On maximizers of a convolution operator in \(L_p\)-spaces. (English. Russian original) Zbl 1435.44003
Sb. Math. 210, No. 8, 1129-1147 (2019); translation from Mat. Sb. 210, No. 8, 67-86 (2019).
Summary: The paper is concerned with convolution operators in \(\mathbb R^d\), whose kernels are in \(L_q\), which act from \(L_p\) into \(L_s\), where \(1/p+1/q=1+1/s\). It is shown that for \(1<q,p,s<\infty\) there exists a maximizer (a function with \(L_p\)-norm 1) at which the supremum of the \(s\)-norm of the convolution is attained. A special analysis is carried out for the cases in which one of the exponents \(q, p\), or \(s\) is \(1\) or \(\infty\).
MSC:
| 44A35 | Convolution as an integral transform |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 49J99 | Existence theories in calculus of variations and optimal control |
Keywords:
convolution; Young inequality; existence of extremal function; tight sequence; concentration compactnessReferences:
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