Majorization, 4G Theorem and Schrödinger perturbations

Schrödinger perturbations of transition densities by singular potentials may fail to be comparable with the original transition density. For instance, this is so for the transition density of a subordinator perturbed by any time-independent unbounded potential. In order to estimate such perturbations, it is convenient to use an auxiliary transition density as a majorant and the 4G inequality for the original transition density and the majorant. We prove the 4G inequality for the 1/2-stable and inverse Gaussian subordinators, discuss the corresponding class of admissible potentials and indicate estimates for the resulting transition densities of Schrödinger operators. The connection of the transition densities to their generators is made via the weak-type notion of fundamental solution.


Introduction and preliminaries
Schrödinger perturbation consists of adding to a given operator an operator of multiplication by a function q. On the level of inverse operators the addition results in the perturbation series. We focus on transition densities p perturbed by functions q ≥ 0. Our main goal is to give pointwise estimates for the resulting perturbation series p under suitable integral conditions on p and q. For instance, bounded potentials q produce transition densitiesp comparable with the original p in finite time. In a series of recent papers, integral conditions leading to comparability ofp and p were proposed which allow for rather singular potentials q, if p satisfies the 3G Theorem [2,4]. The integral conditions compare the second term in the perturbation series (that which is linear in q) with p (the first term of the series). The comparison is meant to prevent the instantaneous blowup and to control the long-time accumulation of mass. The first property may be secured by smallness conditions, like 0 ≤ η < 1 below, and the second is accomplished by using a subadditive function Q. The results render p an approximate majorant forp in finite time [4]. They may also be considered as analogues of the Gronwall inequality [3]. We note that similar estimates for Green-type kernels were recently obtained in [10,11,13].
The 3G Theorem, which is related to the quasi-metric condition [10], is common for transition densities with power-type decay, e.g., the transition density of the fractional Laplacian. However, many transition densities fail to satisfy 3G, for instance, the Gaussian kernel. In [5] and [3], a more flexible majorization technique is proposed, p(s, x, t, y) = ∞ n=0 p n (s, x, t, y), (1.2) where p 0 (s, x, t, y) = p(s, x, t, y) and, for n = 1, 2, . . ., p n (s, x, t, y) = t s X p(s, x, u, z)q(u, z) p n−1 (u, z, t, y) dzdu. (1. 3) The above is an explicit method of constructing new transition densities. In particular, p satisfies the Chapman-Kolmogorov equations [ Since q ≥ 0, we trivially havep ≥ p, so we focus on the upper bounds forp. These may be obtained under suitable conditions on p 1 . In [4] (see also [2,15]  where 0 ≤ η < ∞ and Q is superadditive: 0 ≤ Q(s, u) + Q(u, t) ≤ Q(s, t). The following estimates follow: for all s < t, x, y ∈ X , provided 0 < η < 1, and for η = 0, we even havẽ The condition (1.6) may be considered as property of relative boundedness of q, or Miyadera-type condition for bridges [2,16]. It is convenient to use (1.6), e.g., for the transition density of the isotropic α-stable Lévy process with α ∈ (0, 2), because the so-called 3G inequality holds in this case: 3G simplifies the verification of (1.6) allowing for a simple description of the acceptable growth of q, cf. [2,Corollary 11], [4,Section 4]. In general, however, condition (1.6) may be troublesome. For instance, the transition density of the Brownian motion fails to satisfy 3G and (1.6) is difficult to characterize in a simpler way. Moreover, as we see below, for some transition densities, (1.6) holds for q(u, z) = q(z) (i.e., timeindependent q) only if q is bounded. This explains the need for modifications of [4]. The approach of [5] is based on the assumption that for all s < t, x, y ∈ X , Here it is furthermore assumed that 0 ≤ η < ∞, Q(s, t) is superadditive, rightcontinuous in s and left-continuous in t (in short: regular superadditive), and p * is a majorizing transition density, i.e., there is a constant C ≥ 1 such that for all s < t and x, y ∈ X , p(s, x, t, y) ≤ C p * (s, x, t, y). (1.10) The above assumptions are abbreviated to q ∈ N ( p, p * , C, η, Q). By [5, Theo- (1.11) For instance, p * (s, x, t, y) = p(s/c, x, t/c, y) = c d p(cs, cx, ct, cy) with c ∈ (0, 1) is convenient for the Gaussian kernel in R d [5], and Q(s, t) = β(t − s) with a constant β ≥ 0 is a common choice. In this work, we use similar dilations to produce p * . In principle, (1.9) relaxes (1.6) and allows for more functions q. This is seen in [5] and again in Sect. 3 below, where we consider applications to transition densities of subordinators. We should note that the flexibility comes at the expense of the sharpness of the resulting estimate, as seen when comparing (1.7) and (1.8) with (1.11). Also, the methods of [5] and the present paper are restricted to transition densities, while the methods of [4] handle the more general so-called forward integral kernels. Last but not least, it may be cumbersome to point out p * suitable for p and q, because this essentially requires guessing the rate of inflation ofp. In this connection, we note that (1.5) trivially yields   (7) and Lemma 5]. Similar connections exist for (1.9), parabolic Kato conditions and 4G, but we leave the details to the interested reader (see also the proof of Proposition 2.4).
Of particular interest here is the special case of convolution semigroups of probability measures { p t } t≥0 on X = R d , which are defined by the generating (Lévy) triplets (A, b, ν) [17] and correspond to the generators The identity is essentially a consequence of the fundamental theorem of calculus. It is proved in Sect. 4 in the generality of strongly continuous operator semigroups. We also provide a uniqueness result there. A special case of L is the Weyl derivative of order 1/2 on the real line: We then have the distribution of the 1/2-stable subordinator [17] (also called the Lévy subordinator). More generally, we let λ ≥ 0, δ > 0, z ∈ R, t > 0, and We note that p(t, z) is the density function of the distribution of the inverse Gaussian subordinator ξ t = inf{s > 0 : The generator corresponding to the inverse Gaussian subordinator is calculated for Here Γ λ (a, z) = ∞ z e −λy y a−1 dy for λ, z > 0, a ∈ R, is the incomplete gamma function. For the readers's convenience, we prove (1.17) and (1.20) in Sect. 4. Some further discussion can be found in [6]. We also note that the Laplace exponent of ξ t is

4G inequality for the inverse Gaussian subordinator
Our main goal is to give conditions for and discuss consequences of (1.11). Let λ ≥ 0 and δ > 0. Using (1.19) we define if s < t and x, y ∈ R, and we let p = 0 otherwise. It is a transition density on X = R with respect to the Lebesgue measure. We observe that 3G inequality does not hold and the second expression decays exponentially faster as θ → ∞. For c > 0, we consider auxiliary (inverse Gaussian) transition density In view toward (1.10), we note that for 0 < a < b, We shall consider the Schrödinger perturbationp of p = ρ 1 by q. Clearly, if q ∈ N (ρ 1 , ρ a , (1/a) 1/2 , Q, η), with 0 < a < 1, η ∈ [0, 1), thenp is finite; in fact, it satisfies (1.11). Here is a connection to generators.
Following [8, Theorem 1.1] and [5], the identity in the statement of Lemma 2.1 is interpreted by saying thatp is a fundamental solution of ∂ s + L + q or, in short, for L + q. The identity also means thatp as integral operator is the left inverse of ∂ s + L + q. We refer to [5,Remark 4.10] for further discussion. We point yet another aspect of the relationship betweenp and L +q. By Lemma 2.1 and Chapman-Kolmogorov, The identity is an analogue of classical formulas for strongly continuous operator semigroups, and so is (1.5). Further discussion of the connection to generators is given in Sect. 4. We now investigate the class N (ρ b , ρ a , (b/a) 1/2 , η, Q), where 0 < a < b; namely, we propose conditions sufficient for (1.9). We first recall results of [5, Section 3] on the Gaussian kernel where c > 0, 0 < s < t,x,ȳ ∈ R d and d ∈ N. We denote . Then, we have inequality is used in [5] to obtain Gaussian estimates for Schrödinger perturbations of transition densities of the second-order parabolic differential operators. In this section, we prove a similar inequality for the transition density ρ c defined in (2.1).

y).
We are ready to give sufficient conditions for (1.9). First comes an immediate consequence of Theorem 2.2.
The condition lim h→0 N c h (q) = 0 defines the parabolic Kato class for ρ c , cf. Sect. 1, and if it is satisfied, then Proposition 2.4 applies. A thorough discussion of the Kato condition for arbitrary Lévy processes on R d is given in [12]. For the considered inverse Gaussian subordinator (1.19), including the 1/2-stable subordinator, if q(u, z) = q(z) is time independent, then the Kato condition is equivalent to We refer to [12,Example 3] for the result. A characteristic example here is q(z) = |z| ε−1/2 for ε ∈ (0, 1/2]. In the remainder of this section, we focus on the case λ = 0 and δ = 1 in (1.19), i.e., on the density of the 1/2-stable subordinator, with emphasis on honest constants in estimates. EXAMPLE. We consider q(u, z) = q(z) on R. Let r > 2 and q ∈ L r (R). Observe that for all s < u, x ∈ R and c > 0, Thus, for every c > 0, with arbitrary η > 0 and Q(s, t) = η(t − s)/ h, provided h satisfies Indeed, (2.8) implies (2.7).
A direct consequence is that for every α-stable subordinator and for all s < t, x < y and h > 0, we have For α = 1/2, we may use Theorem 2.2 to get for all s < t, x < y and h > 0, s, x, t, y).  ∈ N (ρ b , ρ a , (b/a) 1/2 , η, Q) with Summarizing this section, we see that 4G for the inverse Gaussian subordinator yields (1.9) for a large class of functions q characterized by simpler Kato-type conditions, and then,p satisfies (1.11) and Lemma 2.1.

Relative boundedness for subordinators with transition density
In this section, we consider a general subordinator with transition density p. Thus, p is space time homogeneous, p(s, x, t, y) = 0 whenever t ≤ s or y ≤ x, and p(s, x, t, y) > 0 otherwise. We first discuss time-independent functions q, aiming at the condition (1.6).
Proof. By the assumption, there is M > 0 such that for some fixed s < t, By Lemma 3.2, q ∈ L 1 loc (R). For s < t and n ∈ N, we let Corollary 3.4 shows that the methods of [4] cannot deliver estimates of Schrödinger perturbations of transition densities p of subordinators by unbounded time-independent q. In contrast, we saw in Sect. 2 that the methods based on majorants p * and 4G inequality handle such situations.
If we allow q to depend on time, the statements of the corollary are no longer valid. Indeed, let q(u, z) = u for some η ≥ 0 and for all s < t, x < y such that (s, x), (t, y) ∈ F := {(u, z) : q(u, z) > 0}. Then, we claim that for all s < t and x < y, For the proof, we consider a Borel nondecreasing function ω : Otherwise, we consider σ = inf{u : u ∈ T (ω)} and τ = sup{u : u ∈ T (ω)}. There are s n ≤ t n such that (s n , ω(s n )), (t n , ω(t n )) ∈ F, s n ↓ σ and t n ↑ τ , hence Finally, let {Y u } u≥0 be the subordinator. Given s < t, x < y we denote by {Z u } s≤u≤t the bridge corresponding to {Y u } u≥0 , which starts from x at time s and reaches y at time t. Since the trajectories of {Z u } u≥0 are almost surely nondecreasing, we have for all s < t, x < y, which is a contradiction.

Auxilary results
In this section, we prove (1.16) and its analogues in the setting of general semigroup theory. We consider a Banach space (Y, ||·||). Let T = (T t ) t≥0 be a strongly continuous semigroup of linear operators on Y . Let L be the corresponding infinitesimal generator with domain D(L) [18,IX].
t → ξ(t) has compact support in R. where the integral is the Riemann integral of a Banach space valued function.
For t = 0, the derivative is understood as the right-hand derivative. The lemma is a version of the differentiation rule for products.
By (4.7)-(4.9), the limit on the right-hand side exists as h → 0 + and equals In fact, the assumptions (4.7)-(4.11) only need to hold on [t, t + ε), ε > 0. We now focus on Lévy semigroups discussed in the Introduction.