Weighted norm inequalities for integral transforms with splitting kernels. (English) Zbl 07900720
Summary: We obtain necessary and sufficient conditions on weights for a wide class of integral transforms to be bounded between weighted \(L^p\)-\(L^q\) spaces, with \(1\leq p\leq q\leq \infty\). The kernels \(K(x,y)\) of such transforms are only assumed to satisfy upper bounds given by products of two functions, one in each variable.
The obtained results are applicable to a number of transforms, some of which are included here as particular examples. Some of the new results derived here are the characterization of weights for the boundedness of the \(\mathscr{H}_\alpha\) (or Struve) transform in the case \(\alpha>\frac{1}{2}\), or the characterization of power weights for which the Laplace transform is bounded in the limiting cases \(p=1\) or \(q=\infty\).
The obtained results are applicable to a number of transforms, some of which are included here as particular examples. Some of the new results derived here are the characterization of weights for the boundedness of the \(\mathscr{H}_\alpha\) (or Struve) transform in the case \(\alpha>\frac{1}{2}\), or the characterization of power weights for which the Laplace transform is bounded in the limiting cases \(p=1\) or \(q=\infty\).
MSC:
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 26D15 | Inequalities for sums, series and integrals |
Keywords:
weighted norm inequalities; weighted Lebesgue spaces; integral operators; integral transformsReferences:
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