Weighted local estimates for fractional type operators. (English) Zbl 1301.42035
Summary: In this note we prove the estimate \(M^{\sharp}_{0,s}(Tf)(x)\leq cM_\gamma f(x)\) for general fractional type operators \(T\), where \(M^{\sharp}_{0,s}\) 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 \(L^p_v\)-\(L^q_w\) 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.
MSC:
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B35 | Function spaces arising in harmonic analysis |
| 26A33 | Fractional derivatives and integrals |
| 31B10 | Integral representations, integral operators, integral equations methods in higher dimensions |
Keywords:
fractional operators; maximal fractional function; local maximal function; Orlicz-Morrey spaces; Orlicz-Campanato spacesReferences:
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