domains considered includes John domains and a class of Holder domains. ... monotonically (sufficiently quickly) near the boundary of a John domain, and.
CONTRACTIVITY PROPERTIES OF SCHRODINGER SEMIGROUPS ON BOUNDED DOMAINS FABIO CIPRIANI AND GABRIELE GRILLO
ABSTRACT
We study intrinsic ultracontractivity (IUC) for the Schrodinger operator H = — A + V with Dirichlet boundary conditions on bounded domains Q, in U". The potential is not assumed either to belong to the Kato class, or to be relatively form-bounded with respect to the Dirichlet Laplacian on Q. The class of domains considered includes John domains and a class of Holder domains. We also give an example of a bounded domain Q on which the Dirichlet Laplacian is not IUC, but on which — A + V is IUC for a suitable potential V.
1. Introduction Let Q be a bounded domain in Un. In this paper we discuss intrinsic ultracontractivity (IUC in the sequel) for the Schrodinger operator H = — A+ V, for V ^ 0, with Dirichlet boundary conditions in Cl. The potential V is assumed to be locally bounded, but it is not assumed to be small (in the quadratic form sense) with respect to the Dirichlet Laplacian — A^ on Q. A negative potential V_ belonging to the Kato class Kn could be added to the above operators without difficulties. When V =0, IUC is known to hold for a large class of domains, including uniformly Holder domains of order a < 2, classes of Leveraging domains, and domains whose boundary is an isotropic self-similar fractal (cf. [2, 5, 9]). The notion of intrinsic ultracontractivity was originally introduced by E. B. Davies in [6], while our results extend the results of E. B. Davies and B. Simon [8], of E. B. Davies [7], and of R. Banuelos [2]. In fact, to the best of our knowledge, the class of potentials considered in those papers includes only potentials which are relatively form-bounded with respect to the Dirichlet Laplacian on Q, and which are therefore considered as (small) perturbations of — AD (cf. in particular [2, p. 196]); in this respect our results are new also for C°° domains. In particular, we discuss potentials which increase monotonically (sufficiently quickly) near the boundary of a John domain, and potentials of the above form which moreover have mild oscillations, in a class of Holder domains of order zero (cf. [19] for the relevant definitions). We establish our results through a probabilistic technique, which goes back to the papers of R. Carmona, and of R. Carmona and B. Simon (cf. [3, 4] and references quoted therein), inspired by previous work of S. Varadhan [20], A. D. Ventcel and M . I . Freidlin [21]. Precisely, we prove pointwise lower bounds on the ground state eigenfunction i//0 of HD (which always exists for the class of domains and potentials considered), via a Harnack inequality, a Feynman-Kac formula and certain large deviations results for Brownian motion conditioned to stay in Q. While our method does not give sharp results when V is small, it is effective when V diverges sufficiently fast as x approaches the boundary of Q. In fact, we obtain bounds which allow us to compare y o (x) with a regularization of the potential V Received 4 January 1994; revised 19 May 1994. 1991 Mathematics Subject Classification 35J10. J. London Math. Soc. (2) 52 (1995) 583-593
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inside a tube contained in Q to which both x and a fixed point xoeQ belong, and which depend on geometric properties of the domain Q as well. These bounds are sufficient to prove that the assumptions of Rosen's Lemma (cf. [7]) are satisfied, and the passage from this to IUC is by now standard under mild technical assumptions. The class of domains for which the method seems to be effective includes, as mentioned above, John domains, and hence uniform, NTA and Lipschitz domains, and a subclass of the class of Holder domains of order zero (cf. [19]). We stress that no other regularity assumption is required on Q, although regularity in the sense of Davies [7] is useful in one of our results (Theorem 4.3). The paper is organized as follows: in Section 2 we collect the basic definitions concerning the Schrodinger operators considered, intrinsic ultracontractivity, and about some class of domains which will be used in the sequel. Section 3 contains the main probabilistic estimates, which are obtained in a class of bounded domains which we call/-regular, while in Section 4 we discuss pointwise lower bounds for the ground state eigenfunction of i//0 from which our main results concerning ultracontractivity follow. Finally, in Section 5 we consider a domain Q on which the Dirichlet Laplacian is not IUC [8], but on which HD = — A + V with Dirichlet boundary conditions is IUC for a suitable choice of V. 2. Notation and basic definitions n
Let Q c IR be bounded, and consider a locally integrable non-negative function V: Q. -* R. Then one can consider the quadratic form for/eC0«>(Q),
(2.1)
which is closable and non-negative. Its closure has domain given by the intersection of the quadratic form domain of - A o and of V. The operator / / = // D = - A + K o n U(Q.,dx) associated with the closure will be called the Schrodinger operator with Dirichlet boundary conditions. Since Q is bounded, the resolvent of HD is compact, and hence the spectrum of HD is purely discrete. Consider the lowest eigenvalue Eo = inf(T(HD) of HD, and the corresponding (unique, positive, normalized in L2(Q, dx)) eigenfunction y/0. We shall be concerned with pointwise lower bounds on if/0, with applications to what is usually called intrinsic ultracontractivity (IUC for short). In particular, we shall denote by HD the ground state representation of HD (cf. [18]), that is, the operator on L2(Q, dfiQ) with dfi0 = y/l dx defined by HD = UVo o (HD - Eo) o C/-o\
Urt: L 2 (Q; dx)
• L 2 (Q; fa)
being the unitary operator given by £/, g = y/^1 g with geL 2 (Q;d.x). DEFINITION 2.1. The Schrodinger operator HD with Dirichlet boundary conditions is said to be intrinsically ultracontractive if HD is ultracontractive, that is if the heat semigroup e x p [ - / i / J is bounded as an operator from V(Q,dfi0) to L°°(Q, 0. The situations in which IUC occurs are precisely those in which the heat kernel pD(t, x,y) of exp[ — tHD] is pointwise comparable with the product y/0(x) y/0(y). IUC in turn is equivalent to a family of (weighted) logarithmic Sobolev inequalities for the
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Dirichlet form of HD (cf. [12, 7]), provided that some additional upper bounds on the divergence of what is known as the local norm function P(e) as £ -> 0 hold. The local norm function is related to the norm of exp [ — tHD] as an operator from L2(Q, d/x0) to L°°(Q, d/i0) (cf. [8, 7] and references quoted therein). Among the many consequences of IUC, we quote, for example, pointwise bounds on the nth eigenfunction y/n of HD, Sobolev estimates up to the boundary, mixing properties for Brownian motion killed when hitting dQ, and estimates for the lifetime of Doob n-conditional diffusion in Q (cf. [7, 2] and references quoted therein). Now we turn to the class of domains that we shall consider. We denote by d(x) for xeQ, the euclidean distance of JC from the boundary of Q. DEFINITION 2.2 (cf. [10,16] and references quoted therein). The domain Q. is an a-John domain, with (John) centre x0, if there exists x o e Q and a family of rectifiable paths yx joining jceQ to x0, with the property that l{yx 2) ^ ad(x) for some a > 0 and for all zsyx; here yxz is the subarc of y^ joining x to z, and l(y) is the length of a curve y.
The above condition can be seen as a twisted internal cone condition. The class of John domains includes that of uniform domains (cf. [1]), and in particular every NTA domain and every Lipschitz domain is a John domain. Finally, we define the quasi-hyperbolic distance between x,yeQ as (cf. [11])
•
(22)
the infimum being taken over all rectifiable paths in Q joining x to y. Geodesies in the metric Kn always exist by a result of Martin [15]. DEFINITION 2.3 (cf. [19]). The domain O is a Holder domain (of order zero), if there exists x o e f i such that, setting Kn(x) = Kn(x,x0), one has
^ j for suitable constants A^\,
(2.3)
BeU.
A John domain is necessarily a Holder domain, but the converse need not be true; it is also possible to prove [19] that any Holder domain, and hence any John domain, is bounded. The class of Holder domains has been studied in [19] in connection with the L p -Poincare inequality. It should be noted that a Holder domain may have a very irregular boundary; for example the Hausdorff dimension of eod(xo)e-Bd(xy.
(3.2)
Proof, (i) The statement on the uniform boundedness of the length of the John paths is immediate from the definition. To prove (3.1),fixee (0,1) and xeQ. Let ybe the subarc of yx lying in B(x,ed(x)). Then infze?d(z) ^ (1 —e)d(x) and in addition, for all z€B(x,ed(x))c (] yx, one has by definition of John domain, so that d(z) ^ ((e/c) A (1 — e))d(x), whence (3.1) follows by maximizing the coefficient of d(x) as £6(0,1). (ii) The uniform boundedness of the length of the quasi-hyperbolic geodesies in 0-H61der domains follows as in [19, Corollary 3]. To prove (3.2), let us note that, by general properties of the quasi-hyperbolic distance Kn [22, p. 34], it follows that, for any zeyx n B(xo,eod(xo))c, one has
so that, by the inequality Kn(x0, z) ^ Kn(x0, x), which is valid by construction, and by definition of 0-H61der domains d(Z)
>
i*0- Z l
>
£
O^o)
>
fiQ^o)^^
>
£ d(x
)e~Bd(x)A
Motivated by the previous proposition, we give the following definition. DEFINITION 3.2. Let / : [0, oo) -* [0, oo) be a strictly increasing function, continuous in x — 0 and with /(0) = 0. We say that a bounded domain OelR n is f-regular if there exists xoeQ. (the/-centre) and, for all xeCl, a path yx joining x and x0 with the properties that
(0 Kfx) < « < oo for all x e Q; (ii) for all zeyx, one has d(z) ^J{d(x)). Part (ii) in this definition has some similarity with Definition 7.1 in Hurri's paper [14], since an argument similar to the one used in Proposition 3.1(ii) shows that, if a domain satisfies a ^-quasi-hyperbolic boundary condition in the sense of Herron and Vuorinen [13] and Hurri [14], then if zeyx, \xQ — z\ ^ const., and d(z) < d(x), one has d(z) ^ const, exp - 0 ——^ Part (i) can be weakened in some situations: however we assume the given condition because it simplifies the calculations and it is satisfied in the examples considered in this paper. Proposition 3.1 and Definition 3.2 immediately imply the following.
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COROLLARY 3.3. (i)IfQisa c-John domain, then it is f-regular with paths given by the John paths yx and with J{t) = const, t. ii) If Q is a O-Holder domain, than it is f-regular with paths given by the quasihyperbolic geodesies yx and with fit) = const. tA.
The definition of an/-regular domain will be crucial in the following theorem, the aim of which is to give short time estimates (up to second order) on the Wiener measure of the set of Brownian paths which start at xeQ, which at time / > 0 belong to a fixed compact set K czQ, and which, for 5 6 [0, t], do not hit the boundary of Qe(x) for a suitable function e(x) with e(x) - ^ 0 a s x - > dQ.. Here we define Qe as {xsQ\d(x) i) > ^[-cJt-c2t/Ad{x)f],
(3.3)
x
where P is Wiener measure with initial point xeQ. Proof
Let y be a path belonging to H1. Choose e, rj > 0 and let = {xeUn,d(x,y)
