Two weight $\Phi$-inequalities for the Hardy operator, Hardy-Littlewood maximal operator, and fractional integrals

Abstract:

Suppose $\Phi$ is an appropriate Young’s function and $w(x),v(x)$ are nonnegative locally integrable functions. Let $T$ denote one of three linear operators of special importance that map suitable functions on ${R^n}$ into functions on ${R^n}$. For the Hardy operator $T$, we study the inequality \[ \int _0^\infty {\Phi (|Tf(x)|)w(x) dx \leqslant C\int _0^\infty {\Phi (|f(x)|)v(x) dx} } \] and for the Hardy-Littlewood maximal operator or fractional integrals $T$, we discuss the inequalities \[ \int _{{R^n}} {\Phi (|T(fv)(x)|)w(x) dx \leqslant C\int _{{R^n}} {\Phi (|f(x)|)v(x) dx.} } \] In all cases we obtain the necessary and sufficient conditions.