Belgacem Selmi, Chahiba Khelifi
In this paper n-dimensional Sobolev type spaces $ E_α^{s,p}(\R^n_+)$ $(α\in \R^n,\;\;α_1> -\frac{1}{2},...,α_n>-\frac{1}{2}, s\in \R, p\in [1,+\infty])$ are defined on $\R^n_+$ by using Fourier-Bessel transform. Some properties including completeness and embedding results for these spaces are obtained, Poincaré's inequality and Reillich theorem are proved.
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Uncertainty principle for the Poly-axially L 2 α -multiplier operators
该研究建立了 Poly-axially L²_α 乘子算子的两类不确定性原理，即 Heisenberg-Pauli-Weyl 与 Donoho-Stark 不等式。
A unified approach to the Dirac fine structures on the $S$-spectrum and a connection with Gegenbauer polynomials
本文提出了一种统一方法研究 $S$-谱上狄拉克精细结构，并揭示了其与雅可比多项式的联系。
We study the partial sum operator for a Sobolev-type inner product related to the classical Gegenbauer polynomials. A complete characterization of the partial sum operator in an appropriate Sobolev space is given. Moreover, we analyze the convergence of the partial sum operators.
A variation of the inverse Fueter theorem and the generalized polyanalytic Cauchy-Kovalevskaya extension of order 2
本文研究二阶轴向多解析函数的广义Cauchy-Kovalevskaya延拓，建立其与Fueter定理的联系，推导Fueter映射分解的可逆性及相关积分表示公式，是超复分析的重要成果。
Let $\{q_n^{(α,β,m)}(x)\}_{n\ge 0}$ be the orthonormal polynomials respect to the Sobolev-type inner product \begin{equation*} \langle f,g\rangle_{α,β,m}=\sum_{k=0}^m \int_{-1}^{1}f^{(k)}(x)g^{(k)}(x)\, dw_{α+k,β+k}(x), \quad α,β>-1, \quad m\ge 1, \end{equation*} where $dw_{a,b}(x)=(1-x)^{a}(1+x)^b\, dx$. We obtain necessary and sufficient conditions for the uniform boundedness of the partial sum operators related to this sequence of polynomials in the Sobolev space $W_{α,β}^{p,m}$. As a consequence we deduce the convergence of such partial sums in the norm of $W_{α,β}^{p,m}$.
The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solutions of the Cauchy-Riemann operator in $ \mathbb{R}^4$, denoted by $ \mathcal{D}$. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operator in four real variables ($Δ$) to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e. $Δ= \mathcal{\overline{D}} \mathcal{D}$ to split the previ…
In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials. We also obtain some non-trivial resul…
We prove a scale of generalized $L^p$-Poincaré inequalities and Sobolev type inequalities on graphs with polynomial volume growth. They are optimal on Vicsek graphs.
In this paper, we investigate the boundedness of Toeplitz product $T_{f}T_{g}$ and Hankel product $H_{f}^{*} H_{g}$ on Fock-Sobolev space for two polynomials $f$ and $g$ in $z,\overline{z}\in\mathbb{C}^{n}$. As a result, the boundedness of Toeplitz operator $T_{f}$ and Hankel operator $H_{f}$ with the polynomial symbol $f$ in $z,\overline{z}\in\mathbb{C}^{n}$ is characterized.
Higher-order chain rules for tensor fields, generalized Bell polynomials, and estimates in Orlicz-Sobolev-Slobodeckij and bounded variation spaces
本文研究张量场高阶链式法则，利用嵌套集合划分与广义贝尔多项式推广 Faà di Bruno 公式，并给出 Sobolev-Slobodeckij、Musielak-Orlicz 范数及全变差半范数估计。
We provide an extension operator for weighted Sobolev spaces on bounded polyhedral cones $K$ involving a mixture of weights, which measure the distance to the vertex and the edges of the cone, respectively. Our results are based on Stein's extension operator for Sobolev spaces.
We survey a few trace theorems for Sobolev spaces on $N$-dimensional Euclidean domains. We include known results on linear subspaces, in particular hyperspaces, and smooth boundaries, as well as less known results for Lipschitz boundaries, including Besov's Theorem and other characterizations of traces on planar domains, polygons in particular, in the spirit of the work of P. Grisvard. Finally, we present a recent approach, originally developed by G. Auchmuty in the case of the Sobolev space $H^1(Ω)$ on a Lipschitz domain $Ω$, and which we have further developed for the trace spaces of $H^k(Ω)…
Translation Invariant Operators on Polyanalytic Sobolev-Fock Spaces
本文利用时频分析刻画多解析 Sobolev-Fock 空间上平移不变算子的积分形式，并借助伪微分算子符号类给出其有界性的充分条件。
A note that points out the possibility to have p<1 in Sobolev type of inequalities by a use of the momomial structure of polynomials or power series. The proof is simple: Triangle angle inequality p*>1, monomial estimate from p* to exponent p<1, then the inverse triangle inequality.
The Fueter-Sce mapping and the Clifford-Appell polynomials
本文利用Fueter-Sce映射与广义Cauchy-Kovalevskaya延拓的联系，显式计算其作用并刻画值域与核，定义新乘积，研究相关空间，并推广得到高维多解析单演多项式。
Contains a further development of the V.G. Maz'ya concept of polynomial capacities.A different different approach is given together with new results. The primary application is for Hardy inequalities. This is treated in a later report. Here is given an application which consists of a reformulation of the Spectral synthesis question in terms of polynomial capacities. However so far this does not give a new proof, though a similar but different question for the nonnegative cone in Sobobolev space of order 2 is resolved.
The $H^\infty$-functional calculi for the quaternionic fine structures of Dirac type
本文基于Fueter-Sce扩张定理的积分表示，针对四元数扇形算子构造了四类泛函演算，并利用相应乘积法则建立了它们的$H^\infty$-版本。
Linear completeness of trajectories in Sobolev spaces and the symmetrised polydisk
本文通过无限分析块Toeplitz算子框架，研究了Hilbert空间中非线性轨迹的线性完备性，证明了Weierstrass函数族和Gross-Pitaevskii方程的本征函数族在Sobolev空间中的线性完备性，连接了非线性分析与线性逼近理论。
We study in this article some Sobolev-type inequalities on polynomial volume growth Lie groups. We show in particular that improved Sobolev inequalities can be extended without the use of the Littlewood-Paley decomposition to this general framework.
We establish sharp Sobolev embedding properties within a broad class of compact matrix quantum groups of Kac type under the polynomial growth or the rapid decay property of their duals. Main examples are duals of polynomially growing discrete quantum groups, duals of free groups and free quantum groups $O_N^+,S_N^+$. In addition, we generalize sharpend Hausdorff-Young inequalities, compute degrees of the rapid decay property for $\widehat{O_N^+},\widehat{S_N^+}$ and prove sharpness of Hardy-Littlewood inequalities on duals of free groups.
We describe the $(p,q)$ Fock--Carleson measures for weighted Fock--Sobolev spaces in terms of the objects $(s,t)$-Berezin transforms, averaging functions, and averaging sequences on the complex space $\mathbb{C}^n$. The main results show that while these objects may have growth not faster than polynomials to induce the $(p,q)$ measures for $q\geq p$, they should be of $L^{p/(p-q)}$ integrable against a weight of polynomial growth for $q<p$. As an application, we characterize the bounded and compact weighted composition operators on the Fock--Sobolev spaces in terms of certain Berezin type inte…
We study some spectral properties of Volterra-type integral operators $V_g$ and $I_g$ with holomorphic symbol $g$ on the Fock--Sobolev spaces $\mathcal{F}_{ψ_m}^p$. We showed that $V_g$ is bounded on $\mathcal{F}_{ψ_m}^p$ if and only if $g$ is a complex polynomial of degree not exceeding two, while compactness of $V_g$ is described by degree of $g$ being not bigger than one. We also identified all those positive numbers $p$ for which the operator $V_g$ belongs to the Schatten $\mathcal{S}_p$ classes. Finally, we characterize the spectrum of $V_g$ in terms of a closed disk of radius twice the c…
On some Sobolev and Pólya-Szegö type inequalities with weights and applications
本文研究了三维空间中退化椭圆方程的边值问题，通过建立加权Sobolev空间的嵌入定理和新的Pólya-Szegö型不等式，扩展了现有结果。
In the setting of the Fock space over the complex plane, Bauer and Lee have recently characterized commutants of Toeplitz operators with radial symbols, under the assumption that symbols have at most polynomial growth at infinity. Their characterization states: If one of the symbols of two commuting Toeplitz operators is nonconstant and radial, then the other must be also radial. We extend this result to the Fock-Sobolev spaces.
Let S be a Sobolev or Orlicz-Sobolev space of functions not necessarily vanishing at the boundary of the domain. We give sufficient conditions on a nonnegative function in S in order that its spherical rearrangement ("Schwartz symmetrization") still belongs to S. These results are obtained via relative isoperimetric inequalities and somewhat generalize a well-known Polya-Szego's theorem. We also prove that the rearrangement of any function in S is locally in S.
We prove some general Sobolev-Orlicz, Nash and Faber-Krahn inequalities for positive operators of given ultracontractive norms of the spectral projectors on ]0, lambda]. For invariant operators on coverings of finite simplicial complexes this "ultracontractive spectral decay" is equivalent to von-Neumann's spectral density function. This allows in the polynomial decay case to relate the Novikov-Shubin numbers of such coverings to Sobolev inequalities on exact $\ell^2$-cochains, and to the vanishing of the torsion of the $\ell^{p,2}$-cohomology for some $p \geq 2$.
Numerical Analysis of differential equations on weighted Sobolev spaces: beyond classical orthogonal polynomials
本文为加权Sobolev空间上的微分方程数值分析奠定数学基础，通过分解主导线性分量实现基于正交多项式的求解，并提出新的Sobolev正交多项式概念，应用于Gross-Pitaevskii方程和随机共振的严格证明。
Gelfand and Kolmogorov numbers of Sobolev embeddings of weighted function spaces
本文研究了带多项式权重的Besov和Triebel-Lizorkin型加权函数空间之间Sobolev嵌入的Gelfand数和Kolmogorov数，在非极限情形下得到了精确的渐近估计。
The Dirichlet-Neumann Operator for Taylor's Cone
本文通过新坐标系和仿微分理论，分析了轴对称锥形域中的Dirichlet-Neumann算子，建立了Sobolev估计并求得形状导数等性质，为泰勒锥扰动控制奠定数学基础。
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