On the strong type multiplier norms of rational functions in several variables (Q1280722)

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.