$H^{p}$ spaces associated with Schrödinger operators with potentials from reverse Hölder classes
Volume 98 / 2003
Colloquium Mathematicum 98 (2003), 5-38
MSC: Primary 42B30, 42B25, 35J10; Secondary 47D03.
DOI: 10.4064/cm98-1-2
Abstract
Let $A=-{\mit \Delta } +V$ be a Schrödinger operator on ${{\mathbb R}}^d$, $d\geq 3$, where $V$ is a nonnegative potential satisfying the reverse Hölder inequality with an exponent $q>d/2$. We say that $f$ is an element of $H^p_A$ if the maximal function $\mathop {\rm sup}_{t>0} |T_tf(x)|$ belongs to $L^p({{\mathbb R}}^d)$, where $\{ T_t\} _{t>0}$ is the semigroup generated by $-A$. It is proved that for $d/(d+1)< p\leq 1$ the space $H^p_A$ admits a special atomic decomposition.
