Mikhlin-H“Ormander型Fourier乘子的极大函数

Weighted Estimates for Operator-Valued Fourier Multipliers

We establish Littlewood-Paley decompositions for Muckenhoupt weights in the setting of UMD spaces. As a consequence we obtain two-weight variants of the Mikhlin multiplier theorem for operator-valued multipliers. We also show two-weight estimates for multipliers satisfying Hörmander type conditions.

由具有分数维膨胀的 Fourier 乘子给定的极大算子

Maximal operators given by Fourier multipliers with dilation of fractional dimensions

本文研究了分数维膨胀下极大 Fourier 乘子算子的 $L^p$ 界，提出了基于膨胀集维数的准则，确保对任意 $p$ 均有 $L^p$ 界，并涵盖 Mikhlin 型、有限衰减与慢衰减乘子。

Fourier multiplier theorems for Triebel-Lizorkin spaces

In this paper we study sharp generalizations of $\dot{F}_p^{0,q}$ multiplier theorem of Mikhlin-Hörmander type. The class of multipliers that we consider involves Herz spaces $K_u^{s,t}$. Plancherel's theorem proves $\widehat{L_s^2}=K_2^{s,2}$ and we study the optimal triple $(u,t,s)$ for which $\sup_{k\in\mathbb{Z}}{\big\Vert \big( m(2^k\cdot)φ\big)^{\vee}\big\Vert_{K_u^{s,t}}}<\infty$ implies $\dot{F}_p^{0,q}$ boundedness of multiplier operator $T_m$ where $φ$ is a cutoff function. Our result also covers the $BMO$-type space $\dot{F}_{\infty}^{0,q}$.

Smooth Fourier multipliers in group algebras via Sobolev dimension

We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new Hörmander-Mikhlin criteria for spectral and non-spectral multipliers. The key novelties which shape our approach are three. First, we control a broad class of Fourier multipliers by certain maximal operators in noncommutative $L_p$ spaces. This general principle ---exploited in Euclidean harmonic analysis during the last 40 years--- is of independent interest and might admit further applications. Second, we replace the formerly used cocycle dimensio…

Generic Multilinear Multipliers Associated to Degenerate Simplexes

For each $1 \leq p \leq \infty$, let $W_{p}(\mathbb{R}) = \left\{ f \in L^p(\mathbb{R}): \hat{f} \in L^{p^\prime}(\mathbb{R}) \right\}$ with norm $||f||_{W_{p}(\mathbb{R})} = ||\hat{f}||_{L^{p^\prime}(\mathbb{R})}$. Moreover, let $ Γ= \left\{ ξ_1 + ξ_2 =0\right\} \subset \mathbb{R}^2$ and $a_1,a_2 : \mathbb{R}^2 \rightarrow \mathbb{C}$ satisfy the Hörmander-Mikhlin condition \begin{eqnarray*} \left| \partial^{\vecα} a_j \left(\vecξ\right) \right| \lesssim_{\vecα} \frac{1}{dist(\vecξ, Γ)^{|\vecα|}}~~~\forall \vecξ \in \mathbb{R}^2, j \in \{1, 2\} \end{eqnarray*} for sufficiently many multi-indi…

关于畸变傅里叶变换的Littlewood-Paley理论的一点注记

A remark on Littlewood-Paley theory for the distorted Fourier transform

我们证明了通常的Mikhlin乘子定理相对于扭曲的傅里叶变换至少对径向函数和势在（3/2，3）范围内成立。限制的范围是由于引起畸变傅里叶变换的薛定谔算子可能具有零能量共振的事实。因此，Littlewood-Paley定理也仅在（3/2，3）范围内得到证明。

近期小沃德-帕利理论在啁啾函数中的进展

Recent progress of Littlewood-paley Theory with chirp function

本文基于啁啾函数发展了分数傅里叶变换下的小沃德-帕利框架，提出了多种估计和理论，包括乘法恒等式、平方函数估计、Marcinkiewicz和Mihlin-Hörmander乘法结果等。

关于二阶差商函数 Schur 乘子的最佳常数

On the best constants of Schur multipliers of second order divided difference functions

本文基于双线性转移和Hörmander-Mikhlin-Schur乘子定理，给出了二阶差商函数双线性Schur乘子有界性的新证明，显著改进了其渐近界的上下界并证明了其在特定条件下的最优性。