Hardy Inequalities and Non-explosion Results for Semigroups

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

embedding theorems and interpolation theorems, e.g. Gagliardo-Nirenberg interpolation inequalities [23]. The connection of Hardy-type inequalities to the theory of superharmonic functions in analytic and probabilistic potential theory was studied, e.g., in [1,6,13,16]. A general rule stemming from the work of P. Fitzsimmons [16] 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. This is one advantage of our approach, as compared to [16] -we propose a direct method for obtaining 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. This is rather sharp, and has no analogue in [16]. For instance we recover the famous non-explosion result of Baras and Goldstein for + (d/2 − 1) 2 |x| −2 , cf. [3] and [27]. 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 nonexplosion 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 [15], Frank, Lieb and Seiringer [17] and Frank and Seiringer [18]. 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 Corollaries 6 and 8 and Remarks 2, 5 and 6.
Currently our methods in Section 3 are confined to the (bilinear) L 2 setting. We refer to [18,29] 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, and to transition densities corresponding to specific boundary conditions (cf. the case of the censored processes in [12]), 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.

Non-explosion for Schrödinger Perturbations
and let p satisfy the Chapman-Kolmogorov equations: and assume that for every t > 0, x ∈ X, p t (x, y)m(dy) is (σ -finite) integral kernel. Let f : R → [0, ∞) be non-decreasing, and let f = 0 on (−∞, 0]. We have f ≥ 0 a.e., and Further, let μ be a nonnegative σ -finite measure on (X, M). We put We denote, as usual, p t h(x) = X p t (x, y)h(y)m(dy). By Fubini-Tonelli and Chapman-Kolmogorov, for t > 0 and x ∈ X we have In this sense, h is supermedian. We define q : X → [0, ∞] as follows: For all x ∈ X we thus have We define the Schrödinger perturbation of p by q [8]: where p (0) t (x, y) = p t (x, y), and It is well-known thatp is a transition density [8].
Proof For n = 0, 1, . . . and t > 0, x ∈ X, we consider and we claim that By Eq. 7 this holds for n = 0. By Eq. 11, Fubini-Tonelli, induction and Eq. 8, where in the last passage we used Eqs. 2 and 4. By Eq. 3, because f (s) = 0 if s ≤ 0. By this and Eq. 6 we obtain The claim (12) is proved. The theorem follows by letting n → ∞.
Remark 1 Theorem 1 asserts that h is supermedian forp. This is much more than Eq. 7, but Eq. 7 may also be useful in applications [22,Lemma 5.2].
We shall see in Section 4 that the above construction gives integral finiteness (nonexplosion) 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 [25,Theorem 3,p. 176], we then 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 [21,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 [19, 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) < ∞ [19].
The following is a Hardy-type inequality with a remainder. If then for every u ∈ L 2 (m) We note that |vh| ≤ |u|, thus vh ∈ L 2 (m) and by Eq. 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 Eq. 13 and Fatou's lemma, Since (vh, vh) for all t > 0, and so E(u, u) = E(vh, vh). From Eq. 18 we obtain Eq. 14.
If f is absolutely continuous on R, then Eq. 3 becomes equality, and we return to Eq. 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 Eq. 15 holds, then we can apply Lebesgue dominated convergence theorem to I t . In view of Eqs. 13 and 17, the limit of J t then also exists, and we obtain Eq. 16. If X u(x) 2 q(x) m(dx) = ∞, then Eq. 18 trivially becomes equality. Finally, 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 Eq. 14 applies when sharp estimates of p are known. Then In the remainder of the paper we discuss applications of the results in Sections 2 and 3 to transition densities with certain scaling properties.

Applications to (Fractional) Laplacian
For 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 [17,Proposition 4.1] (see [4] for another proof; see also [20]). In fact, [17, 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 [17,Proposition 4.1] because we do not use the Fourier transform.
We get a maximal C = 2 α d+α Proof Equation 21 is the Dirichlet form of the convolution semigroup of functions defined by subordination, that is we let p t (x, y) = p t (y − x), where g is the Gaussian kernel defined in Eq. 20 and η t ≥ 0 is the density function of the distribution of the α/2-stable subordinator at time t, see, e.g., [5] and [19]. Thus, η t (s) = 0 for s ≤ 0, and ∞ 0 e −us η t (s) ds = e −tu α/2 , u ≥ 0.

q is given by Eq. 26 andp is given by Eq. 10, then
R dp t (y − x)|y| −r dy ≤ |x| −r .
Proof By Eq. 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 Eq. 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, Eq. 29 below is the optimal classical Hardy equality with remainder, and Eq. 28 is its slight extension, in the spirit of Proposition 5. For the equality (29) 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 Eq. 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 Eq. 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 Eq. 28. It is cumbersome to directly prove the convergence of Eq. 30 to Eq. 28 1 . Here is an approach based on calculus.
Since div(|x| −2 x) = (d − 2)|x| −2 , the divergence theorem yields Eq. 28. We then extend We note the local integrability of |x| −2 in R d with d ≥ 3. We let n → ∞ and have Eq. 28 hold for u by using the convergence in L 2 (|x| −2 dx), inequality |∇u n (x)| ≤ |∇u(x)| + c 1 |u(x)||x| −1 , the identity h(x)∇(u n / h)(x) = ∇u n (x) − u n (x)[∇h(x)]/h(x) for x = 0, the observation that |∇h(x)|/h(x) ≤ c|x| −2 and the dominated convergence theorem. We can now extend Eq. 28 to u ∈ W 1,2 (R d ). Indeed, assume that C ∞ c (R d ) v n → u and ∇v n → g in L 2 (R d ) as n → ∞, so that g = ∇u in the sense of distributions. We have that The latter limit is h∇(u/ h), as we understand it. We obtain the desired extension of Eq. 28. As a byproduct we actually see the convergence of the last term in Eq. 30. Taking γ = (d − 2)/2 in Eq. 28 yields Eq. 29.
We note that Eq. 32 holds for all u ∈ L 2 (R d ).
q is given by Eq. 31, andg is the Schrödinger perturbation of g by q as in Eq. 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 [11], and for unimodal Lévy processes recently estimated in [7]. In what follows we assume that φ : [0, ∞) → [0, ∞) is nondecreasing and left-continuous, We denote, as usual, Here is a simple observation, which we give without proof.
We see that φ −1 is upper semicontinuous, hence right-continuous, and if φ is strictly increasing, then φ −1 (φ(s)) = s for s, u ≥ 0. Both these conditions typically hold in our applications, and then φ −1 is the genuine inverse function.

Theorem 10 Let p be a symmetric subprobability transition density on F , with Dirichlet form E, and assume that
Proof Let y ∈ F and u ∈ L 2 (F, m). The constants in the estimates below are independent of y and u. Let 0 < β < A/α − 1 and define We shall prove that To this end, we first verify Indeed, letting r = ρ(x, y) > 0 we first note that Lemma 9 yields t ≥ φ(r) equivalent To estimate I , we observe that the assumption φ ∈ WUSC(α, c) implies φ −1 ∈ WLSC(1/α, c −1/α ) [7,Remark 4]. If r > 0 and t ≥ φ(r), then The claim (40) now follows because The function This follows by recalculating (40) for β − 1. We get by choosing any β ∈ (0, A/α − 1). The theorem follows from Eq. 16.
Remark 3 Interestingly, the Chapman-Kolmogorov equations and Eq. 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 [7,Theorem26].
Remark 4 We note that [14, Theorem 1, the "thin" case (T)] gives Eq. 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 [14, Theorems 1 and 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 and wherep is given by Eq. 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 [10] and [7]. 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 every0 < 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 J (x, y)dy =: κ 2 < ∞.
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 [9] from the jump kernel J (x, y).