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A Class of Marcinkiewicz Type Integral Operators

Commun. Math. Anal.
Volume 13, Number 2 (2012), 56 - 81

A Class of  Marcinkiewicz Type Integral Operators

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Abstract

We introduce a class of integral operators related to parametric Marcinkiewicz operators. We present a multiplier formula characterizing the L2 boundedness of such class of operators. Also, we prove Lpβ (inhomogeneous Sobolev space)→Lp estimates provided that the kernels are in L(logL)(Sn−1). In fact, we show that the global parts of the introduced operators are bounded on the Lebesgue spaces Lp(1<p<∞) while the local parts are bounded on certain Sobolev spaces Lpβ.