Two Hardy-Littlewood theorems of Fourier series on classical groups. (Chinese. English summary) Zbl 0613.43006
This paper deals with Fourier series on classical groups. The spherical summability including spherical Riesz means at the critical index is considered. Let \(U_ n\) denote the unitary group of order n. In the case \(n=1\) Hardy and Littlewood established the following theorems. Theorem A. Let \(u\in L(U_ 1)\) and \(x_ 0\in {\mathbb{R}}\). If
\[
(1)\quad \lim_{t\to 0^+}(1/t)\int^{t}_{-t}[u(x_ 0+\tau)-u(x_ 0)] d\tau =0
\]
and (2) \(| a_ k| \leq C/| k|\) (k\(\neq 0)\) where \(a_ k\) (k\(\in {\mathbb{Z}})\) are Fourier coefficients of u and C is a constant independent of k, then the Fourier series of u converges to u at the point \(x_ 0.\)
Theorem B. If the conditions (1) and (2) in Theorem A are replaced respectively by \(\lim_{t\to 0}(u(x_ 0+t)-u(x_ 0))/\log t=0\) and \(| a_ k| \leq C/| k|^{\alpha}\) (k\(\neq 0)\) where \(\alpha >0\) is independent of \(k\in {\mathbb{Z}}\), then the conclusion still holds.
The authors generalize these theorems to the general case of classical groups, stating the proofs only on \(U_ n\). The Riemann principle of localization is also generalized for Riesz spherical means.
Theorem B. If the conditions (1) and (2) in Theorem A are replaced respectively by \(\lim_{t\to 0}(u(x_ 0+t)-u(x_ 0))/\log t=0\) and \(| a_ k| \leq C/| k|^{\alpha}\) (k\(\neq 0)\) where \(\alpha >0\) is independent of \(k\in {\mathbb{Z}}\), then the conclusion still holds.
The authors generalize these theorems to the general case of classical groups, stating the proofs only on \(U_ n\). The Riemann principle of localization is also generalized for Riesz spherical means.
Reviewer: Kunyang Wang
MSC:
43A50 | Convergence of Fourier series and of inverse transforms |
43A55 | Summability methods on groups, semigroups, etc. |
42B25 | Maximal functions, Littlewood-Paley theory |