Brandolini, Luca; Colzani, Leonardo Fourier transform, oscillatory multipliers and evolution equations in rearrangement invariant function spaces. (English) Zbl 0913.35014 Colloq. Math. 71, No. 2, 273-286 (1996). 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) Cited in 1 ReviewCited in 3 Documents 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 Keywords:rearrangement invariant Banach function space PDFBibTeX XMLCite \textit{L. Brandolini} and \textit{L. Colzani}, Colloq. Math. 71, No. 2, 273--286 (1996; Zbl 0913.35014) Full Text: DOI EuDML