\(C^ 1\) change of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers (Q1321628)

We recall that \(A_ p (R)\) is the Banach space of all distributions \(f\) with the Fourier transform \({\mathcal F} f \in L^ p(R)\); and \({\mathcal M}_ p (R)\) is the Banach space of Fourier multipliers in \(L^ p (R)\), where \(1 \leq p<\infty\). Among others, the authors prove the following. Theorem 3. If \(\varphi \in C^ 1 (R)\) is a real function and the composition operator \(f \to f \circ \varphi\) is bounded in \(A_ p (R)\) for some \(p\), \(1<p<\infty\), \(p \neq 2\), then \(\varphi\) is linear. Theorem 4. If \(\varphi \in C^ 1 (R)\) is a real nonlinear function, then for every \(p\), \(1<p<\infty\), \(p \neq 2\), we have \[ \| e^{i \lambda \varphi} \|_{{\mathcal M}_ p (R)} \to \infty \quad \text{ as } \quad | \lambda | \to \infty,\;\lambda \in R. \] In both theorems, the condition \(\varphi \in C^ 1(R)\) is essential; it cannot be replaced by the Lipschitz condition. Analogous results are proved when the torus \(T\) is substituted for \(R\). The last section of the paper is devoted to generalizations to \(R^ m\) and \(T^ m\), where \(m=2,3,\dots\).