Weighted Norm Inequalities for a Maximal Operator in Some Subspace of Amalgams

http://dx.doi.org/10.4153/CMB-2010-015-x
Canad. Math. Bull. 53(2010), 263-277

  Published:2009-12-04
 Printed: Jun 2010

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 superspaces 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.