Integral operators and weighted amalgams

For large classes of indices, we characterize the weights u, v for which the Hardy operator is bounded from ${\displaystyle {\ell }^{q̅}(L^{p̅}\_\left\{v\right\})}$ into ${\displaystyle {\ell }^q(L^p\_\left\{u\right\})}$. For more general operators of Hardy type, norm inequalities are proved which extend to weighted amalgams known estimates in weighted ${\displaystyle L^p}$-spaces. Amalgams of the form ${\displaystyle {\ell }^q(L^p\_\left\{w\right\})}$, 1 < p,q < ∞ , q ≠ p, ${\displaystyle w\in A_p}$, are also considered and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and local maximal operator in these spaces are obtained.