A weak type $(p,a)$ criterion for operators, and applications

Abstract

Let $(X,d,\hat{I}¼)$ be a space of homogeneous type and $\hat{I}\text{c◯}$ an open subset of $X$. Given a bounded operator $T:L^p(\hat{I}\text{c◯})\to L^q(\hat{I}\text{c◯})$ for some $1\le p\le q<\infty$, we give a criterion for $T$ to be of weak type $(p_0,a)$ for $p_0$ and $a$ such that $\frac{1}{p_0}-\frac{1}{a}=\frac{1}{p}-\frac{1}{q}$. These results are illustrated by several applications including estimates of weak type $(p_0,a)$ for Riesz potentials $L^{-\frac{\hat{I}}{\pm }2}$ or for Riesz transform type operators $\nabla \hat{I}{”}^{-\frac{\hat{I}}{\pm }2}$ as well as $L^p-L^q$ boundedness of spectral multipliers $F(L)$ when the heat kernel of $L$ satisfies a Gaussian upper bound or an off-diagonal bound. We also prove boundedness of these operators from the Hardy space $H_L^1$ associated with $L$ into $L^a(X)$. By duality this gives boundedness from $L^{a^{\prime }}(X)$ into ${\text{BMO}}_L$.

25 pages. Revised version