H^pspaces associated with Schrödinger operators with potentials from reverse Hölder classes

Abstract

Abstract. Let A = −∆ + V be a Schrödinger operator on R d , d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element ofA admits a special atomic decomposition. Introduction. Let k t (x, y) be the integral kernels of the semigroup of linear operators {T t } t>0 generated by a Schrödinger operatorThroughout this paper we assume that V is a nonnegative potential on R d that belongs to the reverse Hölder class RH q , q > d/2, that is, there exists a constant C > 0 such that

Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type

“…We wish to point out that the atomic space in (5) is a modification of the atomic spaces of Coifman and Weiss -see Definition 2.9. The spaces in (5) and their identification were originally studied in [15,16,19] for X = R n , while variations have since been considered in say [17,18,30,42].…”

“…For the second direction (in generalizing the Laplacian to some other operator L) we cite the body of work in [13,12,14,15,16,19,25,26,27]. The starting point here is to replace the semigroup e −t 2 ∆ in (i) and (ii) by some other semigroup e −t 2 L , but one can define an adaptation of (iii) by encoding the cancellation of atoms using L in a certain way (see [25] and also Definition 2.1 below).…”

Hardy type estimates for commutators of fractional integrals associated with Schrodinger operators

“…We recall the Hardy space associated with Schrödinger operator L which had been studied by Dziubański and Zienkiewicz in [4] and [5]. Because V ∈ L q 1 loc (R n ), the Schrödinger operator L generates a (C 0 ) contraction semigroup {T L s : s > 0} = {e −sL : s > 0}.…”

Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrüdinger operators

“…On the other hand, recall that { T t 2 } t>0 satisfies that for all t ∈ (0, ∞), T t 2 (1) = 1 (see [6], [9]). Thus {T t 2 } t>0 satisfies assumptions (3.1)-(3.3).…”

“…Denote by B q (R d ) the class of functions satisfying the reverse Hölder inequality of order q. For V ∈ B d/2 (R d ) with d ≥ 3, Dziubański et al ([8], [9], [7]) studied the BMO (bounded mean oscillation)-type space BMO L (R d ) and the Hardy space H …”