Weighted norms inequalities for a maximal operator in some subspace of amalgams

Abstract

We give weighted norm inequalities for the maximal fractional operator ${\mathcal{M}}_{q,\beta }$ of Hardy-Littlewood and the fractional integral $I_{\gamma }$. These inequalities are established between $(L^q,L^p{)}^{\alpha }(X,d,\mu )$ spaces (which are super spaces of Lebesgue spaces $L^{\alpha }(X,d,\mu )$, and subspaces of amalgams $(L^q,L^p)(X,d,\mu )$) and in the setting of space of homogeneous type $(X,d,\mu )$. The conditions on the weights are stated in terms of Orlicz norm.

17 pages, accepted for publication