We study Hardy–Sobolev spaces H_n^p(C^+) on the upper half-plane for 1<=p<=infty and n is a nonnegative integer, from both function-theoretic and operator-theoretic viewpoints. We establish an isometric boundary characterization of H_n^p(C^+) via nontangential limits, together with a Sobolev-type embedding theorem, a Cauchy integral representation, a direct-sum decomposition of W_n^p(R) for 1<p<infty, and a generalized Banach algebra structure under pointwise multiplication. We also obtain a finer Fourier-analytic description in the Hilbert case p=2 by proving a Paley–Wiener theorem and deriving the reproducing kernel of H_n^2(C^+).On the operator-theoretic side, we prove the spectral formula for multiplication operators and establish two verifiable sufficient conditions for the boundedness of weighted composition operators. These results provide a systematic theory of Hardy–Sobolev spaces on the upper half-plane beyond the Hilbert setting.

We introduce Hilbertian Hardy–Sobolev spaces on tube domains over convex cones and develop their structural theory from a Fourier-analytic point of view. We first establish a Paley–Wiener type representation, which identifies these spaces with weighted L^2 spaces on the dual cone and reveals their intrinsic Fourier structure. This representation leads naturally to a Hardy–Sobolev decomposition theorem for boundary Sobolev spaces on ℝ^d. Building on these structural results, we derive explicit reproducing kernels and characterize Carleson measures for the Hilbertian Hardy–Sobolev spaces. As a preliminary operator-theoretic application, we also derive basic consequences for multipliers and weighted composition operators on these spaces.

This paper investigates the properties of the difference of composition operators on the Korenblum space over tubular domains, characterizing their boundedness and compactness. Using the result on boundedness, we show that all bounded differences of composition operators on such space are absolutely summing operators.

This paper investigates a class of multidimensional p -adic Hardy-Hilbert-type integral operators with homogeneous kernels of degree -n . We establish the boundedness of these operators on various weighted function spaces, including weighted p -adic Lebesgue spaces, weighted p -adic Morrey spaces, and weighted p -adic mixed Morrey spaces.

In this paper, we introduce and study two classes of multiparameter Forelli–Rudin type operators from L^p⃗( 𝒟) to L^q⃗( 𝒟) , especially on their boundedness, where L^p⃗( 𝒟) and L^q⃗( 𝒟) are both weighted Lebesgue spaces over the Cartesian product of two tubular domains T_B , with mixed-norm and appropriate weights. We completely characterize the boundedness of these two operators when 1≤p⃗≤q⃗<∞ . Moreover, we provide the necessary and sufficient condition of the case that q⃗=(∞ ,∞ ) . As an application, we obtain the boundedness of three common classes of integral operators, including the weighted multiparameter Bergman-type projection and the weighted multiparameter Berezin-type transform.

In this paper, we characterize Bounded Mean Oscillation (BMO) and establish their connection with Hankel operators on weighted Bergman spaces over tubular domains. By utilizing the space BMO, we provide a new characterization of Bloch spaces on tubular domains. Next, we define a modified projection operator and prove its boundedness. Furthermore, we introduce differential operators and demonstrate that these operators belong to Lebesgue spaces on tubular domains. Finally, we establish an integral representation for Bergman functions using these differential operators.

We mainly investigate some results of Hardy-Sobolev spaces on the upper half-plane. In this paper, the Paley-Wiener theorem of Hardy-Sobolev spaces is established. The theorem asserts that the reproducing kernel of the Hardy-Sobolev spaces can be found. On account of this, the estimation of the reproducing kernel is produced. We also consider the multiplication operators on the spaces, and the spectrum of multiplication operators is characterized. In addition, the condition for the boundedness of weighted composition operators can be founded by applying the property of self-maps and the reproducing kernel.

In the present paper, we study the boundedness and compactness of Toeplitz operators and Berezin-type operators between different weighted Bergman spaces over tubular domains in ℂ^n. We establish their connection with Carleson measures and provide some characterizations.

In this paper, we obtain the Gehring-Hayman type theorem on smoothly bounded pseudoconvex domains of finite type in $\mathbb{C}^2$. As an application, we provide a quantitative comparison between global and local Kobayashi distances near a boundary point for these domains.

Following M. Abate and A. Saracco's work on strongly pseudoconvex domains in Cn$\mathbb {C}<^>n$, we characterize Carleson measures of A2(D)$A<^>2(D)$ in bounded convex domains in Cn$\mathbb {C}<^>n$ with smooth boundary of finite type. We also give examples of Carleson measures with uniformly discrete (with respect to the Kobayashi distance) sequences.

As continuation of the study of polynomial approximation and composition operators on Dirichlet spaces of unit disk, which has settled a problem posed by Cima in 1976, the present paper aims to consider the case of the unbounded domains, such as the half-plane. Specifically, we may obtain the rational approximations in the Dirichlet spaces and characterize the composition operators which has dense range on the Dirichlet spaces over the half-plane. Moreover, this paper also considers the relationship between the Dirichlet spaces and Hardy spaces on half-plane.

Flavonol synthase (FLS) serves as a pivotal enzyme in the flavonol biosynthesis pathway, contributing to the abundance of flavonol compounds found in Brassica vegetables. Nevertheless, there are limited investigations on the identification and analysis of FLS gene family and their precise roles in responding to biotic and abiotic stresses remain elusive in Brassica vegetables. Herein, a total of 6 (Brassica rapa), 9 (Brassica oleracea), and 14 (Brassica napus) FLS genes were identified and distributed on 4, 5, and 8 chromosomes, respectively. Notably, while FLS genes exhibited considerable variation in sequence, exon-intron structure, and molecular weight, they displayed highly conserved domain and motif compositions. Of the six identified BrFLSs, namely BrFLS1, BrFLS2, BrFLS3.1, BrFLS3.2, BrFLS3.3, and BrFLS4, all were located in the nucleus and cell membrane except for BrFLS4 exclusively in the nucleus. Specifically, BrFLS1, BrFLS3.2, and BrFLS3.3 showed up-regulation after 6-BA, ABA, BR, and GA3 treatments, while BrFLS2 exhibited down-regulation. Intriguingly, BrFLS members displayed distinct expression patterns under diverse abiotic and biotic stresses. Employing the virus-induced gene silencing (VIGS) technology, we validated BrFLS1/BrFLS3.2, BrFLS1/BrFLS3.3, BrFLS2, BrFLS1, and BrFLS1/BrFLS2/BrFLS3.1/BrFLS3.3 as vital candidate genes, contributing to the flavonol biosynthesis pathway under drought, salt, high temperature, low temperature, and anthracnose stresses in Brassica vegetables, respectively, and they enhanced their resistance to stresses by bolstering antioxidant capacity. Overall, our findings shed light on the biological roles of FLS gene family in the flavonoid biosynthesis pathway, offering valuable genetic resources for crop enhancement.

The Korenblum space, often referred to as a growth space, is a special type of analytic function space. This paper investigates the properties of the difference of composition operators on the Korenblum space over the product of upper half planes, characterizing their boundedness and compactness. Using the result on boundedness, we show that all bounded differences of composition operators are absolutely summable operators.

We examine how the square-integrable function subspaces are transformed using the holomorphic Fourier transform. On account of this, the extended Paley-Wiener theorem over the Hardy-Sobolev spaces is produced. The theorem also asserts that the reproducing kernel of the Hardy-Sobolev spaces can be found. We discuss the relationship between the disc and the upper half-plane.

In this paper, we obtain a more precise estimate of Catlin-type distance for smoothly bounded pseudoconvex domain of finite type in ℂ^2 . As an application, we get an alternative proof of the Gromov hyperbolicity of this domain equipped with the Kobayashi distance.

Let H be a reproducing kernel Hilbert space of analytic functions on the unit disk D. The best kernel approximation problem for H is the following: given any positive integer n and any function f∈H find the best norm approximation of f by a linear combination of no more than n kernel functions K(z,zk), 1≤k≤n. The purpose of this paper is to prove the existence of best kernel approximation for weighted Bergman spaces with standard weights.

Following M.Abate and A.Saracco's work on strongly pseudoconvex domains in ℂ^n, we characterize Carleson measures of A^2(D) in bounded convex domains with smooth boundary of finite type. We also give examples of Carleson measures with uniformly discrete (with respect to the Kobayashi distance) sequences.

This paper aims to obtain decompositions of higher dimensional L-p(R-n) functions into sums of non-tangential boundary limits of the corresponding Hardy space functions on tubes for the index range 0 < p < 1. In the one-dimensional case, Deng and Qian recently obtained such a Hardy space decomposition result: for any function f is an element of L-p(R), 0 < p < 1, there exist functions f(1) and f(2) such that f = f(1) + f(2), where f(1) and f(2) are, respectively, the non-tangential boundary limits of some Hardy space functions in the upper-half and lower-half planes. In the present paper, we generalize the one-dimensional Hardy space decomposition result to the higher dimensions and discuss the uniqueness issue of such decomposition.

As continuation of the study of Fourier spectrum characterization of higher-dimensional Hardy spaces $$H^p(T_{\Gamma })$$ on tubes for $$1\le p\le \infty $$ , this paper aims to obtain analogous Fourier spectrum characterizations and integral representation formulas of higher-dimensional Hardy spaces $$H^p(T_{\Gamma })$$ on tubes for the index range $$0< p < 1$$ . For $$1\le p\le \infty $$ , the $$H^p(T_{\Gamma })$$ are well understood via the Poisson and conjugate Poisson integrals. However, for $$0< p < 1$$ , those integrals are no longer defined that requires more delicate analysis.