In this paper, we investigate the r-summing Carleson embed-dings on weighted Fock spaces F-alpha,w(p). By using duality arguments, translating techniques and block diagonal operator skills, we completely characterize the r-summability of the natural embeddings I-d : F-alpha,w(p) -> L-alpha(p)(mu) for any r >= 1 and p > 1, where w is a weight on the complex plane C that satisfies an A(p)-type condition. As applications, we establish some results on the r-summability of differentiation and integration operators, Volterra-type operators and composition operators. Especially, we completely characterize the boundedness of Volterra-type operators and composition operators on vector-valued Fock spaces for all 1 < p < infinity, which were left open before for the case 1 < p < 2.

Let Ω ⊂ ℂn be a bounded domain covered by the polydisc 𝔻^n through a proper holomorphic mapping Φ. In this paper, we study the Lp regularity and give the Lp-norm estimate for the Bergman projection of Ω. As applications, we obtain the upper bounds of Lp-norm for the Bergman projections on the symmetrized polydisc and monomial polyhedra.

This paper is devoted to an in-depth study of the minimal commutant property for composition operators acting on Hilbert spaces of holomorphic functions in several complex variables, such as the Fock space on ℂ^d, the Hardy space on the Euclidean ball, and the Hardy space on the unit polydisc.

In this paper, we investigate various square functions on the complex unit ball. We prove weighted inequalities for the Lusin area integral associated with the Poisson integral in terms of A(p) weights for all 1 < p < infinity; this gives an affirmative answer to an open question raised by Segovia and Wheeden. In addition, we provide an equivalent characterization of weighted Hardy spaces via the Lusin area integral in the context of holomorphic functions. We also establish weighted inequalities for Volterra integral operators. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

ABSTRACT In this paper, we study properties of the infinitesimal generators of ‐semigroups of composition operators on Hardy spaces of Dirichlet series. We also characterize the ‐semigroups of weighted composition operators on by using abelian intertwiners of multiplication operators. Moreover, we establish a necessary and sufficient condition for embedding a weighted composition operator into a ‐semigroup on .

We study the area operators 𝔸_μ,l, 0<l<∞, induced by positive Borel measures on the right half-plane and acting on the Hardy spaces of Dirichlet series ℋ^p, 0<p<∞. We first disprove a conjecture proposed by the present authors in an earlier work by constructing a probability measure, valid for all 0<p,l<∞, for which the associated area operator is bounded although the measure fails the proposed Carleson conditions. We next investigate compactness of these operators. For every 0<p<∞, we characterize boundedness and compactness of 𝔸_μ,p on both ℋ^p and the Hardy space ℋ^p_0 of Dirichlet series vanishing at +∞; in particular, boundedness and compactness coincide for these operators. For general 0<p,l<∞, we further establish sufficient conditions for compactness in terms of vanishing Carleson measures and compact H_i^p-Carleson embeddings. As an application, we also give a different proof of a known compactness result for Volterra operators on ℋ^p with Dirichlet series symbols in VMOA(ℂ_0).

The C_0 -semigroups for the weighted composition operators on the Hardy spaces of Dirichlet series are completely characterized in this paper. This generalizes previous results in a recent paper of Contreras et al. regarding C_0 -semigroup of (unweighted) composition operators. As applications, we calculate the infinitesimal generator of the C_0 -semigroups and its point spectrum.

We introduce area operators A(mu ,l )in the Dirichlet series setting for l > 0 and positive Borel measures mu on the right half-plane C0. It is proved that if mu is a Carleson measure on C-0, then for 0 < p < infinity, the area operator A(mu ,l )is bounded from the Hardy space H(0)(p )of Dirichlet series vanishing at +infinity to some L-p-space. We also give an application of our methods to Volterra operators.

The 𝒥 -symmetric C_0 -semigroups of weighted composition operators on McCarthy-Bergman spaces of Dirichlet series are completely characterized in this paper. As applications, we calculate the point spectrum of infinitesimal generators for the 𝒥 -symmetric C_0 -semigroups and their dual semigroups. Moreover, we prove that such 𝒥 -symmetric C_0 -semigroups are not hypercyclic. In addition, we further study the commutants of 𝒥 -symmetric weighted composition operators.

We consider the Carleson embeddings from weighted Bergman spaces A^p_α(ℍ) over the upper half-plane ℍ into L^p(μ ) , where μ is a positive Borel measure on ℍ . We characterize the r-summability of such operators for all p≥ 1 , α >-1 and r≥ 1 . In particular, the case p=1 is solved, which was left open in He et al. (Adv Math 439:109495, 2024) in the setting of the unit disk. As applications, we show that for p≥ 1 and α >-1 , there are no r-summing composition operators on A^p_α(ℍ) for any r≥ 1 . Moreover, we obtain the description of r-summing weighted composition operators on A^p_α(ℍ) .

In the setting of the standard weighted Bergman spaces over the unit disk, compactness characterizations for linear combinations of composition operators have been known. One of those characterizations asserts that degenerate double differences, compared with each single difference, do not improve the compactness at all in the sense that a degenerate double difference is compact only when each difference is individually compact. Such a rigid phenomenon is actually known to hold for a certain broader class of linear combinations. In this paper we investigate into similar properties for Hilbert-Schmidtness with main focus on double differences. We first obtain a complete characterization for Hilbert-Schmidt double differences of composition operators. We then observe that double differences, compared with each single difference, can improve the Hilbert-Schmidtness even in the degenerate case, by constructing concrete examples of Hilbert-Schmidt double differences with each difference not being Hilbert-Schmidt. We also include some remarks concerning connection between Hilbert-Schmidtness on the standard weighted Bergman spaces and weak-to-strong boundedness on certain vector-valued weighted Bergman spaces. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Monomial polyhedra are a class of bounded singular Reinhardt domains defined as sublevel sets of holomorphic monomials. The purpose of this paper is twofold. We first establish an L^p -norm estimate for the Bergman projection on the monomial polyhedra 𝒰_B , which can be viewed as a complement of the recent work of the L^p regularity for the Bergman projection on monomial polyhedra by Bender et al. (Can J Math 74:732–772, 2022). Then, if 𝒰_B is a monomial polyhedron associated to the matrix B∈ℤ^n× n satisfying B=1 , we obtain a sharp weighted version of L^p -norm estimate for the Bergman projection on 𝒰_B .

In this paper, we investigate Hermitian weighted composition operators on the Hardy space H^2(𝔻^2) over the bidisk 𝔻^2 . Concretely, we characterize Hermitian weighted composition operators Cψ,φ on H^2(𝔻^2) into two classes. To our surprise, we find that φ1 and φ2 are depending only on one variable in each class, where φ = (φ1, φ2). Moreover, spectra and spectral decompositions of Hermitian weighted composition operators are described. In addition, semigroups of weighted composition operators over the bidisk are studied. Our results extend those of Cowen and Ko [Trans. Amer. Math. Soc., 362, 5771–5801 (2010)].

This paper is devoted to the study of area operators, which originated from research in complex analysis. Using the geometric property of the Carleson measures, we show that the area operators induced by Carleson measures are of weak type (1, 1) and of strong type (p, p) for harmonic functions in terms of the Muckenhoupt conditions. Our results also include the case of real variable. The main difficulty lies in the fact that the ideas in complex analysis are not applicable to our setting. The main tool is the generalized Littlewood-Paley functions.

We completely give the solution of the problem of Littlewood-type randomization in the Hardy and Bergman spaces of Dirichlet series. The Littlewood-type theorem for Bergman spaces of Dirichlet series is very different from the corresponding version for Hardy spaces of Dirichlet series; but also exhibits various pathological phenomena compared with the setting of analytic Bergman spaces over the unit disk, due to the fact that Dirichlet series behave as power series of infinitely many variables. Finally, we completely characterize the superposition operators between Bergman spaces of Dirichlet series relying on our techniques.

In this paper, we study composition operators on weighted Bergman spaces of Dirichlet series. We first establish some Littlewood-type inequalities for generalized mean counting functions. Then we give sufficient conditions for a composition operator with zero characteristic to be bounded or compact on weighted Bergman spaces of Dirichlet series. The corresponding sufficient condition for compactness in the case of positive characteristics is also obtained.

Infinite matrix theory is an important branch of function analysis. Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space, but not every infinite matrix corresponds to an operator. The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators, which is considered a respectable mathematical accomplishment. In this paper, we prove the compact version of the Schur test. Moreover, we provide the Schur test for the Schatten class S2. It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers. We finally provide the Schur test for compact operators from lp into lq.

We study linear combinations of two composition operators induced by linear symbols on the Hilbert space of Dirichlet series. Based on partial reproducing kernels, we obtain an equivalent inscription of the compactness of a single composition operator and describe the compact linear combinations of composition operators.

In this paper, we characterize the d× d matrix weights W on ℂ^n such that the Fock projection P_α is bounded on the vector-valued spaces L^p_α,W(ℂ^n;ℂ^d) induced by W and the Gaussian measures. It is proved that for 1≤ p≤∞, the Fock projection P_α is bounded on L^p_α,W(ℂ^n;ℂ^d) if and only if W satisfies a restricted 𝒜_p-condition. Our result is new even in the scalar setting at the endpoint p=∞.