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

Let $(X,d,\mu )$ be a space of homogeneous type and $\mathrm{\Omega }$ an open subset of $X$. Given a bounded operator $T:L^p(\mathrm{\Omega })\to L^q(\mathrm{\Omega })$ for some $1\le p\le q<\mathrm{\infty }$, 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 $L^{-\frac{\alpha }{2}}$ or for Riesz transform type operators $\mathrm{\nabla }{\mathrm{\Delta }}^{-\frac{\alpha }{2}}$ 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 $L^{a^{\mathrm{\prime }}}(X)$ into ${\text{BMO}}_L$.