A Note on Rough Parametric Marcinkiewicz Functions (Analysis in Theory and Applications, 2022)
In this note, we obtain sharp $L^{p}$ estimates of parametric Marcinkiewicz integral operators. Our result resolves a long standing open problem. Also, we present a class of parametric Marcinkiewicz integral operators that are bounded provided that their kernels belong to the sole space $L^{1}\left( \mathbb{S}^{n-1}\right) $.
Keywords: Marcinkiewicz integrals; parametric Marcinkiewicz functions; rough kernels; Fourier transform; Marcinkiewicz interpolation theorem.