Weighted local estimates for fractional type operators

Abstract

In this note we prove the estimate $M_{0,s}^{♯}(Tf)(x)\le c{M}_{\gamma }f(x)$ for general fractional type operators $T$, where $M_{0,s}^{♯}$ is the local sharp maximal function and $M_{\gamma }$ the fractional maximal function, as well as a local version of this estimate. This allows us to express the local weighted control of $Tf$ by $M_{\gamma }f$. Similar estimates hold for $T$ replaced by fractional type operators with kernels satisfying Hörmander-type conditions or integral operators with homogeneous kernels, and $M_{\gamma }$ replaced by an appropriate maximal function $M_T$. We also prove two-weight, $L_v^p$-$L_w^q$ estimates for the fractional type operators described above for $1<p<q<\infty$ and a range of $q$. The local nature of the estimates leads to results involving generalized Orlicz-Campanato and Orlicz-Morrey spaces.

arXiv admin note: substantial text overlap with arXiv:1308.1134

Cited by in corpus (1)

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