Hardy inequalities and non-explosion results for semigroups

We prove non-explosion results for Schr\"odinger perturbations of symmetric transition densities and Hardy inequalities for their quadratic forms by using explicit supermedian functions of their semigroups.


Introduction
Hardy-type inequalities are important in harmonic analysis, potential theory, functional analysis, partial differential equations and probability. In PDEs they are used to obtain a priori estimates, existence and regularity results [24] and to study qualitative properties and asymptotic behaviour of solutions [28]. In functional and harmonic analysis they yield embedding theorems and interpolation theorems, e.g. Gagliardo-Nirenberg interpolation inequalities [21]. The connection of Hardy-type inequalities to the theory of superharmonic functions in analytic and probabilistic potential theory was studied, e.g., in [1], [14], [5], [11]. A general rule stemming from the work of P. Fitzsimmons [14] may be summarized as follows: if L is the generator of a symmetric Dirichlet form E and h is superharmonic, i.e. h ≥ 0 and Lh ≤ 0, then E(u, u) ≥ u 2 (−Lh/h). The present paper gives an analogous connection in the setting of symmetric transition densities. When these are integrated against increasing weights in time and arbitrary weights in space, we obtain suitable (supermedian) functions h. The resulting analogues q of the Fitzsimmons' ratio −Lh/h yield explicit Hardy inequalities which in many cases are optimal. The approach is very general and the resulting Hardy inequality is automatically valid on the whole of the ambient L 2 space.
We also prove non-explosion results for Schrödinger perturbations of the original transition densities by the ratio q, namely we verify that h is supermedian, in particular integrable with respect to the perturbation. For instance we recover the famous non-explosion result of Baras and Goldstein for ∆ + (d/2 − 1) 2 |x| −2 , cf. [2] and [25].
The results are illustrated by applications to transition densities with certain scaling properties.
The structure of the paper is as follows. In Theorem 1 of Section 2 we prove the non-explosion result for Schrödinger perturbations. In Theorem 2 of Section 3 we prove the Hardy inequality. In fact, under mild additional assumptions we have a Hardy equality with an explicit remainder term. Sections 4, 5 and 6 present applications. In Section 4 we recover the classical Hardy equalities for the quadratic forms of the Laplacian and fractional Laplacian. For completeness we also recover the best constants and the corresponding remainder terms, as given by Filippas and Tertikas [13], Frank, Lieb and Seiringer [15] and Frank and Seiringer [16]. In Section 5 we consider transition densities with weak global scaling in the setting of metric spaces. These include a class of transition densities on fractal sets (Theorem 10 and Corollary 11) and transition densities of many unimodal Lévy processes on R d (Corollary 12). We prove Hardy inequalities for their quadratic forms. In Section 6 we focus on transition densities with weak local scaling on R d . The corresponding Hardy inequality is stated in Theorem 13.
The calculations in Sections 4, 5 and 6, which produce explicit weights in Hardy inequalities, also give non-explosion results for specific Schrödinger perturbations of the corresponding transition densities by means of Theorem 1. These are stated in Corollary 6 and 8 and Remark 2, 5 and 6.
Currently our methods are confined to the (bilinear) L 2 setting. We refer to [16], [27] for other frameworks. Regarding further development, it is of interest to find relevant applications with less space homogeneity and scaling than required in the examples presented below, extend the class of considered time weights, prove explosion results for "supercritical" Schrödinger perturbations, and understand more completely when equality holds in our Hardy inequalities.
Below we use ":=" to indicate definitions, e.g. a ∧ b := min{a, b} and a ∨ b := max{a, b}. For two nonnegative functions f and g we write f ≈ g if there is a positive number c, called constant, such that c −1 g ≤ f ≤ c g. Such comparisons are usually called sharp estimates. We write c = c(a, b, . . . , z) to claim that c may be so chosen to depend only on a, b, . . . , z. For every function f , let f + := f ∨ 0. For any open subset D of the d-dimensional Euclidean space R d , we denote by C ∞ c (D) the space of smooth functions with compact supports in D, and by C c (D) the space of continuous functions with compact supports in D. In statements and proofs, c i denote constants whose exact values are unimportant. These are given anew in each statement and each proof.
for comments, suggestions and encouragement. We also thank Rupert L. Frank and Georgios Psaradakis for remarks on the literature related to Section 4.
We define q : X → [0, ∞] as follows: For all x ∈ X we thus have (9) q We define the Schrödinger perturbation of p by q [7]: t (x, y) = p t (x, y), and t−s (z, y) m(dz) ds, n ≥ 1.
It is well-known thatp is a transition density [7].
We shall see in Section 4 that the above construction gives integral finiteness (non-explosion) results for specific Schrödinger perturbations with rather singular q, cf. Corollaries 6 and 8. In the next section q will serve as an admissible weight in a Hardy inequality.

Hardy inequality
Throughout this section we let p, f , µ, h and q be as defined in Section 2. Additionally we shall assume that p is Markovian, namely X p t (x, y)m(dy) ≤ 1 for all x ∈ X. In short, p is a subprobability transition density. By Holmgren criterion [23, Theorem 3, p. 176], we then have Since the semigroup of operators (p t , t > 0) is self-adjoint and weakly measurable, we have where P λ is the spectral decomposition of the operators, see [19,Section 22.3]. For u ∈ L 2 (m) and t > 0 we let By the spectral decomposition, t → E (t) (u, u) is nonnegative and nonincreasing [17,Lemma 1.3.4], which allows to define the quadratic form of p, The domain of the form is defined by the condition E(u, u) < ∞ [17].
The following is a Hardy-type inequality with a remainder.
or, more generally, if f is absolutely continuous and there are δ > 0 and c < ∞ such that for all s > 0 and 0 < t < δ, then for every u ∈ L 2 (m) We note that |vh| ≤ |u|, thus vh ∈ L 2 (m) and by (7), vp t h ∈ L 2 (m). We then have By the definition of J t and the symmetry (1) of p t , To deal with I t , we let x ∈ X, assume that h(x) < ∞, and consider Thus, By (13) and Fatou's lemma, for all t > 0, and so E(u, u) = E(vh, vh). From (18) we obtain (14).
If f is absolutely continuous on R, then (3) becomes equality, and we return to (17) to analyse I t and J t more carefully. If X u(x) 2 q(x) m(dx) < ∞, which is satisfied in particular when E(u, u) < ∞, and if (15) holds, then we can apply Lebesgue dominated convergence theorem to I t . In view of (13) and (17), the limit of J t then also exists, and we obtain (16). If X u(x) 2 q(x) m(dx) = ∞, then (18) trivially becomes equality. Finally, (15) We are interested in non-zero quotients q. This calls for lower bounds of the numerator and upper bounds of the denominator. The following consequence of (14) applies when sharp estimates of p are known.
In the remainder of the paper we discuss applications of the results in Section 2 and Section 3 to transition densities with certain scaling properties.

Applications to (fractional) Laplacian
The important case β = (d − α)/(2α) in the following Hardy equality for the Dirichlet form of the fractional Laplacian was given by Frank, Lieb and Seiringer in [15,Proposition 4.1] (see [3] for another proof; see also [18]). In fact, [15, formula (4.3)] also covers the case of (d − α)/(2α) ≤ β ≤ (d/α) − 1 and smooth compactly supported functions u in the following Proposition. Our proof is different from that of [15,Proposition 4.1] because we do not use the Fourier transform.
q is given by (26) and p is given by (10), then R dp t (y − x)|y| −r dy ≤ |x| −r .
Proof. By (7) and the proof of Proposition 5, we get the first estimate. The second estimate is stronger, becausep ≥ p, cf. (10), and it follows from Theorem 1, cf. the proof of Proposition 5. We do not formulate the second estimate for r = 0 and d−α, because the extension suggested by (26) reduces to a special case of the first estimate.
For completeness we now give Hardy equalities for the Dirichlet form of the Laplacian in R d . Namely, (29) below is the optimal classical Hardy equality with remainder, and (28) is its slight extension, in the spirit of Proposition 5. For the equality (29), see for example [13, formula (2.3)], [16, Section 2.3] or [3]. Equality (28) may also be considered as a corollary of [16, Section 2.3].
where h(x) = |x| γ+2−d . In particular, Proof. The first inequality is trivial for γ = d − 2, so let 0 ≤ γ < d − 2. We first prove that for u ∈ L 2 (R d , dx), where g is the Gaussian kernel defined in (20), and C is the corresponding quadratic form. Even simpler than in the proof of Proposition 5, we let f (t) = t γ/2 and µ = δ 0 , obtaining By Theorem 2 we get (30). Since the quadratic form of the Gaussian semigroup is the classical Dirichlet integral, taking γ = (d − 2)/2 and q(x) = (d − 2) 2 /(4|x| 2 ) we recover the classical Hardy inequality: We, however, desire (28). It is cumbersome to directly prove the convergence of (30) to (28) 1 . Here is an approach based on calculus.
We note that (32) holds for all u ∈ L 2 (R d ).
q is given by (31), andg is the Schrödinger perturbation of g by q as in (10), then The proof is similar to that of Corollary 6 and is left to the reader.

Applications to transition densities with global scaling
In this section we show how sharp estimates of transition densities satisfying certain scaling conditions imply Hardy inequalities. In particular we give Hardy inequalities for symmetric jump processes on metric measure space studied in [10], and for unimodal Lévy processes recently estimated in [6]. In what follows we assume that φ : [0, ∞) → [0, ∞) is nondecreasing and leftcontinuous, φ(0) = 0, φ(x) > 0 if x > 0 and φ(∞ − ) := lim x→∞ φ(x) = ∞. We denote, as usual, Here is a simple observation, which we give without proof.
We call V the volume function.
Remark 3. Interestingly, the Chapman-Kolmogorov equations and (36) imply that φ in Theorem 10 satisfies a lower scaling, too. We leave the proof of this fact to the interested reader because it is not used in the sequel. An analogous situation occurs in [6,Theorem26].
In [10] a wide class of transition densities are constructed on locally compact separable metric measure spaces (F, ρ, m) with metric ρ and Radon measure m of infinite mass and full support on F . Here are some of the assumptions of [10] (for details see [10,Theorem 1.2]). The func- for every r > 0.
Remark 4. We note that [12, Theorem 1, the "thin" case (T)] gives (48) for continuous functions u of compact support in R d . Here we extend the result to all functions u ∈ L 2 (R d ), as typical for our approach. We note in passing that [12, Theorem 1, Theorem 5] offers a general framework for Hardy inequalities without the remainder terms and applications for quadratic forms on Euclidean spaces.
Here is an analogue of Remark 2.
Remark 5. Using the notation above, for every 0 < β < (d − α)/α, there exist constants c 1 , c 2 such that wherep is given by (10) with q(x) = c 2 ψ(1/|x|) on R d . The result is proved as Remark 2. In particular we obtain non-explosion of Schrödinger perturbations of such unimodal transition densities with q(x) = c 2 ψ(1/|x|). Naturally, the largest valid c 2 is of further interest.

Weak local scaling on Euclidean spaces
In this section we restrict ourselves to the Euclidean space and apply Theorem 2 to a large class of symmetric jump processes satisfying two-sided heat kernel estimates given in [9] and [6]. Let φ : R + → R + be a strictly increasing continuous function such that φ(0) = 0, φ(1) = 1, and c R r α ≤ φ(R) φ(r) ≤ c R r α for every 0 < r < R ≤ 1.
Let J be a symmetric measurable function on R d × R d ∩ {x = y} and let κ 1 , κ 2 be positive constants such that We consider the quadratic form with the Lebesgue measure dx as the reference measure, for the symmetric pure-jump Markov processes on R d constructed in [8] from the jump kernel J(x, y).
Theorem 13. If d ≥ 3, then Proof. Let Q and p t (x, y) be the quadratic form and the transition density corresponding to the symmetric pure-jump Markov process in R d with the jump kernel J(x, y)1 {|x−y|≤1} instead of J(x, y), cf. [9, Theorem 1.4]. Thus, We define h as h(x) =