Integral convolution operators. (English. Russian original) Zbl 0659.46034
Math. Notes 38, 554-556 (1985); translation from Mat. Zametki 38, No. 1, 74-79, 169 (1985).
Let \(E_ 1,E_ 2\) and \(E_ 3\) be symmetric function spaces on an interval [0,a]. The following problem is under investigation: under which conditions is the convolution operator continuous from \(E_ 1\times E_ 2\) into \(E_ 3?\)
The author gives some sufficient conditions for the continuity in terms of the Boyd indices \(\alpha_ E\) and \(\beta_ E\) of a space E. For example, if \(\beta_{E_ 3}+\beta_{E_ 1}<1+\alpha_{E_ 2}\) and \(\alpha_{E_ 1}+\beta_{E_ 3}>1\), then the convolution operator maps \(E_ 1\times E_ 2\) continuously into \(E_ 3\).
The author gives some sufficient conditions for the continuity in terms of the Boyd indices \(\alpha_ E\) and \(\beta_ E\) of a space E. For example, if \(\beta_{E_ 3}+\beta_{E_ 1}<1+\alpha_{E_ 2}\) and \(\alpha_{E_ 1}+\beta_{E_ 3}>1\), then the convolution operator maps \(E_ 1\times E_ 2\) continuously into \(E_ 3\).
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 47Gxx | Integral, integro-differential, and pseudodifferential operators |
| 44A35 | Convolution as an integral transform |
Keywords:
symmetric function spaces on an interval; convolution operator; sufficient conditions for the continuity in terms of the Boyd indicesReferences:
| [1] | L. Hörmander, Estimates for Translation-Invariant Operators [Russian translation], IL, Moscow (1962). |
| [2] | S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978). |
| [3] | E. A. Pavlov, ?On convolution operator in symmetric spaces,? Usp. Mat. Nauk,31, No. 1(187), 257-258 (1976). · Zbl 0331.44006 |
| [4] | R. O’Neil, ?Convolution operators and L(p, q) spaces,? Duke Math. J.,30, 129-142 (1965). · Zbl 0178.47701 · doi:10.1215/S0012-7094-63-03015-1 |
| [5] | E. A. Pavlov, ?On boundedness of convolution operators in symmetric spaces,? Izv. Vyssh. Uchebn. Zaved., Mat., No. 2(237), 36-40 (1982). · Zbl 0496.47044 |
| [6] | E. A. Pavlov, ?Some properties of the Hardy-Littlewood operator,? Mat. Zametki,26, No. 6, 909-912 (1979). |
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