Fourier integral inequalities and applications (Q1065306)

Let T denote one-dimensional Hardy operator, Laplace transform and Fourier transform. The question considered by the author is what are necessary and sufficient conditions on non-negative weight functions u and v such that \[ (\int_{\Omega}u(x)| T(f)(x)|^ qdx)^{1/q}\leq C(\int_{\Omega}v(x)| f(x)|^ pdx)\quad^{1/p}, \] where \(1\leq p,q\leq \infty\), \(\Omega\) \(\subset {\mathbb{R}}\) and C denotes a constant independent of f. The author states the results which have been obtained by himself and by some other authors in the references. These results include the solution of the above question for the Hardy operator in the case \(p=q\geq 1\) and the principal but partial results for the Laplace and Fourier transform. The author proves a lemma by which the sufficiency part of the result for the Hardy operator in the case \(p=q\) extends easily to the case \(p<q\), and for Laplace transform the case \(p<q\) may also follow from the \(p=q\) case. Meanwhile some applications are given.