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Characterizations are obtained for those pairs of weight functions u and v for which the operators $Tf\left(x\right)={ʃ}_{a\left(x\right)}^{b\left(x\right)}f\left(t\right)dt$ with a and b certain non-negative functions are bounded from $L_u^p(0,\infty )$ to $L_v^q(0,\infty )$, 0 < p,q < ∞, p≥ 1. Sufficient conditions are given for T to be bounded on the cones of monotone functions. The results are applied to give a weighted inequality comparing differences and derivatives as well as a weight characterization for the Steklov operator.

The object of this note is to generalize some Fourier inequalities.

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