Carleson Measures Research Articles

Joint Carleson measure for the difference of composition operators on the polydisks

ABSTRACT In Koo and Wang (Joint Carleson measure and the difference of composition operators on . J Math Anal Appl. 2014;419:1119–1142), the authors introduced a concept of joint Carleson measure and used it to characterize when the difference of two composition operators on weighted Bergman space over the unit ball is bounded or compact. In this paper, we extend the concept of joint Carleson measure to the polydisk setting and obtain analogue characterizations of the boundedness (compactness, resp.) of the difference of two composition operators on the weighted Bergman spaces over the unit polydisk, which may provide a unified approach for various ad hoc studies on the boundedness or the compactness of the difference of composition operators on polydisk. Moreover, we construct a concrete example to show that both the boundedness and the compactness depend on the index p when the dimension , which is in sharp contrast with the one-variable case where the boundedness and the compactness of the difference of two composition operators are independent of p>0. Due to the complexity of the Carleson measure on the unit polydisk, some new techniques are required in the polydisk setting.

Carleson perturbations of elliptic operators on domains with low dimensional boundaries

We prove an analogue of a perturbation result for the Dirichlet problem of divergence form elliptic operators by Fefferman, Kenig and Pipher, for the degenerate elliptic operators of David, Feneuil and Mayboroda, which were developed to study geometric and analytic properties of sets with boundaries whose co-dimension is higher than 1. These operators are of the form −divA∇, where A is a weighted elliptic matrix crafted to weigh the distance to the high co-dimension boundary in a way that allows for the nourishment of an elliptic PDE theory. When this boundary is a d-Alhfors-David regular set in Rn with d∈[1,n−1) and n≥3, we prove that the membership of the harmonic measure in A∞ is preserved under Carleson measure perturbations of the matrix of coefficients, yielding in turn that the Lp-solvability of the Dirichlet problem is also stable under these perturbations (with possibly different p). If the Carleson measure perturbations are suitably small, we establish solvability of the Dirichlet problem in the same Lp space. One of the corollaries of our results together with a previous result of David, Engelstein and Mayboroda, is that, given any d-ADR boundary Γ with d∈[1,n−2), n≥3, there is a family of degenerate operators of the form described above whose harmonic measure is absolutely continuous with respect to the d-dimensional Hausdorff measure on Γ.

Beurling–Ahlfors extension by heat kernel, A∞‐weights for VMO, and vanishing Carleson measures

We investigate a variant of the Beurling-Ahlfors extension of quasisymmetric homeomorphisms of the real line that is given by the convolution of the heat kernel, and prove that the complex dilatation of such a quasiconformal extension of a strongly symmetric homeomorphism (i.e. its derivative is an ${\rm A}_\infty$-weight whose logarithm is in VMO) induces a vanishing Carleson measure on the upper half-plane.

A new Carleson measure adapted to multi-level ellipsoid covers

<p style='text-indent:20px;'>We develop highly anisotropic Carleson measure over multi-level ellipsoid covers <inline-formula><tex-math id="M1">\begin{document}$ \Theta $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> that are highly anisotropic in the sense that the ellipsoids can change rapidly from level to level and from point to point. Then we show that the Carleson measure <inline-formula><tex-math id="M3">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> is sufficient for which the integral defines a bounded operator from <inline-formula><tex-math id="M4">\begin{document}$ H^p(\Theta) $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M5">\begin{document}$ L^p(\mathbb{R}^{n+1}, \, \mu),\ 0<p\leq 1 $\end{document}</tex-math></inline-formula>. Finally, we give several equivalent Carleson measures adapted to multi-level ellipsoid covers and obtain a specific Carleson measure induced by the highly anisotropic BMO functions.</p>

Weak products of complete Pick spaces

Let H be the Drury-Arveson or Dirichlet space of the unit ball of Cd. The weak product H ☉ H of H is the collection of all functions h that can be written as h =∑∞n=1 fngn, where ∑∞n=1 ||fn|| ||gn|| < ∞. We show that H ☉ H is contained in the Smirnov class of H; that is, every function in H ☉ H is a quotient of two multipliers of H, where the function in the denominator can be chosen to be cyclic in H . As a consequence, we show that the map N → closH ☉H N establishes a one-to-one and onto correspondence between the multiplier invariant subspaces of H and of H ☉ H . The results hold for many weighted Besov spaces H in the unit ball of Cd provided the reproducing kernel has the complete Pick property. One of our main technical lemmas states that, for weighted Besov spaces H that satisfy what we call the multiplier inclusion condition, any bounded column multiplication operator H → ⊕∞n=1 H induces a bounded row multiplication operator ⊕∞n=1 H → H . For the Drury-Arveson space Hd2 this leads to an alternate proof of the characterization of interpolating sequences in terms of weak separation and Carleson measure conditions. (Less)

The Dirichlet problem in domains with lower dimensional boundaries

The present paper pioneers the study of the Dirichlet problem with $L^q$ boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in $\Omega := \mathbb{R}^n \setminus \mathbb{R}^d$, with $d < n-1$. Following results of David, Feneuil and Mayboroda, we introduce an appropriate degenerate elliptic operator and show that the Dirichlet problem is solvable for all $q > 1$, provided that the coefficients satisfy the small Carleson norm condition. Even in the context of the classical case $d = n-1$, (the analogues of) our results are new. The conditions on the coefficients are more relaxed than the previously known ones (most notably, we do not impose any restrictions whatsoever on the first $n-1$ rows of the matrix of coefficients) and the results are more general. We establish local rather than global estimates between the square function and the non-tangential maximal function and, perhaps even more importantly, we establish new Moser-type estimates at the boundary and improve the interior ones.

Converse growth estimates for ODEs with slowly growing solutions

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.

English

We study Carleson measures and Toeplitz operators on the class of so-called small weighted Bergman spaces, introduced recently by Seip. A characterization of Carleson measures is obtained which extends Seip’s results from the unit disk of ℂ to the unit ball of ℂn. We use this characterization to give necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators. Finally, we study the Schatten p classes membership of Toeplitz operators for 1 < p < ∞.

Tent space well-posedness for parabolic Cauchy problems with rough coefficients

We study the well-posedness of Cauchy problems on the upper half space R+n+1 associated to higher order systems ∂tu=(−1)m+1divmA∇mu with bounded measurable and uniformly elliptic coefficients. We address initial data lying in Lp (1<p<∞) and BMO (p=∞) spaces and work with weak solutions. Our main result is the identification of a new well-posedness class, given for p∈(1,∞] by distributions satisfying ∇mu∈Tmp,2, where Tmp,2 is a parabolic version of the tent space of Coifman–Meyer–Stein. In the range p∈[2,∞], this holds without any further constraints on the operator and for p=∞ it provides a Carleson measure characterization of BMO with non-autonomous operators. We also prove higher order Lp well-posedness, previously only known for the case m=1. The uniform Lp boundedness of propagators of energy solutions plays an important role in the well-posedness theory and we discover that such bounds hold for p close to 2. This is a consequence of local weak solutions being locally Hölder continuous with values in spatial Llocp for some p>2, what is also new for the case m>1.

Toeplitz operators and skew Carleson measures for weighted Bergman spaces on strongly pseudoconvex domains

In this paper we study properties of Toeplitz operators on weighted Bergman spaces of bounded strongly pseudoconvex domains. We prove that a Toeplitz operator built using a weighted Bergman kernel of weight β and integrating against a measure μ maps continuously a weighted Bergman space Ap1α1(D) into Ap2α2(D) if and only if μ is a (λ,γ)-skew Carleson measure, where λ=1+1p1−1p2 and γ=1λ(β+α1p1−α2p2). This generalizes results obtained by Pau and Zhao on the unit ball, and by Abate, Raissy and Saracco on a smaller class of Toeplitz operators on strongly pseudoconvex domains.

Boundary value problems for second-order elliptic operators with complex coefficients

The theory of second order complex coefficient operators of the form $\mathcal{L}=\mbox{div} A(x)\nabla$ has recently been developed under the assumption of $p$-ellipticity. In particular, if the matrix $A$ is $p$-elliptic, the solutions $u$ to $\mathcal{L}u = 0$ will satisfy a higher integrability, even though they may not be continuous in the interior. Moreover, these solutions have the property that $|u|^{p/2-1}u \in W^{1,2}_{loc}$. These properties of solutions were used by Dindo\v{s}-Pipher to solve the $L^p$ Dirichlet problem for $p$-elliptic operators whose coefficients satisfy a further regularity condition, a Carleson measure condition that has often appeared in the literature in the study of real, elliptic divergence form operators. This paper contains two main results. First, we establish solvability of the Regularity boundary value problem for this class of operators, in the same range as that of the Dirichlet problem. The Regularity problem, even in the real elliptic setting, is more delicate than the Dirichlet problem because it requires estimates on derivatives of solutions. Second, the Regularity results allow us to extend the previously established range of $L^p$ solvability of the Dirichlet problem using a theorem due to Z. Shen for general bounded sublinear operators.

Perturbations of elliptic operators in 1-sided chord-arc domains. Part II: Non-symmetric operators and Carleson measure estimates

We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness) and which have an Ahlfors regular boundary, a result of Kenig-Kirchheim-Pipher-Toro, in which Carleson measure estimates for bounded solutions of the equation L u = − div ⁡ ( A ∇ u ) = 0 Lu=-\operatorname {div}(A\nabla u) = 0 with A A being a real (not necessarily symmetric) uniformly elliptic matrix imply that the corresponding elliptic measure belongs to the Muckenhoupt A ∞ A_\infty class with respect to surface measure on the boundary. We present two applications of this result. In the first one we extend a perturbation result recently proved by Cavero-Hofmann-Martell presenting a simpler proof and allowing non-symmetric coefficients. Second, we prove that if an operator L L as above has locally Lipschitz coefficients satisfying certain Carleson measure condition, then ω L ∈ A ∞ \omega _L\in A_\infty if and only if ω L ⊤ ∈ A ∞ \omega _{L^\top }\in A_\infty . As a consequence, we can remove one of the main assumptions in the non-symmetric case of a result of Hofmann-Martell-Toro and show that if the coefficients satisfy a slightly stronger Carleson measure condition the membership of the elliptic measure associated with L L to the class A ∞ A_\infty yields that the domain is indeed a chord-arc domain.

A criterion for uniform approximability of individual functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients

A natural counterpart of Vitushkin’s criterion is obtained in the problem of uniform approximation of functions by solutions of second-order homogeneous elliptic equations with constant complex coefficient on compact subsets of , . It is stated in terms of a single (scalar) capacity connected with the leading coefficient of the Laurent series. The scheme of approximation uses methods in the theory of singular integrals and, in particular, constructions of certain special Lipschitz surfaces and Carleson measures. Bibliography: 23 titles.

On Two-Weight Norm Inequalities for Positive Dyadic Operators

Let σ and ω be locally finite Borel measures on ℝd, and let pin (1,infty ) and qin (0,infty ). We study the two-weight norm inequality lVert T(fsigma ) rVert _{L^{q}(omega )}leq C lVert f rVert _{L^{p}(sigma )}, quad text {for all} f in L^{p}(sigma ), for both the positive summation operators T = Tλ(⋅σ) and positive maximal operators T = Mλ(⋅σ). Here, for a family {λQ} of non-negative reals indexed by the dyadic cubes Q, these operators are defined by T_{lambda }(fsigma ):={sum }_{Q} lambda _{Q} langle {f}rangle ^{sigma }_{Q} 1_{Q} quad text { and } quad M_{lambda }(fsigma ):=sup _{Q} lambda _{Q} langle {f}rangle ^{sigma }_{Q} 1_{Q}, where langle {f}rangle ^{sigma }_{Q}:=frac {1}{sigma (Q)} {int limits }_{Q} |f| d sigma . We obtain new characterizations of the two-weight norm inequalities in the following cases: (1) For T = Tλ(⋅σ) in the subrange q < p. Under the additional assumption that σ satisfies the A_{infty } condition with respect to ω, we characterize the inequality in terms of a simple integral condition. The proof is based on characterizing the multipliers between certain classes of Carleson measures. (2) For T = Mλ(⋅σ) in the subrange q < p. We introduce a scale of simple conditions that depends on an integrability parameter and show that, on this scale, the sufficiency and necessity are separated only by an arbitrarily small integrability gap. (3) For the summation operators T = Tλ(⋅σ) in the subrange 1 < q < p. We characterize the inequality for summation operators by means of related inequalities for maximal operators T = Mλ(⋅σ). This maximal-type characterization is an alternative to the known potential-type characterization. The subrange of the exponents q < p appeared recently in applications to nonlinear elliptic PDE with lambda _{Q} = sigma (Q) |Q|^{frac {alpha }{d}-1}, α ∈ (0, d). In this important special case Tλ is a discrete analogue of the Riesz potential I_{alpha }=(-{Delta })^{-frac {alpha }{2}}, and Mλ is the dyadic fractional maximal operator.

Strongly Quasiconformal Extension of Harmonic Mappings

In terms of pre-Schwarzian and Schwarzian derivatives, we obtain some sufficient conditions to ensure that a sense-preserving locally univalent harmonic mapping f in the unit disk can be extended to a quasiconformal mapping in the complex plane so that its complex dilatation satisfies some Carleson measure conditions.

Difference of composition operators over the half-plane

To overcome the unboundedness of the half-plane, we use Khinchine’s inequality and atom decomposition techniques to provide joint Carleson measure characterizations when the difference of composition operators is bounded or compact from standard weighted Bergman spaces to Lebesgue spaces over the half-plane for all index choices. For applications, we obtain direct analytic characterizations of the bounded and compact differences of composition operators on such spaces. This paper concludes with a joint Carleson measure characterization when the difference of composition operators is Hilbert-Schmidt.

On BMO and Carleson Measures on Riemannian Manifolds

Abstract Let $\mathcal{M}$ be a Riemannian $n$-manifold with a metric such that the manifold is Ahlfors regular. We also assume either non-negative Ricci curvature or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. We characterize BMO-functions $u: \mathcal{M} \to \mathbb{R}$ by a Carleson measure condition of their $\sigma $-harmonic extension $U: \mathcal{M} \times (0,\infty ) \to \mathbb{R}$. We make crucial use of a $T(b)$ theorem proved by Hofmann, Mitrea, Mitrea, and Morris. As an application, we show that the famous theorem of Coifman–Lions–Meyer–Semmes holds in this class of manifolds: Jacobians of $W^{1,n}$-maps from $\mathcal{M}$ to $\mathbb{R}^n$ can be estimated against BMO-functions, which now follows from the arguments for commutators recently proposed by Lenzmann and the 2nd-named author using only harmonic extensions, integration by parts, and trace space characterizations.

Carleson measure characterizations of the Campanato type space associated with Schrödinger operators on stratified Lie groups

Abstract Let ℒ = - Δ 𝔾 + V {\mathcal{L}=-{\Delta}_{\mathbb{G}}+V} be a Schrödinger operator on the stratified Lie group 𝔾 {\mathbb{G}} , where Δ 𝔾 {{\Delta}_{\mathbb{G}}} is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class B q 0 {B_{q_{0}}} with q 0 &gt; 𝒬 / 2 {q_{0}&gt;\mathcal{Q}/2} and 𝒬 {\mathcal{Q}} is the homogeneous dimension of 𝔾 {\mathbb{G}} . In this article, by Campanato type spaces Λ ℒ α ⁢ ( 𝔾 ) {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})} , we introduce Hardy type spaces associated with ℒ {\mathcal{L}} denoted by H ℒ p ⁢ ( 𝔾 ) {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})} and prove the atomic characterization of H ℒ p ⁢ ( 𝔾 ) {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})} . Further, we obtain the following duality relation: Λ ℒ 𝒬 ⁢ ( 1 / p - 1 ) ⁢ ( 𝔾 ) = ( H ℒ p ⁢ ( 𝔾 ) ) ∗ , 𝒬 / ( 𝒬 + δ ) &lt; p &lt; 1 for ⁢ δ = min ⁡ { 1 , 2 - 𝒬 / q 0 } . \Lambda_{\mathcal{L}}^{\mathcal{Q}(1/p-1)}(\mathbb{G})=(H^{{p}}_{\vphantom{% \varepsilon}{\mathcal{L}}}(\mathbb{G}))^{\ast},\quad\mathcal{Q}/(\mathcal{Q}+% \delta)&lt;p&lt;1\quad\text{for}\ \delta=\min\{1,2-\mathcal{Q}/q_{0}\}. The above relation enables us to characterize Λ ℒ α ⁢ ( 𝔾 ) {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})} via two families of Carleson measures generated by the heat semigroup and the Poisson semigroup, respectively. Also, we obtain two classes of perturbation formulas associated with the semigroups related to ℒ {\mathcal{L}} . As applications, we obtain the boundedness of the Littlewood–Paley function and the Lusin area function on Λ ℒ α ⁢ ( 𝔾 ) {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})} .

Volterra type integration operators from Bergman spaces to Hardy spaces

We completely characterize the boundedness of the Volterra type integration operators Jb acting from the weighted Bergman spaces Aαp to the Hardy spaces Hq of the unit ball of Cn for all 0<p,q<∞. A partial solution to the case n=1 was previously obtained by Z. Wu in [35]. We solve the cases left open there and extend all the results to the setting of arbitrary complex dimension n. Our tools involve area methods from harmonic analysis, Carleson measures and Kahane-Khinchine type inequalities, factorization tricks for tent spaces of sequences, as well as techniques and integral estimates related to Hardy and Bergman spaces.

Localization, Carleson measure and BMO Toeplitz operators on the Bergman space

In this paper we prove that a Toeplitz operator with complex Borel measure symbol, whose total variation is Carleson, is weakly localized on the Bergman space. We define strongly localized and sufficiently localized operators on the Bergman space, and show that they are also weakly localized. Finally, we show that bounded Toeplitz operators with BMO1 symbols are strongly (and therefore also weakly) localized, and that the Bergman space Toeplitz algebra has yet another characterization as the C⁎-algebra generated by this class of operators.