In this paper, we introduce a kind of weighted Fock space F^p_φ on the complex plane ℂ , and obtain some estimates on the Bergman kernel function. As an application, we discuss the Bergman projection, density, and L^p_φ -regularity of the canonical solution to ∂ .

In this paper, we establish a sharp comparison between Carleson-cube and Bergman-metric-ball conditions on the open unit ball $\B$ and combine it with a Berezin-type characterization to prove embedding theorems for Besov spaces and Bergman spaces on $\B$ into logarithmic tent spaces in the Bergman metric. As applications, we characterize the boundedness, compactness, and essential norms of the Volterra-type integral operators $T_g$ and $I_g$ acting from the Besov space $B_t(\B)$ to the general function space $F(p,q,s)$.

In this paper, for 1 ≤ p, r < ∞ we characterize those symbols f so that the induced Hankel operators H_f are r-summing from Fock spaces F^p_α to L^p_α. The main result shows that the r-summing norm of H_f is equivalent to the IDA^κ, p-norm of f, where κ is a positive number determined by p and r, and the IDA space is as in [13]. As some application, we discuss the Berger-Coburn phenomenon for r-summing Hankel operators on Fock spaces.

Let 0 = 0) with entries mu[i+j] = integral(D)z(i+j)d mu(z), which formally induces the operator S-p[mu] defined as S-p[mu](f)(z)=& sum;(infinity)(n=0)(& sum;(infinity)(k=0)(n+k+1)(p-1)mu[n+k]a(k))z(n), z is an element of D, where f(z)=& sum;(infinity)(n=0)a(n)z(n) is an analytic function in D. We characterize the positive Borel measures mu supported on (-1,1) for which S-p[mu] is bounded (resp. compact) from one Dirichlet space D-alpha to another D-beta. Additionally, we investigate the boundedness (resp. compactness) of S-p[mu] on Dirichlet-type spaces D-alpha(l). Our results generalize those of Bao and Wulan when alpha=beta or l=2.

In this paper, we prove embedding theorems for the Möbius invariant space Q p \mathcal {Q}_p on the open unit ball of C n \mathbb {C}^n into logarithmic tent spaces in the Bergman metric.

In this paper, we prove embedding theorems for the Mobius invariant space Q p on the open unit ball of C n into logarithmic tent spaces in the Bergman metric.

Let A(omega)(p) denote the Bergman space in the unit disc induced by a radial weight omega with the doubling property integral(1)(r)omega(s)ds <= C integral(1)((1+r)/2) omega(s)ds. The tent space T-s(q) (nu, omega) consists of functions such that integral(D) (integral(Gamma(zeta)) |f(z)|(s)d nu(z)(q/s) . omega(zeta)dA(zeta) < infinity, where Gamma(zeta) is a non-tangential approach region with vertex zeta in the punctured unit disc D\ {0}. We characterize the positive Borel measures nu such that A(omega)(p) is embedded into the tent space T-s(q)(nu, omega) for 0 < q < p < infinity, by considering a generalized area operator.

Suppose ϕ is a real-valued subharmonic function on C with ΔϕdA is a doubling measure. The doubling Fock space Fϕp is the family of holomorphic functions on C such that f(⋅)e−ϕ(⋅)∈Lp. We introduce the function space IDArs,q,α and discuss the decomposition theorem for this space. We use it to characterize the boundedness and compactness of Hankel operators from a doubling Fock space Fϕp to a weighted Lesbegue space Lϕq for all possible 1≤p,q<∞, which extends the results of [9] from the special case ρ≍1. We also obtain the relationship between the solution operators to ∂‾-equation and Hankel operator. As some applications, we obtain the characterizations on f for which Hankel operators Hf and Hf‾ are both bounded (or compact) from Fϕp to Lϕq.

We completely characterize those positive Borel measures $\mu$ on the unit ball $\mathbb{B}_ n$ such that the Carleson embedding from Hardy spaces $H^p$ into the tent-type spaces $T^q_ s(\mu)$ is bounded, for all possible values of $0

We completely characterize the boundedness of area operators from the Bergman spaces A_α ^p(𝔹_n) to the Lebesgue spaces L^q(𝕊_n) for all 0 < p,q < ∞. For the case n = 1, some partial results were previously obtained by Wu in [Wu, Z.: Volterra operator, area integral and Carleson measures, Sci. China Math. , 54 , 2487–2500 (2011)]. Especially, in the case q < p and q < s , we obtain some characterizations for the area operators to be bounded. We solve the cases left open there and extend the results to n -complex dimension.

Stemming from the Pythagorean Identity sin(2) z + cos(2) z = 1 and Hormander's L-2-solution of the Cauchy-Riemann's equation (partial derivative) over baru = f on C, this article demonstrates a corona-type principle which exists as a somewhat unexpected extension of the analytic Hilbert's Nullstellensatz on C to the quadratic Fock-Sobolev spaces on C.

Stemming from the Pythagorean Identity $ \sin^2z+\cos^2z = 1 $ and Hörmander's $ L^2 $-solution of the Cauchy-Riemann's equation $ \bar{\partial}u = f $ on $ \mathbb C $, this article demonstrates a corona-type principle which exists as a somewhat unexpected extension of the analytic Hilbert's Nullstellensatz on $ \mathbb C $ to the quadratic Fock-Sobolev spaces on $ \mathbb C $.

We completely characterize the boundedness of the area operators from the Bergman spaces A α (Bn) to the Lebesgue spaces L q(Sn) for all 0 < p, q < ∞. For the case n = 1, some partial results were previously obtained by Wu in [19]. Especially, in the case q < p and q < s, we obtain the new characterizations for the area operators to be bounded. We solve the cases left open there and extend the results to n-complex dimension.

Given 0 < p < infinity, the Bergman space A(phi)(p) consists of all holomorphic functions on D such that parallel to integral parallel to(p,phi) = (integral(D)vertical bar integral(z)e(-phi(z))vertical bar(p) dA(z))(1/p) < infinity, where phi belongs to a large class W-0 which cover those defined by Borichev, Dhuez and Kellay (2007) [4]. For 0 < p < 1, A(phi)(p) is neither a self-adjoint nor Banach space. By new approaches, we study the characterizations on positive Borel measures mu on D for which the induced Toeplitz operators T-mu are bounded or compact from one Bergman space A(phi)(p) to another A(phi)(q) for p or q is an element of (0, 1]. (C) 2021 Published by Elsevier Masson SAS.

In this paper, we characterize the mapping properties of Hankel operators $$H_{g}$$ and $$ H_{\overline{g}}$$ associated to some restricted function g on the complex space $$\mathbf{C}^n$$. We, in particular, describe the boundedness and compactness of operators $$H_{g}$$ and $$ H_{\overline{g}}$$ acting between Fock spaces in terms of Berezin transforms of their inducing function g. Our results extend a recent work of Z. Hu and E. Wang and fills the remaining gap when the largest Fock spaces are taken into account. And for $$1 \le s, p \le \infty $$, we also obtain the characterization on $$IMO^{s,p}$$, the space of functions satisfying an integral condition for the mean oscillation, via Berezin transform.

In this paper,we discuss the complex interpolations for Bergman space with exponential weights Apφ using the knowledge of Complex Functions and Functional Analysis.As an application,the houndedness of Berezin transforms on Lebesgue space Lp is obtained.

In this article, given some positive Borel measure mu, we define two integration operators to be I-mu(f)(z) = integral(D)f(w)K(z, w)e(-2 phi(w))d mu(w) and J(mu)(f)(z) = integral(D)vertical bar f(w)K(z, w)vertical bar e(-2 phi(w))d mu(w). We characterize the boundedness and compactness of these operators from the Bergman space A(phi)(p) to L-phi(q) for 1 < p, q < infinity, where phi belongs to a large class W-0, which covers those defined by Borichev, Dhuez, and Kellay in 2007. We also completely describe those mu's such that the embedding operator is bounded or compact from A(phi)(p) to L-phi(q)(d mu), 0 < p, q < infinity.

In this paper we introduce a kind of Bergman space Aφp on the unit disk D with exponential weights, which cover those defined by Borichev et al. (2007) [6]. We obtain upper and lower bound estimates on the Bergman kernel. As an application, we discuss the Bergman projection and duality.

Given the subharmonic function phi is an element of C-2(C-n), withM-1 omega(0) <= dd(c)phi <= M-2 omega(0),we define the Fock space F-p(phi), which was first introduced by A. P. Schuster and D. Varolin. In this paper, we characterize the symbols g for which the induced Hankel operators H-g and H-(g) over bar are simultaneously bounded (or compact) from Fock space F-p(phi) to Lebesgue space F-q(phi) for all possible 1 <= p, q < infinity.

We introduce and study a family of Banach spaces $$IMO^{s,\sigma }_p$$ of functions on the open unit ball $${\mathbb {B}}_n$$ in $${\mathbb {C}}^n$$ , defined by an integrability condition of the mean oscillation in the Bergman metric. Our main results include a decomposition theorem for $$IMO^{s,\sigma }_p$$ , a characterization of holomorphic functions in $$IMO^{s,\sigma }_p$$ , and an application of $$IMO^{s,\sigma }_p$$ to the boundedness and compactness of Hankel operators between weighted Bergman spaces.