A Class of Marcinkiewicz Type Integral Operators

Abstract

We introduce a class of integral operators related to parametric Marcinkiewicz operators. We present a multiplier formula characterizing the $ L^{2}$ boundedness of such class of operators. Also, we prove $\mathcal{L} _{-\beta}^{p}$ (inhomogeneous Sobolev space)$\rightarrow L^{p}$ estimates provided that the kernels are in $L(\log L)(\mathbf{S}^{n-1})$. In fact, we show that the global parts of the introduced operators are bounded on the Lebesgue spaces $L^{p}(1\lt p \lt \infty )$ while the local parts are bounded on certain Sobolev spaces $\mathcal{L}_{-\beta }^{p}$.