Weighted Local Estimates for Fractional Type Operators

Abstract

In this note we prove the estimate \(M^{\sharp }_{0,s}(Tf)(x) \le c\,M_{\gamma } f(x)\) for general fractional type operators T, where \(M^{\sharp }_{0,s}\) is the local sharp maximal function and M γ the fractional maximal function, as well as a local version of this estimate. This allows us to express the local weighted control of T f by M γ 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 γ replaced by an appropriate maximal function M T . We also prove two-weight, \({L^{p}_{\text {\textit {v}}}}\)-\({L^{q}_{\text {\textit {w}}}}\) estimates for the fractional type operators described above for 1 < p < q < ∞ and a range of q. The local nature of the estimates leads to results involving generalized Orlicz-Campanato and Orlicz-Morrey spaces.