Estimates of perturbation series for kernels

We consider Schrödinger-type perturbations of integral kernels on space–time. Under suitable conditions on the first nontrivial term of the perturbation series, we prove that the perturbed kernel is locally in time comparable with the original kernel.


Introduction
Estimates of the Green function and heat kernel of Schrödinger operators + q were studied, for example, in [9,8,14,20]. Local integral smallness of the function q, defined via Kato-type conditions [8,20], played an important role in these considerations. Similar Schrödinger-type operators based on the fractional Laplacian α/2 were studied in [1,2,6] (see also [7]), with focus on comparability of the resulting Green functions. The heat kernel estimates for α/2 + q, in fact Schrödinger-type perturbations of general transition densities, were then studied in [3] under the following integral condition on q, t s X p(s, x, u, z)|q(u, z)| p(u, z, t, y)dzdu ≤ [η + β(t − s)] p(s, x, t, y), (1) where p is a finite jointly measurable transition density, β and η are fixed nonnegative numbers, while times s < t and states x, y are arbitrary. Given (1), the following estimate was obtained in [3], provided η < 1. Herep denotes the Schrödinger perturbation series defined by p and q (see below for details). The approach of [3] depends on nontrivial combinatorics of the perturbation series. Further combinatorial arguments were used in [12] to refine the above result by skipping the Chapman-Kolmogorov condition on p, relaxing the assumptions on q, and strengthening the estimate, as in (25) below. Meanwhile, a more straightforward method was proposed in [13] for gradient perturbations of the transition density of the fractional Laplacian. It was suggested in [13] that the technique may be applied to Schrödinger perturbations to produce the main results of [12]. In the present paper we develop this observation and estimate Schrödinger-type perturbations of Markovian semigroups, potential kernels, and general forward integral kernels on space-time by rather singular functions q. We obtain local in time and global in space comparability of the original and perturbed kernels under suitable conditions on the first nontrivial term of the perturbation series. We say that f, g ≥ 0 are comparable if a number c > 0 exists such that c −1 f ≤ g ≤ c f . We wish to mention a related paper [4] on the von Neumann series of general integral kernels with a certain transience-type property. Both papers were inspired by [3,12], but their methods and results are different. In particular, the present estimates are more convenient and precise for forward kernels in continuous time perturbed by functions.
In what follows we will assume that q is nonnegative, since the absolute value of the perturbation with signed q is bounded by the perturbation with |q|, if finite. In this connection we also note that a discussion of the positive lower bound for signed perturbations of transition densities is given in [3].
Our main results are given in Sect. 3. Examples of applications and further comments are given in Sect. 4. In particular, we estimate the inverse kernel of Schrödinger perturbations of Weyl fractional derivatives on the real line.

Preliminaries
Following [10], we recall some basic properties of kernels. DEFINITION 1. Let (E, E) be a measurable space. A kernel on E is a map K from E × E to [0, ∞] with the following properties: Consider kernels K and L on E. The map is a kernel on E, called the composition of K and L, and denoted K L. Composition of kernels is associative [10]. We write q ∈ E + if q : E → [0, ∞] and q is E-measurable. We will denote by the same symbol the kernel q(x, A) = q(x)1 A (x). Here 1 A is the indicator function of A. We let K n = (K q) n K , n = 0, 1, . . .. Associativity yields the following. LEMMA 1. K n = K n−1−m q K m for all n ∈ N and m = 0, 1, . . . , n − 1.
We will consider the perturbation of K by q, defined as the kernel Vol. 12 (2012) Estimates of perturbation series for kernels 975 Of course, K ≤K . In what follows we will prove upper bounds forK under additional conditions on K and K 1 = K q K .

Estimates for kernels on space-time
In what follows we consider a set X (the state space) with σ -algebra M, the real line R (the time) equipped with the Borel sets B R , and E = R × X (the space-time) with the product σ -algebra E = B R × M. We also fix q ∈ E + , a number η ∈ [0, ∞), and a function Q : R × R → [0, ∞) satisfying the following condition of super-additivity: Let K be a kernel on E. We will assume that K is a forward kernel, that is, REMARK 1. In the language of [4], (s, ∞) × X is absorbing for forward kernels.
We note that the comparability in Theorem 1 is local in time as the factors in (10) and (11) are bounded if so are s and t. The comparability is global in space, meaning that the factors are independent of x and dy.
Theorem 1 has two fine or pointwise variants, which we will state under suitable conditions. We fix a (nonnegative) σ -finite, nonatomic measure is jointly measurable. We will call k a transition kernel if the following Chapman-Kolmogorov identity holds For instance, if p is a transition probability, and we let k(s, x, t, A) = p s,t (x, A), then k is a transition kernel, provided it is jointly measurable. We let k 0 = k, and for n = 1, 2, . . ., we define Proof. If m = 0, then the equality (13) holds by the definition of k n . In particular, this proves our claim for n = 1. If n ≥ 1 is such that (13) holds for all m < n, then so for every m = 1, 2, . . . , n, we obtain Vol. 12 (2012) Estimates of perturbation series for kernels 977 We definek We will assume that for all or k 1 (s, x, t, dy) ≤ [η + Q(s, t)]k(s, x, t, dy). Thus, (15) is a fine version of (6).
For the finest variant of Theorem 1, we fix a σ -finite measure dz = m(dz) on (X, M).We will consider function κ(s, x, t, y) defined for s < t and x, y ∈ X , such that (s, We will call such κ a (forward) kernel density, because {(t,y)∈E:s<t} κ(s, x, t, y) f (t, y)dtdy is a forward kernel on E. For instance, we may take k(s, x, t, y) = p s,t (x, y), if measurable and finite, where p is a transition probability density function.We define κ 0 (s, x, t, y) = κ(s, x, t, y), Proof. The result was stated in [3, Lemma 3] under stronger conditions, so for the comfort of the reader we repeat the arguments of [3]. If m = 0, then the equality (20) holds by the definition of κ n . In particular, this proves our claim for n = 1. If n ≥ 1 is such that (20) holds for all m < n, then for every m = 1, 2, . . . , n, by Fubini we indeed obtain Vol. 12 (2012) Estimates of perturbation series for kernels 979 The Schrödinger perturbation of κ by q is defined as follows: We will assume that for all s < t ∈ R and x, y ∈ X , Q(s, t)]. This is a fine analogue of (6) and (15).
The following is a fine version of Theorems 1 and 2. We note that (24, 25, 26), but not (23), were first proved in [12] by involved combinatorics.
Proof. We proceed as in the proof of Theorem 1, using Lemma 3 and (22).

Discussion and applications
The proofs of Theorems 1, 2, and 3 indicate that our estimates are rather tight. The observation is supported by the exact formulas for Schrödinger perturbations of transition densities by Dirac measures (not directly manageable by the methods of the present paper), see [4]. We like to note that the iterated integrals defining K n , k n , and κ n exhibit similarity to the expectations of powers of the additive functional in Khasminski's lemma [1,8], to Wiener chaoses and the multiple integrals in the theory of rough paths [15]. In fact, our results offer a far-reaching extension and strengthening of Khasminski's lemma for transition kernels and densities. On a formal level, a unique feature of our estimates is the combinatorics triggered by η, Q, and the assumptions (6), (15), (22). As we will see below, the room given by η is quite convenient in applications, and Q is often chosen linear.
We note that such affine upper bounds are an important special case of (22), in particular (29) allows for an application of Theorem 3. We can handle some unbounded functions q, too. For s < u < t we have hence the following 3P Theorem holds for κ, .
REMARK 2. Let κ be a (forward) kernel density. We will say that q is of relative Kato class [3] for κ, if inf{c : t s X κ(s, x, u, z)q(u, z)κ(u, z, t, y)dzdu ≤ cκ(s, x, t, y) for all s<t<s+h and x, y ∈ X } → 0 as h → 0. In short, sup{κ 1 (s, x, t, y)/κ(s, x, t, y) : s < t < s + h, x, y ∈ X → 0 as h → 0. where the supremum is taken over all s < t < s + h and x, y ∈ X . The latter condition was proposed in [19] under the name of parabolic Kato condition, and the former was essentially used already in [20] to estimate Schrödinger perturbations of the Gaussian kernel. We note that the latter condition is usually weaker and easier to verify. As indicated by Example 2, when κ satisfies the 3P Theorem, the Kato condition implies the relative Kato condition. Accordingly, the two are equivalent for the transition density of the fractional Laplacian α/2 with 0 < α < 2, but not α = 2, because 3P fails for the Gaussian kernel. The details and further references are given in [3] for transition densities, see also [4] for the special case of Schrödinger perturbations of the Cauchy transition density.
We will make a connection to Schrödinger operators analogous to + q, as mentioned in Introduction. Consider a kernel K on E, function q ∈ E + , and real-valued E-measurable functions φ and ψ on E such that K ψ = −φ. Here we assume absolute integrability: K |ψ| < ∞. Then, provided the integrals are absolutely convergent for all arguments. For forward kernels we can give rather explicit sufficient conditions for the absolute integrability. We will say K is locally finite in time if for all real s < t, u ∈ R and z ∈ X , we have K 1 (s,t) (u, z) = K (u, z, (s, t) × X ) < ∞. LEMMA 4. Consider a forward kernel K locally finite in time. Let q ∈ E + satisfy (6) with η < 1 and some superadditive function Q. Let ψ and φ be real-valued Emeasurable functions such that K ψ = −φ, and |ψ| ≤ c1 (a,b) for some a, b, c ∈ R. ThenK (ψ + qφ) = −φ.
Proof. We have |φ| ≤ K |ψ| < ∞, by the local finiteness of K . By the preceding discussion it suffices to prove that K q K |ψ|,K q K |ψ| andK q K q K |ψ| are finite. In bounded time, by our assumptions and Theorem 1, K q K ≤ C K ,K ≤ C K , and K q K q K ≤ C K , with some C ∈ R, which ends the proof.