Operators with Gaussian kernel bounds on Mixed Morrey spaces

Abstract

Let $\mathscr{L}$ be an analytic semigroup on $L^{2}(\R^{n})$ with Gaussian kernel bound, and let $\mathscr{L}^{-\tfrac{\a}{2}}$ be the fractional
operator associated to $\mathscr{L}$ for $0 < \alpha < n$. In this paper, we prove some boundedness properties for the commutator $[b,\mathscr{L}^{-\tfrac{\a}{2}}]$ on Mixed Morrey spaces $L^{q,\m} \left( 0, T, L^{p, \l} \left( \mathbb{R}^{n} \right) \right)$, when $b$ belongs to $BMO ( \mathbb{R}^{n})$ or to suitable homogeneous Lipschitz spaces.