Inequalities of Calderón-Zygmund type for trigonometric polynomials. (English) Zbl 0981.42006

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 \[ ||P_m(D)t||_p\leq Cn^{m-2\beta}||\sum|k|^{2\beta}c_ke^{ik\cdot}||_p \] 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 \(t.\)
Among 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.
Note that the paper by 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.


42B05 Fourier series and coefficients in several variables
42B15 Multipliers for harmonic analysis in several variables
42A05 Trigonometric polynomials, inequalities, extremal problems
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type


Zbl 0826.42003
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