On the Boundedness of Singular Integral Operators in Symmetric Spaces

Abstract

We consider the integral convolution operators $T_\varepsilon f\left( x \right) = \int\limits_{|x - y| > \varepsilon } {k\left( {x - y} \right)f\left( y \right)dy}$ defined on spaces of functions of several real variables. For the kernels k(x) satisfying the Hörmander condition, we establish necessary and sufficient conditions under which the operators {T _ε} are uniformly bounded from Lorentz spaces into Marcinkiewicz spaces.