Inequalities of Calderón-Zygmund type for trigonometric polynomials (Q2726704)

The main result of the paper reads as follows. Let \(P_m\) be a \(d\)-dimensional homogeneous polynomial of order \(m\) and \(t=\sum c_k e^{ikx}\) be a trigonometric polynomial spanned by harmonics \(e^{ikx}\) with \(|k|^2=k_1^2+...+k_d^2\leq n^2.\) Let \(P_m(D)\) be a differential operator defined by \(P_m.\) If \(P_m(x)(x_1^2+...+ x_d^2)^{-\beta}\) is not identical on \(\mathbf R^d\setminus\{0\}\) with a polynomial, then the inequality NEWLINE\[NEWLINE||P_m(D)t||_p\leq Cn^{m-2\beta}||\sum|k|^{2\beta}c_ke^{ik\cdot}||_pNEWLINE\]NEWLINE is valid if and only if \({d\over d+m-2\beta}<p\leq+\infty\) for \(m>2\beta\) and if and only if \(1<p<+\infty\) for \(m=2\beta.\) If \(\beta=0,\) the inequality is valid for all \(0<p\leq+\infty.\) The constant \(C\) is independent of \(t.\) NEWLINENEWLINENEWLINEAmong special cases are such known inequalities as those of Calderón-Zygmund and of Bernstein. The proof is based on estimates of the Fourier transform of certain functions. The result itself is applied to the sharpness problem of the image of the Fourier transform.NEWLINENEWLINENEWLINENote that the paper by \textit{E. Belinskij} and the reviewer [``Approximation properties in \(L_p,\) \(0<p<1\)'', Funct. Approximatio, Comment. Math. 22, 189-199 (1993; Zbl 0826.42003)], not mentioned in the list of references, is related to the paper under review both by results and by method.