Conditions on the nonnegative weight functions $u(x)$ and $v(x)$ are given which ensure that an inequality of the form (∫ | (Tf)(x)u(x) |q dx )^1 /q ≤ C(∫ | f(x)v(x) |ᵖ dx )^1 /p holds where T is an integral operator of the form ∫- ∞ˣ K (x,y)f(y)dy or ∫ₓ^∞ K (x,y)f(y)dy and C is a constant depending on $K,p,q$ but independent of f; the inequality being reversed in case $p,q < 1$. In particular, new inequalities and a unified treatment of several known inequalities are obtained for a class of convolution operators, various fractional integrals and the Laplace transform.
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Andersen et al. (1983) studied this question.