Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂_t-Δ_x)u(t,x)=0 on subject to u(0,x)=f(x) on Rⁿ. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t²,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞).     

Ahlfors-regular curves and Carleson measures

Science China Mathematics - We study the relation between the boundary of a simply connected domain Ω being Ahlfors-regular and the invariance of Carleson measures under the push-forward...     

Carleson measures and vector-valued BMO martingales

We study the relationship between vector-valued BMO martingales and Carleson measures. Let ${(\Omega,\mathcal {F} ,P)}$ be a probability space and 2 ≤ q < ∞. Let X be a Banach space. Given a stopping time τ, let ${\widehat{\tau}}$ denote the tent over τ: $$\widehat{\tau}=\{(w,k)\in \Omega\times \mathbb {N}: \tau(w)\leq k, \tau(w) < \infty\}.$$ We prove that there exists a positive constant c such that $$\sup_{\tau}\frac{1}{P(\tau < \infty)}\int \limits_{{\widehat{\tau}}}\|df_k\|^qdP\otimes dm\leq c^q\|f\|_{BMO(X)}^q$$ for any finite martingale with values in X iff X admits an equivalent norm which is q-uniformly convex. The validity of the converse inequality is equivalent to the existence of an equivalent p-uniformly smooth norm. And then we also give a characterization of UMD Banach lattices.     

Carleson measures on planar sets

In this paper, we investigate what are Carleson measures on open subsets in the complex plane. A circular domain is a connected open subset whose boundary consists of finitely many disjoint circles. We call a domain G multi-nicely connected if there exists a circular domain W and a conformal map ψ from W onto G such that ψ is almost univalent with respect the arclength on δW. We characterize all Carleson measures for those open subsets so that each of their components is multinicely connected and harmonic measures of the components are mutually singular. Our results suggest the extension of Carleson measures probably is up to this class of open subsets     

Carleson measures for the area Nevanlinna spaces and applications

Let 1 ≤ p < ∞ and let μ be a finite positive Borel measure on the unit disk D. The area Nevanlinna-Lebesgue space N ^p (μ) consists of all measurable functions h on D such that log⁺ |h| ∈ L ^p (μ), and the area Nevanlinna space N _α ^p is the subspace consisting of all holomorphic functions, in N ^p ((1−|z|²)^α dv(z)), where α > −1 and ν is area measure on D. We characterize Carleson measures for N _α ^p , defined to be those measures μ for which N _α ^p ⊂ N ^p (μ). As an application, we show that the spaces N _α ^p are closed under both differentiation and integration. This is in contrast to the classical Nevanlinna space, defined by integration on circles centered at the origin, which is closed under neither. Applications to composition operators and to integral operators are also given. The second author was supported in part by KRF-2004-015-C00019.     

Characterization of Carleson measures by the Hausdorff-Young property

It is shown that the Laplace transform of an L ^p (1 < p ≤ 2) function defined on the positive semiaxis satisfies the Hausdorff-Young type inequality with a positive weight in the right complex half-plane if and only if the weight is a Carleson measure. In addition, Carleson’s weighted L ^p inequality for the harmonic extension is given with a numeric constant.     

Carleson measures and the generalized Campanato spaces of vector-valued martingales

In this paper, the so-called(p, φ)-Carleson measure is introduced and the relationship between vector-valued martingales in the general Campanato spaces Lp,φ(X) and the(p, φ)-Carleson measures is investigated. Specifically, it is proved that for q ∈ [2, ∞), the measure dμ := ||dfk||~qdP ? dm is a(q, φ)-Carleson measure on ? × N for every f ∈ L_q,φ(X)if and only if X has an equivalent norm which is q-uniformly convex; while for p ∈(1, 2], the measure dμ :=||dfk||~pdP ? dm is a(p, φ)-Carleson measure on ? × N implies that f ∈ L_p,φ(X)if and only if X admits an equivalent norm which is p-uniformly smooth. This result extends an earlier result in the literature from BMO spaces to general Campanato spaces.     

Carleson measures and some classes of meromorphic functions

For let be the Möbius transformation defined by , and let be the Green's function of the unit disk . We construct an analytic function belonging to for all , , but not belonging to meromorphic in and for any , . This gives a clear difference as compared to the analytic case where the corresponding function spaces ( and ) are same.

     

Carleson measures for weighted holomorphic Besov spaces

We obtain characterizations of positive Borel measures μ on B ⁿ so that some weighted holomorphic Besov spaces B _s ^p (ω,B ⁿ ) are embedded in L ^p (d μ).     

Strong type estimate and Carleson measures for Lipschitz spaces

We establish a capacitary strong type estimate for Lipschitz space and characterize the related Carleson measures.

     

Carleson measures for Besov spaces on the ball with applications

Carleson and vanishing Carleson measures for Besov spaces on the unit ball of are characterized in terms of Berezin transforms and Bergman-metric balls. The measures are defined via natural imbeddings of Besov spaces into Lebesgue classes by certain combinations of radial derivatives. Membership in Schatten classes of the imbeddings is considered too. Some Carleson measures are not finite, but the results extend and provide new insight to those known for weighted Bergman spaces. Special cases pertain to Arveson and Dirichlet spaces, and a unified view with the usual Hardy-space Carleson measures is presented by letting the order of the radial derivatives tend to 0. Weak convergence in Besov spaces is also characterized, and weakly 0-convergent families are exhibited. Applications are given to separated sequences, operators of Forelli–Rudin type, gap series, characterizations of weighted Bloch, Lipschitz, and growth spaces, inequalities of Fejér–Riesz and Hardy–Littlewood type, and integration operators of Cesàro type.     

Carleson measures on product domains and 2-parameter Brownian martingales

Archiv der Mathematik -     

Carleson measures and conformal self‐mappings in the real unit ball

Bounded and compact Carleson measures in the unit ball B of R ⁿ , n ≥ 2, are characterized by means of global Dirichlet integrals of the conformal self‐map T_a taking a ∈ B to the origin. The same proof applies in the unit ball of C ⁿ . It is also proved that the powers of the Jacobian of T_a satisfy the weak Harnack inequality and even Harnack's inequality with a constant independent of a. As an application of these results it is shown that the two different definitions for Carleson measures in the existing literature are equivalent for a certain range of parameter values. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Carleson measures and balayage for Bergman spaces of strongly pseudoconvex domains

Given a bounded strongly pseudoconvex domain D in with smooth boundary, we characterize ‐Bergman Carleson measures for , , and . As an application, we show that the Bergman space version of the balayage of a Bergman Carleson measure on D belongs to BMO in the Kobayashi metric.     

Carleson and sampling measures on Bernstein spaces on Siegel CR manifolds

In this paper, we introduce and study Carleson and sampling measures on Bernstein spaces on a class of quadratic CR manifold called Siegel CR manifolds. These are spaces of entire functions of exponential type whose restrictions to the given Siegel CR manifold are $L^p$-integrable with respect to a natural measure. For these spaces, we prove necessary and sufficient conditions for a Radon measure to be a Carleson or a sampling measure. We also provide sufficient conditions for sampling sequences.     

Carleson measures, multipliers and integration operators for spaces of Dirichlet type

For 0<p<∞ and α>−1, we let denote the space of those functions f which are analytic in the unit disc and satisfy . In this paper we characterize the positive Borel measures μ in D such that , 0<p<q<∞. We also characterize the pointwise multipliers from to (0<p<q<∞) if p−2<α<p. In particular, we prove that if the only pointwise multiplier from to (0<p<q<∞) is the trivial one. This is not longer true for and we give a number of explicit examples of functions which are multipliers from to for this range of values.