On the strong type multiplier norms of rational functions in several variables
type: Item
instance of: scholarly article
title: On the strong type multiplier norms of rational functions in several variables
zbMATH Open document ID: 0933.42005
author: Michal Wojciechowski
published in: Illinois Journal of Mathematics
publication date: 1999-04-22T00:00:00Z
review text: Let \(\phi\) be a function whose gradient tends to zero at infinity, but which itself does not have a limit at infinity. Specifically, assume that there exists \(a \neq b\) and sequences \(n_j, k_j \to \infty\) such that \(\phi(x \pm k_j) \to a\) and \(\phi(x \pm n_j) \to b\) as \(j \to \infty\) for all \(x\). Then the author shows that the Fourier multiplier corresponding to \(\phi\) is unbounded at \(L^1\) and \(L^\infty\), and moreover the \(L^p\) norm must grow by at least \(\max(p,p')\). The same result is obtained if we replace the above limiting conditions by \(\phi(x + k_j) \to a\), \(\phi(x - k_j) \to b\). The proof uses a Riesz product construction. As a corollary, one can show that a multiplier which is a rational function (a quotient of two multipliers) is either a Fourier transform of an integrable measure (and hence the multiplier operator is bounded on all \(L^p\)), or else the \(L^p\) norm of the operator grows by at least \(\max(p,p')\), providing that the denominator obeys a (mild) ellipticity condition. Applications are discussed for the canonical projection of the jet representation of a general anisotropic Sobolev space.
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 60G46
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1262677
zbMATH Keywords: \(L^p\) multipliers
zbMATH Keywords: rational functions
zbMATH Keywords: multiplier norms
zbMATH Keywords: Riesz product
zbMATH Keywords: Sobolev space
reviewed by: Terence Tao
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1262677