Schrödinger type singular integrals: weighted estimates for p=1

A. CABRAL; B. BONGIOANNI; E. HARBOURE

MATHEMATISCHE NACHRICHTEN

WILEY-V C H VERLAG GMBH

Lugar: Weinheim; Año: 2016 vol. 289 p. 1341 - 1369

0025-584X

A critical radius function $ho$ assigns to each $xinRR^d$ a positive number in a way that its variation at different points is somehow controlled by a power of the distance between them. This kind of function appears naturally in the harmonic analysis  related to a Schr{"o}dinger operator $-Delta+V$ with $V$ a non-negative potential satisfying some specific reverse H"older condition. For a family of singular integrals associated to such critical radius function, we prove boundedness results in the extreme case $p=1$. On one side we obtain weighted weak $(1,1)$ results for a class of weights larger than Muckenhoupt class $A_1$. On the other side, for the same weights, we prove continuity from appropriate weighted Hardy spaces into weighted $L^1$. To achieve the latter result we define weighted Hardy spaces by means of a $ho$-localized maximal heat operator. We obtain a suitable atomic decomposition and a characterization via $ho$-localized Riesz Transforms for these spaces. For the case of $ho$ derived from a Schr"odinger operator, we obtain new estimates for of many of the operators appearing in~cite{Shen}.