Javad Mashreghi, Thomas Ransford
We give an elementary proof of an analogue of Fejér's theorem in weighted Dirichlet spaces with superharmonic weights. This provides a simple way of seeing that polynomials are dense in such spaces.
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In this paper, we completely characterize the order boundedness of weighted composition operators between different weighted Dirichlet spaces and different derivative Hardy spaces.
The Carleson measures for weighted Dirichlet spaces had been characterized by Girela and Peláez, who also characterized the boundedness of Volterra type operators between weighted Dirichlet spaces. However, their characterizations for the boundedness are not complete. In this paper, we completely characterize the boundedness and compactness of Volterra type operators from the weighted Dirichlet spaces $D_α^p$ to $D_β^q$ ($-1<α,β$ and $0<p<q<\infty$), which essentially complete their works. Furthermore, we investigate the order boundedness of Volterra type operators between weighted Dirichlet s…
The Hadamard product of two power series is obtained by multiplying them coefficientwise. In this paper we characterize those power series that act as Hadamard multipliers on all weighted Dirichlet spaces on the disk with superharmonic weights, and we obtain sharp estimates on the corresponding multiplier norms. Applications include an analogue of Fejér's theorem in these spaces, and a new estimate for the weighted Dirichlet integrals of dilates.
Higher order weighted Dirichlet type spaces with poly-superharmonic weights and Dirichlet type operators of finite order
本文研究了与多项超调和权重相关的一类单位圆上高阶加权狄利克雷型空间，通过建立高阶Littlewood-Paley公式，将权重的迭代拉普拉斯算子与移位算子的高阶缺陷算子联系起来，引入了有限阶狄利克雷型算子，扩展了循环完全超膨胀和完全超收缩算子及m-等距算子的模型理论。
We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $(C,α)$-summable to the function in the norm of the space, provided that $α>1/2$. We further show that the constant $1/2$ is sharp, in marked contrast with the classical case of the disk algebra.
In this paper, we introduce a general class of weighted spaces of holomorphic Dirichlet series (with real frequencies) analytic in some half-plane and study composition operators on these spaces. In the particular case when the symbol inducing the composition operator is an affine function, we give criteria for boundedness and compactness. We also study the cyclicity property and as a byproduct give a sufficient condition so that the direct sum of the identity plus a weighted forward shift operator on the Hardy space H^2 is cyclic.
Mean ergodicity of multiplication operators in weighted Dirichlet spaces
本文刻画了加权 Dirichlet 空间中乘法算子及其伴随算子的有界性条件，证明了均值遍历性与 Cesàro 有界性的等价性，并构造出一致均值遍历但非幂有界的算子特例。
We show that if $u$ is a compactly supported distribution on the complex plane such that, for every pair of entire functions $f,g$, \[ \langle u,f\overline{g}\rangle=\langle u,f\rangle\langle u,\overline{g}\rangle, \] then $u$ is supported at a single point. As an application, we complete the classification of all weighted Dirichlet spaces on the unit disk that are de Branges-Rovnyak spaces by showing that, for such spaces, the weight is necessarily a superharmonic function.
Boundedness of weighted composition operators $W_{u,φ}$ acting on the classical Dirichlet space $\mathcal{D}$ as $W_{u,φ}f= u\, (f\circ φ)$ is studied in terms of the multiplier space associated to the symbol $φ$, i.e., ${\mathcal{M}(ϕ)}=\{ u \in {\mathcal D}: W_{u,ϕ} \hbox{ is bounded on } {\mathcal D} \}$. A prominent role is played by the multipliers of the Dirichlet space. As a consequence, the spectrum of $W_{u,φ}$ in $\mathcal{D}$ whenever $φ$ is an automorphism of the unit disc is studied, extending a recent work of Hyvärinen, Lindström, Nieminen and Saukko to the context of the Dirichl…
We characterize polynomials that are cyclic in Dirichlet-type spaces in the unit ball in $\mathbb C^2$
We extend Carleson's formula to radially polynomially weighted Dirichlet spaces.
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 …
Composition operators on weighted Hilbert spaces of Dirichlet series
本文研究了Dirichlet级数加权Hilbert空间上特征为零的复合算子，通过证明加权均值计数函数的存在性并给出变量替换公式，导出了有界性与紧性的必要条件，并在Bergman型空间中利用Schwarz型引理证明了紧性条件也是充分的。
We introduce and study a generalization of the classical weighted Bergman and Dirichlet spaces on the unit ball in high dimension, the Bergman-Dirichlet spaces. Their counterparts on the whole $n$-complex space, the Bargmann-Dirichlet spaces, are also introduced and studied. Mainly, we give a complete description of the considered spaces, including orthonormal basis and the explicit formulas for their reproducing kernel functions. Moreover, we investigate their asymptotic behavior when the curvature goes to $0$.
We introduce new functional spaces that generalize the weighted Bergman and Dirichlet spaces on the disk D(0,R) in the complex plane and the Bargmann-Fock spaces on the whole complex plane. We give a complete description of the considered spaces. Mainly, we are interested in giving explicit formulas for their reproducing kernel functions and their asymptotic behavior as R goes to infinity.
We treat the problem of characterizing the cyclic vectors in the weighted Dirichlet spaces, extending some of our earlier results in the classical Dirichlet space. The absence of a Carleson-type formula for weighted Dirichlet integrals necessitates the introduction of new techniques.
Let $1 \leq p < q < \infty$. We show that \[ \sup{\frac{\left\| D\right\|_{\mathcal{H}_{q}}}{\left\| D\right\|_{\mathcal{H}_{p}}}} = \exp{\left( \frac{\log{x}}{\log{\log{x}}} \left(\log{\sqrt{\frac{q}{p}}} + \left(\frac{\log{\log{\log{x}}}}{\log{\log{x}}}\right)\right) \right)} \,,\] where the supremum is taken over all non-zero Dirichlet polynomials of the form $D(s)=\sum_{n \leq x}{a_{n} n^{-s}}$. An aplication is given to the study of multipliers between Hardy spaces of Dirichlet series.
Application of Meyer's theorem on quasicrystals to exponential polynomials and Dirichlet series
本文利用 Meyer 准晶定理，给出了绝对收敛 Dirichlet 级数（虚指数、实零点）为有限正弦乘积的充要条件。
We establish necessary and sufficient conditions for boundedness of composition operators on the most general class of Hilbert spaces of entire Dirichlet series with real frequencies. Depending on whether or not the space contains any nonzero constant function, different criteria for boundedness are developed. Thus, we complete the characterization of bounded composition operators on all known Hilbert spaces of entire Dirichlet series of one variable.
Using Carleson measure theorem of weighted Bergman spaces, we provide a complete characterization of embedding theorem for Dirichlet type spaces. As an application, we study the Volterra integral operator and multipliers for Dirichlet type spaces.
Integration type operators and point evaluation on weighted Bergman spaces of Dirichlet series
本研究针对狄利克雷级数加权Bergman空间的点赋值泛函范数估计问题，引入作用于该空间的Riemann-Liouville半群版本作为新工具，完善了该空间理论中的核心问题研究。
Norm inequalities for weighted Dirichlet spaces with applications to conformal maps
本文研究了加权狄利克雷空间的范数不等式，并将其应用于共形映射的最大模增长与图像面积的几何关系，揭示了新的几何特征。
We consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight $λ$ depends on the independent variable $z$. We prove that for an increasing radial weight $λ(z)$ the infimal energy within the class of all Sobolev homeomorphisms is the same as in the class of radially symmetric maps. For a general radial weight $λ(z)$ we establish the same result in the case when the target is conformally thin compared to the domain. Fixing the admissible homeomorphisms on the outer boundary we establish the radial s…
Polynomial Counterexamples to Shanks' Conjecture in Dirichlet-Type Spaces over the Bidisc
本文在双圆盘Dirichlet型空间中构造对称多项式反例，证明对任意正参数，线性最优逼近在域内有零点而函数本身无零点，从而否定弱Shanks猜想，并推广至多圆盘及各向异性情形。
We study Dirichlet-type spaces $\mathfrak{D}_α$ of analytic functions in the unit bidisk and their cyclic elements. These are the functions $f$ for which there exists a sequence $(p_n)_{n=1}^{\infty}$ of polynomials in two variables such that $\|p_nf-1\|_α\to 0$ as $n\to \infty$. We obtain a number of conditions that imply cyclicity, and obtain sharp estimates on the best possible rate of decay of the norms $\|p_nf-1\|_α$, in terms of the degree of $p_n$, for certain classes of functions using results concerning Hilbert spaces of functions of one complex variable and comparisons between norms …
For functions $f$ in Dirichlet-type spaces we study how to determine constructively optimal polynomials $p_n$ that minimize $\|p f-1\|_α$ among all polynomials $p$ of degree at most $n$. Then we give upper and lower bounds for the rate of decay of $\|p_{n}f-1\|_α$ as $n$ approaches $\infty$. Further, we study a generalization of a weak version of the Brown-Shields conjecture and some computational phenomena about the zeros of optimal polynomials.
On the equality of De Branges-Rovnyak and Dirichlet spaces
本文完全刻画了当测度为有限原子和时，哪些 de Branges-Rovnyak 空间与调和加权 Dirichlet 空间相等，推广了 Costara-Ransford 的结果，无需对 Pythagorean 对作假设。
The problem of describing the analytic functions $g$ on the unit disc such that the integral operator $T_g(f)(z)=\int_0^zf(ζ)g'(ζ)\,dζ$ is bounded (or compact) from a Banach space (or complete metric space) $X$ of analytic functions to the Hardy space $H^\infty$ is a tough problem and remains unsettled in many cases. For analytic functions $g$ with non-negative Maclaurin coefficients, we describe the boundedness and compactness of $T_g$ acting from a weighted Dirichlet space $D^p_ω$, induced by an upper doubling weight $ω$, to $H^\infty$. We also characterize, in terms of neat conditions on $ω…
Zero-free regions of the Riemann zeta function and approximation in weighted Dirichlet spaces
本文研究黎曼Zeta函数在加权Dirichlet空间中的无零点区域，证明了对标准加权空间D_α(α∈(-3,-2))的逼近问题可给出半平面无零点条件，并将结果推广至解析函数空间ℓ^p_α，特例情形给出了素数定理的新等价表述。
To each weighted Dirichlet space $\mathcal{D}_p$, $0<p<1$, we associate a family of Morrey-type spaces ${\mathcal{D}}_p^λ$, $0< λ< 1$, constructed by imposing growth conditions on the norm of hyperbolic translates of functions. We indicate some of the properties of these spaces, mention the characterization in terms of boundary values, and study integration and multiplication operators on them.
Non-cyclicity and polynomials in Dirichlet-type spaces of the unit ball
本文通过描述单位球中无零点多项式的零点集与单位球面的交集，提出Dirichlet型空间中循环多项式刻画的猜想，并部分解决了任意多项式非循环性的判定问题。
Let $Ω\subset\mathbb{C}$ be a bounded domain such that there exists an algebraic harmonic function of degree two vanishing on the boundary of $Ω.$ Then we show that the Khavinson-Shapiro conjecture holds for $Ω:$ if the Dirichlet problem on $Ω$ with all polynomial boundary data have polynomial solutions, then $Ω$ must be an ellipse. We also prove that if there exists a rational function with a singularity in $Ω$, such that the Dirichlet problem for its restriction on $\partialΩ$ along with all polynomial functions have rational solutions, then $Ω$ must be a disc. This generalizes a well-known …
In the context of a theorem of Richter, we establish a similarity between $C_0$-semigroups of analytic 2-isometries $\{T(t)\}_{t\geq0}$ acting on a Hilbert space $\mathcal H$ and the multiplication operator semigroup $\{M_{ϕ_t}\}_{t\geq 0}$ induced by $ϕ_t(s)=\exp (-st)$ for $s$ in the right-half plane $\mathbb{C}_+$ acting boundedly on weighted Dirichlet spaces on $\mathbb{C}_+$. As a consequence, we derive a connection with the right shift semigroup $\{S_t\}_{t\geq 0}$ $$ S_tf(x)=\left \{ \begin{array}{ll} 0 & \mbox { if }0\leq t\leq x, \\ f(x-t)& \mbox { if } x>t, \end{array} \right . $$ ac…
Infinitely supported harmonically weighted Dirichlet spaces which are de Branges Rovnyak spaces
本文研究调和加权Dirichlet空间与de Branges-Rovnyak空间的相等问题，突破有限支撑限制，刻画了无限支撑测度下两类空间相等的条件，还建立了相关T(1)型结果与扰动定理，并证明了Brown-Shields猜想。
We study the supremum of some random Dirichlet polynomials and obtain sharp upper and lower bounds for supremum expectation that extend the optimal estimate of Halász-Queffélec and enable to cunstruct random polynomials with unusually small maxima. Our approach in proving these results is entirely based on methods of stochastic processes, in particular the metric entropy method.
We study the supremum of some random Dirichlet polynomials with independent coefficients and obtain sharp upper and lower bounds for supremum expectation thus extending the results from our previous work (see http://arXiv.org/abs/math/0703691). Our approach in proving these results is entirely based on methods of stochastic processes, in particular the metric entropy method.
We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [4].
The $L^q$ norm of a Dirichlet polynomial $F(s)=\sum_{n=1}^{N} a_n n^{-s}$ is defined as \[\| F\|_q:=(\lim_{T\to\infty}\frac{1}{T}\int_{0}^T |F(it)|^qdt)^{1/q}\] for $0<q<\infty$. It is shown that \[ (\sum_{n=1}^{N} |a_n|^2|μ(n)|[d(n)]^{\frac{\log q}{\log 2} -1})^{1/2}\le \| F\|_q \] when $0<q<2$; here $μ$ is the Möbius function and $d$ the divisor function. This result is used to prove that the $L^q$ norm of $D_N(s):=\sum_{n=1}^{N} n^{-1/2-s}$ satisfies $\|D_N\|_q\gg (\log N)^{q/4}$ for $0<q<\infty$. By Helson's generalization of the M. Riesz theorem on the conjugation operator, the reverse in…
The problem of representation of elements of weighted space of infinitely differentiable functions on real line by exponential series is considered.
Consider the Dirichlet-type space on the bidisk consisting of holomorphic functions $f(z_1,z_2):=\sum_{k,l\geq 0}a_{kl}z_1^kz_2^l$ such that $\sum_{k,l\geq 0}(k+1)^{α_1} (l+1)^{α_2}|a_{kl}|^2 <\infty.$ Here the parameters $α_1,α_2$ are arbitrary real numbers. We characterize the polynomials that are cyclic for the shift operators on this space. More precisely, we show that, given an irreducible polynomial $p(z_1,z_2)$ depending on both $z_1$ and $z_2$ and having no zeros in the bidisk: if $α_1+α_2\leq 1$, then $p$ is cyclic; if $α_1+α_2>1$ and $\min\{α_1,α_2\}\leq 1$, then $p$ is cyclic if and…
The definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs triangle which includes some weighted Bergman spaces, a candidate Hardy space and a candidate Dirichlet space. For the weighted Bergman spaces and the Hardy space we study the $L^p$ mapping properties of Bergman and Szegő projection respectively, whereas for the Dirichlet space we prove it is isometric to the Dirichlet space on the bidisc.
We characterize the uniform limits of Dirichlet polynomials on a right half plane. In the Dirichlet setting, we find approximation results, with respect to the Euclidean distance and {to} the chordal one as well, analogous to classical results of Runge, Mergelyan and Vitushkin. We also strengthen the notion of universal Dirichlet series.
Ellipses and Polynomial-to-Polynomial Mapping of Weighted Szego Projections
该研究结合位势理论相关结果，探究椭圆与椭球上的加权Szego投影，证明平面椭圆上存在能将多项式映射为同次数多项式的加权Szego投影。
We study the relation between weighted pluripotential on a compact set E in C^N and the pluripotential theory of an associated circed set Z in C^(N+1)
A Polynomial Rate of Convergence for the Dirichlet Problem on Orthodiagonal Maps
本文推广了先前工作，证明了正交对角映射上带Hölder边界数据的Dirichlet问题具有多项式收敛速率，关键方法是利用光滑磨光子使离散调和函数近似调和，并由此得到介观尺度上的Lipschitz性质。
Radiall symmetry of minimizers to the weighted $p-$Dirichlet energy
本文研究了复平面上环面间 Sobolev 同胚的加权 $p-$Dirichlet 能量极小化问题，证明了在存在径向微分同胚极小值的条件下，该能量积分的极小值由径向微分同胚取得，并揭示了 $p=1$ 时与标准 Nitsche 现象相反的共形模条件。
We prove effective results on when a function can be approximated by a Dirichlet polynomial with bounded coefficients. Assuming that Φ(n) is an increasing function we prove that the set of polynomials {\sum_{n=2}^N a_n n^{it-1}: N \geq 2, |a_n| \leq Φ(n)}, is dense in L^2(0,H) if and only if \sum_{n=2}^\infty \frac{\log Φ(n)} {n \log^2 n} = \infty. We also prove variants of this result for generalized Dirichlet polynomials. The main tools are theorems of Paley and Wiener related to quasianalyticity and the Pechersky rearrangement theorem. We use this result to give precise conditions on when a…
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