Relativistic stable operators with critical potentials

We give local in time sharp two sided estimates of the heat kernel associated with the relativistic stable operator perturbed by a critical (Hardy) potential.


Introduction
Let d ∈ N := {1, 2, . ..} and α ∈ (0, 2 ∧ d).For δ ∈ (0, d − α) we consider the operator where Γ denotes the Gamma function and K ν is the modified Bessel function of the second kind.The potential V δ arises naturally as a critical (Hardy) potential for the relativistic operator when implementing the approach developed in Bogdan et al. [11].For δ = d−α 2 it was derived and investigated by Roncal [59] from three different perspectives.
The main purpose of the present paper is to analyse the heat kernel corresponding to L, that is, the fundamental solution to the parabolic equation ∂ t u = Lu.Analogous problems were studied for the classical Hardy operator ∆+κ|x| −2 and its fractional counterpart −(−∆) α/2 + κ|x| −α with α ∈ (0, 2), see Subsection 1.3 for related literature.Contrary to the latter two, the operator −(−∆ + 1) α/2 + V δ (x) is not homogeneous; the kinetic term −((−∆ + 1) α/2 − 1) manifests properties of both operators: ∆ and −(−∆) α/2 .This generates new challenges and requires the development of an appropriate methodology.The methods we propose allow us to treat in a common framework other important operators, e.g., the relativistic operator with Coulomb potential 1.1.Main results.In our first result we address the question of the existence of the heat kernel corresponding to the operator L. We use the notation The function p is constructed in Subsection 2.1 by the use of the perturbation technique of kernels.Our main result is Theorem 1.2, which is concerned with the estimates of p.For t > 0, x ∈ R d 0 we define Let p be the heat kernel for the operator −(−∆ + 1) α/2 (see (2.7) with m = 1).Here, and in what follows, we write f ≈ g on D if f, g ≥ 0 and there is a (comparability) constant c ≥ 1 such that c −1 g ≤ f ≤ cg holds on D.
The exact value of the constant c d,α,δ > 0 is given in Lemma 2.3, but it does not matter in Corollary 1.3 due to cancellations taking place in the double integral expression.The role of the function h δ and its properties are important also in other places of our reasoning.For instance, in Theorem 4.9, we prove that for each δ ∈ (0, (d − α)/2], for all t > 0 and x ∈ R d 0 , The latter is crucial in the proof of the lower bound in Subsection 4.3.In light of that equality, the identity in Corollary 1.3 may be interpreted as the ground state representation for the quadratic form of the operator L.
The methods used in [12] cannot be transferred to our problem and, for this reason, we have to find a new strategy.Below, we outline main steps that led to the sharp two-sided estimates in Theorem 1.2.We start with the upper estimates: We take the heat kernel estimates from [12] as the starting point, and after a careful analysis of the potential V δ (x), in particular of the difference V δ (x) − κ δ |x| −α (see Lemma 2.10), by applying perturbation technique we arrive at (U 1 ).The latter is again non-trivial, not only because ), but also since the difference is unbounded (singular) in the neighbourhood of zero if δ < α, while the heat kernel of −(−∆) α/2 + κ δ |x| −α is already singular at zero.The standard realisation of the perturbation procedure fails, because the kernel of this operator does not satisfy the 3G inequality typically used in the first step of the method.When proceeding towards (U 2 ), one encounters certain technical difficulties.For instance, the heat kernel p of −(−∆ + 1) α/2 does not have scaling, and has exponential decay in the spatial variable, which is harder to retain compared to the power-type nature of the heat kernel of the fractional Laplacian, which has scaling.We develop a new integral method to take into account that faster decay, and combine it with (U 1 ), to finally obtain the proper outcome in (U 2 ).From that we also get: 2 ) upper heat kernel estimates for −(−∆ + 1) α/2 + κ δ |x| −α (Theorem 5.1).Since sharp bounds of p are only known locally in time, this is reflected in our results.
Here are the key steps for the lower estimates: First of all, the formulae for h δ and V δ are not accidental, see Subsection 2.3.They are computed according to a general procedure proposed in [11], which we used specifically to the operator −(−∆ + 1) α/2 .From [11] it is known that h δ is the so-called super-median.
In (L 1 ) we prove more, namely that h δ is actually invariant for the relativistic semigroup perturbed by V δ (x).As far as (L 1 ) is intuitively expected, the proof is technical, especially if δ = d−α 2 (in view of blow-up result this value of δ may be regarded as critical).The step (L 1 ) is inevitable in order to carry out an exact integral analysis and prove (L 2 ).Let us point out that in (L 1 ), as in the whole Section 4, we require δ ∈ (0 2. We comment on the steps for the lower heat kernel estimates of the relativistic operator with Coulomb potential: We note that the potential V δ (x) is tailor-made for −(−∆ + 1) α/2 , which manifests in (L 1 ) and (L 2 ).Clearly, (L * 1 ) is a consequence of (U 2 ) and (L 2 ).In order to prove (L * 2 ), we perturb the heat kernel of −(−∆ + 1) α/2 + V δ (x) by (possibly singular) potential κ δ |x| −α − V δ (x) < 0.
To succeed, we heavily rely on (L * 1 ) and adapt the new integral method used to prove (U 2 ).In Section 5 we summarize all the estimates and explain the blow-up phenomenon, that is, the criticality of the parameter δ * := d−α 2 .
1.3.Historical and bibliographical comments.The analysis of heat kernels corresponding to operators with the so-called critical potentials goes back to Baras and Goldstein [7], where the existence of non-trivial non-negative solutions of the heat equation , and non-existence (explosion) for bigger constants κ.The operator was also studied by Vazquez and Zuazua [68] in bounded subsets of R d as well as in the whole space.Sharp estimates of the heat kernel were obtained by Liskevich and Sobol [51] for 0 < κ < (d − 2) 2 /4, and by Milman and Semenov [55, Theorem 1] for 0 < κ ≤ (d − 2) 2 /4, see also [1].Sharp estimates in bounded domains were given by Moschini and Tesei [57] in the subcritical case, and by Filippas et al. [33] for the critical value (d − 2) 2 /4.Further generalizations for local operators were given in a series of papers by Metafune et al. that include [54,53].Asymptotics of solutions to the Cauchy problem by self-similar solutions were proved by Pilarczyk [58].
Another operator that drew attention in this context was the fractional Laplacian with Hardy potential −(−∆) α/2 + κ|x| −α .Abdellaoui et al. [2,3] , then the operator has no weak positive supersolution, while for 0 < κ ≤ κ * non-trivial non-negative solutions exist.In the latter case Bogdan et al. [12] obtained sharp two-sided estimates of the heat kernel, namely that it is comparable on (0, ∞) × R d 0 × R d 0 to the expression p 0 (t, x, y)H 0 (t, x)H 0 (t, y) .
Here p 0 is the heat kernel of the fractional Laplacian and satisfies on (0, The estimates from [12] were a key ingredient in the analysis of Sobolev norms by Frank et al. [36], Merz [52], and Bui and D'Ancona [20].They were also used by Bui and Bui [19] to study maximal regularity of the parabolic equation, and by Bhakta et al. [9] to represent weak solutions.Hardy spaces of the operator were investigated by Bui and Nader in [21].Bogdan et al. [15] found asymptotics of the heat kernel by studying self-similar solutions.Cholewa et al. [27] studied the parabolic equation (also of order greater than 2) in the context of homogeneity.We also refer to BenAmor [8], Chen and Weth [23], Jakubowski and Maciocha [44] for the fractional Laplacian with Hardy potential on subsets of R d , and to Frank et al. [35], where the operator |x| −β (−(−∆) α/2 + κ|x| −α − 1), for certain β > 0, was treated.Perturbations of the fractional Laplacian, and more general operators, by negative critical potentials were considered by Jakubowski and Wang [45], Cho et al. [26], Song et al. [65].
The relativistic operator √ −∆ + m 2 is an important object in physical studies, because it describes the kinetic energy of a relativistic particle with mass m.In quantum mechanics it was used in problems concerning the stability of relativistic matter, in particular, the relativistic operator with Coulomb potential √ −∆ + m 2 − κ|x| −1 is of interest, see Weder [70,71], Herbst [41], Daubechies and Lieb [28,29], Fefferman and Llave [32], Carmona et al. [22], Frank et al. [34,37], Lieb and Seiringer [50].We note in passing that in Subsection 5.1 we provide local in time sharp estimates of the heat kernel corresponding to that operator and in [43] we prove pointwise estimates of its eigenfunctions.
The relativistic stable operators −((−∆ + m 2/α ) α/2 − m), α ∈ (0, 2), m > 0, were investigated by Ryznar [60], who obtained Green function and Poisson kernel estimates on bounded domains as well as Harnack inequality.Kulczycki and Siudeja [49] studied intrinsic ultracontractivity of the associated Feynman-Kac semigroup.After these two papers the topic was intensely studied and resulted in rich literature concerning such operators and corresponding stochastic processes, see e.g.[48,40,24,25,64,66,6,46,5].The undertaken topics involve also linear or non-linear mostly elliptic equations or systems of equations with or without critical potentials, and with certain focus on the unique continuation properties, see for instance Fall and Felli [30,31], Secchi [63], Ambrosio [4], Bueno et al. [18] and the references therein.The list is far from being complete.Results more closely related to the present paper can be found in Grzywny et al. [39], where perturbations of non-local operators by a proper Kato class potentials are considered, and include relativistic stable operators.
1.4.Notation and organization.We use := to indicate the definition.As usual a ∧ b := min{a, b}, a ∨ b := max{a, b}.For a function f (x) which is radial, i.e. its value depends only on r = |x|, we use the same letter to denote its profile f (r) := f (x).We write c = c(a, . ..) to indicate that the constant c depends only on the listed parameters.We also recall that In certain parts of the presented theory the functions, series or integrals are allowed to attain the infinite value.On the other hand, we often avoid it by restricting the domain to R d 0 .It is though sometimes replaced by R d , for instance in the integration regions, since one point is of the Lebesgue measure zero.For n ∈ N we denote by C 0 (R n ) the space of continuous functions f : R n → R that vanish at infinity, and The paper is organized as follows.The preliminary Section 2 is divided into four parts.First in Subsection 2.1 we introduce the general framework of Schrödinger perturbations of transition densities, which is used in the paper.In Subsection 2.2 we provide the context for the relativistic stable operator.Next, in Subsection 2.3 we present computations that give rise to V δ and h δ and we prove Corollary 1.3.Finally, in Subsection 2. 4 we study properties of the potential V δ .Section 3 is mainly devoted to the analysis of the heat kernel corresponding to the fractional Laplacian perturbed by V δ .In the same section we prove Proposition 1.1.In Section 4, which consists of four parts, we focus on the heat kernel corresponding to the relativistic stable operator perturbed by V δ for δ ∈ (0, d−α 2 ].In subsequent subsections we show upper bounds, invariance of h δ with respect to perturbed semigroup and lower bounds.In Subsection 4.4 we prove Theorem 4.12.In Section 5 we extend Theorem 4.12 to other transition densities and potentials, and we discuss blow-up phenomenon.In Appendix A we collect known properties of the modified Bessel function of the second kind and of the heat kernel and the Lévy measure corresponding to the relativistic stable operator.

Preliminaries
2.1.Schrödinger perturbation.The following subsection is general and independent of more specific framework of Section 1.Let We call p a transition density.For a Borel function q : R d 0 → [0, ∞] we define the Schrödinger perturbation of p by q as where, for t > 0, x, y ∈ R d 0 , we let p 0 (t, x, y) = p(t, x, y) and From the general theory developed in [13], based solely on the algebraic structure of the above series and the Fubini-Tonelli theorem, the Duhamel's formula, and the Chapman-Kolmogorov equation, hold.In particular, p is a transition density.Furthermore, for every q 1 , q 2 ≥ 0 we have Namely, the perturbation of p by q 1 + q 2 may be realized in two steps: first by obtaining pq 1 , and then by perturbing pq 1 by q 2 .Suppose that ρ 1 and ρ 2 are two transition densities such that We also consider (similarly to above) perturbations by singed q.In that case we have to make sure that the series converges properly and that it is non-negative.For convenience we merge arguments used in [13] in such a way that fits well our setting and applications.
Lemma 2.1.Suppose that q 1 ≥ 0 and pq 1 (t, x, y) < ∞ for every t > 0, x, y ∈ R d 0 .Assume that there are ϵ ∈ [0, 1/2) (for q 2 ≤ 0 we only require that ϵ ∈ [0, 1)) and τ > 0 such that for all t ∈ (0, τ ], x, y ∈ R d 0 , Then pq 1 +q 2 = (p q 1 ) q 2 is a finite transition density.Furthermore, for every T > 0 there exists Proof.Clearly, pq 1 is a finite transition density, therefore by [13, Theorem 2] the series (p q 1 ) q 2 is a transition density.By (2.3) and [13, Theorem 2] we have pq This guarantees the absolute convergence of the series and by [13,Lemma 8] gives pq 1 +q 2 = (p q 1 ) q 2 .Now, for q 2 ≤ 0, like in [13, (25)] we have and for general (signed) q 2 we get The relativistic stable operator.We briefly recall fundamental properties of the relativistic stable operator −((−∆ + m 2/α ) α/2 − m), where α ∈ (0, 2), m ≥ 0. In fact, we focus on the operator where The function −ψ m is called the symbol or the Fourier multiplier of the operator (2.5).The value ψ m (0) = m is known as the killing rate.We refer the reader to [17], [61], [62] for a broader perspective and details of the material presented below.It is well known that the operator (2.5) uniquely generates a translation invariant Feller semigroup (P m t ) t≥0 or (equivalently) vaguely continuous convolution semigroup of measures p m t (dx) or (equivalently) a Lévy process (X m t ) t≥0 with exponential killing rate m.For f ∈ C 0 (R d ) we have that Due to the latter equality, ψ m is also referred to as the characteristic exponent and admits the Lévy-Khintchine representation and may be obtained by taking the limit as m → 0 + , see (A.1).The measure ν m (x)dx is called the Lévy measure.Since p m t is integrable we have p m t (dx) = p m t (x)dx and the heat kernel p m (t, x, y) = p m t (y − x) corresponding to the operator (2.5) may be recovered from the symbol by using the inverse Fourier transform (2.7) It will be convenient for us to use an alternative equivalent approach to the semigruop (P m t ) t≥0 (or the process (X m t ) t≥0 ) by the subordination technique.Before we move further, we note that (P m t ) t≥0 is a strongly continuous contraction semigroup on C 0 (R d ), its infinitesimal generator has C ∞ c (R d ) as a core and for f ∈ C ∞ c (R d ) it coincides with the operator (2.5) which is a non-local integro-differential operator Let η t (s) be the probability density of the α/2-stable subordinator and 1 s>0 s 1+α/2 ds be the corresponding Lévy measure.The Laplace transform of e −m 2/α s η t (s) is equal to ∞ 0 e −λs e −m 2/α s η t (s) ds = e −tϕ m (λ) , where the Laplace exponent ϕ m is a Bernstein function and admits the following Lévy-Khintchine representation Let g t (x) = (4πt) −d/2 exp{−|x| 2 /(4t)} be the Gauss-Weierstrass kernel.The Bochner subordination of the Gaussian semigroup with respect to ϕ m results in the following relations Note that it is a sub-probabilistic kernel, namely R d p m (t, x, y)dy = e −mt , t > 0. Clearly, it is also a finite transition density as defined in Subsection 2.1.In Subsection A.2 we collect further important properties of ν 1 (x) and p 1 (t, x, y).
Together with (2.8) this leads to the second line of equalities above.□ In view of Lemma 2.2, in what follows we fix m = 1 and we remove it from the notation, The function h β (r) is decreasing in r > 0.
Recall that p is the Schrödinger perturbation of p by V δ for δ ∈ (0, d − α) according to Subsection 2.1.We show how to recover the case of m > 0 from that with m = 1.Proof.By Lemma 2.2 the first equality holds for n = 0. Then by induction Since by (A.8) and Lemma A.4 we have p(t, x, y)/t ≤ cν(x − y) and p(t, x, y)/t → ν(x − y) as t → 0 + , the equality (2.9) holds by the dominated convergence theorem if the right hand side of (2.9) is finite, and by Fatou's lemma in the opposite case.□ 2.4.Analysis of the potential.We assume that α ∈ (0, 2 ∧ d).In view of Lemmas 2.2 and 2.3, and Remark 2.1 we have for β ∈ (0, d − α), m > 0 and x ∈ R d 0 , where Note that κ β |x| −α is the Hardy potential for the fractional Laplacian.We write The following two properties stem from [42, Lemma 2.6] and [72, Theorem 2.9], respectively, is radial increasing in x, and decreasing in β ∈ (0, d − α). (2.11) We proceed with the analysis of V β .
We will later on need the following technical result that provides the upper bound of the difference V β (x) − κ β |x| −α , which is uniform in the parameter β.
Then we use the third limit from the proof of Lemma 2.10.
In this section we assume that α ∈ (0, 2 ∧ d).Before treating the heat kernel for the operator (1.1), we first analyse the one corresponding to −(−∆) α/2 + V δ .Namely, we consider the heat kernel p 0 of −(−∆) α/2 , see (2.8) for the definition, and we concentrate on p 0 V δ that is the Schrödinger perturbation of p 0 by V δ .As an auxiliary function we use p 0 V 0 δ that is the Schrödinger perturbation of p 0 by V 0 δ (z) = κ δ |z| −α , which was thoroughly investigated in [12].By Corollary 2.7 we have for δ ∈ (0, d − α) that In Proposition 3.4 below we show that the converse of the latter inequality holds up to multiplicative constant.We first prove two auxiliary results.
Hence, for |x| α ≥ t, The same steps lead to the second estimate for γ = β − α.
There is a constant c > 0 such that for all t > 0, Proof.Recall from [12, Theorem 1.1 and (2.4)] that where H 0 (t, x) = 1 + t δ/α |x| −δ as introduced in Section 1.Note that for 0 < s < t we have H 0 (t − s, y) ≤ H 0 (t, y) and and for k ∈ {0, 1}, Similarly, Finally, the result follows from the latter two inequalities combined with ≤ cp 0 (t, x, y) p 0 (s, x, z) + p 0 (t − s, z, y) H 0 (s, x)H 0 (s, z)H 0 (t − s, y)H 0 (t − s, z) , which holds by (3.1) and 3G-inequality for p 0 (t, x, y), see [14,Theorem 4].□ We show that V δ − V 0 δ can be conveniently used to perturb p 0 . For every T > 0 there is a constant c > 0 such that for all t ∈ (0, T ] and x, y ∈ R d 0 , If δ ≥ α the inequality is trivial, because due to Lemmas 2.8 and 2.10 the function q is bounded, and p 0 V 0 δ satisfies Chapman-Kolmogorov equation (2.2).If 0 < δ < α, then q is bounded for |z| ≥ 1/2, while for |z| ≤ 1/2 we can use Lemmas 2.10 and 3.2 to obtain the desired inequality.□ Proposition 3.4.Let δ ∈ (0, d − α).For every T > 0 there is a constant c > 0 such that for all t ∈ (0, T ] and x, y ∈ R d 0 , p 0 Proof.According to (2.3) we treat p 0 V δ as the perturbation of p 0 the result follows from [13, Theorem 2] since q is relatively Kato for p 0 The lower bound follows from the trivial inequality p 0 V 0 δ ≤ p 0 V δ , see the comment preceding Lemma 3.1.□ We will use the following observation several times throughout the paper.
Remark 3.1.Let δ ∈ (0, d−α).Recall that p := pV δ is the Schrödinger perturbation of p := p 1 by V δ := V 1 δ .Directly from (2.8) we have p ≤ p 0 , and by Proposition 3.4, and T > 0 are fixed.We are ready to show Proposition 1.1.The essence of the proof is to take a similar equality for the original transition density p and the operator ∂ u − (−∆ z + 1) α/2 , and to use the algebraic structure of the perturbed transition density p.That idea goes back to [13, (39)], [16,Lemma 4] and [10,Lemma 2.1].In doing so one should ensure that certain integrals converge absolutely.This can be guaranteed using Proposition 3.4 and [12, Proposition 3.2].
Proof of Proposition 1.1.Recall that p := pV δ that is the perturbation of p := p 1 by for a jointly measurable function f such that the integrals converge absolutely.Let Hence, P ψ = −ϕ.Then, which is the desired equality, but we need to make sure that the integrals converge absolutely.We consider Note that the function ψ is bounded [61, p. 211] and zero if |u| is large.By Remark 3.1, for every s ∈ R and x ∈ R d 0 there is c such that P |ψ|(s, x) ≤ c P 0 |ψ|(s, x) < ∞ .
The finiteness follows from [12, Proposition 3.2] and because P 0 is the same for δ and δ ′ = d − α − δ, see Remark 2.2.This implies that P |ψ|, P |ψ|, P V δ P |ψ| and P V δ |ϕ| are finite.Indeed, by (2.1) we have Proof.The convergence and monotonicity of p m follows from (2.8).The convergence of V m δ stems from Lemmas 2.6 and 2.2, and the monotonicity from (2.11).To prove the third convergence we use the dominated convergence theorem.To justify its use we note that , and we let p 0 V δ (the Schrödinger perturbation of p 0 by V δ = V 1 δ ) to be the majorant, see also Proposition 3.4 for summability or integrability.□ 4. Heat kernel of −(−∆ + 1) α/2 + V δ Under the constraint in the whole section that α ∈ (0, 2 ∧ d) and δ ∈ (0, d−α 2 ] we finally consider the heat kernel p for the operator (1.1).As specified after Theorem 4.12 and in the comment preceding Lemma 2.4, we investigate p := pV δ that is the Schrödinger perturbation of p := p 1 by V δ := V 1 δ .4.1.Upper bound.We consider the following function p(t − s, x, z)V δ (z) dzds .
Note that by Lemma 2.6 there is c 1 such that V −1 δ ( 1 2t ) ≤ c 1 t 1/α for all t ∈ (0, T ].Thus, by (A.9) there is c 2 > 0 such that for all 0 < s < t ≤ T and |z| ≤ V −1 δ ( Finally, note that h δ (2c In what follows, for T > 0 we set We shall also use ν(x) defined in Corollary 1.3.Lemma 4.4.Let T > 0. There exists a constant c such that for all t ∈ (0 Proof.Since |x| ≤ 2R 0 , |y| ≥ R 0 + 1, using (A.8) and (A.6) we get p(s, z, y) ≤ csν(x − y) for all s ∈ (0, T ] and |z| ≤ R 0 .Thus, by ( The first equality above follows from the Chapman-Kolmogorov equation and the definition of hβ .In the second equality we used integration by parts and that p(s + r, x, 0)f (r) ≤ p 0 (s + r, x, 0)f (r) vanishes at zero and at infinity as a function of r.In the third equality we used Fubini's theorem, which was justified because That would require the finiteness of these integrals.We also note that the case δ = d−α 2 is more challenging than that of δ < d−α 2 .We prove both cases at once by a limiting procedure.Theorem 4.9.For all t > 0 and x ∈ R d 0 , Proof.By the definition of h m β in Subsection 2.3 we have Thus, by Remark 3.1 and [12, (3.3)], for 0 < β < δ the integrals on the left hand side of the equality in Proposition 4.8 are finite, hence the one on the right hand side is also finite, and we can rewrite the equality as where and the value of r 0 ∈ (0, 1] will be specified later.Clearly, I 2 ≥ 0 and I 3 ≥ 0. All the following inequalities will be uniform in β ∈ [β 0 , δ].By Lemma 2.11 for all 0 < |y| ≤ 1, Fix ε > 0. Since by [12, (3.3)] with β = δ − β 0 the latter expression is integrable against p 0 V 0 δ (s, x, y) dyds on (0, t] × R d , by Remark 3.1 there is r 0 ∈ (0, 1] such that −I 4 ≤ ε.Now, for that choice of r 0 ∈ (0, 1] and all |y| ≥ r 0 , |V δ (y) − V β (y)|h β (y) ≤ sup β∈[β 0 ,δ] V δ (r 0 ) + V β (r 0 ) h β (r 0 ) < ∞ .
Combined with (4.3) it justifies the usage of the dominated convergence theorem, and we get lim β↑δ I 1 = 0. Finally, we have R d p(t, x, y)h δ (y) dy ≥ h δ (x) − ε .

Further analysis and consequences
In the whole section we assume that α ∈ (0, 2 ∧ d).
5.1.Four transition densities.Note that we have two heat kernels p 0 and p := p 1 as well as two potentials V 0 δ and V δ := V 1 δ .That amounts to four possible transition densities as Schrödinger perturbations.In [12] the authors studied p 0 , x, y)f (y) dy andE(f, f ) = lim t→0 + 1 t ⟨f − P t f, f ⟩ .

Lemma 2 . 4 . 1 α x, m 1 α 1 α x, m 1 α
Let p m be the Schrödinger perturbation of p m by V m δ , and (p m ) n be the summand of the corresponding series.Then for all t > 0, x, y ∈ R d 0 , n = 0, 1, . . .and m > 0, (p m ) n (t, x, y) = m d α p n (mt, m y) , and p m (t, x, y) = m d α p(mt, m y) .
By the symmetry of p we may and do assume that |y| ≥ 3R 0 .The case of |x| ≤ 2R 0 is covered by Lemma 4.4 with R = 2R 0 .From now on we consider |x| ≥ 2R 0 .Using the symmetry of p again, we get by (4.2) that δ (z)p(s, z, x) dzds .Furthermore, by Lemma 4.4 with R = R 0 , we have by the monotonicity of h δ that for 0 < s < t ≤ T and |z| ≤ R 0 , p(t − s, z, y) ≤ cT ν(z − y)H(T, z) .