Boundedness of Singular Integral Operator in Variable Exponent Lebesgue Spaces

This manuscript establishes the boundedness results for a class of singular integral operators to a more general framework by imposing conditions on the Hardy–Littlewood maximal function. We establish the results where we construct a suitable range space Lt(·)(R+) for a given domain space Ls(·)(R+) such that the operator maps Ls(·)(R+) into Lt(·)(R+) under appropriate assumptions. Conversely, for a prescribed range space Lt(·)(R+), we determine a corresponding domain space Ls(·)(R+) ensuring that the operator maps Ls(·)(R+) into Lt(·)(R+). In both settings, we provide illustrative examples to explicitly construct the respective spaces. Since our approach fundamentally relies on the boundedness of the maximal operator, the main results are valid only when the essential infimum s−>1. In the limiting case s−=1, we derive weak-type boundedness results of the form (1,t(·)). Additionally, we present analogous formulations of these results in the classical setting.