Weighted norm inequalities for singular and fractional integrals

Abstract

Inequalities of the form ‖ | x | α T f ‖ q ≦ C ‖ | x | α f ‖ p {\left \| {{{\left | x \right |}^\alpha }Tf} \right \|_q} \leqq C{\left \| {{{\left | x \right |}^\alpha }f} \right \|_p} are proved for certain well-known integral transforms, T, in E n {E^n} . The transforms considered include Calderón-Zygmund singular integrals, singular integrals with variable kernel, fractional integrals and fractional integrals with variable kernel.