For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal expansions with respect to $L^\infty$ norm, which generalize analogous results obtained for little $q$-Legendre, little $q$-Jacobi and little $q$-Laguerre polynomials, by the authors.
In this paper, we completely characterize the order boundedness of weighted composition operators between different weighted Dirichlet spaces and different derivative Hardy spaces.
The main result of this paper refers to the boundedness of the orthogonal projection $P_α:L^{2}(\mathbb{R}^{n},dμ_α)\rightarrow \mathcal{H}_α^{2}, n\geq2 $ associated to the harmonic Fock space $\mathcal{H}_α^{2},$ where $dμ_α(x)=(πα)^{-n/2}e^{-\frac{|x|^2}α}dx.$ We prove that the operator $P_α$ is not bounded on $L^{p}(\mathbb{R}^{n},dμ_β)$ when $0<p< 1$ and we found a necessary and sufficient condition for the boundedness when $1\leq p<\infty$ and $n$ is an even integer.
For a Dirichlet series g, we study the Volterra operator Tg of symbol g, acting on a class of weighted Hilbert spaces of Dirichlet series. We obtain sufficient / necessary conditions for Tg to be bounded (resp. compact), involving BMO and Bloch type spaces on some half-plane. We also investigate the membership of Tg in Schatten classes. We also relate the boundedness of Tg to the boundedness of a multiplicative Hankel form of symbol g.
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…
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 study boundedness and compactness of composition operators on weighted Bergman spaces of Dirichlet series. Particularly, we obtain in some specific cases, upper and lower bounds of the essential norm of these operators and a criterion of compactness on classicals weighted Bergman spaces. Moreover, a sufficient condition of compactness is obtained using the notion of Carleson's measure.
We show that the decay of approximation numbers of compact composition operators on the Dirichlet space $\mathcal{D}$ can be as slow as we wish, which was left open in the cited work. We also prove the optimality of a result of O.~El-Fallah, K.~Kellay, M.~Shabankhah and A.~Youssfi on boundedness on $\mathcal{D}$ of self-maps of the disk all of whose powers are norm-bounded in $\mathcal{D}$.
Let $\mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators $\mathscr{C}_φ$ on $\mathscr{H}^2$ which are generated by symbols of the form $φ(s) = c_0s + \sum_{n\geq1} c_n n^{-s}$, in the case that $c_0 \geq 1$. If only a subset $\mathbb{P}$ of prime numbers features in the Dirichlet series of $φ$, then the operator $\mathscr{C}_φ$ admits an associated orthogonal decomposition. Under sparseness assumptions on $\mathbb{P}$ we use this to asymptotically estimate the approximation numbers of $\mathscr{C}_φ$. Furthermore, in the…
We consider the problem of $L^1$ (un)boundedness for a wide class of orthogonal projections, including Bergman projections on domains in complex manifolds and Szegö projections on abstract CR manifolds.
We generalize the classical Muckenhoupt inequality with two measures to three under appropriate conditions. As a consequence, we prove a simple characterization of the undedness of the multiplication operator and thus of the boundedness of the zeros and the asymptotic behavior of the Sobolev orthogonal polynomials, for a large class of measures which includes the most usual examples in the literature.
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.
Weak type $(1,1)$ boundedness of noncommutative singular integral operators with rough kernels
Institute for Advanced Study in Mathematics Harbin Institute of Technology（哈尔滨工业大学数学高等研究院）；Harbin Institute of Technology（哈尔滨工业大学）；Zhengzhou Research Institute Harbin Institute of Technology（哈尔滨工业大学郑州研究院）
Necessary and sufficient conditions are given for boundedness of Hausdorff operators on generalized Hardy spaces $H^p_E(G)$, real Hardy space $H^1_{\mathbb{R}}(G)$, $BMO(G)$, and $BMOA(G)$ for compact Abelian group $G$. Surprisingly, these conditions turned out to be the same for all groups and spaces under consideration. Applications to Dirichlet series are given. The case of the space of continuous functions on $G$ and examples are also considered.
Hardy-Littlewood type phenomena and the Girela-Peláez conjecture for the Möbius invariant Laplacian operator
Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院）；Center for Applied Mathematics of Guangxi, Guangxi Normal University（广西师范大学应用数学中心）；Industry-Academia Co-innovation and Research Promotion Headquarters, Kyushu Sangyo University（九州产业大学产学协同创新与研究推进总部）
Orthogonal polynomials $P_{n}(λ)$ are oscillating functions of $n$ as $n\to\infty$ for $λ$ in the absolutely continuous spectrum of the corresponding Jacobi operator $J$. We show that, irrespective of any specific assumptions on coefficients of the operator $J$, amplitude and phase factors in asymptotic formulas for $P_{n}(λ)$ are linked by certain universal relations found in the paper. Our approach relies on a study of operators diagonalizing Jacobi operators. Diagonalizing operators are constructed in terms of orthogonal polynomials $P_{n}(λ)$. They act from the space $L^2 (\Bbb R)$ of func…
Let $v(r)=\exp\left(-\fracα{1-r}\right)$ with $α>0$, and let $\mathbb{D}$ be the unit disc in the complex plane. Denote by $A^p_v$ the subspace of analytic functions of $L^p(\mathbb{D},v)$ and let $P_v$ be the orthogonal projection from $L^2(\mathbb{D},v)$ onto $A^2_v$. In 2004, Dostanic revealed the intriguing fact that $P_v$ is bounded from $L^p(\mathbb{D},v)$ to $A^p_v$ only for $p=2$, and he posed the related problem of identifying the duals of $A^p_v$ for $p\ge 1$, $p\neq 2$. In this paper we propose a solution to this problem by proving that $P_v$ is bounded from $\,L^p(\D,v^{p/2})$ to $…
We study boundedness, compactness, and Schatten-class membership of the canonical solution operator to dbar, restricted to (0,1)-forms with holomorphic coefficients, on L^2(d mu) where mu is a measure with the property that the monomials form an orthogonal family in L^2 (d mu). The characterizations are formulated in terms of moment properties of mu. Our results generalize the results of the first author to several variables, contain some known results for several variables, and also cover new ground.
