BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality

Abstract

We identify the dual space of the Hardy-type space H 1 L related to the time independent Schrödinger operator L = − + V , with V a potential satisfying a reverse Hölder inequality, as a BMO-type space BMO L . We prove the boundedness in this space of the versions of some classical operators associated to L (Hardy-Littlewood, semigroup and Poisson maximal functions, square function, fractional integral operator). We also get a characterization of BMO L in terms of Carlesson measures.

Regularity estimates in Hölder spaces for Schrödinger operators via a $$T1$$ theorem

“…For α = 0 the result is already contained in [11, p. The duality of the L-Hardy space H 1 L with BM O L was proved in [11]. As already mentioned in the paper by Bongioanni, Harboure and Salinas [6], the BM O α L spaces are the duals of the H p L spaces defined in [12,13,14].…”

“…For the definitions of the operators see subsections 4.1 to 4.5. In [11] it was proved that the maximal operator of the heat semigroup, the maximal operator of the Poisson semigroup and the square function of the heat semigroup are bounded in BM O L , and that the fractional integral L −γ/2 maps L n/γ (R n ) into BM O L , 0 < γ < n. The square function was also studied in [1]. In [25] it was proved that the fractional integral in the case of the harmonic oscillator has similar boundedness properties in the scale of spaces BM O α H , or more generally, C k,α H (R n ).…”

“…The Spectral Theorem implies that g W is an isometry on L 2 (R n ), see [11,Lemma 3]. As before, to get the boundedness of g W from BM O α L into itself it is sufficient to prove the following result.…”

“…The definition of space BM O L was given in [11]. The space BM O α L , 0 < α ≤ 1, was introduced in [6].…”

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

“…It is well known that the dual space of the Hardy space H p (R d ) with p ∈ (0, 1) is the Morrey-Campanato space E 1/p−1,1 (R d ). Notice that Morrey-Campanato spaces on R d are essentially related to the Laplacian Δ, where Δ ≡ On the other hand, there exists an increasing interest in the study of Schrödinger operators on R d and the sub-Laplace Schrödinger operators on connected and simply connected nilpotent Lie groups with nonnegative potentials satisfying the reverse Hölder inequality (see, e.g., [10], [34], [25], [18], [8], [7], [19], [33], [16]). Let L ≡ −Δ + V be the Schrödinger operator on R d , where the potential V is a nonnegative locally integrable function.…”