Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Studia Mathematica / All issues

Abstract

We study the $L^\infty (\mathbb {R}^d)$ boundedness for Riesz transforms of the form ${V^{a}\bigl(-\frac 12\Delta +V\bigr)^{-a}},$ where $a \gt 0$ and $V$ is a non-negative potential with power growth acting independently on each coordinate. We factorize the semigroup $e^{-tL}$ into one-dimensional factors, estimate them separately and combine the results to estimate the original semigroup. Similar results with the additional assumption $a \leq 1$ are obtained on $L^1(\mathbb {R}^d)$.