Simplified Inequalities for Fractional Wavelet Frames on L^2(R)

Norm related inequalities for fractional integrals

Fractional spline wavelet systems are considered in the work. Molecular structure of elements of such systems admits estimates connecting norms of fractional integrals' images and pre--images in Besov spaces.

Wavelet characterization of local Muckenhoupt weighted Lebesgue spaces with variable exponent

Our aim in this paper is to characterize local Muckenhoupt weighted Lebesgue spaces with variable exponent by compactly supported smooth wavelets. We also investigate necessary and sufficient conditions for the corresponding modular inequalities to hold. One big achievement is that the weights with exponetial growth can be handled in the framework of variable exponents.

外推法在各类函数空间小波刻画及延拓算子中的应用

Applications of extrapolations to wavelet characterization of various function spaces and extension operators

本文应用不依赖凸化的外推方法，通过小波刻画了考虑光滑性的球 Banach 函数空间，证明向量值不等式，并改进了 Zhu、Yang 和 Yuan 的延拓算子结果。

Hardy-Littlewood inequalities for norms of positive operators on sequence spaces

We consider estimates of Hardy and Littlewood for norms of operators on sequence spaces, and we apply a factorization result of Maurey to obtain improved estimates and simplified proofs for the special case of a positive operator.

Greedy bases in variable Lebesgue spaces

We compute the right and left democracy functions of admissible wavelet bases in variable Lebesgue spaces defined on $R^n$. As an application we give Lebesgue type inequalities for these wavelet bases. We also show that our techniques can be easily modified to prove analogous results for weighted variable Lebesgue spaces and variable exponent Triebel-Lizorkin spaces.

集中不等式的一致稳定性及其应用

Uniform stability of concentration inequalities and applications

本文针对一类柯西小波窗的连续小波变换证明了参数一致且双向渐近精确的 Faber-Krahn 型集中不等式，并由此恢复短时傅里叶变换的已知结果，同时给出哈代空间中泊松延拓最优集中性的新结果及其稳定性版本。

Restricted non-linear approximation in sequence spaces and applications to wavelet bases and interpolation

Restricted non-linear approximation is a type of N-term approximation where a measure $ν$ on the index set (rather than the counting measure) is used to control the number of terms in the approximation. We show that embeddings for restricted non-linear approximation spaces in terms of weighted Lorentz sequence spaces are equivalent to Jackson and Bernstein type inequalities, and also to the upper and lower Temlyakov property. As applications we obtain results for wavelet bases in Triebel-Lizorkin spaces by showing the Temlyakow property in this setting. Moreover, new interpolation results for …

A general wavelet-based profile decomposition in the critical embedding of function spaces

We characterize the lack of compactness in the critical embedding of functions spaces $X\subset Y$ having similar scaling properties in the following terms : a sequence $(u_n)_{n\geq 0}$ bounded in $X$ has a subsequence that can be expressed as a finite sum of translations and dilations of functions $(ϕ_l)_{l>0}$ such that the remainder converges to zero in $Y$ as the number of functions in the sum and $n$ tend to $+\infty$. Such a decomposition was established by Gérard for the embedding of the homogeneous Sobolev space $X=\dot H^s$ into the $Y=L^p$ in $d$ dimensions with $0<s=d/2-d/p$, and t…

关于混合正则性的 Sobolev 和 Besov 空间

On Sobolev and Besov Spaces of Hybrid Regularity

本文研究单位 n-立方体上 Sobolev 空间 H^q 中函数关于张量积或双曲小波基的非线性逼近，推导 Jackson 和 Bernstein 不等式，证明逼近类包含混合正则性 Besov 空间，并表明张量积小波逼近能力不弱于经典小波。

Fourier型积分变换及其衍生变换的数学综述：窗口Fourier变换、小波变换与Stockwell变换

A mathematical survey on Fourier type integral transform and their offshoots: windowed Fourier transform, wavelet transform and Stockwell transform

本文综述了Fourier型积分变换（如FrFT、LCT、QPFT）及其衍生变换的卷积性质、不确定性原理和多分辨率分析，并详细给出了四元数情形下的对应不等式。

A simplified and unified generalization of some majorization results

We consider positive, integral-preserving linear operators acting on $L^1$ space, known as stochastic operators or Markov operators. We show that, on finite-dimensional spaces, any stochastic operator can be approximated by a sequence of stochastic integral operators (such operators arise naturally when considering matrix majorization in $L^1$). We collect a number of results for vector-valued functions on $L^1$, simplifying some proofs found in the literature. In particular, matrix majorization and multivariate majorization are related in $\mathbb{R}^n$. In $\mathbb{R}$, these are also equiva…

Universal bounds for the Hardy--Littlewood inequalities on multilinear forms

The Hardy--Littlewood inequalities for multilinear forms on sequence spaces state that for all positive integers $m,n\geq2$ and all $m$-linear forms $T:\ell_{p_{1}}^{n}\times\cdots\times\ell_{p_{m}}^{n}\rightarrow\mathbb{K}$ ($\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$) there are constants $C_{m}\geq1$ (not depending on $n$) such that \[ \left( \sum_{j_{1},\ldots,j_{m}=1}^{n}\left\vert T(e_{j_{1}},\ldots,e_{j_{m}})\right\vert ^ρ\right) ^{\frac{1}ρ}\leq C_{m}\sup_{\left\Vert x_{1}\right\Vert ,\dots,\left\Vert x_{m}\right\Vert \leq 1}\left\vert T(x_{1},\dots,x_{m})\right\vert, \] where $ρ=\frac{2m}{…

矩阵加权的Poincaré型不等式及其在Logarithmic Hajłasz--Besov空间上的应用

Matrix-Weighted Poincaré-Type Inequalities with Applications to Logarithmic Hajłasz--Besov Spaces on Spaces of Homogeneous Type

本文通过矩阵加权Poincaré型不等式和小波再现公式，建立了同型空间上新的Logarithmic Hajłasz--Besov空间的点wise刻画，摆脱了反向倍增条件的依赖。

带应用的精确加权Cohen--Dahmen--Daubechies--DeVore不等式：（加权）临界Sobolev空间、Gagliardo--Nirenberg不等式与Muckenhoupt权

Sharp Weighted Cohen--Dahmen--Daubechies--DeVore Inequality with Applications to (Weighted) Critical Sobolev Spaces, Gagliardo--Nirenberg Inequalities, and Muckenhoupt Weights

本文建立了Cohen等人2003年提出的不等式的精确加权变体，通过发展好立方体方法追踪对A_p权常数的精确依赖，得到加权Sobolev空间刻画、插值及Gagliardo-Nirenberg不等式等应用，还给出了Muckenhoupt权的新刻画并解决了一个公开问题。

离散图上拟线性算子的Hardy权空间

The space of Hardy-weights for quasilinear operators on discrete graphs

本文在一般加权图上研究p-Schrödinger算子的Hardy不等式，通过广义容量给出Hardy权空间的Maz'ya型刻画，并利用临界性理论探讨极小化元存在性及其与Cheeger型估计的联系。

与Sierpinski垫片上Dirichlet型相关的抛物型方程

Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket

本文首次研究Sierpinski垫片上的非线性抛物方程，建立了含两个相互奇异测度$L^p$范数的新型Sobolev不等式，推导了解的时空正则性，并得到了垫片上Burgers方程解的最大值原理、存在唯一性及正则性。