Carleson-type embeddings with closed range

Besov spaces and Carleson measures on the ball

Carleson measures and interpolating and sampling sequences for weighted Bergman spaces on the unit disk are described for weights that are radial and grow faster than the standard weights (1 − |z|)−α, 0 < α < 1. These results make the Hardy space H 2 appear naturally as a “degenerate” endpoint case for the class of Bergman spaces under study.

1. IntroductionLet D = fz : jzj< 1gbe the unit disk in the complex plane and letH(D) denote the space of all analytic functions on D. Recall that a positiveBorel measure on Dis called a Carleson measure if there is a positive niteconstant Ksuch that(1) (S(I)) KjIjfor all arcs Iˆ@D, where jIjdenotes the normalized arc length of I(so thatj@Dj= 1) and S(I) is the Carleson square de ned byS(I) = fz: 1 j Ij<jzj<1; z=jzj2Ig:Carleson measures are ubiquitous in the study of function-theoretic operatortheory. A fundamental property of Carleson measures due to L. Carlesonaddresses the issue of when the inclusion map is bounded from the Hardyspace H

Let $\mathcal {M}$ be an Ahlfors $n$-regular Riemannian manifold such that either the Ricci curvature is non-negative or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. In the paper [IMRN, 2022, no. 2, 1245-1269] of Brazke–Schikorra–Sire, the authors characterised the BMO function $u : \mathcal {M} \to \mathbb {R}$ by a Carleson measure condition of its $\sigma$-harmonic extension $U:\mathcal {M}\times \mathbb {R}_+ \to \mathbb {R}$. This paper is concerned with the similar problem under a more general Dirichlet metric measure space setting, and the limiting behaviours of BMO &amp; Carleson measure, where the heat kernel admits only the so-called diagonal upper estimate. More significantly, without the Ricci curvature condition, we relax the Ahlfors regularity to a doubling property, and remove the pointwise bound on the gradient of the heat kernel. Some similar results for the Lipschitz function are also given, and two open problems related to our main result are considered.

Bergman Projections and Operators on Hardy Spaces

Recently J. Miao proved that if is a holomorphic function with Hadamard gaps on the open unit disc D then f ∈ Xp if and only if f ∈ Bp if and only if if and if only if where Xp, Bp and denote respectively the class of holomorphic functions on D which satisfy |f′(z)|p(1 − |z|2)p − 1dxdy is a finite measure, a Carleson measure and a little Carleson measure on D. In this paper we give a higher-dimensional version of Miao's result.

Recently Choe et al. have introduced the notion of dual Berezin transforms and used it to obtain new characterizations of the Carleson measures for the weighted Bergman spaces over the unit ball in C n . Continuing our investigation on the Hardy spaces, we obtain new characterizations of the Carleson measures for the Hardy spaces by means of the dual Poisson–Szego transforms introduced by Koosis. Compared with the results for the weighted Bergman spaces, our results for the Hardy spaces not only show an similarity, but also reveal a new characterization.

In this article, given some positive Borel measure μ, we define two integration operators to be Iμ(f)(z)=∫Df(w)K(z,w)e−2φ(w)dμ(w) and Jμ(f)(z)=∫D|f(w)K(z,w)|e−2φ(w)dμ(w). We characterize the boundedness and compactness of these operators from the Bergman space Aφp to Lφq for 1<p,q<∞, where φ belongs to a large class W0, which covers those defined by Borichev, Dhuez, and Kellay in 2007. We also completely describe those μ’s such that the embedding operator is bounded or compact from Aφp to Lφq(dμ), 0<p,q<∞.

We introduce area operators Aμ,l in the Dirichlet series setting for l&gt;0 and positive Borel measures μ on the right half-plane C0. It is proved that if μ is a Carleson measure on C0, then for 0&lt;p&lt;∞, the area operator Aμ,l is bounded from the Hardy space Hp0 of Dirichlet series vanishing at +∞ to some Lp-space. We also give an application of our methods to Volterra operators.

Carleson measures and vanishing Carleson measures for Hardy spaces are characterized by using product of Hardy functions. The results are applied to obtain characterizations of bounded and compact Riemann–Stieltjes operators and pointwise multiplication operators from Hardy spaces into the general family of spaces F(p, q, s), which include many classical function spaces, such as BMOA and the Bloch space.

Carleson measures on the generalized Hartogs triangles

COMPACT TOEPLITZ OPERATORS ON Ap (ϕ) (1⩽p<∞)

Let f_1,f_2 be linearly independent solutions of f''+Af=0, where the coefficient A is an analytic function in the open unit disc {mathbb {D}} of the complex plane {mathbb {C}}. It is shown that many properties of this differential equation can be described in terms of the subharmonic auxiliary function u=-log , (f_1/f_2)^{#}. For example, the case when sup _{zin {mathbb {D}}} |A(z)|(1-|z|^2)^2 < infty and f_1/f_2 is normal, is characterized by the condition sup _{zin {mathbb {D}}} |nabla u(z)|(1-|z|^2) < infty . Different types of Blaschke-oscillatory equations are also described in terms of harmonic majorants of u. Even if f_1,f_2 are bounded linearly independent solutions of f''+Af=0, it is possible that sup _{zin {mathbb {D}}} |A(z)|(1-|z|^2)^2 = infty or f_1/f_2 is non-normal. These results relate to sharpness discussion of recent results in the literature, and are succeeded by a detailed analysis of differential equations with bounded solutions. Analogues for the Nevanlinna class are also considered, by taking advantage of Nevanlinna interpolating sequences. It is shown that, instead of considering solutions with prescribed zeros, it is possible to construct a bounded solution of f''+Af=0 in such a way that it solves an interpolation problem natural to bounded analytic functions, while |A(z)|^2(1-|z|^2)^3, dm(z) remains to be a Carleson measure.

We extend a discrete version of an extension of Carleson’s theorem proved in [5] to a large class of trees that have certain radial properties. We introduce the geometric notion of s-vanishing Carleson measure on such a tree T (with s ≥ 1) and give several characterizations of such measures. Given a measure σ on T and p ≥ 1, let Lp(σ) denote the space of functions g defined on T such that |g|p is integrable with respect to σ and let Lp(∂T) be the space of functions f defined on the boundary of T such that |f|p is integrable with respect to the representing measure of the harmonic function 1.We prove the following extension of the discrete version of a classical theorem in the unit disk proved by Power.

Toeplitz and asymptotic Toeplitz operators on [formula omitted