Boundedness of a class of rough maximal functions

Abstract

Abstract In this work, we obtain appropriate sharp bounds for a certain class of maximal operators along surfaces of revolution with kernels in Lq(Sn−1) $L^{q}(\mathbf{S}^{n-1})$, q>1 $q > 1$. By using these bounds and using an extrapolation argument, we establish the Lp $L^{p}$ boundedness of the maximal operators when their kernels are in L(logL)α(Sn−1) $L(\log L)^{\alpha}(\mathbf{S}^{n-1})$ or in the block space Bq0,α−1(Sn−1) $B^{0,\alpha-1}_{q} (\mathbf{S}^{n-1})$. Our main results represent significant improvements as well as natural extensions of what was known previously.

Keywords

• Maximal functions
• L p $L^{p}$ boundedness
• Rough kernels
• Surfaces of revolution
• Extrapolation