{−(n+α)}dx < ∞ is said to be superharmonic respect to X^{D} if f is lower semicontinuous in D and for each x ∈ D and each ball B(x, r) with \overline{B(x, r)} ⊂ D,

f(x) ≥ E_{x}[f(X_{τ_{B(x,r)}}); τ_{B(x,r)} < τ_{D}].

A function h is said to be harmonic with respect to X^{D} if both h and −h are superharmonic respect to X^{D}.

Suppose that h ≥ 0 is superharmonic in D with respect to X^{D}. Then by Theorem 2.3 of [13], for any domain D_{1} ⊂ \overline{D_{1}} ⊂ D, E_{x}[h^{−}(X_{τ_{D_{1}}}^{D})] < ∞ and

h(x) ≥ E_{x}[h(X_{τ_{D_{1}}}^{D})] for every x ∈ D_{1}.

This implies (cf. e.g. page 11 of Dynkin [21]) that

h(x) ≥ E^{x}[h(X_{t}^{D})].

Therefore one can define the h–conditioned stable process. Note that for each fixed y ∈ D and z ∈ ∂D, G_{D}(x, y) and M_{D}(x, z) are harmonic functions in x with respect to X^{D\{y}} and X^{D}, respectively. For any w ∈ \overline{D}^{c}, K_{D}(x, w) is superharmonic in x with respect to X^{D}. Therefore we can define the G_{D}(·, y)–conditioned stable process, K_{D}(·, w)–conditioned stable process and the M_{D}(·, z)–conditioned stable process. The probability laws corresponding to these conditional stable processes will be denoted by P_{y}^{x}, P_{w}^{x} and P_{z}^{x}, respectively. The following conditional gauge theorem is established in Chen and Song [12].

Theorem 1.1 Assume that D is a bounded C^{1,1} smooth domain in R^{n}, q ∈ K_{n,α} and that (D, q) is gaugeable. Then exists a constant c > 1 such that

c^{−1} ≤ inf_{(x,z)∈D×(R^{n}\∂D)} E_{z}^{x} [e_{q}(ζ)] ≤ sup_{(x,z)∈D×(R^{n}\∂D)} E_{z}^{x} [e_{q}(ζ)] ≤ c.

Recently it is shown in [13] that under the condition of Theorem 1.1, there is a constant c > 1 such that

c^{−1} ≤ inf_{(x,z)∈D×∂D} E_{z}^{x} [e_{q}(ζ)] ≤ sup_{(x,z)∈D×∂D} E_{z}^{x} [e_{q}(ζ)] ≤ c.

Gauge theorem for Brownian motion was first proved by Chung and Rao in [15] for bounded q and later was generalized to more general q by various authors. The conditional gauge theorem for Brownian motion was first proved by Falkner [22] for bounded q and a class of domains including bounded C^{2} domains. Extensions of this result to q belonging to the Kato class and to bounded C^{1,1} domains were given by Zhao in [32] and [33]. The conditional gauge theorem has also been generalized by Cranston, Fabes and Zhao [18] to diffusion processes whose infinitesimal generators are uniformly elliptic divergence form operators and to bounded Lipschitz domain in R^{n}. For a more detailed story about gauge and conditional gauge theorem for Brownian motion, the interested reader is referred to the recent book of Chung and Zhao [17].

In this paper, we extend the above conditional gauge theorem for symmetric stable processes from bounded C^{1,1} domains to bounded Lipschitz domains. We follow the idea from Chen and Song [12], proving conditional gauge theorem through intrinsic ultracontractivity, but with substantial improvements, motivated by Bañuelos [3]. In [12], sharp estimates on Green functions of bounded C^{1,1} domains obtained in [11] were used to establish conditional gauge theorem. However these estimates are no longer available for bounded Lipschitz domains. We are able to circumvent it in this paper. We first show that, under the assumption that (D, q) is gaugeable, the Green function V_{q} of L^{D} + q on D is bounded by a constant multiple of G_{D}. The latter is then used to prove the conditional gauge theorem on bounded Lipschitz domains. In order to establish V_{q} ≤ c G_{D} on D × D for some c > 0, we show that operator L^{D} + q is intrinsic ultracontractive on D. In fact we show that L^{D} + q is intrinsic ultracontractive for any Hölder domain D of order 0. For definitions of intrinsic ultracontractivity and Hölder domain of order 0, see Definitions 3.1 and 3.2 below. We mention here that John domains, particularly bounded NTA domains and Lipschitz domains, are Hölder domains of order 0. Under the assumption that D has finite Lebesgue measure such that L^{D} is intrinsic ultracontractive, we show that there is a constant c > 0 such that for each nontrivial nonnegative superharmonic function h in D with respect to X^{D}, sup_{x∈D} E_{h}^{x}[τ_{D}] ≤ c. This especially implies that for a bounded Lipschitz domain D,

sup_{x∈D, z∈R^{n}} E_{z}^{x}[τ_{D}] < ∞.

Previously the boundedness of conditional lifetimes was proved for bounded C^{1,1} smooth domains for symmetric stable processes in [11] and [13]. Conditional lifetime estimates for Brownian motion in planar domains was first studied by M. Cranston and T. McConnell, in answering a question of K. L. Chung. The first extension to several dimensions for Brownian motion was done by M. Cranston, followed by many works on important extensions to more general domains and to elliptic diffusions (see [3], [4] and the references therein).

Acknowledgement. We are grateful to Rodrigo Bañuelos, Rich Bass and Chris Burdzy for very helpful discussions. We thank Krzysztof Bogdan for very interesting and helpful discussions at the MSRI, Berkeley, and University of Washington, Seattle. Some of our results have been independently and simultaneously obtained by him and will appear in [10]. Part of the research for this paper were conducted while the authors were visiting MSRI in Berkeley. Financial support from MSRI are gratefully acknowledged.

2 Preliminaries

Throughout this paper, we assume that n ≥ 2 and 0 < α < 2. Let X be a symmetric α-stable process in R^{n}. It is well known that the Dirichlet form (E, F^{R^{n}}) associated with X is given by

E(u, v) = A(n, α) ∫_{R^{n}} ∫_{R^{n}} \frac{(u(x) − u(y))(v(x) − v(y))}{|x − y|^{n+α}} dxdy
F^{R^{n}} = \{u ∈ L^{2}(R^{n}) : ∫_{R^{n}} ∫_{R^{n}} \frac{(u(x) − u(y))^{2}}{|x − y|^{n+α}} dxdy < ∞\} ,

where A(n, α) is the constant in (1.1). As usual, we use {P_{t}}_{t≥0} to denote the transition semigroup of X and G to denote the potential of {P_{t}}_{t≥0}; that is,

Gf(x) = ∫_{0}^{∞} P_{t}f(x)dt.

FROM now on, we assume that D is a domain in R^{n}. Adjoin an extra point ∂ to D and set

X_{t}^{D}(ω) = \{ \begin{matrix}X_{t}(ω) & if t < τ_{D}(ω), \\ ∂ & if t ≥ τ_{D}(ω). \end{matrix}

The process X^{D} is called the symmetric α-stable process killed upon leaving D, or simply the killed symmetric α-stable process on D. It is well known (cf. [23]) that the Dirichlet form corresponding to the killed symmetric α-stable process X^{D} on D is (E, F) where

F = {u ∈ F^{R^{n}} : ũ = 0 quasi everywhere on D^{c}},

where ũ denotes a quasi continuous version of u.

We are going to use P_{t}^{D} and p^{D}(t, x, y) to denote the transition semigroup and transition density of X^{D} respectively. L^{D} will used to denote the non–positive definite infinitesimal generator of X^{D} on L^{2}(D, dx). G_{D}, K_{D} and M_{D} will be used to denote the Green function, Poisson kernel and Martin kernel of X on D respectively. From [11] we know that when D satisfies the uniform exterior cone condition, G_{D} and K_{D} are related by

K_{D}(x, z) = A(n, α) ∫_{D} \frac{G_{D}(x, y)}{|y − z|^{n+α}} dy,  (2.1)

where A(n, α) is the constant in (1.1). Suppose that h > 0 is a positive superharmonic function with respect to X^{D}. We define

p_{h}^{D}(t, x, y) = h(x)^{−1}p^{D}(t, x, y)h(y), t > 0, x, y ∈ D.

It is easy to check that p_{h}^{D} is a transition density and it determines a Markov process. This process is called the h–conditioned symmetric stable process.

For any y ∈ D, the G_{D}(·, y)–conditioned symmetric stable process is a Markov process with state space (D \ {y}) ∪ {∂}, with lifetime ζ = τ_{D\{y}}. We will use P_{y}^{x} and E_{y}^{x} to denote the probability and expectation with respect to this process.

For any w ∈ \overline{D}^{c}, the K_{D}(·, w)–conditioned symmetric stable process is a Markov process with state space D ∪ {∂}, with lifetime ζ = τ_{D}. We will use P_{w}^{x} and E_{w}^{x} to denote the probability and expectation with respect to this process.

For any z ∈ ∂D, the M_{D}(·, z)–conditioned symmetric stable process is a Markov process with state space D ∪ {∂}, with lifetime ζ = τ_{D}. We will use P_{z}^{x} and E_{z}^{x} to denote the probability and expectation with respect to this process.

In [11] we proved the following 3G Theorem.

Theorem 2.1 Suppose that D is a bounded C^{1,1} domain in R^{n} with n ≥ 2. Then there exists a constant c = c(D, n, α) > 0 such that

\frac{G_{D}(x, y)G_{D}(y, w)}{G_{D}(x, w)} ≤ \frac{c |x − w|^{n−α}}{|x − y|^{n−α}|y − w|^{n−α}} , x, y, w ∈ D,  (2.2)
\frac{G_{D}(x, y)K_{D}(y, z)}{K_{D}(x, z)} ≤ \frac{c |x − z|^{n−α}}{|x − y|^{n−α}|y − z|^{n−α}} , x, y ∈ D, z ∈ D^{c}.  (2.3)

Using Theorem 2.1 above and the scaling property we easily get the following result.

Corollary 2.2 There exists a constant c = c(n, α) > 0 such that for any ball B in R^{n} one has

\frac{G_{B}(x, y)G_{B}(y, w)}{G_{B}(x, w)} ≤ \frac{c |x − w|^{n−α}}{|x − y|^{n−α}|y − w|^{n−α}} , x, y, w ∈ B,  (2.4)
\frac{G_{B}(x, y)K_{B}(y, z)}{K_{B}(x, z)} ≤ \frac{c |x − z|^{n−α}}{|x − y|^{n−α}|y − z|^{n−α}} , x, y ∈ B, z ∈ B^{c}.  (2.5)

3G Theorem actually holds for bounded Lipschitz domains.

Theorem 2.3 Suppose that D is a bounded Lipschitz domain in R^{n}. Then there exists a constant c = c(D, n, α) > 0 such that

\frac{G_{D}(x, y)G_{D}(y, w)}{G_{D}(x, w)} ≤ \frac{c |x − w|^{n−α}}{|x − y|^{n−α}|y − w|^{n−α}} , x, y, w ∈ D  (2.6)
\frac{G_{D}(x, y)M_{D}(y, z)}{M_{D}(x, z)} ≤ \frac{c |x − z|^{n−α}}{|x − y|^{n−α}|y − z|^{n−α}} , x, y ∈ D, z ∈ ∂D.  (2.7)

Proof. Note that the boundary Harnack inequality holds on D for positive harmonic functions in D with respect to the symmetric stable process X, due to Bogdan [8]. One can then prove (2.6) by repeating the argument in Section 6.2 of [17] so we omit the details here. Recalling (1.2), inequality (2.7) follows from (2.6) by passing w → z.

We need the following result on the decomposition of Kato class functions later on.

Lemma 2.4 Let q have compact support. Then q ∈ K_{n,α} if and only if, for any ε > 0, there is a function q_{ε} such that q − q_{ε} is bounded and

sup_{x∈R^{n}} ∫_{R^{n}} \frac{|q_{ε}(y)|}{|x − y|^{n−α}} dy ≤ ε

Proof. The proof is the same as that of Theorem 4.16 in [1].

In what follows, q is an arbitrary but fixed function in K_{n,α}. For a domain D in R^{n}, define

T_{t}f(x) = E^{x} [e_{q}(t)f(X(t)); t < τ_{D}] , x ∈ D.

The semigroup T_{t} admits an integral kernel u_{q}(t, x, y) (cf. [12]). The infinitesimal generator of the semigroup T_{t} on L^{2}(D, dx) is L^{D}+q. If D has finite Lebesgue measure, then it is known (see Theorem 3.3 of [12]) that L^{D} +q has discrete spectrum. Let {λ_{k}, k = 0, 1, ⋯} be all the eigenvalues of L^{D} +q written in decreasing order, each repeated according to its multiplicity. Then λ_{k} ↓ −∞ and the corresponding eigenfunctions {φ_{k}, k = 0, 1, ⋯} can be chosen that they form an orthonormal basis of L^{2}(D, dx). We know that all the eigenfunctions φ_{k} are bounded and the first eigenfunction φ_{0} can be chosen to be strictly positive in D (cf. [12]).

Definition 2.1 A bounded Borel function f defined on R^{n} is said to a solution to the equation

(L^{D} + q)f(x) = 0,  x ∈ D  (2.8)

if it is continuous in D and for any open domain D_{0} ⊂ \overline{D_{0}} ⊂ D,

f(x) = E^{x}[e_{q}(τ_{D_{0}})f(X_{τ_{D_{0}}})], x ∈ D_{0}.

Clearly the first eigenfunction φ_{0} of T_{t} is a positive solution of

(L^{D} + q + λ_{0})f(x) = 0, x ∈ D.

For positive solutions of (2.8) we have the following uniform local Harnack inequality, which is applicable to φ_{0} with q + λ_{0} in place of q.

Theorem 2.5 There exist two positive constants r_{0} = r_{0}(q) and C = C(q) > 0 such that for any solution f of (2.8) which is strictly positive on D and for any ball B(x_{0}, r) with 0 < r ≤ r_{0} and B(x_{0}, 2r) ⊂ D, one has

sup_{x∈B(x_{0},r)} f(x) ≤ C inf_{x∈B(x_{0},r)} f(x).

Proof. It follows from (2.5) and the assumption of q ∈ K_{n,α} that there exists a positive number R_{0} such that for any ball of radius r ≤ R_{0} in R^{n},

sup_{x∈B,z∈\overline{B}} ∫_{B} \frac{G_{B}(x, y)|q(y)|K_{B}(y, z)}{K_{B}(x, z)}dy ≤ \frac{1}{2}.

By Jensen’s inequality and Khas’minskii’s lemma,

e^{−1/2} ≤ inf_{x∈B,z∈\overline{B}^{c}} E_{z}^{x}[e_{q}(τ_{B})] ≤ sup_{x∈B,z∈\overline{B}^{c}} E_{z}^{x}[e_{q}(τ_{B})] ≤ 2.

For any ball B(x_{0}, r) with 0 < r ≤ R_{0}/2 and \overline{B(x_{0}, 2r)} ⊂ D, we have for any x ∈ B(x_{0}, 2r),

f(x) = E^{x}[e_{q}(τ_{B(x_{0},2r)})f(X_{τ_{B(x_{0},2r)}})]
= ∫_{B(x_{0},2r)^{c}} E_{z}^{x}[e_{q}(τ_{B(x_{0},2r)})] f(z) K_{B(x_{0},2r)}(x, z) dz
≤ 2E^{x}[f(X_{τ_{B(x_{0},2r)}})].

By Harnack inequality in [5], there exists a constant c = c(n, α) > 0 such that

sup_{x∈B(x_{0},r)} E^{x}[f(X_{τ_{B(x_{0},2r)}})] ≤ c inf_{x∈B(x_{0},r)} E^{x}[f(X_{τ_{B(x_{0},2r)}})].

Therefore for any x, y ∈ B(x_{0}, y),

f(x) ≤ 2c E^{y}[f(X_{τ_{B(x_{0},2r)}})]
≤ 2c e^{1/2} ∫_{B(x_{0},2r)^{c}} E_{z}^{y}[e_{q}(τ_{B(x_{0},2r)})] f(z) K_{B(x_{0},2r)}(y, z) dz
= 2c e^{1/2}f(y),

and the proof is now complete.

From the Theorem above we immediately get the following Harnack inequality by a standard chain argument.

Theorem 2.6 Suppose that K is a compact subset of D. There exists a constant C = C(D, q, n, α) > 0 such that for any solution f of (2.8) which is strictly positive on D one has

sup_{x∈K} f(x) ≤ C inf_{x∈K} f(x).

3 Intrinsic Ultracontractivity

Let us first recall the definition of intrinsic ultracontractivity, due to Davies and Simon [20]. Suppose that H is a semibounded self-adjoint operator on L^{2}(D) with D being a domain in R^{n} and that {e^{Ht}, t > 0} is an irreducible positivity-preserving semigroup with integral kernel a(t, x, y). Assume that the top of the spectrum μ_{0} of H is an eigenvalue. In this case, μ_{0} has multiplicity one and the corresponding eigenfunction φ_{0}, normalized by ∥φ_{0}∥_{2} = 1, can be chosen to be positive almost everywhere on D. φ_{0} is called the ground state of H.

Let U be the unitary operator U from L^{2} (D, φ_{0}^{2}(x)dx) to L^{2}(D) given by Uf = φ_{0}f and define H̃ on L^{2} (D, φ_{0}^{2}(x) dx) by

H̃ = U^{−1} (H − μ_{0}) U.

Then e^{H̃t} is an irreducible symmetric Markov semigroup on L^{2} (D, φ_{0}^{2}(x) dx) whose integral kernel with respect to the measure φ_{0}^{2}(x)dx is given by

\frac{e^{−μ_{0}t}a(t, x, y)}{φ_{0}(x)φ_{0}(y)}.

Definition 3.1 H is said to be ultracontractive if e^{Ht} is a bounded operator from L^{2}(D) to L^{∞}(D) for all t > 0. H is said to be intrinsically ultracontractive if H̃ is ultracontractive; that is, e^{H̃t} is a bounded operator from L^{2} (D, φ_{0}^{2}(x) dx) to L^{∞} (D, φ_{0}^{2}(x) dx) for all t > 0.

Ultracontractivity is connected to logarithmic Sobolev inequalities. The connection between logarithmic Sobolev inequalities and L^{p} to L^{q} bounds of semigroups was first given by L. Gross [24] in 1975. E. Davies and B. Simons [1] adopted L. Gross’s approach to allow q = ∞ and therefore established the connection between logarithmic Sobolev inequalities and ultracontractivity. (For an updated survey on the subject of logarithmic Sobolev inequalities and contractivity properties of semigroups, see [2], [25].) In [3], R. Bañuelos proved the intrinsic ultracontractivity for Shrödinger operators on uniformly Hölder domains of order α ∈ (0, 2) using logarithmic Sobolev inequality characterization. We will use the same strategy in this section; that is, establishing intrinsic ultracontractivity through logarithmic Sobolev inequalities.

In the rest of this section, unless otherwise specified, D is a domain in R^{n} with finite Lebesgue measure, q is a fixed function in the Kato class K_{n,α}. Recall that the semigroup {T_{t}, t > 0}) is defined as follows.

T_{t}f(x) = E^{x} [e_{q}(t)f(X(t)); t < τ_{D}] , x ∈ D.

{λ_{k} : k = 0, 1, ⋯} are all the eigenvalues of L^{D} +q written in decreasing order, each repeated according to its multiplicity. {φ_{k} : k = 0, 1, ⋯} are the corresponding eigenfunctions, normalized so that they form an orthonormal basis of L^{2}(D, dx) and φ_{0} > 0 on D.

The following result is proven in [12].

Theorem 3.1 The logarithmic Sobolev inequality holds for functions in (E, F). That is, for any η > 0 and f ∈ F ∩ L^{∞}(D, dx), we have

∫_{D} f^{2} log |f|dx ≤ ηE(f, f) + β(η)∥f∥_{2}^{2} + ∥f∥_{2}^{2} log ∥f∥_{2},

with

β(η) = −\frac{n}{2α} log η + c

for some constant c > 0.

Recall that for any domain D in R^{n}, the quasi–hyperbolic distance between any two points x_{1} and x_{2} in D is defined by

ρ_{D}(x_{1}, x_{2}) = inf_{γ} ∫_{γ} \frac{ds}{δ(x, ∂D)}

where the infimum is taken over all rectifiable curves γ joining x_{1} to x_{2} in D and δ(x, ∂D) is the Euclidean distance between x and ∂D. Fix a point x_{0} ∈ D which we call the center of D and we may assume without loss of generality that δ(x_{0}, ∂D) = 1.

Definition 3.2 A domain D in R^{n} is a Hölder domain of order 0 if for a fixed x_{0} ∈ D, there exist constants C_{1} and C_{2} such that for all x ∈ D,

ρ_{D}(x_{0}, x) ≤ C_{1} log (\frac{1}{δ(x, ∂D)}) + C_{2}.

It is shown in Smith and Stegenga [29] that a Hölder domain of order 0 is bounded. It is well known that John domains, in particular bounded NTA domains and Lipschitz domains, are Hölder domains of order 0 (cf. [3]).

Lemma 3.2 If D is a Hölder domain of order 0, then for any β > 0, there exists a constant C = C(D) > 0 such that

∫_{D}(ρ_{D}(x_{0}, x))^{β}u^{2}(x)dx ≤ CE(u, u), u ∈ F

Proof. From [29] we know that for any β > 0 we have

∫_{D}(ρ_{D}(x_{0}, x))^{β}dx < ∞.

It follows from the Sobolev inequality (see formula (1.5.20) of [23] or Theorem 1 on page 119 of [31]) that there is constant C_{1} > 0 such that for any u ∈ F,

∥u∥_{p_{0}} ≤ C_{1}\sqrt{E(u, u)},

where p_{0} is such that 1/p_{0} = 1/2 − α/(2n). Let p = n/(n − α) and p′ = n/α. Applying Hölder’s inequality we get that for any u ∈ F,

∫_{D}(ρ_{D}(x_{0}, x))^{β}u^{2}(x)dx ≤ (∫_{D}(ρ_{D}(x_{0}, x))^{βp′}dx)^{1/p′} (∫_{D} |u|^{p_{0}}(x)dx)^{(n−α)/n}
≤ C_{1} (∫_{D}(ρ_{D}(x_{0}, x))^{βp′}dx)^{1/p′} E(u, u).

The proof is now complete.

Theorem 3.3 If D is a Hölder domain of order 0, then for any ε > 0 and any σ > 0 we have

∫_{D} f^{2} log \frac{1}{φ_{0}} dx ≤ εE(f, f) + β(ε)∥f∥_{2}^{2}, f ∈ F

with

β(ε) = c_{1}ε^{−σ} + c_{2}

for some positive constants c_{1} and c_{2}. Here φ_{0} is the ground state of L^{D} + q.

Proof. Let W = {Q_{j}} be a Whitney decomposition of D. This is a decomposition of D into closed cubes Q with the following three properties (see [31] for details)

(1) for j ≠ k, the interior of Q_{j} and the interior of Q_{k} are disjoint;

(2) if Q_{j} and Q_{k} intersects, then

\frac{1}{4} ≤ \frac{diam(Q_{j})}{diam(Q_{k})} ≤ 4;

(3) for any j,

1 ≤ \frac{δ(Q_{j}, ∂D)}{diam(Q_{j})} ≤ 4.

Let x_{0} be a fixed point in D and x_{0} ∈ Q_{0}. If Q_{k} ∈ W, we say that Q_{0} = Q(0) → Q(1) → ⋯ → Q(m) = Q_{k} is a Whitney chain connecting Q_{0} and Q_{k} of length m if Q(i) ∈ W for all i and Q(i) and Q(i + 1) have touching edges for all i. We define the Whitney distance d(Q_{0}, Q_{k}) to be the length of the shortest Whitney chain connecting Q_{0} and Q_{k}. If x ∈ Q_{k} we define ρ̃(x_{0}, x) = d(Q_{0}, Q_{k}). It is well known and easy to prove that this distance is comparable with ρ_{D}, the quasi–hyperbolic distance.

By Theorem 2.5, property (3) of the Whitney decomposition, the boundedness of D and the equivalence of ρ̃ and ρ_{D}, there is a constant C_{1} = C_{1}(D) > 0 such that for any Q ∈ W we have

sup_{x∈Q} φ_{0}(x) ≤ C_{1} inf_{x∈Q} φ_{0}(x).

Therefore there exists a constant C_{2} = C_{2}(D) > 0 such that

φ_{0}(x) ≥ e^{−C_{2}ρ_{D}(x_{0},x)}φ_{0}(x_{0}),  x ∈ D.  (3.1)

For any p > 1, let p′ be its conjugate. By (3.1) and Lemma 3.2, we get that for any ε > 0 and u ∈ F,

∫_{D} u^{2} log \frac{1}{φ_{0}} dx ≤ C_{2} ∫_{D} ρ_{D}(x_{0}, x)u(x)^{2}dx
= C_{2} ∫_{D} \frac{ε^{1/p}}{ε^{1/p}} ρ_{D}(x_{0}, x)u(x)^{2}dx
≤ εC_{2} ∫_{D}(ρ_{D}(x_{0}, x))^{p}u(x)^{2}dx + C_{2}ε^{−p′/p} ∫_{D} u^{2}(x)dx
≤ ε C_{3} E(u, u) + C_{2} ε^{−p′/p} ∫_{D} u^{2}(x)dx,

where C_{3} is a positive constant depending on D only. The proof is now complete. Combining the two theorems above we get the following result.

Theorem 3.4 If D is a Hölder domain of order 0, then for any ε > 0 and any σ > 0 we have

∫_{D} f^{2} log \frac{|f|}{φ_{0}}dx ≤ ηE(f, f) + β(η)∥f∥_{2}^{2} + ∥f∥_{2}^{2} log ∥f∥_{2}, f ∈ F ∩ L^{∞}(D, dx)

with

β(ε) = −\frac{n}{2α} log ε + c_{1}ε^{−σ} + c_{2}

for some positive constants c_{1} and c_{2}.

With the result above, we can easily get our main result of this section.

Theorem 3.5 Assume that D is a Hölder domain of order 0. Then L^{D} + q is intrinsically ultracontractive. More precisely,

\frac{e^{−λ_{0}t}u_{q}(t, x, y)}{φ_{0}(x)φ_{0}(y)} ≤ e^{2M(t/2)} for all x, y ∈ D and t > 0,

where

M(t) = \frac{1}{t} ∫_{0}^{t} A(ε)dε

with

A(ε) = \{ \begin{matrix}−\frac{n}{2α} log ε + c_{1}ε^{−1/3} + c_{2} & for ε ≤ 1, \\ c_{1} + c_{2} & for ε > 1. \end{matrix}

for some positive constants c_{1} and c_{2}.

Proof. By taking σ = 1/3 in Theorem 3.4, we get that for any ε > 0 and any f ∈ F ∩ L^{∞}(D, dx),

∫_{D} f^{2} log \frac{|f|}{φ_{0}}dx ≤ εE(f, f) + β_{1}(ε)∥f∥_{2}^{2} + ∥f∥_{2}^{2} log ∥f∥_{2},  (3.2)

with

β_{1}(ε) = −\frac{n}{2α} log ε + c_{1}ε^{−1/3} + c_{2}

for some positive constants c_{1} and c_{2}.

Suppose that (T̃, F̃) is the Dirichlet form on L^{2}(m) with m(dx) = φ_{0}^{2}dx associated with the semigroup whose integral kernel with respect to the measure m is given by

\frac{e^{−λ_{0}t}u_{q}(t, x, y)}{φ_{0}(x)φ_{0}(y)}.

Then

F̃ = {f : fφ_{0} ∈ F}

and

T̃(f, h) = E(fφ_{0}, hφ_{0}) − ∫_{D} qfφ_{0} hφ_{0} dx + λ_{0} ∫_{D} fh dm

Since q ∈ K_{n,α}, by Theorem 3.2 of [30] there exists a constant B > 0 such that

∫_{D} |q|u^{2}dx ≤ \frac{1}{2}E(u, u) + B ∫_{D} u^{2}dx, u ∈ F.

Thus

T̃(h, h) ≥ \frac{1}{2}E(hφ_{0}, hφ_{0}) − (B − λ_{0}) ∫_{D} h^{2}dm.

By putting f = hφ_{0} in (3.2) we get that for h ∈ F̃ ∩ L^{∞}(D, dm),

∫_{D} h^{2} log |h| dm ≤ 2ε T̃(h, h) + (β_{1}(ε) + 2(B − λ_{0})) ∫_{D} h^{2}dm + ∥h∥_{L^{2}(m)}^{2} log ∥h∥_{L^{2}(m)}. (3.3)

Therefore for 0 < ε ≤ 1 and h ∈ F̃ ∩ L^{∞}(D, dm),

∫_{D} h^{2} log |h| dm ≤ ε T̃(h, h) + β_{2}(ε) ∫_{D} h^{2}dm + ∥h∥_{L^{2}(m)}^{2} log ∥h∥_{L^{2}(m)},  (3.4)

where

β_{2}(ε) = −\frac{n}{2α} log ε + c_{3}ε^{−1/3} + c_{4},

for some constants c_{3}, c_{4} > 0. For ε > 1, since T̃ is nonnegative and (3.4) holds for ε = 1, we have for any h ∈ F̃ ∩ L^{∞}(D, dm),

∫_{D} h^{2} log |h|dm ≤ T̃(h, h) + β_{1}(1) ∫_{D} h^{2}dm + ∥h∥_{L^{2}(m)}^{2} log ∥h∥_{L^{2}(m)}
≤ εT̃(h, h) + β_{1}(1) ∫_{D} h^{2}dm + ∥h∥_{L^{2}(m)}^{2} log ∥h∥_{L^{2}(m)}.  (3.5)

Combining (3.4) and (3.5) we get that for any ε > 0 and any h ∈ F̃ ∩ L^{∞}(D, dm),

∫_{D} h^{2} log |h|dm ≤ εT̃(h, h) + A(ε)∥h∥_{L^{2}(m)}^{2} + ∥h∥_{L^{2}(m)}^{2} log ∥h∥_{L^{2}(m)},  (3.6)

with

A(ε) = \{ \begin{matrix}−\frac{n}{2α} log ε + c_{5}ε^{−1/3} + c_{6} & for η ≤ 1, \\ c_{5} + c_{6} & for ε > 1. \end{matrix}

for some positive constants c_{5} and c_{6}. By Corollary 2.2.8 of [19], we have

\frac{e^{−λ_{0}t}u_{q}(t, x, y)}{φ_{0}(x)φ_{0}(y)} ≤ e^{2M(t/2)} for all x, y ∈ D and t > 0,

where

M(t) = \frac{1}{t} ∫_{0}^{t} A(ε)dε < ∞.

Using the same argument as that for Theorem 6 in R. Smits [28], we have

Theorem 3.6 Assume that D is a domain in R^{n} with finite Lebesgue measure such that L^{D} + q is intrinsic ultracontractive. Then there exists C > 0 such that for any t > 1,

e^{(λ_{1}−λ_{0})t} ≤ sup_{x,y∈D} |\frac{e^{−λ_{0}t}u_{q}(t, x, y)}{φ_{0}(x)φ_{0}(y)} − 1| ≤ Ce^{(λ_{1}−λ_{0})t}.

4 Conditional Lifetimes

Assume in this section that D is a domain in R^{n} with finite Lebesgue measure such that L^{D} is intrinsic ultracontractive, unless otherwise specified. In particular, we know from Theorem 3.5 that a Hölder domain of order 0 satisfies this assumption. Since (D, 0) is gaugeable, the first eigenvalue μ_{0} of L^{D} is negative. Let ϕ_{0} be the ground state of L^{D}. Recall that P_{D} is the transition density function for the killed symmetric stable process X^{D}. Similar to Corollary 1 of Bañuelos [3], we have

Theorem 4.1 Under the assumption given at the beginning of this section. There is a constant c > 0 such that

(1) e^{μ_{0}t}ϕ_{0}(x)ϕ_{0}(y) ≤ p^{D}(t, x, y) ≤ ce^{μ_{0}t}ϕ_{0}(x)ϕ_{o}(y) for all x, y ∈ D and t > 0;

(2) Let SH^{+} denote all non-trivial nonnegative superharmonic functions in D with respect

to X^{D}. Then

sup_{x∈D, h∈SH^{+}} E_{h}^{x}[τ_{D}] < ∞;

(3) For h ∈ SH^{+},

lim_{t→∞} e^{−μ_{0}t}P_{h}^{x}(τ_{D} > t) = \frac{ϕ_{0}(x)}{h(x)} ∫_{D} ϕ_{0}(y)h(y).

In particular, lim_{t→∞} \frac{1}{t} log P_{h}^{x}(τ_{D} > t) = μ_{0}.

Proof. (1) It follows directly from Theorem 3.6.

(2) Note that for each h ∈ SH^{+}, by (1)

h(x) ≥ ∫_{D} p^{D}(1, x, y)h(y)dy ≥ e^{μ_{0}}ϕ_{0}(x) ∫_{D} ϕ_{0}(y)h(y)dy for x ∈ D.

Therefore

sup_{x∈D, h∈SH^{+}} \frac{ϕ_{0}(x)}{h(x)} ∫_{D} ϕ_{0}(y)h(y) dy ≤ e^{−μ_{0}} < ∞.

Therefore by (1)

sup_{x∈D, h∈SH^{+}} E_{h}^{x}[τ_{D}]

= sup_{x∈D, h∈SH^{+}} \frac{1}{h(x)} ∫_{0}^{∞} ∫_{D} p^{D}(t, x, y)h(y) dy
≤ c ∫_{0}^{∞} e^{μ_{0}t} dt sup_{x∈D, h∈SH^{+}} \frac{ϕ_{0}(x)}{h(x)} ∫_{D} ϕ_{0}(y)h(y) dy < ∞.

(3) By Theorem 3.6

lim_{t→∞} e^{−μ_{0}t}P_{h}^{x}(τ_{D} > t)

= lim_{t→∞} e^{−μ_{0}t}ϕ_{0}(x)^{−1} ∫_{D} p^{D}(t, x, y)h(y)dy
= \frac{ϕ_{0}(x)}{h(x)} ∫_{D} ϕ_{0}(y)h(y).

When D is a bounded Lipschitz domain, G_{D}(x, y), M_{D}(x, z) and K_{D}(x, w) are superharmonic functions in x with respect to X^{D} for each fixed y ∈ D, z ∈ ∂D and w ∈ \overline{D}^{c}, respectively. The above theorem in particular implies that

Corollary 4.2 (Conditional Lifetimes) Assume that D is a bounded Lipschitz domain. Then

sup_{x∈D, z∈R^{n}} E_{z}^{x}[τ_{D}] < ∞.

5 Conditional Gauge Theorem

Throughout this section, D is a bounded Lipschitz domain. Recall that L^{D} is the nonpositive definite infinitesimal generator of the killed α-stable process on D. For q ∈ K_{n,α}, let u_{q}(t, x, y) be the kernel of the following Feynman-Kac semigroup

T_{t}f(x) = E^{x} [e_{q}(t)f(X_{t})1_{{t<τ_{D}}}] .

Note that the semigroup T_{t} only depends on function q through q1_{D} so we may assume that q = 0 off D. The following result is proven in [12].

Theorem 5.1 Suppose that q ∈ K_{n,α} is such that

sup_{x,y∈D} ∫_{D} \frac{G_{D}(x, z)|q(z)|G_{D}(z, y)}{G_{D}(x, y)}dz ≤ \frac{1}{2}.

Then we have

e^{−1/2}G_{D}(x, y) ≤ V_{q}(x, y) ≤ 2G_{D}(x, y),  (5.1)

where

V_{q}(x, y) = ∫_{0}^{∞} u_{q}(t, x, y)dt.

Theorem 5.2 Assume that q ∈ K_{n,α} and (D, q) is gaugeable. Then there is a constant c > 0 such that

V_{q}(x, y) ≤ c G_{D}(x, y) for all x, y ∈ D.

Proof. By Lemma 2.4 and Theorem 2.3, the function q can be decomposed as q = q_{1} + q_{2} with q_{1} bounded and q_{2} ∈ K_{n,α} satisfying

sup_{x,y∈D} ∫_{D} \frac{G_{D}(x, z)|q_{2}(z)|G_{D}(z, y)}{G_{D}(x, y)}dz ≤ \frac{1}{2}.  (5.2)

Therefore by Theorem 5.1

e^{−1/2}G_{D}(x, y) ≤ V_{q_{2}}(x, y) ≤ 2G_{D}(x, y).  (5.3)

Let {ν_{k}, k = 0, 1, ⋯} be all the eigenvalues of L^{D} + q_{2} written in decreasing order, each repeated according to its multiplicity. and let {ψ_{k}, k = 0, 1, ⋯} be the corresponding eigenfunctions with ψ > 0 on D. We can assume that {ψ_{k}, k = 0, 1, ⋯} form an orthonormal basis of L^{2}(D, dx). Since by Khas’minskii’s lemma E_{y}^{x}[e_{q_{2}}(ζ)] < 2 for x, y ∈ D, we know from Theorem 3.11 of [12] that ν_{0} < 0.

By Theorem 3.6, there is a t_{1} > 1 such that for all t ≥ t_{1} and all x, y ∈ D,

\frac{1}{2} ≤ \frac{e^{−ν_{0}t}u_{q_{2}}(t, x, y)}{ψ_{0}(x)ψ_{0}(y)} ≤ \frac{3}{2}.

Therefore for all x, y ∈ D,

V_{q_{2}}(x, y) = ∫_{0}^{∞} u_{q_{2}}(t, x, y)dt
≥ ∫_{t_{1}}^{∞} u_{q_{2}}(t, x, y)dt
≥ \frac{1}{2}ψ_{0}(x)ψ_{0}(y) ∫_{t_{1}}^{∞} e^{ν_{0}t} dt
≥ C_{1} ψ_{0}(x)ψ_{0}(y),  (5.4)

for some positive constant C_{1} > 0.

Since L^{D} + q = (L^{D} + q_{2}) + q_{1} and q_{1} is bounded, it follows from Theorem 3.4 of [20] that the first eigenfunction φ_{0} of L^{D} + q is comparable to the first eigenfunction ψ_{0} of L^{D} + q_{2}, i.e., there exists a constant C_{2} > 1 such that

C_{2}^{−1}ψ_{0} ≤ φ_{0} ≤ C_{2}ψ_{0}.  (5.5)

By Theorem 3.6 again, there is a t_{2} > 1 such that for all t ≥ t_{2} and all x, y ∈ D,

\frac{1}{2} ≤ \frac{e^{−λ_{0}t}u_{q}(t, x, y)}{φ_{0}(x)φ_{0}(y)} ≤ \frac{3}{2}.

Since the gauge function of (D, q) is assumed to be finite, the first eigenvalue λ_{0} is negative by Theorem 3.11 of [12]. Therefore it follows from (5.5), (5.4) and (5.3) that for any x, y ∈ D,

∫_{t_{2}}^{∞} u_{q}(t, x, y)dt ≤ \frac{3}{2}φ_{0}(x)φ_{0}(y) ∫_{t_{2}}^{∞} e^{λ_{0}t}dt
≤ C_{3}φ_{0}(x)φ_{0}(y) ≤ C_{4}ψ_{0}(x)ψ_{0}(y)
≤ C_{5}V_{q_{2}}(x, y) ≤ C_{6}G_{D}(x, y),

where C_{3}, C_{4}, C_{5}, C_{6} are positive constants. Since q_{1} is bounded,

∫_{0}^{t_{2}} u_{q}(t, x, y)dt ≤ e^{∥q_{1}∥_{∞}t_{2}} ∫_{0}^{t_{2}} u_{q_{2}}(t, x, y)dt
≤ e^{∥q_{1}∥_{∞}t_{2}}V_{q_{2}}(x, y) ≤ 2e^{∥q_{1}∥_{∞}t_{2}}G_{D}(x, y).

Hence there exists a constant C > 1 such that

V_{q}(x, y) ≤ CG_{D}(x, y), x, y ∈ D.

Applying the Theorem above and the 3G Theorem we can get the following results which correspond to Theorems 5.3–5.7 of [12]. The proofs are exactly the same as those of the corresponding results in [12] so the details are omitted.

Theorem 5.3 Assume that (D, q) is gaugeable. Then for all x, y ∈ D with x ≠ y,

V_{q}(x, y) = G_{D}(x, y) + ∫_{D} V_{q}(x, u)q(u)G_{D}(u, y)du  (5.6)
V_{q}(x, y) = G_{D}(x, y) + ∫_{D} G_{D}(x, u)q(u)V_{q}(u, y)du.  (5.7)

Theorem 5.4 Assume that (D, q) is gaugeable. Then for all (x, y) ∈ D × D with x ≠ y,

E_{y}^{x} [e_{q}(ζ)] = 1 + G_{D}(x, y)^{−1} ∫_{D} V_{q}(x, w)q(w)G_{D}(w, y)dw.

Theorem 5.5 Assume that (D, q) is gaugeable. Then for all (x, y) ∈ D × D with x ≠ y,

E_{y}^{x} [e_{q}(ζ)] = \frac{V_{q}(x, y)}{G_{D}(x, y)}.

Theorem 5.6 Assume that (D, q) is gaugeable. Then for There exist c > 1 such that

c^{−1} ≤ inf_{x,y∈D} E_{y}^{x} [e_{q}(ζ)] ≤ sup_{x,y∈D} E_{y}^{x} [e_{q}(ζ)] ≤ c.

Theorem 5.7 Assume that (D, q) is gaugeable. There exists a constant c > 1 such that

c^{−1} G_{D}(x, y) ≤ V_{q}(x, y) ≤ c G_{D}(x, y) for x, y ∈ D.

That is, the Green function of D with respect to L^{D} + q is comparable to the Green function of D with respect to L^{D}.

Theorem 5.8 Assume that (D, q) is gaugeable. For all x ∈ D and z ∈ \overline{D}^{c},

E_{z}^{x} [e_{q}(ζ)] = 1 + K_{D}(x, z)^{−1} ∫_{D} V_{q}(x, w)q(w)K_{D}(w, z)dw.

Proof. For any Borel measurable subset A ⊂ D, by 2.1

∫_{A} \frac{G_{D}(x, y)|q(y)|K_{D}(y, z)}{K_{D}(x, z)}dy

= \frac{A(n, α)}{K_{D}(x, z)} ∫_{A} G_{D}(x, y)|q(y)| (∫_{D} \frac{G_{D}(y, w)}{|w − z|^{n+α}} dw) dy

= \frac{A(n, α)}{K_{D}(x, z)} ∫_{D} \frac{G_{D}(x, w)}{|w − z|^{n+α}} (∫_{A} \frac{G_{D}(x, y)|q(y)|G_{D}(y, w)}{G_{D}(x, w)}dy) dw.

The family of functions

\{\frac{G_{D}(x, ·)|q(·)|G_{D}(·, w)}{G_{D}(x, w)}, x, w ∈ D\}

is uniformly integrable by (2.6) and therefore the family of functions

\{\frac{G_{D}(x, ·)|q(·)|K_{D}(·, z)}{K_{D}(x, z)}, x ∈ D, z ∈ \overline{D}^{c}\}

is uniformly integrable. Using Fubini’s theorem,

E_{z}^{x} [∫_{0}^{ζ} e_{q}(t)|q(X_{t})|dt]
= ∫_{0}^{∞} E_{z}^{x} [e_{q}(t)|q(X_{t})|; t < ζ] dt
= K_{D}(x, z)^{−1} ∫_{0}^{∞} E^{x} [e_{q}(t)|q(X_{t})|K_{D}(X_{t}, z); t < τ_{D}] dt
= K_{D}(x, z)^{−1} ∫_{0}^{∞} ∫_{D} u_{q}(t, x, y)|q(w)|K_{D}(w, z)dwdt
= K_{D}(x, z)^{−1} ∫_{D} V_{q}(x, w)|q(w)|K_{D}(w, z)dw < ∞.

Hence

E_{z}^{x}[e_{q}(ζ)] = 1 + E_{z}^{x} [∫_{0}^{ζ} e_{q}(t)q(X_{t})dt]
= 1 + K_{D}(x, z)^{−1} ∫_{D} V_{q}(x, w)q(w)K_{D}(w, z)dw.

Theorem 5.9 Assume that (D, q) is gaugeable. There exists a constant c > 1 such that

c^{−1} ≤ inf_{(x,z)∈D×\overline{D}^{c}} E_{z}^{x} [e_{q}(τ_{D})] ≤ sup_{(x,z)∈D×\overline{D}^{c}} E_{z}^{x} [e_{q}(τ_{D})] ≤ c.

Proof. By (2.1)

E_{z}^{x} [e_{q}(τ_{D})] = 1 + \frac{1}{K_{D}(x, z)} ∫_{D} V_{q}(x, w)q(w) ∫_{D} \frac{G_{D}(w, v)}{|v − z|^{n+α}} dvdw

= \frac{A(n, α)}{K_{D}(x, z)} ∫_{D} \frac{G_{D}(x, v)}{|v − z|^{n+α}} (1 + \frac{1}{G_{D}(x, v)} ∫_{D} V_{q}(x, w)q(w)G_{D}(w, v)dw) dv

= \frac{A(n, α)}{K_{D}(x, z)} ∫_{D} \frac{G_{D}(x, v)}{|v − z|^{n+α}} E_{v}^{x} [e_{q}(ζ)] dv.

The assertion of this theorem then follows from Theorem 5.6. Theorem 5.10 Assume that (D, q) is gaugeable. There exists a constant c > 1 such that

c^{−1} ≤ inf_{(x,z)∈D×∂D} E_{z}^{x} [e_{q}(τ_{D})] ≤ sup_{(x,z)∈D×∂D} E_{z}^{x} [e_{q}(τ_{D})] ≤ c.

Proof. It follows from the (2.7) that the family of functions

\{\frac{G_{D}(x, ·)|q(·)|M_{D}(·, z)}{M_{D}(x, z)}, x ∈ D, z ∈ ∂D\}

is uniformly integrable. A similar calculation as that given in the proof of Theorem 5.8 yields

E_{z}^{x}[e_{q}(ζ)] = 1 + \frac{1}{M_{D}(x, z)} ∫_{D} V_{q}(x, u)q(u)M_{D}(u, z)du.

Let x_{0} be a fixed point in D. Then from [13] we know that

lim_{y→z} \frac{1}{G_{D}(x, y)}V_{q}(x, w)q(w)G_{D}(w, y) = lim_{y→z} V_{q}(x, w)q(w)\frac{G_{D}(w, y)/G_{D}(x_{0}, y)}{G_{D}(x, y)/G_{D}(x_{0}, y)}
= V_{q}(x, w)q(w)\frac{M_{D}(w, z)}{M_{D}(x, z)}.

Now using the uniform integrability of the family of functions

\{\frac{G_{D}(x, ·)|q(·)|G_{D}(·, y)}{G_{D}(x, y)}, x, y ∈ D\}

and letting y → z ∈ ∂D in Theorem 5.4 yield

lim_{y→z} E_{y}^{x}[e_{q}(ζ)] = 1 + \frac{1}{M_{D}(x, z)} ∫_{D} V_{q}(x, u)q(u)M_{D}(u, z)du.

Therefore

lim_{y→z} E_{x}^{y}[e_{q}(ζ)] = E_{x}^{z}[e_{q}(ζ)]  (5.8)

and the theorem now follows from Theorem 5.6.

Combining Theorems 5.6, 5.9 and 5.10 we get the following

Theorem 5.11 (Conditional Gauge Theorem). Assume that (D, q) is gaugeable. There exists a constant c > 1 such that

c^{−1} ≤ inf_{(x,z)∈D×R^{n}} E_{z}^{x} [e_{q}(ζ)] ≤ sup_{(x,z)∈D×R^{n}} E_{z}^{x} [e_{q}(ζ)] ≤ c.

For q ∈ K_{n,α}, let ψ_{0} > be the ground state of L^{D} + q and let u_{q}(t, x, y) be the density kernel of L^{D} + q with respect to the Lebesgue measure in D.

Theorem 5.12 Suppose that (D, q) is gaugeable.

(1) There is a constant c > 1) such that

c^{−1}ϕ_{0}(x) ≤ ψ_{0}(x) ≤ cϕ_{0}(x) for all x ∈ D;

(2) For each t > 0 there is a constant c_{t} > 1 such that

c_{t}^{−1}p^{D}(t, x, y) ≤ u_{q}(t, x, y) ≤ c_{t} p^{D}(t, x, y) for all x, y ∈ D.

Proof. The proof is the same as that for Theorem 2 in Bañuelos [3] and is thus omitted. Remark. It is possible to extend conditional gauge theorem beyond bounded Lipschitz domains to domains having the following properties. Suppose that D is a domain having finite Lebesgue measure and q ∈ K_{n,α} such that L^{D} + q is intrinsic ultracontractive. Assume also that q admits a decomposition q = q_{1} +q_{2} with q_{1} and q_{2} satisfying (5.2). Then Theorem 5.2 remains true by exactly the same argument. Assume further that

\{\frac{G_{D}(x, ·)|q(·)|G_{D}(·, y)}{G_{D}(x, y)} : x, y ∈ D\}  (5.9)

is uniformly integrable on D. Then Theorems 5.3–5.7 hold by the same proof as those for Theorems 5.3–5.7 in [12]. In this case, Theorem 5.12 holds as well. When D is a bounded Lipschitz domain, the above conditions are satisfied due to 3G Theorem 2.3.

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