标题： 变量各向异性Hardy空间的傅里叶变换及其在Hardy-Littlewood不等式中的应用 显示英文标题

标题： Fourier Transform of Variable Anisotropic Hardy Spaces with Applications to Hardy-Littlewood Inequalities

摘要： 设$p(\cdot):\ \mathbb{R}^n\to(0,1]$为满足全局对数霍尔德连续条件的变量指数函数，$A$为$\mathbb{R}^n$上的一个一般扩张矩阵。 设$H_A^{p(\cdot)}(\mathbb{R}^n)$为与$A$相关的通过径向极大函数定义的变系数各向异性 Hardy 空间。 在本文中，通过已知的$H_{A}^{p(\cdot)}(\mathbb{R}^n)$的原子刻画，并建立关于各向异性变量原子的两个有用估计，作者证明了 Fourier 变换$\widehat{f}$与$f\in H_A^{p(\cdot)}(\mathbb{R}^n)$在广义函数的意义下与连续函数$F$相同，且$F$满足一个逐点不等式，该不等式包含关于$A$的阶梯函数以及$f$的 Hardy 空间范数。 作为应用，作者还得到了连续函数$F$在原点的高阶收敛性。 最后，也给出了在变系数各向异性 Hardy 空间设置下的 Hardy--Littlewood 不等式的类似结果。 所有这些结果甚至在经典的各向同性情况下也是新的。 显示英文摘要

摘要： Let $p(\cdot):\ \mathbb{R}^n\to(0,1]$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. Let $H_A^{p(\cdot)}(\mathbb{R}^n)$ be the variable anisotropic Hardy space associated with $A$ defined via the radial maximal function. In this article, via the known atomic characterization of $H_{A}^{p(\cdot)}(\mathbb{R}^n)$ and establishing two useful estimates on anisotropic variable atoms, the author shows that the Fourier transform $\widehat{f}$ of $f\in H_A^{p(\cdot)}(\mathbb{R}^n)$ coincides with a continuous function $F$ in the sense of tempered distributions, and $F$ satisfies a pointwise inequality which contains a step function with respect to $A$ as well as the Hardy space norm of $f$. As applications, the author also obtains a higher order convergence of the continuous function $F$ at the origin. Finally, an analogue of the Hardy--Littlewood inequality in the variable anisotropic Hardy space setting is also presented. All these results are new even in the classical isotropic setting.