On differentiation of integrals in the infinite-dimensional torus

A decomposition of Calderón--Zygmund type and some observations on differentiation of integrals on the infinite-dimensional torus

In this note we will show a Calderón--Zygmund decomposition associated with a function $f\in L^1(\mathbb{T}^ω)$. The idea relies on an adaptation of a more general result by J. L. Rubio de Francia in the setting of locally compact groups. Some related results about differentiation of integrals on the infinite-dimensional torus are also discussed.

Weak-type maximal function estimates on the infinite-dimensional torus

We prove necessary and sufficient conditions for the weak-$L^p$ boundedness, for $p \in (1,\infty)$, of a maximal operator on the infinite-dimensional torus. In the endpoint case $p=1$ we obtain the same weak-type inequality enjoyed by the strong maximal function in dimension two. Our results are quantitatively sharp.

On the absolute divergence of Fourier series in the infinite dimensional torus

We present some simple counterexamples, based on quadratic forms in infinitely many variables, showing that the implication $f\in C^{(\infty}(\mathbb{T}^ω)\Longrightarrow\sum_{\bar{p}\in\mathbb{Z}^\infty}|\widehat{f}(\bar{p})|<\infty$ is false. There are functions of the class $C^{(\infty}(\mathbb{T}^ω)$ (depending on an infinite number of variables) whose Fourier series diverges absolutely. This fact establishes a significant difference from the finite dimensional case.

第2部分。与 Whittaker 函数 $\mathrm{W}_{κ,μ}\left(x\right)$ 关于参数微分相关的无穷级数和对数积分

Part 2. Infinite series and logarithmic integrals associated to differentiation with respect to parameters of the Whittaker $\mathrm{W}_{κ,μ}\left( x\right) $ function

研究 Whittaker 函数关于参数的一阶导数，给出其无穷级数与对数积分表示及特定参数下的闭合形式，并推导积分 Whittaker 函数的表示和约化公式。

积分 Mittag-Leffler 函数与积分 Wright 函数关于参数的微分

Differentiation of integral Mittag-Leffler and integral Wright functions with respect to parameters

本文研究积分 Mittag-Leffler 函数与积分 Wright 函数关于参数的导数，将其表示为 digamma 与 gamma 函数之商的无穷和，并推导约化公式验证和补充 MATHEMATICA 的计算结果。

第1部分。与 Whittaker 函数 $\mathrm{M}_{κ,μ}(x)$ 关于参数微分相关的无穷级数和对数积分

Part 1. Infinite series and logarithmic integrals associated to differentiation with respect to parameters of the Whittaker $\mathrm{M}_{κ,μ}\left( x\right) $ function

研究 Whittaker 函数 $\mathrm{M}_{κ,μ}(x)$ 的参数一阶导数，给出其无穷级数与有限、无穷对数积分表示，并推导相关闭式约化公式。