Peng Gao
We give a condition on weighted mean matrices so that their $l^p$ norms are determined on decreasing sequences when the condition is satisfied. We apply our result to give a proof of a conjecture of Bennett and discuss some related results.
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Weighted Hardy inequalities involving supremum for decreasing sequences
本文对所有正参数，完全刻画了限制在非增序列锥上的、涉及上确界算子的加权 Hardy 不等式，并将其等价转化为非负序列锥上的不等式，从而扩展了分析框架和证明范围。
Range decreasing group homomorphisms and weighted composition operators
本文提出基于值域递减群同态的方法，给出了李群丛光滑截面空间间群同态成为加权复合算子的充要条件，并推广了 Shanks-Pursell 定理。
We characterize the (essentially) decreasing sequences of positive numbers $β$ = ($β$ n) for which all composition operators on H 2 ($β$) are bounded, where H 2 ($β$) is the space of analytic functions f in the unit disk such that $\infty$ n=0 |c n | 2 $β$ n < $\infty$ if f (z) = $\infty$ n=0 c n z n. We also give conditions for the boundedness when $β$ is not assumed essentially decreasing.
Let $A=(a_{j,k})_{j,k \ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\|A\|_{E, F}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences.
Recently, in [Bor4], Bor proved a main theorem dealing with $|\bar{N}, p_{n}|_{k}$ summability factors of infinite series. In the present paper, we have generalized that theorem for $|A, p_{n}|_{k}$ summability method by taking normal matrices in place of weighted mean matrices.
It is a well-known fact in K-theory that the rapidly decreasing matrices of countable size form an associative topological algebra whose set of quasi-invertible elements is open, and such that the quasi-inversion map is continuous. We generalize these conclusions to further algebras of weighted matrices with entries in a Banach algebra.
In this paper, we develop a theory of symmetrization on the one dimensional integer lattice. More precisely, we associate a radially decreasing function $u^*$ with a function $u$ defined on the integers and prove the corresponding Polya-Szegö inequality. Along the way we also prove the weighted Polya-Szegö inequality for the decreasing rearrangement on the half-line, i.e., non-negative integers. As a consequence, we prove the discrete weighted Hardy's inequality with the weight $n^α$ for $1 < α\leq 2$.
We present some results concerning the $l^p$ norms of weighted mean matrices. These results can be regarded as analogues to a result of Bennett concerning weighted Carleman's inequalities.
We give a proof of Cartlidge's result on the $l^{p}$ operator norms of weighted mean matrices for $p=2$ on interpreting the norms as eigenvalues of certain matrices.
In this paper, we study main properties of cone normed spaces, and prove some theorems of weighted means in cone normed spaces.
We study $l^{p}$ operator norms of weighted mean matrices using the approaches of Kaluza-Szegö and Redheffer. As an application, we prove a conjecture of Bennett.
We give another proof of a result of Bennett on the $l^{p}$ operator norms of some weighted mean matrices for the case $p=2$ and we also present some related results.
The purpose of this paper is to establish the weighted norm inequalities of one-sided oscillatory integral operators by the aid of interpolation of operators with change of measures.
We study extension operators on Sobolev spaces with decreasing integrability on the base of set functions associated with the operator norms. Sharp necessary conditions in the terms of the generalized density condition and the terms of weak equivalence of Euclidean and intrinsic metrics are given.
In this paper, weighted norm inequalities with $A_p$ weights are established for the multilinear singular integral operators whose kernels satisfy $L^{r'}$-Hörmander regularity condition. As applications, we recover a weighted estimate for the multilinear Fourier multiplier obtained by Fujita and Tomita, and obtain several new weighted estimates for the multilinear Fourier multiplier as well.
We set up some weighted norm inequalities for fractional oscillatory integral operators. As applications, the corresponding results for commutators formed by $BMO(\mathbb{R}^{n})$ functions and the operators are established.
We give different types of new characterizations for the boundedness and essential norms of generalized weighted composition operators between Zygmund type spaces. Consequently, we obtain new characterizations for the compactness of such operators.
We provide an estimate for the essential norm of a weighted composition operator $W_{ψ,φ}\colon f\mapsto ψ(f\circφ)$ acting on the space $BMOA$ in terms of the weight function $ψ$ and the $n$-th power $φ^n$ of the analytic self-map $φ$ of the open unit disc $\mathbb{D}$. We also provide a new estimate for the norm of the weighted composition operator on $BMOA$.
Let $μ_p(A,B,t)=(tA^p+(1-t)B^p)^{1/p}$ denote the weighted power mean between positive operators $A$ and $B$. We show that the function $t\to \|A-μ_p(A,B,t)\|_2$ is monotonically decreasing whenever $1/2 \leq p \leq 1$. Hence showing that the weighted power means satisfy Audenaert's "in-betweenness" property for positive operators for power satisfying $1/2 \leq p \leq 1$. We also show that when $p>2$ there exist operators for which the weighted power mean does not satisfy this "in-betweenness" property with respect to the Euclidean metric.
Let $A=(a_{j,k})_{j,k \ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\|A\|_{\ell_p,\ell_q}$ are determined by their actions on non-negative decreasing sequences, where one of $p$ and $q$ is 1 or $\infty$. The conditions forcing on $A$ are sufficient and they are also necessary for non-negative finite matrices.
Let $L=-Δ+V$ be a Schrödinger operator, where $Δ$ is the Laplacian operator on $\rz$, while the nonnegative potential $V$ belongs to certain reverse Hölder class. In this paper, we establish some weighted norm inequalities for commutators of Littlewood-Paley functions related to Schrödinger operators.
On the norms of the multiplication operators between weighted Bergman spaces
本文研究乘法算子在加权伯格曼空间间的范数，推导了新的范数估计并探讨了伯恩斯坦猜想与单值函数的Schwarzian导数诱导的乘法算子范数之间的联系。
We obtain weighted $L^p$ inequalities for pseudo-differential operators with smooth symbols and their commutators by using a class of new weight functions which include Muckenhoupt weight functions. Our results improve essentially some well-known results.
In this paper, we establish a good-$\lz$ inequality with two parameters in the Schrödinger settings. As it's applications, we obtain weighted estimates for spectral multipliers and Littlewood-Paley operators and their commutators in the Schrödinger settings.
Let $L=-Δ+V$ be a Schrödinger operator, where $Δ$ is the Laplacian operator on $\rz$, while nonnegative potential $V$ belongs to the reverse Hölder class. In this paper, we establish the weighted norm inequalities for some Schrödinger type operators, which include Riesz transforms and fractional integrals and their commutators. These results generalize substantially some well-known results.
We study holomorphic functions attaining weighted norms and its connections with the classical theory of norm attaining holomorphic functions. We prove that there are polynomials on $\ell_p$ which attain their weighted but not their supremum norm and viceversa. Nevertheless, we also prove that in the context of polynomials of fixed degree both norms are in fact equivalent. This leads us to the main problem of the paper, namely, whether the holomorphic functions attaining their weighted norm are dense. Although we exhibit an example where this does not hold, as the main theorem of our paper, we…
In this article, we estimate the essential norm of weighted composition operator $W_{u, φ}$, acting on the logarithmic Bloch space $\mathcal{B}^{v_{\log}}$, in terms of the $n$-power of the analytic function $φ$ and the norm of the $n$-power of the identity function. Also, we estimate the essential norm of the weighted composition operator from $\mathcal{B}^{v_{\log}}$ into the growth space $H_{v_{\log}}^{\infty}$. As a consequence of our result, we estimate the essential norm of the composition operator $C_φ$ acting on the Logarithmic-Zygmund space.
We extend the classical Hardy-Sobolev-Poincare-Wirtinger inequalities from the ordinary Lebesgue-Riesz spaces into the Grand Lebesgue ones, with exact constants evaluation.
Global and local weighted Gagliardo-Nirenberg inequalities with doubling measures are established. These inequalities are key ingredients for the regularity theory and existence of strong solutions for strongly coupled parabolic and elliptic systems which are degenerate or singular because of the unboundedness of dependent and independent variables.
When \ph\ is an analytic self-map of the unit disk with Denjoy-Wolff point $a \in \D$, and $ρ(\W) = ψ(a)$, we give an exact characterization for when \W\ is normaloid. We also determine the spectral radius, essential spectral radius, and essential norm for a class of non-compact composition operators whose symbols have Denjoy-Wolff point in \D. When the Denjoy-Wolff point is on $\partial \D$, we give sufficient conditions for several new classes of normaloid weighted composition operators.
We obtain sharp inequalities for the k-plane transform, the "j-plane to k-plane" transform, and the corresponding dual transforms, acting on $L^p$ spaces with a radial power weight. The operator norms are explicitly evaluated. Some generalizations and open problems are discussed.
Let $ω$ be a weight on a right cancellative semigroup $S$. Let $\|\cdot\|_ω$ be the weighted norm on the weighted discrete semigroup algebra $\ell^1(S, ω)$. In this paper, we prove that the weight $ω$ satisfies F-property if and only if the operator norm $\| \cdot \|_{ωop}$ of $\| \cdot \|_ω$ is exactly equal to another weighted norm $\| \cdot \|_{\widetildeω_1}$ [Theorem 2.5 ($iii$)]. Though its proof is elementary, the result is unexpectedly surprising. In particular, $\| \cdot \|_{1 op}$ is same as $\| \cdot \|_1$ on $\ell^1(S)$. Moreover, various examples are discussed to understand the re…
Some Classes of Absolutely Norm Attaining Weighted Shift operators on Directed Graphs
研究有向图上绝对范数可达的拟$\ast$-超正规加权移位算子，通过举例揭示其相关性质，还对有向图上的加权复合算子和兰伯特算子展开研究。
Two-side estimates for two-weighted discrete Hardy-type operators on a tree are obtained. For general weights we prove the discrete analogue of Evans - Harris - Pick theorem (it is a quite simple consequence from their result). It gives the order estimate for the norm of the weighted summation operator, but this estimate is rather complicated. Under some conditions on weights, we obtain estimates which are more simple and convenient for applications.
We study the $l^p$ norms of a class of weighted mean matrices whose diagonal terms are given by $n^α/\sum^{n}_{i=1}i^α$ with $α> -1$. The $l^p$ norms of such matrices are known for $p \geq 2, (α+1)p >1$ and $1<p \leq 4/3, 1/p \leq α\leq 1$. In this paper, we determine the $l^p$ norms of such matrices for $p \geq 1.35, 0\leq α\leq 1$.
We complete the different cases remaining in the estimation of the essential norm of a weighted composition operator acting between the Hardy spaces $H^p$ and $H^q$ for $1\leq p,q\leq\infty.$ In particular we give some estimates for the cases $1=p\leq q\leq\infty$ and $1\leq q<p\leq\infty.$
In this paper we prove weighted norm inequalities for Weyl multipliers satisfying Mauceri's condition. As applications of this we obtain some estimates for $L^p$ multipliers on the Heisenberg group and also show in the context of a theorem of Weis on operator valued Fourier multipliers that the R-boundedness of the derivative of the multiplier is not necessary for the boundedness of the multiplier transform.
Two-weight norm estimates for the double Hardy transforms and strong fractional maximal functions are established in variable exponent Lebesgue spaces. Derived conditions are simultaneously necessary and sufficient in the case when the exponent of the right-hand side space is constant.
We prove that the norm of a weighted composition operator on the Hardy space H^2 of the disk is controlled by the norm of the weight function in the de Branges-Rovnyak space associated to the symbol of the composition operator. As a corollary we obtain a new proof of the boundedness of composition operators on H^2, and recover the standard upper bound for the norm. Similar arguments apply to weighted Bergman spaces. We also show that the positivity of a generalized de Branges-Rovnyak kernel is sufficient for the boundedness of a given composition operator on the standard functions spaces on th…
In this paper we present some inequalities involving operator decreasing functions and operator means. These inequalities provide some reverses of operator Aczél inequality dealing with the weighted geometric mean.
We give estimates for the essential norms of a positive Toeplitz operator on the Bergman space of a minimal bounded homogeneous domain in terms of the Berezin transform or the averaging function of the symbol. Using these estimates, we also give necessary and sufficient conditions for the positive Toeplitz operators to be compact.
On the numerical radius of weighted shifts on $\ell^{2}$ as a norm on $\ell^{\infty}$ and connections to Banach limits
本文研究了 $\ell^{2}$ 上加权移位的数值半径作为 $\ell^{\infty}$ 上范数的性质，证明了其与 Banach 极限、谱半径间的取值关系，并比较了不同算子范数的异同。
In this paper, we study several weighted norm inequalities for the Opdam--Cherednik transform. We establish different versions of the Heisenberg--Pauli--Weyl inequality for this transform. In particular, we give an extension of this inequality using weights with different exponents and present a variation of the inequality that incorporates $L^p$-norms for the Opdam--Cherednik transform. Also, we prove a version of the Hardy--Littlewood inequality for this transform. Finally, we give other variations of the Heisenberg--Pauli--Weyl inequality such as the Nash-type and Clarkson-type inequalities…
In this paper, we show that on the weighted Bergman space of the unit disk the essential norm of a noncompact Hankel operator equals its distance to the set of compact Hankel operators and is realized by infinitely many compact Hankel operators, which is analogous to the theorem of Axler, Berg, Jewell and Shields on the Hardy space in Axler et al. (1979); moreover, the distance is realized by infinitely many compact Hankel operators with symbols continuous on the closure of the unit disk and vanishing on the unit circle.
Essential norm and Schatten class of Hankel operators between weighted Fock spaces
本文研究加权Fock空间到加权Lebesgue空间的Hankel算子，完整刻画了其在所有 $1 \leq p, q < \infty$ 下的本质范数，并给出了由连续递增凸函数 $h$ 定义的Schatten-$h$ 类隶属准则。
Very recently, Božin and Karapetrović solved a conjecture by proving that the norm of the Hilbert matrix operator $\mathcal{H}$ on the Bergman space $A^p$ is equal to $\fracπ{\sin(\frac{2π}{p})}$ for $2 < p < 4.$ In this article we present a partly new and simplified proof of this result. Moreover, we calculate the exact value of the norm of $\mathcal{H}$ defined on the Korenblum spaces $H^\infty_α$ for $0 < α\le 2/3$ and an upper bound for the norm on the scale $2/3 < α< 1$.
Hausdorff operators on weighted mixed norm Fock spaces
本文研究加权混合范数Fock空间$F_ϕ^{p,q}$上的Hausdorff算子，刻画其有界性与紧性，并给出该算子为幂有界或一致平均遍历的条件。
Let $u$ be a holomorphic function and $φ$ a holomorphic self-map of the open unit disk $\mathbb{D}$ in the complex plane. We give some new characterizations for the boundedness of the weighted composition operators $uC_φ$ from Bloch type spaces to Zygmund type spaces in $\mathbb{D}$ in terms of $u, φ$, their derivatives and the $n$-th power $φ^n$ of $φ$. Moreover, we obtain some similar estimates for their essential norms. From which the sufficient and necessary conditions of compactness of the operators $uC_φ$ follows immediately.
Semyanistyi's fractional integrals have come to analysis from integral geometry. They take functions on $R^n$ to functions on hyperplanes, commute with rotations, and have a nice behavior with respect to dilations. We obtain sharp inequalities for these integrals and the corresponding Radon transforms acting on $L^p$ spaces with a radial power weight. The operator norms are explicitly evaluated. Similar results are obtained for fractional integrals associated to $k$-plane transforms for any $1\le k<n$.
For an almost radial and typical weight $v$, we characterize the continuity and compactness of the weighted composition operator $u C_φ$ acting on the weighted Banach spaces of analytic functions $H_{v}^{\infty}$ in terms of the $n$th power of the symbol $φ$, where $u: \mathbb{D} \to \mathbb{C}$ is an analytic function and $φ: \mathbb{D} \to \mathbb{D}$ is analytic. More precisely, we extend the results of Hyvärinen, Kemppainen, Lindström, Rautio and Saukko relating to the continuity and essential norm calculus of the operator $u C_φ$.
Wold-type decompositions for norm-increasing $m$-concave weighted shifts on directed trees
本文证明范数递增的 $3$-凹加权位移具有 Wold 型分解，分类了无根有向树上的 $4$-等距，并构造了无游荡子空间性质的严格 $4$-等距，否定了 Shimorin 对所有 $m\geq 4$ 的猜想。
We use techniques from time-frequency analysis to show that the space $\mathcal S_ω$ of rapidly decreasing $ω$-ultradifferentiable functions is nuclear for every weight function $ω(t)=o(t)$ as $t$ tends to infinity. Moreover, we prove that, for a sequence $(M_p)_p$ satisfying the classical condition $(M1)$ of Komatsu, the space of Beurling type $\mathcal S_{(M_p)}$ when defined with $L^{2}\,$norms is nuclear exactly when condition $(M2)'$ of Komatsu holds.
In this paper order estimates for the linear widths of some function classes are obtained; these classes are defined by restrictions on the weighted $L_{p_1}$-norm of the r-th derivative and the weighted $L_{p_0}$-norm of zero derivative.
We give some new estimates for the norm and essential norm of a weighted composition operator on the Bloch space. As corollaries, we obtain some new characterizations of the boundedness and compactness of a weighted composition operator on the Bloch space.
In this paper, we give some estimates for the essential norm and a new characterization for the boundedness and compactness of weighted composition operators from weighted Bergman spaces and Hardy spaces to the Bloch space.
In this paper, estimates for norms of weighted summation operators (discrete Hardy-type operators) on a tree are obtained for $1<p<q<\infty$ and for arbitrary weights and trees.
Weighted composition--differentiation operators on mixed-norm, derivative Hardy and F(p,q,s) spaces
该研究探讨混合范数、导数Hardy空间与F(p,q,s)空间上的加权复合微分算子，通过二进准则得到其有界性等性质，还获乘子、Schatten准则及精确奇异值等结果。
Essential norm and Schatten class difference of weighted composition operators over the ball
研究单位球上加权 Bergman 空间中两个加权复合算子之差，利用 Carleson 测度和 $R$-Berezin 变换估计其本性范数，并刻画其属于 Schatten 类的条件。
An equivalent norm in the weighted Bergman space $A^p_ω$, induced by an $ω$ in a certain large class of non-radial weights, is established in terms of higher order derivatives. Other Littlewood-Paley inequalities are also considered. On the way to the proofs, we characterize the $q$-Carleson measures for the weighted Bergman space $A^p_ω$ and the boundedness of a Hörmander-type maximal function. Results obtained are further applied to describe the resolvent set of the integral operators $T_g(f)(z)=\int_0^z g'(ζ)f(ζ)\,dζ$ acting on $A^p_ω$.
In this note we show that the Taylor series of a function in a weighted Dirichlet space is (generalized) Nörlund summable, provided that the sequence determining the Nörlund operator is non-decreasing and has finite upper growth rate. In particular the Taylor series is Nörlund summable for all $α>1/2$, and the rate of convergence is of the order $O(n^{-1/2})$. The inequality $α>1/2$ is sharp. On the other hand if the Taylor series is Nörlund summable and the partial sums of the determining sequence enjoy a certain growth condition then the determining sequence has finite lower growth rate. An …
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