The Fefferman–Stein Type Inequalities for Strong Fractional Maximal Operators

Abstract

We prove the Fefferman–Stein type inequalities for strong fractional maximal operators by additional compositions of certain maximal operators instead of using the strong Muckenhoupt weight. With an arbitrary weight, in $${{\mathbb {R}}}^2$$ , we establish an endpoint estimate and in $${{\mathbb {R}}}^n$$ , $$n\ge 2$$ , we give a weak (p, p) type estimate for $$p>1$$ . We also investigate the case $$p=1$$ in higher dimensions.