Sharp Gaussian estimates for heat kernels of Schr\"odinger operators

We characterize functions $V\le 0$ for which the heat kernel of the Schr\"o\-dinger operator $\Delta+V$ is comparable with the Gauss-Weierstrass kernel uniformly in space and time. In dimension $4$ and higher the condition turns out to be more restrictive than the condition of the boundedness of the Newtonian potential of $V$. This resolves the question of V.~Liskevich and Y.~Semenov posed in 1998. We also give specialized sufficient conditions for the comparability, showing that local $L^p$ integrability of $V$ for $p>1$ is not necessary for the comparability.

It is well known that g is a time-homogeneous probability transition density and the fundamental solution of the equation ∂ t = ∆. For Borel measurable function V : R d → R we call G the Schrödinger perturbation of g by V , or the fundamental solution of ∂ t = ∆ + V , if the following Duhamel or perturbation formula holds for t > 0, x, y ∈ R d , G(t, x, y) = g(t, x, y) + t 0 R d G(s, x, z)V (z)g(t − s, z, y)dzds.
Given the function V : R d → R we ask if there are numbers 0 < c 1 ≤ c 2 < ∞ such that the following two-sided bound holds, One can also ponder a weaker property-if for a given T ∈ (0, ∞), We call (1) and (2) sharp Gaussian estimates or bounds, respectively global (or uniform) and local in time. We observe that the inequalities in (1) and (2) are stronger than the plain Gaussian estimates: where 0 < ε 1 , c 1 ≤ 1 ≤ ε 2 , c 2 < ∞, which can also be global or local in time.
Berenstein proved the plain Gaussian estimates for V ∈ L p with p > d/2 (see [18]). Simon [25,Theorem B.7.1] resolved them for V in the Kato class, Zhang [30] and Milman and Semenov [21] applied the parabolic Kato class for this purpose. For further discussion we refer the reader to [19], [20], [21], [31]. We also refer to Bogdan and Szczypkowski [9, Section 1, 4] for a survey of the plain Gaussian bounds for Schrödinger heat kernels along with a streamlined approach, new results and explicit constants based on the so-called 4G inequality.
The plain Gaussian estimates are ubiquitous in analysis but (1) and (2) provide precious qualitative information, if they hold for V . It is intrinsically difficult to characterize (1) and (2) for those V that take on positive values, while the case of V ≤ 0 is more manageable. Arsen'ev proved (2) for V ∈ L p + L ∞ with p > d/2, d ≥ 3. Van Casteren [26] proved (2) for V in the intersection of the Kato class and L d/2 + L ∞ for d ≥ 3 (see [21]). Arsen'ev also obtained (1) for V ∈ L p with p > d/2 under additional smoothness assumptions (see [18] (2) and (1) for general V and characterized (2) and (1) for V ≤ 0. It will be convenient to state the conditions by means of The motivation for using this quantity comes from Zhang [31, Lemma 3.1 and Lemma 3.2] and from Bogdan, Jakubowski and Hansen [7, (1)]. We often write S(V ) if we do not need to specify t, x, y. As explained in Section 4, S(V ) is the potential of |V | for the so-called Gaussian bridges. We also note that [7, Section 6] uses S(V ) for general transition densities.
In the next two results we just compile [31, Theorem 1.1] with observations from [7] and [8]. For completeness, the proofs are given in Section 2.
If V ≤ 0, then for each T ∈ (0, ∞), (2) is equivalent to We say that V satisfying (3) or (4) has bounded potential for bridges (is bridge-potential bounded) globally or locally in time, respectively.
The conditions in Lemma 1.1 and 1.2 may be cumbersome to verify for specific functions. For this reason we propose simpler equivalent conditions for (1) and even simpler sufficient conditions for (1) and (2). For clarity we remark that S(V ) is unbounded for every nontrivial V in dimensions d = 1 and 2. Therefore (1) is impossible for nontrivial V ≥ 0 and nontrivial V ≤ 0 in these dimensions. This is explained after Lemma 2.1 below.
For d ≥ 3 and x, y ∈ R d we define and x · y is the usual scalar product. We denote Confronted with S(V ), the quantity K(V ) has less arguments. However, the integro-supremum tests based on K(V ) and S(V ) appear equivalent.
Here by constants we mean positive numbers. The proof of Theorem 1.3 is given in Section 3. By (7) and Lemma 1.1 we get the following result.
For d ≥ 3 we let C d = Γ(d/2 − 1)/(4π d/2 ). The Newtonian potential of nonnegative function f and x ∈ R d is We note that for d = 3 the formula for K simplifies and we easily obtain Thus if d = 3 and V ≤ 0, then the sharp global Gaussian bounds (1) are equivalent to the condition The main focus of the present paper is on the case of d ≥ 4. The next estimate is a variant of [18, Corollary 1] and motivates our development: The following result is proved in Section 3. .
We conclude that for d ≥ 4 neither finiteness nor smallness of ∆ −1 V ∞ are sufficient for (1), so the answer to the question of Liskevich and Semenov is negative. The second question motivated by (9) is whether the finiteness of K(V ) ∞ implies that of V d/2 . The answer is also negative, as follows.
In particular (1) may hold even if V d/2 = ∞. Proposition 1.6 is verified in Section 5 by means of explicit examples of functions V , which are highly anisotropic. They are constructed from tensor products of power functions and to study them we use in a crucial way the tensorization of the Gauss-Weierstrass kernel and its bridges. This is the second main topic of the paper-in Theorem 4.9 below we give new sufficient conditions for the sharp Gaussian estimates, which show that L p integrability is not necessary for (1) or (2). We note in passing that local L 1 integrability is necessary for (2) if V does not change sign, cf. Lemma 1.1 and 2.1.
The structure of the remainder of the paper is the following. In Section 2 we give definitions and preliminaries, in particular we prove Lemma 1.1 and 1.2. In Section 3 we prove Theorem 1.3, (9) and Proposition 1.5. In Theorem 4.9 of Section 4 we propose new sufficient conditions for (1) and (2), with emphasis on those functions V which tensorize. In Section 5 we prove Proposition 1. 6 and give examples which illustrate and comment our results. Appendix A gives auxiliary results on inverse-Gaussian-type integral. We should also note that the present paper merges the results of the two preprints [5] and [6].
Here are a few more comments that relate our result to existing literature. First, in [22, Theorem 1C] Milman and Semenov discuss (1) using The spatial anisotropy introduced by α · ∇ has a similar role as that seen in the integral defining S(V, t, x, y). In fact there are constants c 1 , c 2 depending only on d ≥ 3 such that This follows from (18) and (19)
We start with the following observations on integrability and potentialboundedness (12) of functions V which are bridges potential-bounded.
We see that (12) and thus also (3) fail for all nonzero V in dimensions d = 1 and d = 2, because then ∞ 0 g(s, x, z)ds ≡ ∞. Consequently, (1) fails for nontrivial V ≤ 0 and for nontrivial V ≥ 0 if d = 1 or 2, as noted in Section 1.
Following [7,9] we shall study and use the following functions We fix V and x, y ∈ R d . For 0 < ε < t, we consider By Fatou's lemma we get It follows that f is lower semi-continuous on the left, too. In consequence, We next claim that f is sub-additive, that is, This follows from the Chapman-Kolmogorov equations for g. Indeed, we have S(V, t 1 + t 2 , x, y) = I 1 + I 2 , where and I 2 equals This yields (13).
Then t = θ + (k − 1)h, and by (13) we get [31, p. 470], and the Duhamel formula follows from the discussion after [31, (3.3)] and the finiteness of S(V − ). Then the left-hand side of (5) follows from [31, pp. 467-469], or we can use [7, (41)], which follows therein from Jensen's inequality and the second displayed formula on page 252 of [7]. In general, G is obtained by applying the above procedure with −V − , and then perturbing the resulting kernel by V + , using the perturbation series, cf. [7], and then the Duhamel formula obtains without further conditions. We now prove the right hand side of (5), and without loss of generality we may assume (14) is satis-
As a consequence of Corollary 2.3 we obtain the following result.
for some constants C > 0, c ≥ 0. In fact we can take Proof. Obviously, (16) implies (2) We note in passing that the above proof shows that (2) is determined by the behavior of sup x,y∈R d S(V, t, x, y) for small t > 0. We end our discussion by recalling the connection of G to ∆ + V aforementioned in Abstract. As it is well known, and can be directly checked by using the Fourier transform or by arguments of the semigroup theory [4, Section 4], We refer to [8,Lemma 4] for a general approach to such identities.

Characterization of the sharp global Gaussian estimates
For t > 0, x, y ∈ R d , we consider Clearly, N (V ) = N (|V |) and S(V ) = S(|V |). Because of the work of Zhang [31], N is a proxy for S. Namely, by [31, Lemma 3.1, Lemma 3.2], there are constants m 1 , m 2 depending only on d such that In this section we prove our main result, i.e., Theorem 1.3. We start by using N (V, t), (U) and (L), to estimate S(V, t).
Proof. The first inequality follows by the definition of N (V, t)(x, y). For the proof of the second one we note that We can now make connections to e * (V, 0), cf. (10). Let By definition, Proof. By (L) and Lemma 3.1, By (U) and Lemma 3.1, Proof of Theorem 1.3. We claim (7) holds with M 1 > 0 that depends only on d, and M 2 = m 2 2 d ∞ 0 (1 ∨ r) d/2−3/2 r −1/2 e −r dr. To this end, according to Lemma 3.2, we analyze and thus Finally, by Theorem A.1 with a = |z − x|/2, b = |y|/2, β = d/2 and c = 1, This also gives the explicit constants, as a consequence of Remark A.5. For instance we can take Proof of (9) . The left hand side inequality follows from the identity K(V, x, 0) = C −1 d (−∆ −1 )|V |(x). If y = 0, then the upper bound trivially holds. For y = 0 we consider two domains of integration. We have Furthermore, by a change of variables and the Hölder inequality, The finiteness of κ d follows from Lemma A.6 below.
Proof of Proposition 1.5. We use the notation introduced in the formulation of the theorem. First we prove that and thus also z 1 ≤ |z| ≤ 2z 1 . Then, We now prove that ∆ −1 |V | ∞ < ∞. By the symmetric rearrangement inequality (see [17, Chapter 3]) we have It suffices then to consider x = (x 1 , 0, . . . , 0) and we only need to show that the following three integrals are uniformly bounded for x 1 ≥ 4. The first integral is The second integral we consider is

The remaining integral is
To prove the second statement of Proposition 1.5, for s > 0 we let d s f (x) = sf ( √ sx). Note that the dilatation does not change the norms: where r n is chosen such that K(V 1 Br n ) ∞ ≥ 4 n . Also, supp(V n ) ⊆ B(0, 1). We defineṼ = ∞ n=1 V n /2 n . Then, as n → ∞, and Similarly, (1) fails for −εṼ ≥ 0 with any ε > 0, cf. Lemma 2.1.

Sufficient conditions for the sharp Gaussian estimates
Recall from [10, (2.5)] that for p ∈ [1, ∞], We will give an analogue for the bridges T t,y s . Here t > 0, y ∈ R d , and Clearly, By the Chapman-Kolmogorov equations (the semigroup property) for the kernel g, we have T t,y s 1 = 1. We also note that S(V ) is related to the potential (0-resolvent) operator of T as follows, Proof. We note that As in [29, (3.4)], we have Indeed, (21) obtains from by the triangle and Cauchy-Schwarz inequalities: For p = 1, the assertion of the lemma results from (21). For p ∈ (1, ∞), we let p ′ = p/(p − 1), apply Hölder's inequality and the semigroup property, Here we also use the identity g(s, x, z) For p = ∞, the assertion follows from the identity T t,y s 1 = 1. Zhang [31,Proposition 2.1] showed that (1) and (2) hold for V in specific L p spaces (see also [31, Theorem 1.1 and Remark 1.1]). We can reprove his result as follows.
Proof. Part (a) follows from Lemma 4.1, so we proceed to (b). For t > 2, Estimating the first term of the sum, by Lemma 4.1 we obtain By (20), the second term has the same bound. For t ∈ (0, 2] we use (a).
By Lemma 1.1 and 1.2 we get the following conclusion.
is small enough. We can reduce Proposition 4.2(b) to this result as follows.
In what follows, we propose suitable sufficient conditions for (1) and (2). We let d 1 , d 2 ∈ N and d = d 1 + d 2 . Remark 4.5. The Gauss-Weierstrass kernel g(t, x) in R d can be represented as a tensor product: x 2 ). The kernels of the bridges factorize accordingly: g(s, x, z) g(t − s, z, y) g(t, x, y) Proof. In estimaing S(V, t, x, y) we first use the factorization of the bridges and the boundedness of V 1 , and then the Chapman-Kolmogorov equations and the boundedness of S(V 2 ). Clearly, Proof. We proceed as in the proof of Lemma 4.1, using Remark 4.5.
We extend Proposition 4.2 as follows.
Proof. We follow the proof of Proposition 4.2, replacing Lemma 4.1 by Lemma 4.8.
By Lemma 1.1 and 1.2 we get the following conclusion. Clearly, if |U | ≤ |V |, then S(U ) ≤ S(V ). This may be used to extend the conclusions of Theorem 4.9 and Corollary 4.10 beyond tensor products

Examples
Let 1 A denote the indicator function of A. In what follows, G in (1) is the Schrödinger perturbation of g by V .
Proof of Proposition 1.6. Add the functions from Example 5.2 and 5.3.
, where x 1 ∈ R d 1 and x 2 ∈ R d 2 . Let G 1 (t, x 1 , y 1 ), G 2 (t, x 2 , y 2 ) be the Schrödinger perturbations of the Gauss-Weierstrass kernels on R d 1 and R d 2 by V 1 and V 2 , respectively. Then G(t, (x 1 , x 2 ), (y 1 , y 2 )) := G 1 (t, x 1 , y 1 ) G 2 (t, x 2 , y 2 ) is the Schrödinger perturbation of the Gauss-Weierstrass kernel on R d by V . Clearly, if the sharp Gaussian estimates hold for G 1 and G 2 , then they hold for G. Our next example is aimed to show that such trivial conclusions are invalid for tensor products V (x 1 , x 2 ) = V 1 (x 1 )V 2 (x 2 ).
Indeed, by the symmetric rearrangement inequality [17,Chapter 3], By the comment following (8), we get (1) for the fundamental solutions in R 3 of ∂ t = ∆ + V 1 and ∂ t = ∆+V 2 . However, the fundamental solution in R 6 of ∂ t = ∆+V fails even (2). Indeed, if we let T ≤ 1, a ∈ R 6 , |a| = 1, and c = 1 0 p(s, 0, a)ds, then by [11,Lemma 3.5], By Lemma 2.1, (4) fails, and so does (2), according to the last sentence in Lemma 2.1. Thus, the sharp Gaussian estimates may hold for the Schrödinger perturbations of the Gauss-Weierstrass kernels by V 1 and V 2 but fail for the Schrödinger perturbation of the Gauss-Weierstrass kernel by V (x 1 , x 2 ) = V 1 (x 1 )V 2 (x 2 ). Considering −V 1 and −V 2 , by a comment at the end of Section 1 we get a similar counterexample for perturbations by two nonnegative factors, because 1/2 < 1. Let us also remark that the sharp global Gaussian estimates may hold for V (x 1 , x 2 ) = V 1 (x 1 )V 2 (x 2 ) but fail for V 1 or V 2 . Indeed, it suffices to consider V 1 (x 1 ) = −1 |x 1 |<1 on R 3 and V 2 ≡ 1 on R, and to apply Theorem 4.9. We see that it is indeed the combined effect of the factors V 1 and V 2 that matters-as captured in Section 4.
Appendix A.
In this section we collect auxiliary calculations used in Section 3.
Theorem A.1. Let c > 0, β > 1 and We have Here β,c ≈ means that the ratio of both sides is bounded above and below by constants depending only on β and c. Since for all x, r ≥ 0 we have 1 ∨ r ≥ x 1 + x + r 1 + x ≥ r/2 , for x ∈ (0, 1) , 1/2 , for x ≥ 1 , the last integral in the above is comparable with a positive constant depending only on γ and c. Proof. Observe that 0 ≤ s ≤ √ 4ab + s 2 . Thus with h(x) and γ = β − 3/2 from Lemma A.2 we have The assertion follows by Lemma A.2.  e −cs 2 ds √ 4ab + s 2 .
We now verify the finiteness of κ d from (9). e −(|w|−w·1) |w| −β dw < ∞, therefore we only need to characterize the finiteness of the complementary integral. We will follow the usual notation for spherical coordinates in R d [3]. In particular, w · 1 = r cos ϕ 1 and the Jakobian is r d−1 d−2 k=1 sin k (ϕ d−1−k ). We denote ϕ = ϕ 1 , and we consider