$A_p$ weights and Quantitative Estimates in the Schrödinger Setting

Abstract

Suppose L = −∆ + V is a Schrödinger operator on R n with a potential V belonging to certain reverse Hölder class RH σ with σ ≥ n/2. The aim of this paper is to study the A p weights associated to L, denoted by A L p , which is a larger class than the classical Muckenhoupt A p weights. We first prove the quantitative A L p bound for the maximal function and the maximal heat semigroup associated to L. Then we further provide the quantitative A L p,q bound for the fractional integral operator associated to L. We point out that all these quantitative bounds are known before in terms of the classical A p,q constant. However, since A p,q ⊂ A L p,q , the A L p,q constants are smaller than A p,q constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to L. Next, we prove two-weight inequalities for the fractional integral operator; these have been unknown up to this point. Finally we also have a study on the "exp-log" link between A L p and BMO L (the BMO space associated with L), and show that for w ∈ A L p , log w is in BMO L , and that the reverse is not true in general.

The heat equation with rough boundary conditions and holomorphic functional calculus

“…Moreover, the H ∞ -functional calculus properties of differential operators on weighted space have been treated in several papers as well (see e.g. [6,10,11,51,60].…”