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Fourier transform, oscillatory multipliers and evolution equations in rearrangement invariant function spaces. (English) Zbl 0913.35014

The authors study the properties of operators defined via Fourier transform. Starting from the property that the only rearrangement invariant Banach function space on which the Fourier transform is bounded is the Hilbert space \(L^2(\mathbb R^N)\), they prove an analogous result for an operator \(T\) defined by \(T\widehat{\phi}(\xi)=\exp(itP(\xi))\widehat\phi(\xi)\), where \(P(\xi)\) is a real polynomial in \(N\) variables of degree strictly greater than one and, as usual, \(\widehat\phi(\xi)\) denotes the Fourier transform of \(\phi\). It is shown that the only rearrangement invariant Banach function space on which, for a fixed \(t\), the operator \(T\) is bounded is the space \(L^2(\mathbb R^N)\). Similar results are obtained for operators related to wave equation.
Reviewer: V.Ferone (Napoli)

MSC:

35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
42B15 Multipliers for harmonic analysis in several variables
35L05 Wave equation
35A22 Transform methods (e.g., integral transforms) applied to PDEs
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