A class of marcinkiewicz type integral operators

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 L ^p_-β (inhomogeneous Sobolev space)→L^p estimates provided that the kernels are in L(logL)(S^n-1). In fact, we show that the global parts of the introduced operators are bounded on the Lebesgue spaces L^p(1 < p < ∞) while the local parts are bounded on certain Sobolev spaces L^p_-β.