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We consider the weighted Bergman space A ψ 2 of all holomorphic functions on B n square integrable with respect to an exponential weight measure e - ψ d V on B n , where ψ (z) : = 1 1 - | z | 2 . We characterize boundedness (or compactness) of Toeplitz operators and Hankel operators on A ψ 2 . [ABSTRACT FROM AUTHOR]

To study the boundedness of operators on multi-parameter local Hardy space h p (R n 1 × R n 2) , inhomogeneous Journé class has been introduced. It is well known that operators in the Journé class are bounded on multi-parameter Hardy space H p (R n 1 × R n 2) if and only if T 1 ∗ (1) = T 2 ∗ (1) = 0 for p near 1. Under the same conditions, operators in inhomogeneous Journé class are bounded on h p (R n 1 × R n 2) . In this paper, We give an operator belonging to the inhomogeneous Journé class without T 1 ∗ (1) = T 2 ∗ (1) = 0 and prove its boundedness from h p (R n 1 × R n 2) to h p (R n 1 × R n 2) by almost orthogonality estimates. It implies that T 1 ∗ (1) = T 2 ∗ (1) = 0 is not a necessary condition for the boundedness on h p (R n 1 × R n 2) of a singular operator in the inhomogeneous Journé class. [ABSTRACT FROM AUTHOR]

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

Suppose that Ω ⊂ R n + 1 , n ≥ 1 , is a uniform domain with n-Ahlfors regular boundary and L is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in Ω . We show that the corresponding elliptic measure ω L is quantitatively absolutely continuous with respect to surface measure of ∂ Ω in the sense that ω L ∈ A ∞ (σ) if and only if any bounded solution u to L u = 0 in Ω is ε -approximable for any ε ∈ (0 , 1) . By ε -approximability of u we mean that there exists a function Φ = Φ ε such that ‖ u - Φ ‖ L ∞ (Ω) ≤ ε ‖ u ‖ L ∞ (Ω) and the measure μ ~ Φ with d μ ~ = | ∇ Φ (Y) | d Y is a Carleson measure with L ∞ control over the Carleson norm. As a consequence of this approximability result, we show that boundary BMO functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy L 1 -type Carleson measure estimates with BMO control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis. [ABSTRACT FROM AUTHOR]

Let (X, d, μ) be a metric measure space satisfying the doubling condition and an L2-Poincaré inequality. Consider the nonnegative operator L generalized by a Dirichlet form on X. We will show that a solution u to (− ∂ t 2 + L) u = 0 on X × ℝ+ satisfies an α-Carleson condition if and only if u can be represented as the Poisson integral of the operator L with the trace in the generalized Morrey space L2,α(X), where α is a nonnegative function defined on a class of balls in X. This result extends the analogous characterization founded by R. Jiang, J. Xiao, D. Yang (2016) from the classical Morrey space on Euclidean space to the generalized Morrey space on the metric measure space. [ABSTRACT FROM AUTHOR]

We investigate an integral operator T t , λ which preserves the Carleson measure for the Möbius invariant Besov space B p on the unit ball of C n . A holomorphic function space W β p , associated with the Carleson measure for B p , is introduced. As applications for the operator T t , λ , we estimate the distance from Bloch-type functions to the space W β p , which extends Jones' formula. Moreover, the bounded small Hankel operators on B p and the atomic decomposition of W β p are characterized. [ABSTRACT FROM AUTHOR]

An RD-space X is a space of homogeneous type in the sense of Coifman and Weiss satisfying a certain reverse (volume) doubling condition. Let L be a non-negative self-adjoint operator acting on L 2 (X) . Assume that L generates an analytic semigroup { e - t L } t > 0 whose kernels { h t (x , y) } t > 0 satisfy a generalized Gaussian heat kernel upper estimate. Roughly speaking, the heat kernel behavior is a mixture of locally Gaussian and sub-Gaussian at infinity. With the help of this Gaussian heat kernel, we first introduce a novel Morrey space and then prove that it coincides with the classical Morrey space. As applications, some new characterizations of square Morrey space are established via a Carleson measure condition. [ABSTRACT FROM AUTHOR]

In this paper, we give a universal description of the boundedness and compactness of Toeplitz operator T μ ω between Bergman spaces A η p and A ν q when μ is a positive Borel measure, 1 < p,q < ∞ and ω, η, ν are regular weights. By using Khinchin's inequality and Kahane's inequality, we get a new characterization of the Carleson measure for Bergman spaces induced by regular weights. [ABSTRACT FROM AUTHOR]

Inspired by Xiao’s work about the Hankel measures for the weighted Bergman spaces, in this paper, if β > 0 and the measure μ is a complex Borel measure on the unit disk D , we define the Hankel type operator K μ , β by K μ , β : f ⟼ ∫ D (1 - w z) - (β) f (w) d μ (w). The operator itself has been widely studied when μ is a positive Borel measure supported on the interval [0, 1). We study the boundedness of K μ , 1 acting on Hardy spaces and the boundedness of K μ , α , α > 1 acting on weighted Bergman spaces. Then we raise and answer some questions about the boundedness of those operators. Also, we find some special measures μ ′ s such that s-Hankel measure is equal to s-Carleson measure. [ABSTRACT FROM AUTHOR]

In this paper, we study the corona theorem and Wolff theorem for the multiplier algebra of the Besov–Morrey space $ \mathcal {B}_p^{\lambda }(s) $ B p λ (s) when $ 0 0 < λ , s < 1 < p < ∞. Here the Besov–Morrey space $ \mathcal {B}_p^{\lambda }(s) $ B p λ (s) is the space consisting of those analytic functions on the unit disc $ {\mathbb D} $ D such that $ |f'(z)|^p(1-|z|^2)^{p+s-2}{\rm d}A(z) $ | f ′ (z) | p (1 − | z | 2) p + s − 2 d A (z) is a $ \lambda s $ λs -Carleson measure. [ABSTRACT FROM AUTHOR]

This paper characterizes boundedness and compactness of the identity operator I d : F (p , p - 2 , s) → T t , m q (μ) . Applying this result, we obtain the characterization of the boundedness of the Volterra integral operator T g from F (p , p - 2 , s) spaces to LF (q , q - 2 , t) spaces. Furthermore, the compactness and essential norm of T g : F (p , p - 2 , s) → LF (q , q - 2 , t) are also investigated. [ABSTRACT FROM AUTHOR]

Let D be the unit disc in the complex plane. Given a positive finite Borel measure μ on the radius [0, 1), we denote the n-th moment of μ as μ n , that is, μ n = ∫ [ 0 , 1) t n d μ (t). The Cesàro-like operator C μ , s is defined on H (D) as follows: If f (z) = ∑ n = 0 ∞ a n z n ∈ H (D) then C μ , s (f) is defined by C μ , s (f) (z) = ∑ n = 0 ∞ μ n ∑ k = 0 n Γ (n - k + s) Γ (s) (n - k) ! a k z n , z ∈ D. In this paper, our focus is on the action of the C e s a ` r o -type operator C μ , s on spaces of analytic functions in D . We characterize the boundedness (compactness) of the C e s a ` r o -like operator C μ , s , acting between the Bloch space B and the Bergman space A p . [ABSTRACT FROM AUTHOR]

Let $ 0 \lt p \lt \infty $ 0 < p < ∞ and $ K:[0,\infty)\to [0,\infty) $ K : [ 0 , ∞) → [ 0 , ∞) be a nondecreasing and right-continuous function. A new Dirichlet-Morrey type space $ \mathcal {D}^{p,K}_{p-1} $ D p − 1 p , K is introduced in this paper. The boundedness of the identity operator $ I_d $ I d from $ \mathcal {D}^{p,K}_{p-1} $ D p − 1 p , K to tent spaces $ \mathcal {T}_{K}^{p}(\mu) $ T K p (μ) is characterized. As an application, the boundedness, compactness and essential norm of Volterra integral operators on $ \mathcal {D}^{p,K}_{p-1} $ D p − 1 p , K are investigated. [ABSTRACT FROM AUTHOR]

Recently, Choe et al. obtained characterizations for bounded/compact differences of weighted composition operators acting from a standard weighted Bergman space into another over the unit disk. In this paper we extend those results to the ball setting. By devising a new approach regarding test functions, we improve the characterizations as well as the proofs. Namely, the Reproducing Kernel Thesis is added to the characterizations and our proofs, when restricted to the disk, are new and simpler. [ABSTRACT FROM AUTHOR]

Let (X , d , μ) be a metric measure space satisfying a doubling condition and the L 2 -Poincaré inequality. This paper is concerned with the boundary behavior of harmonic functions on the (open) upper half-space X × R + . We show that the traces of harmonic functions are in the bounded mean oscillation (BMO) space if and only if they satisfy the Carleson condition. This characterization is new even for uniformly elliptic operator on Euclidean spaces. [ABSTRACT FROM AUTHOR]

This paper characterises the boundedness and compactness of Agler–McCarthy monomial operators by reducing them to weighted composition operators and deriving explicit Carleson measure criteria on the half-plane. The results are illustrated by examples. [ABSTRACT FROM AUTHOR]

This article studies the boundedness of the inclusion mapping from weighted Bergman spaces Aαp into a class of tent type space Tsp,n(μ). As an application, the boundedness, compactness and essential norm of generalized integral operators Tgn,k and Sgn,0 from Aαp to general function spaces are also investigated. [ABSTRACT FROM AUTHOR]

A family of Dirichlet-Morrey spaces \(\mathcal{D}_{\lambda,K}\) of functions analytic in the open unit disk \(\mathbb{D}\) are defined in this paper. We completely characterize the boundedness of the Volterra integral operators \(T_g\), \(I_g\) and the multiplication operator \(M_g\) on the space \(\mathcal{D}_{\lambda,K}\). In addition, the compactness and essential norm of the operators \(T_g\) and \(I_g\) on \(\mathcal{D}_{\lambda,K}\) are also investigated.

Let L : = - 1 ω div (A ∇ ·) + μ be the generalized degenerate Schrödinger operator in L ω 2 (R n) ( n ≥ 3 ) with suitable weight ω and nonnegative Radon measure μ . In this article, the authors first introduce the Hardy spaces H L , S h p (R n , ω) and H L p (R n , ω) , respectively, via the Lusin area function and the maximal function associated with L, where p ∈ (0 , 1 ] , and then, show that, when p ∈ (n n + θ , 1 ] , H L , S h p (R n , ω) = H L p (R n , ω) with equivalent quasi-norms, where θ ∈ (0 , 1 ] is the critical index of Hölder continuity for the heat kernels { k t } t > 0 generated by L. As an application, the authors further obtain that the BMO type space BMO L (R n , ω) associated with L can be characterized by the Carleson measure. This result is also new even when L : = - 1 ω div (A ∇ ·) + V , where V ≥ 0 belongs to the reverse Hölder class with respect to the measure ω (x) d x . [ABSTRACT FROM AUTHOR]

We present a new characterization of Muckenhoupt A_{\infty }-weights whose logarithm is in vanishing mean oscillation (\mathbb {R}) in terms of vanishing Carleson measures on \mathbb {R}_+^2 and vanishing doubling weights on \mathbb {R}. This also gives a novel description of strongly symmetric homeomorphisms on the real line by using a geometric quantity. [ABSTRACT FROM AUTHOR]

The boundedness of the maximal operator on the upper half-plane \Pi ^{+} is established. Here \Pi ^+ is equipped with a positive Borel measure d\omega (y)dx satisfying the doubling property \omega ((0,2t))\leq C\omega ((0,t)). This result is connected to the Carleson embedding theorem, which we use to characterize the boundedness and compactness of the Volterra type integral operators on the Bergman spaces A_{\omega }^{p}(\Pi ^{+}). [ABSTRACT FROM AUTHOR]

Let (X , d , μ , E) be a Dirichlet metric measure space satisfying a doubling condition and supporting a scale-invariant L 2 -Poincaré inequality. Assume that L is a non-negative operator on L 2 (X) (similar to the negative Laplace operator on L 2 (R n) ) generalized by a Dirichlet form E . This paper is concerned with the boundary behavior of the caloric functions u(x, t) (i.e., the solution to the heat equation ∂ t u + L u = 0 ) on the upper half-space X × R + . We characterize all caloric functions with boundary value in bounded mean oscillation (BMO) space by means of certain Carleson measure condition. This extends the known result of Fabes–Neri [Duke Math. J., 1975, 725-734] from the Euclidean space to the metric measure space. As an application, we further consider the limiting behavior of the caloric function with the vanishing mean oscillation (CMO) trace, which is new even for the Laplace operator on Euclidean space. [ABSTRACT FROM AUTHOR]

Let μ\mu be a positive Borel measure on the interval [0,1)\left[0,1). The Hankel matrix Hμ=(μn,k)n,k≥0{{\mathcal{ {\mathcal H} }}}_{\mu }={\left({\mu }_{n,k})}_{n,k\ge 0} with entries μn,k=μn+k{\mu }_{n,k}={\mu }_{n+k}, where μn=∫[0,1)tndμ(t){\mu }_{n}={\int }_{\left[0,1)}{t}^{n}{\rm{d}}\mu \left(t), induces formally the operator as follows: DHμ(f)(z)=∑n=0∞∑k=0∞μn,kak(n+1)zn,z∈D,{{\mathcal{D {\mathcal H} }}}_{\mu }(f)\left(z)=\mathop{\sum }\limits_{n=0}^{\infty }\left(\mathop{\sum }\limits_{k=0}^{\infty }{\mu }_{n,k}{a}_{k}\right)\left(n+1){z}^{n},\hspace{1em}z\in {\mathbb{D}}, where f(z)=∑n=0∞anznf\left(z)={\sum }_{n=0}^{\infty }{a}_{n}{z}^{n} is an analytic function in D{\mathbb{D}}. In this article, we characterize those positive Borel measures on [0,1)\left[0,1) for which DHμ{{\mathcal{D {\mathcal H} }}}_{\mu } is bounded (resp. compact) from Dirichlet spaces Dα(0

Let $ \mu $ be a positive Borel measure on the interval $ [0, 1) $. The Hankel matrix $ \mathcal{H}_{\mu} = (\mu_{n, k})_{n, k\geq 0} $ with entries $ \mu_{n, k} = \mu_{n+k} $, where $ \mu_{n} = \int_{[0, 1)}t^nd\mu(t) $, formally induces the operator as follows: $ \mathcal{DH}_\mu(f)(z) = \sum\limits_{n = 0}^\infty\left(\sum\limits_{k = 0}^\infty \mu_{n,k}a_k\right)(n+1)z^n , \; z\in \mathbb{D}, $ where $ f(z) = \sum_{n = 0}^\infty a_nz^n $ is an analytic function in $ \mathbb{D} $. In this article, we characterize those positive Borel measures on $ [0, 1) $ such that $ \mathcal{DH}_\mu $ is bounded (resp., compact) from Bergman spaces $ \mathcal{A}^p $ into Hardy spaces $ H^q $, where $ 0 < p, q < \infty $.

Let A be a general expansive matrix and let X be a ball quasi-Banach function space on R n , whose certain power (namely its convexification) supports a Fefferman–Stein vector-valued maximal inequality and the associate space of whose other power supports the boundedness of the powered Hardy–Littlewood maximal operator. The authors first introduce some anisotropic ball Campanato-type function spaces associated with both A and X, prove that these spaces are dual spaces of anisotropic Hardy spaces H X A (R n) associated with both A and X, and obtain various anisotropic Littlewood–Paley function characterizations of H X A (R n) . Also, as applications, the authors establish several equivalent characterizations of anisotropic ball Campanato-type function spaces, which, combined with the atomic decomposition of tent spaces associated with both A and X, further induce their Carleson measure characterization. All these results have a wide range of generality and, particularly, even when they are applied to Morrey spaces and Orlicz-slice spaces, some of the obtained results are also new. The novelties of this article are reflected in that, to overcome the essential difficulties caused by the absence of both an explicit expression and the absolute continuity of the quasi-norm ‖ · ‖ X under consideration, the authors embed X under consideration into the anisotropic weighted Lebesgue space with certain special weight and then fully use the known results of this weighted Lebesgue space. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundedness and compactness of the Volterra integral operators from Dirichlet type spaces D p - 2 + s p into Hardy spaces H q when 1 < p < q < ∞ , q ≥ 2 and 0 < s < 1. The multipliers from D p - 2 + s p to H q are also investigated. [ABSTRACT FROM AUTHOR]

Let \mathcal {L}=-\Delta _{\mathbb {H}^n}+V be a Schrödinger operator with the nonnegative potential V belonging to the reverse Hölder class B_{Q}, where Q is the homogeneous dimension of the Heisenberg group \mathbb {H}^n. In this paper, we obtain pointwise bounds for the spatial derivatives of the heat and Poisson kernels related to \mathcal {L}. As an application, we characterize the space BMO_{\mathcal {L}}(\mathbb {H}^n), associated to the Schrödinger operator \mathcal {L}, in terms of two Carleson type measures involving the spatial derivatives of the heat kernel of the semigroup \{e^{-s\mathcal {L}}\}_{s>0} and the Poisson kernel of the semigroup \{e^{-s\sqrt {\mathcal {L}}}\}_{s>0}, respectively. At last, we pose a conjecture about the converse characterization of BMO_{\mathcal {L}}(\mathbb {H}^n). [ABSTRACT FROM AUTHOR]

In this paper, the boundedness and compactness of the identity operator from Qx spaces into tent spaces Tqλ,s are completely characterized in the unit ball of Cn when q > 2. As an application, the boundedness of the extended Cesaro operator Tg from Qx to the space F(p;q; s; p) is obtained. Moreover, the essential norm and compactness of Tg are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weighted Hardy space H p (ω) induced by an A p weight ω. We characterize the positive Borel measures μ such that the identity operator I d : H p (ω) → L q (d μ) is bounded when 1 ≤ p , q < ∞. As an application, we obtain necessary and sufficient conditions for the boundedness of the generalized area operator A μ , ν from H p (ω) to L q (ω). [ABSTRACT FROM AUTHOR]

This paper focuses on the structure of the subspace Tc of the BMO Teichmüller space Tb corresponding to chord-arc curves, which contains the VMO Teichmüller space Tv. We prove that Tc is not a subgroup with respect to the group structure of Tb, but it is preserved under the inverse operation and the left and the right translations by any element of Tv. Moreover, we show that Tb has a fiber structure induced by Tv, and the complex structure of Tb can be projected down to the quotient space Tv\Tb. Then, we see that Tc consists of fibers of this projection, and its quotient space also has the induced complex structure. [ABSTRACT FROM AUTHOR]

A family of Besov-Morrey type spaces in the open unit disc are introduced in this paper. The boundedness of the embedding from Besov-Morrey type spaces to a class tent spaces is studied. As an application, the boundedness, compactness and essential norm of the Volterra integral operator from Besov-Morrey type spaces to a general function space are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we determine the range of a Cesàro-like operator acting on H ∞ by describing characterizations of Carleson type measures on [0, 1). A special case of our result gives an answer to a question posed by P. Galanopoulos, D. Girela and N. Merchán recently. [ABSTRACT FROM AUTHOR]

We study weighted mixed norm spaces of harmonic functions defined on smoothly bounded domains in ℝn. Our principal result is a characterization of Carleson measures for these spaces. First, we obtain an equivalence of norms on these spaces. Then we give a necessary and sufficient condition for the embedding of the weighted harmonic mixed norm space into the corresponding mixed norm space. [ABSTRACT FROM AUTHOR]

We obtain the norm and essential norm estimates for the difference of two weighted composition operators from a weighted Bergman space A α p into a Lebesgue space L q (μ) , where 0 < p ≤ q , α > - 1 and μ is a positive Borel measure on the unit disk of C . The weights of the weighted maps in this paper are not necessarily analytic or bounded. Consequently, characterizations for the boundedness and compactness of the difference operator are established. [ABSTRACT FROM AUTHOR]

In this paper, the boundedness and compactness of the inclusion mapping from $Q_p$ spaces into tent spaces $\mathcal{T}_{\frac{qp}{2},s}^{q}$ are completely characterized when $q>2$. As an application, the boundedness of the Volterra integral operator $T_g$ from $Q_p$ to the space $\mathcal{LF}(q,q-2,\frac{qp}{2})$ is obtained. Moreover, the essential norm and compactness of $T_g$ are also investigated.

In this paper, the boundedness and compactness of embedding from Dirichlet-Morrey spaces Dʎp into tent spaces Ts(µ) are investigated. As an application, we characterize the boundedness of the Volterra integral operator Jg from Dʎp to general function spaces F(2, p(1 - ʎ), s). Meanwhile, the operator Ig and the multiplication operator Mg from Dʎp to F(2, p(1 - ʎ), s) are studied. Furthermore, the essential norm of Jg and Ig from Dʎp to F(2, p(1 - ʎ), s) are also considered. [ABSTRACT FROM AUTHOR]

If μ is a positive Borel measure on the interval [0, 1), we let 퓗μ be the Hankel matrix 퓗μ = ( μn,k)n,k≥0 with entries μn,k = μn+k and μn = ∫[0,1) tnd μ(t). Using 퓗μ, Ye and Zhou first defined the Derivative-Hilbert operator as D퓗μ(f)(z) = ... (n+1)zn, z ∈ 픻, where f (z)= ... is an analytic function in 픻. In this paper, we characterize the measure μ for which D퓗μ is a bounded (resp., compact) operator from Besov space Bp into Bloch space B with 1 < p < ∞. [ABSTRACT FROM AUTHOR]

We study monomial operators on L²[0, 1], that is bounded linear operators that map each monomial xn to a multiple of xpn for some pn. We show that they are all unitarily equivalent to weighted composition operators on a Hardy space. We characterize what sequences pn can arise. In the case that pn is a fixed translation of n, we give a criterion for boundedness of the operator. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of properties of second-order elliptic equation solutions. The main content of the paper coincides with the report made by the author at the international conference dedicated to the 75th anniversary of I. V. Volovich. The solution behavior near the boundary and the Dirichlet problem formulation, which is closely related to this issue, are studied. At the end of the paper, we will briefly discuss the results obtained in elegant and extremely important works by E. De Giorgi and J. Nash regarding Hölder continuity of the equation solutions within the considered domain. We present results that combine and complement the belonging of the solution to the Hölder and Sobolev spaces. Note that all the concepts and statements under consideration are united by a common approach and are formulated in close terms. [ABSTRACT FROM AUTHOR]

Let ψ1 and ψ2 be analytic functions on the open unit disk D and f an analytic self map on D. Let Mψ, Cψ and D denote the multiplication, composition and differentiation operators. We consider operators Mψ1Cf, Mψ2CfD and the Stević-Sharma operator Tψ1,ψ2,f(f) = Mψ1Cf(f)+Mψ2CfD(f) on α-Besov space Bp,α and weak vector valued a-Besov space wBp,α(X) for complex Banach space X and find some equivalent statements for boundedness of these operators. Also, boundedness and compactness of composition operator Cf on Bp,α(D) and wBp,α(D) are given. [ABSTRACT FROM AUTHOR]

Weakly harmonic maps from a domain Ω ⊂ R d , d ≥ 2 (the upper half-space R + d or a bounded C 1 , α domain, α ∈ (0 , 1 ] ) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes L ∞ (∂ Ω) or BMO (∂ Ω) , we establish solvability of the resulting boundary value problems by means of a nonvariational method. As a by-product, solutions are shown to be locally smooth, C loc ∞. Moreover, we show that boundary data can be chosen large in the underlying topologies if Ω is smooth and bounded by perturbing strictly stable smooth harmonic maps. [ABSTRACT FROM AUTHOR]

A family of Bergman-Morrey type spaces in the unit disc are introduced. The boundedness of the embedding from Bergman-Morrey type spaces to a class of tent spaces is studied. The boundedness, compactness, norm and essential norm of Volterra integral operators on Bergman-Morrey type spaces are also investigated in this paper. [ABSTRACT FROM AUTHOR]

Let (X, d, μ) be a metric measure space satisfying a Q-doubling condition (Q > 1) and an L2-Poincaré inequality. Let ℒ = ℒ + V be a Schrödinger operator on X, where ℒ is a non-negative operator generalized by a Dirichlet form, and V is a non-negative Muckenhoupt weight that satisfies a reverse Hölder condition RHq for some q ⩾ (Q + 1)/2. We show that a solution to (ℒ − ∂ t 2) u = 0 on X × ℝ+ satisfies the Carleson condition sub B (x B , r B) 1 μ (B (x B , r B)) ∫ 0 r B ∫ B (x B , r B) | t ∇ u (x , t) | 2 d μ d t t > ∞ if and only if u can be represented as the Poisson integral of the Schrödinger operator ℒ with the trace in the BMO (bounded mean oscillation) space associated with ℒ. [ABSTRACT FROM AUTHOR]