Sharp bounds for fractional type operators with \(L^{\alpha,s}\)-Hörmander conditions. (English) Zbl 1509.42023
Summary: We provide the sharp bound for a fractional type operator given by a kernel satisfying the \(L^{\alpha,s}\)-Hörmander condition and certain fractional size condition, \(0 < \alpha < n\) and \(1 < s\leq \infty\). In order to prove this result we use a new appropriate sparse domination. Examples of these operators include the fractional rough operators. For the case \(s=\infty\) we recover the sharp bound of the fractional integral, \(I_{\alpha}\), proved by M. T. Lacey et al. [J. Funct. Anal. 259, No. 5, 1073–1097 (2010; Zbl 1196.42014)].
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 26A33 | Fractional derivatives and integrals |
Keywords:
fractional operators; fractional \(L^s\)-Hörmander’s condition; sharp weights inequalities; sparse operatorsCitations:
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