Weighted inequalities for negative powers of Schrödinger operators

Abstract

In this article we obtain boundedness of the operator (− + V ) −α/2 from L p,∞ (w) into weighted bounded mean oscillation type spaces BMO β L (w) under appropriate conditions on the weight w. We also show that these weighted spaces also have a point-wise description for 0 < β < 1. Finally, we study the behaviour of the operator (− + V ) −α/2 when acting on BMO β L (w).

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

“…In the Schrödinger case the analogous result was proved by B. Bongioanni, E. Harboure and O. Salinas in [6]. They identified the Hölder space associated to L with a Campanato type BM O α L space, see Proposition 2.4 below.…”

“…They also studied the corresponding boundedness results for the negative powers, see [6], and L p -boundedness for the commutators with a function, see [4]. Following the pattern of the proof of Theorem 1.3 we can recover the results from [6] and [5]. We state them as a theorem for further reference.…”

“…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]. In fact, if s > n and 0 [15], references in [20] and [26].…”

“…The definition of space BM O L was given in [11]. The space BM O α L , 0 < α ≤ 1, was introduced in [6]. We collect from there the following facts.…”

Extrapolation for Classes of Weights Related to a Family of Operators and Applications

“…To this end, in Section 4, we prove a Fefferman-Stein type inequality that takes into account the structure of the appropriate version of BMO space introduced in [2,8].…”