Heat kernel estimates of fractional Schr\"odinger operators with negative hardy potential

We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schr\"odinger operator with negative Hardy potential $$\Delta^{\alpha/2} -\lambda |x|^{-\alpha}$$ on $\RR^d$, where $\alpha\in(0,d\wedge 2)$ and $\lambda>0$. The proof is purely analytical but elementary. In particular, for upper bounds of heat kernel we use the Chapman-Kolmogorov equation and adopt self-improving argument.

Theorem 1.1. For any δ ∈ (0, α), the Schrödinger operator L given by (1.1) has the heat kernelp(t, x, y), which is jointly continuous on (0, ∞) × R d × R d , and satisfies two-sided estimates as follows We note that the last expression in (1.3) may be replaced by the heat kernel p(t, x, y) of ∆ α/2 (see Subsection 2.1 and (2.2)). As pointed out before Lemma 2.3 below, the function δ → κ δ is strictly decreasing on (0, α) with lim δ→0 κ δ = 0 and lim δ→α κ δ = −∞. Hence, Theorem 1.1 essentially gives us two-sided estimates and the joint continuity of heat kernel associated with the operator ∆ α/2 − λ|x| −α for all λ > 0. It is well known that the fractional Laplacian ∆ α/2 is the infinitesimal generator of the rotationally symmetric α-stable process, which now has been attracted a lot of interests in the field of probability and potential theory (see [6] and references therein). Recently there are also a few works concerning on gradient perturbations and Schrödinger perturbations of fractional Laplacian (see e.g. [9,10,11,12,15,16,25,27,28,33,34]). In particular, according to [33,Theorem 3.4], when the potential belongs to the so-called Kato class, heat kernel estimates for Schrödinger perturbations of fractional Laplacian are comparable with these for fractional Laplacian (at least for any fixed finite time). Note that q(x) = −λ|x| −α does not belong to the Kato class. As shown in Theorem 1.1, the heat kernelp(t, x, y) associated with the Schrödinger operator L given by (1.1) exhibits behaviour which is different from that of the case that q(x) = −λ|x| −γ with γ ∈ (0, α), which is in the Kato class. The study of heat kernel estimates for Schrödinger-type perturbations by the Hardy potential of fractional Laplacian is much more delicate.
In the classical case α = 2, the Schrödinger-type perturbations by the Hardy potential were considered for the first time by Baras and Goldstein [4]. They proved the existence of nontrivial nonnegative solutions of the classical heat equation ∂ t = ∆ + κ|x| −2 in R d for 0 ≤ κ ≤ (d − 2) 2 /4, and nonexistence of such solutions, that is explosion, for bigger constants κ. Sharp upper and lower bounds for the heat kernel of the Schrödinger operator ∆ + κ|x| −2 were obtained by Liskevich and Sobol [29, p. 365 and Examples 3.8, 4.5 and 4.10] for 0 < κ < (d − 2) 2 /4. Milman and Semenov proved the upper and lower bounds for κ ≤ (d − 2) 2 /4, see [30,Theorem 1] and [31]. In this paper, they also allowed κ < 0 and obtained the sharp upper and lower bounds for the perturbed kernel (see [30,Theorem 2 and Corollary 4]). See [24] and the references therein for the recent works of this topic.
For α ∈ (0, d∧2) the Schrödinger operator L with κ ≥ 0 attains recently more and more interest. In [1,2] for κ > κ * := 2 α Γ((d+α)/4) 2 Γ((d−α)/4) 2 the phenomenon of instantaneous blow up of heat kernel was proven. In [5], the author gives the upper bound for the heat kernel of L with the Dirichlet conditions on bounded open subsets of R d . In the recent paper [8], the following sharp estimates for the heat kernelp(t, x, y) of L were obtained. For 0 ≤ κ ≤ κ * , there is a unique constant δ ∈ [0, (d − α)/2] such that for all t > 0 and x, y ∈ R d 0 := R d \{0}, Note that since the singularity of the function R d ∋ x → κ|x| −α at the origin is critical,p(t, x, y) is not comparable with the unperturbed kernel p(t, x, y). Like in Theorem 1.1, the choice of κ influences the growth rate or the decay rate of the heat kernel at the origin. This rate is represented by the function |x| −δ , where δ is connected with κ via the formula κ = ) .
In this setting, Theorem 1.1 may be treated as both a fractional counterpart of the result obtained in [30] and the extension of (1.4) to negative values of κ. Here, we would like to stretch out one difference between the cases α = 2 and α < 2 for κ < 0. The general form of the estimate in both cases is similar, i.e., the perturbed kernelp(t, x, y) is comparable with the unperturbed kernel p(t, x, y) multiplied by some weighted functions. However, in [30, Theorem 2 and Corollary 4], for α = 2, the exponent of the weighted function is equal to δ = √ and converges to infinity as κ → −∞. In our case α < 2, as it was mentioned below the statement of Theorem 1.1, δ → α for κ → −∞. Since q(x) = κ|x| −α is negative and does not belong to any Kato class on R d , the construction and proofs of the estimates of p(t, x, y) are very delicate. In particular, we cannot use the perturbation series (at least for large values of −κ) to constructp(t, x, y) as used in [7,8,9]. That is why we will consider the Dirichlet fractional Laplacian operator Hence, the operator L with negative values of κ also enjoys some probabilistic meaning. Roughly speaking, it is connected with a symmetric α-stable process with the killing rate e −κ|x| −α , which strongly affects the behaviour ofp(t, x, y) for x and y near 0. It turns out that due to the strong singularity of q(x) at 0, the heat kernel (or the transition density function)p(t, x, y) is equal to 0 when x = 0 or y = 0. In consequence, the kernelp(t, x, y) defined on (0, We note that Theorem 1.1 was proved independently in a very recent paper [17]. In the proofs, the authors use generally probabilistic tools. In our paper we propose a different method. Although the perturbed kernelp(t, x, y) is defined by the Feyman-Kac formula, in the proofs we apply only analytical tools. For upper bounds, we generally use the Chapman-Kolmogorov equation and the method of "self-improving estimates" (see the proofs of Proposition 3.1 and Theorem 3.5, see also the proof of [26, Theorem 1.1]). Roughly speaking, to show the inequality is in some sense small. Next, by plugging this estimate to the proper functional inequality on f , we get the improved estimate of the form f (x) ≤ g n (x) + c n F (x), where g n (x) → 0 as n → ∞ and sup n∈N + c n < ∞. By passing with n to infinity we obtain the desired estimate. To obtain lower bounds we use the generally well known estimate from Lemma 3.8 and upper bound estimates. Although the estimate from Lemma 3.8 is generally well known, we couldn't find the proper reference with the assumptions on the potential satisfied by q(x). We note that the setting of [17] is more general than the present paper. From the other side, we give more details about the kernelp(t, x, y), see e.g. Theorem 2.4. We also note that in our paper we show the straightforward dependence between the exponent δ and the potential q(x), while in [17,Theorem 3.9] this dependence, given by double integral, is much more complicated.
The paper is organized as follows. In Section 2, we constructp(t, x, y) and prove some basic properties of this kernel. In Section 3, we give the proof of Theorem 1.1. First, we prove upper bounds in Theorem 3.5. Next, we show lower bounds in Theorem 3.13 and joint continuity (Theorem 3.16). We end this section with short discussion on Dirichlet forms associated with the Schrödinger operator L given by (1.1). Finally, in the Appendix, we present the proof Lemma 3.8.
Throughout the paper, we write f ≈ g for f, g ≥ 0, if there is a constant c ≥ 1 such that c −1 f ≤ g ≤ cf on their common domain. The constants c, C, c i , whose exact values are unimportant, are changed in each statement and proof. Let B(x, r) be the open ball with center x ∈ R d and radius r > 0. As usual we write a ∧ b := min(a, b) and a ∨ b := max(a, b).

Preliminary estimates
2.1. Fractional Laplacian and rotationally symmetric α-stable Lévy process. Let For (smooth and compactly supported) test function ϕ ∈ C ∞ c (R d ), we define the fractional Laplacian by In terms of the Fourier transform (see [21, Section 1.1.2]), ∆ α/2 ϕ(ξ) = −|ξ| αφ (ξ). Denote by p(t, x, y) the heat kernel (or the fundamental function) of ∆ α/2 (or equivalently, the transition density function of a (rotationally) symmetric α-stable Lévy process (X t ) t≥0 ). It is well known that p(t, x, y) is symmetric in the sense that p(t, x, y) = p(t, y, x) for any t > 0 and x, y ∈ R d , and enjoys the following scaling property p(t, x, y) = t −d/α p(1, t −1/α x, t −1/α y), t > 0, x, y ∈ R d . Moreover, We also note that p(t, x, y) is a function of t and x − y, so sometimes we also write it as p(t, x − y), i.e. p(t, x, y) = p(t, x − y). See [6] for more details.

2.2.
Fractional Laplacian Schrödinger operator and Feynman-Kac formula. In this part, we apply some results from [20,Chapter 2] to the operator L = ∆ α/2 + q given by ( A nonnegative Borel measurable function V on R d 0 is said to belong to the local Kato class K α,loc , if V 1 D ∈ K α for all compact subsets D of R d 0 . A Borel measurable function V on R d 0 is said to belong to the Kato-Feller class, if its positive part V + := max{V, 0} ∈ K α and its negative part V − := max{−V, 0} ∈ K α,loc . (Different from [20], in the present setting we start from the nonpositive definite operator ∆ α/2 + q, and so we make the corresponding changes in the definition of the Kato-Feller class.) It is easily seen from [8, Lemma 2.3] that −q / ∈ K α , but always we have −q ∈ K α,loc . In particular, q belongs to the Kato-Feller class.
In the following, we will restrict ourselves on the killed subprocess of the symmetric α-stable Lévy process (X t ) t≥0 upon exiting R d 0 (or hitting the origin), i.e., . By the strong Markov property of the process (X t ) t≥0 , it is easy to see that the process (X x, y), which enjoys the following relation with p(t, x, y): It is well known that for every t > 0 the function p(t, ·, ·) is continuous on R d ×R d , and p(t, x, y) satisfies the following Chapman-Kolmogorov equation ; that is, we consider a negative perturbation of the fractional Laplacian on R d 0 (with the Dirichlet boundary condition at {0}). Therefore, according to [20,Theorem 2.5], the operator L = ∆ α/2 + q can generate a strongly continuous and positivity preserving semigroup , which is given bỹ where the kernelp(t, x, y) satisfies the Chapman-Kolmogorov equation too, i.e., Additionally, for t > 0, we putp(t, x, y) = 0, whenever x = 0 or y = 0. Moreover, (P t ) t≥0 also acts as a strongly continuous semigroup in is given via the Feynman-Kac formula: (2.6) According to [20, Propositions 5.2 and 5.3] and their proofs,p(t, x, y) will satisfy the following Duhamel's formula: for all t > 0 and x, y ∈ R d 0 . Next, we show thatp(t, x, y) enjoys the same scaling property as p(t, x, y).
Proof. We only consider the case that x, y ∈ R d 0 ; otherwise, the statement holds trivially. Recall that for the symmetric α-stable process (X t ) t≥0 , the processes (X ut ) t≥0 and (t 1/α X u ) t≥0 enjoy the same law for any fixed u > 0. For fixed t > 0, setX u = X ut for u ≥ 0. Then, by (2.7), for any where in the last equality we used the fact that q(x) = κ δ |x| −α . Hence, the desired assertion follows from the equality above.
Lemma 2.2. Let β ∈ (0, 2). Then, Proof. We follow the method used in the proof of [7,Proposition 5]. First, let η t (s) be the density function of the distribution of the α/2-stable subordinator at time t.

Then, by integrating by parts, for
Note that, for any γ > −1, , we get the assertion of the lemma. We recall from [7, (25)] that for any β ∈ (0, d), Thus, (2.9) may be treated as an extension of the formula (2.10) to negative β. Note that in the proof of (2.9) we have to use a compensated kernel p(t, 0) to ensure convergence of the integral involved. Now, let β ∈ (0, α). By (2.10), On the other hand, let f (r) = cr (d−α+β)/α with Then, according to (2.9), Combining two equations above together, we will find that ) .
Proof. Let f (r) = cr (d−α+β)/α with the constant c given by (2.11). By (2.13) and (2.12), for any t > 0 and x ∈ R d , where in the second equality we used the fact that This completes the proof.
Set h β (x) = |x| β . Letting t → 0 in (2.14), informally it holds that for all x ∈ R d . That is, the function h β is harmonic with respect to the operator ∆ α/2 + κ β |x| −α . From now, we will fix δ ∈ (0, α), and write κ δ as κ for simplicity. The following theorem is an analog of [8, Theorem 3.1]. Since there is no problem with convergence of the integrals involved, the proof is much simpler than that of [8, Theorem 3.1].
Although the following lemma is not used in the proofs, we state it as one of the results. From this lemma we see that the right-hand side of (2.18) behaves near 0 as − log |x|. Lemma 2.5. For any t > 0 and x ∈ R d , it holds that R dp (s, x, y)|y| δ−α dy ds, Proof. By (2.16), (2.17) and the dominated convergence theorem, proving the desired assertion.
3. Two-sided estimates and joint continuity ofp(t, x, y) 3.1. Upper bounds ofp(1, x, y). For any t > 0 and x ∈ R d , define Note that, by Lemma 2.1, for all t > 0 and x ∈ R d , we have On the other hand, by the fact 0 ≤p(t, x, y) ≤ p(t, x, y) for any t > 0 and x, y ∈ R d , it also holds that Proof. By the Chapman-Kolmogorov equation (2.5) (which holds true for all x, y ∈ R d ) and (3.1), for any x, y ∈ R d ,  where the constant c comes from the estimatep(1/3, x, y) ≤ p(1/3, x, y) ≤ c.
Denote by |B(0, r)| the Lebesgue measure of B(0, r). Fix r > 0 small enough such that η := c|B(0, r)| < 3 −δ/α . According to (2.16), (3.3) and (3.2), for any x ∈ R d , we have where M = r −δ . Now, we can iterate the inequality (3.4) to obtain that for all x ∈ R d , By (3.2), taking n → ∞ in the inequality above, we get that for any x ∈ R d , yielding the desired assertion.
Hence, for any t > 0 and x, y ∈ R d , ≤ Mh(t, x)p(t, x, y), thus we get the assertion of the lemma.

3.2.
Lower bounds ofp(1, x, y). We first begin with the following lemma, which is a consequence of Theorem 3.5.
The following estimate is generally well known (see e.g. [9, Section 6] for further background).
Proof. Since the proof is a little long, we will postpone it to the appendix.
We note that the estimate in Lemma 3.8 is not sharp. More precisely, one may show that lim . Hence, by (1.3), for fixed y = 0, p(1, x, y)e −κ p 1 (1,x,y) p(1,x,y) → ∞ as x → 0. However, we still can get the following useful estimate.
Corollary 3.9. There are constants c, γ > 0 such that for all t > 0 and x, y ∈ R d 0 , we havep Proof. Lemmas 3.8 and 3.7 along with (3.11) yield that for any x, y ∈ R d 0 , thus we get (3.13) with γ = αC and c = 4 −C .
Lemma 3.10. For any r > 0, there is a constant C r > 0 such that for all x, y ∈ R d with |x| ∧ |y| ≥ r,p (1, x, y) ≥ C r p(1, x, y).
Proof. For r > 0 and x, y ∈ R d with |x| ∧ |y| ≥ r, by (3.13), we get where c and γ are the constants from Corollary 3.9.
Remark 3.12. Instead of applying Lemma 3.10, we can make use of the Feynman-Kac formula (2.6) for the semigroup (P t ) t≥0 and Dirichlet heat kernel estimates for fractional Laplacian obtained in [14] to achieve (3.15).
3.3. Joint continuity ofp(t, x, y). To prove the joint continuity ofp(t, x, y), we just follow the same argument of [7,Subsection 4.3]. For the sake of completeness, we present the proof here.
Combining with all the estimates above, we prove the desired assertion.
Proposition 3.15. The functionp(t, x, y) is jointly continuous with respect to t > 0 and x, y ∈ R d 0 . Proof. By the scaling property ofp(t, x, y), it suffices to show the continuity of p(1, x, y) with respect to x, y ∈ R d 0 . As indicated in the proof of Lemma 3.14, we only need to verify that for any x, y,x,ỹ ∈ R d 0 withx → x andỹ → y. In addition to (3.17), we have

FRACTIONAL SCHRÖDINGER OPERATORS WITH NEGATIVE HARDY POTENTIAL 19
For any ε < s < 1 − ε and x, y, z,x,ỹ ∈ R d 0 with x →x and y →ỹ, p(s, z,ỹ) ≈ p(s, z, y), andp(1 − s,x, z) ≈ p(1 − s, x, z), thanks to Lemma 3.14. Then, by the dominated convergence theorem, it holds that for any x, y,x,ỹ ∈ R d 0 withx → x andỹ → y. Hence, according to all the estimates above, we prove the desired assertion. Proof. According to Proposition 3.15 and the scaling property ofp(t, x, y), we only need to verify that p(1, x, y) is jointly continuous with respect to x, y ∈ R d when x = 0 or y = 0. Sincep(1, x, y) = 0 when x = 0 or y = 0, the desired assertion for the joint continuity is a direct consequence of the fact thatp(1, x, y) ≥ 0 and two-sided estimates forp(1, x, y) on R d 0 × R d 0 . 3.4. Dirichlet forms. Finally, we discuss the Dirichlet form associated with the Schrödinger operator L given by (1.1); see [22] for the theory of Dirichlet forms. According to [20,Theorem 2.5], the Feynman-Kac semigroup (P t ) t≥0 in L 2 (R d 0 ; dx) coincides with the semigroup corresponding toẼ with the domain for any f, g ∈ D(Ẽ ), where ν is defined by (2.1). Clearly, the quadratic form (Ẽ , D(Ẽ )) is equivalently given bỹ which are extended to be defined on L 2 (R d ; dx). and Then, (E , D(E )) is a symmetric Dirichlet form on L 2 (R d ; dx) associated with fractional Laplacian; moreover, C ∞ c (R d ) ⊂ D(∆ α/2 ) (here D(∆ α/2 ) denotes the L 2domain of ∆ α/2 on L 2 (R d ; dx)) and (E , D(E )) is regular with core C ∞ c (R d ); see [13, Section 2.2.2] for more details. On the other hand, due to α < d, we can verify that In particular, . It is easy to prove that (Ẽ , D(Ẽ )) is a symmetric Dirichlet form on L 2 (R d ; dx).
. According to the Hardy inequality for fractional Laplacian (see [7,Proposition 5]), there is a constant C 0 > 0 such that for all f ∈ L 2 (R d ; dx), Thus, the norms Ẽ + · L 2 (R d ;dx) and √ E + · L 2 (R d ;dx) are equivalent. Therefore, the desired assertion above immediately follows from the fact that (E , D(E )) is regular with core C ∞ c (R d ). Let h(x) = |x| δ , and definē Proof. Denote by ·, · the inner product of L 2 (R d ; dx). Recall that (P t ) t≥0 is well defined on L 2 (R d ; dx) by settingP t f (0) = 0 for any f ∈ L 2 (R d ; dx). According to (2.17), Hence, by the symmetry, R dp (t − s, x, z)q(z)p(s, z, y) dz ds.
Combining with both inequalities, we prove the desired assertion.
Remark 3.19. The construction of (Ē , D(Ē )) can be deduced from Doob's theory of h-transformations; see [18,Chapter 11] for more details. Indeed, as shown by (2.15), the function h = |x| δ is harmonic with respect to the operator L given by (1.1). Define L h f (x) := h(x) −1 L(f h)(x) for all f ∈ L 2 (R d ; h(x) 2 dx). It is easy to see that the operator L h is symmetric on L 2 (R d ; h(x) 2 dx), and the associated symmetric regular Dirichlet form (E h , D(E h )) on L 2 (R d ; h(x) 2 dx) is given by for all f ∈ D(E h ), where in the fourth equality we used (2.15) and the last equality follows form the property that ν(x − y) = ν(y − x). Note that Combining both equalities above together, we arrive at Proof. We use induction. When k = 0, (4.1) holds trivially. For k = 1, p η−λ (t − s, x, z)|q 0 (z)|p η−λ (s, z, y) dz ds nλ n−1 p η n (t, x, y) .