Search

In this paper, the boundedness and compactness of embedding from Dirichlet-Morrey spaces Dpλ into tent spaces Ts(μ) are investigated. As an application, we characterize the boundedness of the Volte...

A class of Dirichlet-Morrey spaces $ D_{\beta, \lambda} $ is introduced in this paper. For any positive Borel measure $ \mu $, the boundedness and compactness of the identity operator from $ D_{\beta, \lambda} $ into the tent space $ \mathcal{T}_s^1(\mu) $ are characterized. As an application, the boundedness of the Volterra integral operator $ T_g: D_{\beta, \lambda} \to F(1, \beta-s, s) $ is studied. Moreover, the essential norm and the compactness of the operator $ T_g $ are also investigated.

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 > 𝒬 / 2 {q_{0}>\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 ) < p < 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) 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})} .

Let $$\vec {p}\in (0,\infty )^n$$ and A be a general expansive matrix on $${\mathbb {R}}^n$$ . In this article, the authors first introduce some new anisotropic mixed-norm Campanato-type space associated with A. Then the authors prove that this Campanato-type space is the dual space of the anisotropic mixed-norm Hardy space $$H^{\vec {p}}_A({\mathbb {R}}^n)$$ for any given $$\vec {p}\in (0,\infty )^n$$ , which further implies several equivalent characterizations of this Campanato-type space. Finally, as further applications, the authors establish the Carleson measure characterization of this Campanato-type space via first introducing the anisotropic mixed-norm tent space and establishing its atomic decomposition. In particular, even when the expansive matrix A is a diagonal matrix, all these results are new and, even in this case, the obtained dual result gives a complete answer to one open question proposed by Cleanthous et al. (J Geom Anal 27: 2758–2787, 2017).

In Nagel et al. (J Funct Anal 181:29–118, 2001), Nagel–Ricci–Stein established the relationships between product kernels and flag kernels on the Euclidean space, that is, product kernels are finite sums of flag kernels. The main purpose of this paper is to characterize the product Hardy space as the intersection of flag Hardy spaces, and characterize the product Carleson measure space as the sum of flag spaces.

Let $$L=-\Delta +|x|^2$$ be the Hermite operator, where $$\Delta $$ is the Laplacian on $${\mathbb {R}}^{d}$$. In this paper, we define fractional Carleson measure associated with Hermite operator, which is adapted to the operator L. Then, we will use it to characterize the dual spaces and predual spaces of the Hardy spaces $$H_L^p({\mathbb {R}}^d)$$ associated with L.

Let $\mathcal{L}=-\Delta+\mu$ be the generalized Schrödinger operator on $\mathbb{R}^{n},n\geq3$ , where $\Delta$ is the Laplacian and $\mu\notequiv0$ is a nonnegative Radon measure on $\mathbb{R}^{n}$ . In this article, we introduce two families of Carleson measures $\{d\nu_{h,k}\}$ and $\{d\nu_{P,k}\}$ generated by the heat semigroup $\{e^{-t\mathcal{L}}\}$ and the Poisson semigroup $\{e^{-t\sqrt{\mathcal{L}}}\}$ , respectively. By the regularities of semigroups, we establish the Carleson measure characterizations of BMO-type spaces $\mathrm{BMO}_{\mathcal{L}}(\mathbb{R}^{n})$ associated with the generalized Schrödinger operators.

If $$\,\mu \,$$ is a finite positive Borel measure on the interval $$\,[0,1)$$ , we let $$\,\mathcal {H}_\mu \,$$ be the Hankel matrix $$\,(\mu _{n, k})_{n,k\ge 0}\,$$ with entries $$\,\mu _{n, k}=\mu _{n+k}$$ , where, for $$\,n\,=\,0, 1, 2, \ldots $$ , $$\mu _n\,$$ denotes the moment of order $$\,n\,$$ of $$\,\mu $$ . This matrix induces formally the operator $$\,\mathcal {H}_\mu (f)(z)= \sum _{n=0}^{\infty }\left( \sum _{k=0}^{\infty } \mu _{n,k}{a_k}\right) z^n\,$$ on the space of all analytic functions $$\,f(z)=\sum _{k=0}^\infty a_kz^k\,$$ , in the unit disc $$\,\mathbb {D} $$ . When $$\,\mu \,$$ is the Lebesgue measure on $$\,[0,1)\,$$ the operator $$\,\mathcal {H}_\mu \,$$ is the classical Hilbert operator $$\,\mathcal {H}\,$$ which is bounded on $$\,H^p\,$$ if $$\,1

Let $\mathcal{L}=-\Delta+\mu$ be the generalized Schrödinger operator on $\mathbb{R}^{n}$ , $n\geq3$ , where $\Delta$ is the Laplacian and $\mu\nequiv0$ is a nonnegative Radon measure on $\mathbb{R}^{n}$ . In this article, we give a characterization of $\mathrm{BMO}_{\mathcal{L}}$ in terms of Carleson measures, where $\mathrm{BMO}_{\mathcal{L}}$ is the $\mathrm{BMO}$ -type space associated with the generalized Schrödinger operator.

Let D μ , p be a Dirichlet type space induced by a positive parameter p and a positive Borel measure μ on the open unit disk. Denote by M ( D μ , p ) the Mobius invariant function space generated by D μ , p . It is known that if the measure μ is finite, then M ( D μ , p ) is equal to the well-known Mobius invariant space Q p . In this paper, we investigate D μ , p and M ( D μ , p ) spaces when the measures μ are not necessarily finite. We give the relation between M ( D μ , p ) and the Bloch space. We characterize inner functions in M ( D μ , p ) spaces. We also consider a Carleson measure problem for D μ , p spaces.

We discuss random interpolation in weighted Dirichlet spaces $\mathcal{D}_\alpha$, $0\leq \alpha\leq 1$. While conditions for deterministic interpolation in these spaces depend on capacities which are very hard to estimate in general, we show that random interpolation is driven by surprisingly simple distribution conditions. As a consequence, we obtain a breakpoint at $\alpha=1/2$ in the behavior of these random interpolating sequences showing more precisely that almost sure interpolating sequences for $\mathcal{D}_\alpha$ are exactly the almost sure separated sequences when $0\le \alpha, Comment: With respect to previous versions of this paper we have clarified the situation of the breakpoint $\alpha=1/2$ as well as the endpoint case of the classical Dirichlet space $\mathcal{D}$. All the main results are now sharp

We define Hardy spaces $\mathcal{H}^p$ for quasiregular mappings in the plane, and show that for a particular class of these mappings many of the classical properties that hold in the classical setting of analytic mappings still hold. This particular class of quasiregular mappings can be characterised in terms of composition operators when the symbol is quasiconformal. Relations between Carleson measures and Hardy spaces play an important role in the discussion. This program was initiated and developed for Hardy spaces of quasiconformal mappings by Astala and Koskela in 2011 in their paper $\mathcal{H}^p$-theory for Quasiconformal Mappings., 11 pages

In this paper, we investigate the Dirichlet type space $$D_{\log }$$ , which is closely associated with the analytic version of $$\mathcal Q_1(\partial {\mathbb {D}})$$ space. We show that the space $$D_{\log }$$ has the Pick Property. A characterization of interpolating sequence for multipliers of $$D_{\log }$$ is given.

We characterize interpolating sequences for multiplier algebras of spaces with the complete Pick property. Specifically, we show that a sequence is interpolating if and only if it is separated and generates a Carleson measure. This generalizes results of Carleson for the Hardy space and of Bishop, Marshall, and Sundberg for the Dirichlet space. Furthermore, we investigate interpolating sequences for pairs of Hilbert function spaces.

Let (X, d, μ) be a metric measure space with doubling property. The Hardy spaces associated with operators L were introduced and studied by many authors. All these spaces, however, were first defined by L 2(X) functions and finally the Hardy spaces were formally defined by the closure of these subspaces of L 2(X) with respect to Hardy spaces norms. A natural and interesting question in this context is to characterize the closure. The purpose of this paper is to answer this question. More precisely, we will introduce ${CMO}_{L}^{p}(X)$ , the Carleson measure spaces associated with operators L, and characterize the Hardy spaces associated with operators L via $({CMO}_{L}^{p}(X))'$ , the distributions of ${CMO}_{L}^{p}(X)$ .

We study parabolic chord arc domains, introduced by Hofmann, Lewis and Nystr\"om, and prove a free boundary regularity result below the continuous threshold. More precisely, we show that a Reifenberg flat, parabolic chord arc domain whose Poisson kernel has logarithm in VMO must in fact be a vanishing chord arc domain (i.e. satisfies a vanishing Carleson measure condition). This generalizes, to the parabolic setting, a result of Kenig and Toro and answers in the affirmative a question left open in the aforementioned paper of Hofmann et al. A key step in this proof is a classification of "flat" blowups for the parabolic problem., Comment: 91 pages. Comments welcome

It follows, from a generalised version of Paley-Wiener theorem, that the Laplace transform is an isometry between certain spaces of weighted $L^2$ functions defined on $(0, \infty)$ and (Hilbert) spaces of analytic functions on the right complex half-plane (for example Hardy, Bergman or Dirichlet spaces). We can use this fact to investigate properties of multipliers and multiplication operators on the latter type of spaces. In this paper we present a full characterisation of multipliers in terms of a generalised concept of a Carleson measure. Under certain conditions, these spaces of analytic functions are not only Hilbert spaces but also Banach algebras, and are therefore contained within their spaces of multipliers. We provide some necessary as well as sufficient conditions for this to happen and look at its consequences., Comment: Keywords: Banach algebras, Banach spaces, Bergman spaces, Carleson measures, Dirichlet spaces, Hardy spaces, Hardy-Sobolev spaces, Hilbert spaces, Laplace transform, maximal ideal spaces, multiplication operators, multipliers, reproducing kernels, spaces of analytic functions, weighted $L^2$ spaces, Zen spaces

The notion of a Carleson measure was introduced by Lennart Carleson in his proof of the Corona Theorem for $H^\infty(\mathbb{D})$. In this paper we will define it for certain type of reproducing kernel Hilbert spaces of analytic functions of the complex half-plane, $\mathbb{C}_+$, which will include Hardy, Bergman and Dirichlet spaces. We will obtain several necessary or sufficient conditions for a positive Borel measure to be Carleson by preforming tests on reproducing kernels, weighted Bergman kernels, and studying the tree model obtained from a decomposition of the complex half-plane. The Dirichlet space will be investigated in detail as a special case. Finally, we will present a control theory application of Carleson measures in determining admissibility of controls in well-posed linear evolution equations., Keywords: Carleson measures, reproducing kernel Hilbert spaces, Dirichlet space, control operators, admissibility, Laplace transform

We give a condition under which a divisor \(\hat{X}\) in a bounded convex domain of finite type D in \(\mathbb {C}^n\) is the zero set of a function in a Hardy space \({\mathcal {H}}^p(D)\) for some \(p>0\). This generalizes Varopoulos’ result (Pac J Math 88:189–246, 1980) on zero sets of \({\mathcal {H}}^p\)-functions in strictly convex domains of \(\mathbb {C}^n\).

Suppose $\mu$ be a Beltrami coefficient on the unit disk, which is compatible with a convex co-compact Fuchsian group $G$ of the second kind. In this paper we show that if $\displaystyle\frac{|\mu|^{2}}{1-|z|^{2}}dxdy $ satisfies the Carleson condition on the infinite boundary boundary of the Dirichlet domain of $G$, then $\displaystyle\frac{|\mu|^{2}}{1-|z|^{2}}dxdy$ is a Carleson measure on the unit disk., Comment: 10 pages

In this paper we characterize off-diagonal Carleson embeddings for both Hardy-Orlicz spaces and Bergman-Orlicz spaces of the upper-half plane. We use these results to obtain embedding relations and pointwise multipliers between these spaces., Comment: 41 pages

In this paper, we introduce the Carleson measure spaces with variable exponents $CMO^{p(\cdot)}$. By using discrete Littlewood$-$Paley$-$Stein analysis as well as Frazier and Jawerth's $\varphi-$transform in the variable exponent settings, we show that the dual space of the variable Hardy space $H^{p(\cdot)}$ is $CMO^{p(\cdot)}$. As applications, we obtain that Carleson measure spaces with variable exponents $CMO^{p(\cdot)}$, Campanato space with variable exponent $\mathfrak{L}_{q,p(\cdot),d}$ and H\"older-Zygmund spaces with variable exponents $\mathcal {\dot{H}}_d^{p(\cdot)}$ coincide as sets and the corresponding norms are equivalent. Via using an argument of weak density property, we also prove the boundedness of Calder\'{o}n-Zygmund singular integral operator acting on $CMO^{p(\cdot)}$., Comment: 26 pages, submit to a journal on 02 Dec 2018

A characterisation is given of bounded embeddings from weighted $L^2$ spaces on bounded intervals into $L^2$ spaces on the half-plane, induced by isomorphisms given by the Laplace transform onto weighted Hardy and Bergman spaces (Zen spaces). As an application necessary and sufficient conditions are given for the boundedness of truncated Hankel and Toeplitz integral operators, including the weighted case., 19 pages. Some minor revisions

In the present paper, we introduce a class of $\alpha$-Carleson measures $\mathcal{C}_{\alpha,v}(\mathbb{R}^{n+1}_+)$, which is called by the vanishing $\alpha$-Carleson measures. We prove that $\mathcal{C}_{1/p,v}(\mathbb{R}^{n+1}_+)$ is just a predual of the tent space $\widetilde{T}_{\infty}^p$ ($0 \lt p \lt 1$). Furthermore, we construct the $\alpha$-Carleson measures and the vanishing $\alpha$-Carleson measures by the Campanato functions and its a subclass, respectively. Moreover, a characterization of the vanishing $\alpha$-Carleson measure by the compactness of Poisson integral is given in this paper. Finally, as some applications, we give the $(L^{2/\alpha},L^2)$ boundedness and compactness for some paraproduct operators.

We develop highly anisotropic Carleson measure over multi-level ellipsoid covers \begin{document}$ \Theta $\end{document} of \begin{document}$ \mathbb{R}^n $\end{document} that are highly anisotropic in the sense that the ellipsoids can change rapidly from level to level and from point to point. Then we show that the Carleson measure \begin{document}$ \mu $\end{document} is sufficient for which the integral defines a bounded operator from \begin{document}$ H^p(\Theta) $\end{document} to \begin{document}$ L^p(\mathbb{R}^{n+1}, \, \mu),\ 0. 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.

In this paper, we introduce a weighted Carleson measure \(d\nu _{\mathbb {E}, w}\) associated with the family \(\mathbb {E}\), where \(\mathbb {E}=\{E_r(x)\}_{r\in \mathcal {I}, x\in X}\) is a family of open subsets of a topological space X endowed with a nonnegative Borel measure \(\mu \) satisfying certain basic conditions. Using Calderon–Zygmund theory, we show that the weighted BMO associated with the family \(\mathbb {E}\) can be characterized by the weighted Carleson measure \(d\nu _{\mathbb {E}, w}\).

We show that a Carleson measure satisfies the reverse Carleson condition if and only if its Berezin symbol is bounded below on the unit disk D . We provide new necessary and sufficient conditions for the composition operator to have closed range on the Bergman space. The pull-back measure of area measure on D plays an important role. We also give a new proof in the case of the Hardy space and conjecture a condition in the case of the Dirichlet space.

We consider the Weinstein type equation L λ u = 0 on ( 0 , ∞ ) × ( 0 , ∞ ) , where L λ = ∂ t 2 + ∂ x 2 − λ ( λ − 1 ) x 2 , with λ > 1 . In this paper we characterize the solutions of L λ u = 0 on ( 0 , ∞ ) × ( 0 , ∞ ) representable by Bessel–Poisson integrals of BMO-functions as the ones satisfying certain Carleson properties.

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a measurable function satisfying some decay condition and some locally log-H\"older continuity. In this article, via first establishing characterizations of the variable exponent Hardy space $H^{p(\cdot)}(\mathbb R^n)$ in terms of the Littlewood-Paley $g$-function, the Lusin area function and the $g_\lambda^\ast$-function, the authors then obtain its intrinsic square function characterizations including the intrinsic Littlewood-Paley $g$-function, the intrinsic Lusin area function and the intrinsic $g_\lambda^\ast$-function. The $p(\cdot)$-Carleson measure characterization for the dual space of $H^{p(\cdot)}(\mathbb R^n)$, the variable exponent Campanato space $\mathcal{L}_{1,p(\cdot),s}(\mathbb R^n)$, in terms of the intrinsic function is also presented., Comment: Bull. Malays. Math. Sci. Soc. (2) (to appear)

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} $ induces the operator $ {\mathcal H}_\mu(f)(z) = \sum\limits_{n = 0}^\infty\left(\sum\limits_{k = 0}^\infty\mu_{n,k}a_k\right)z^n $ on the space of all analytic functions $ f(z) = \sum^\infty_{n = 0}a_nz^n $ in the unit disk $ {\mathbb{D}} $. In this paper, we characterize the boundedness and compactness of $ {\mathcal H}_\mu $ from Bloch type spaces to the BMOA and the Bloch space. Moreover we obtain the essential norm of $ {\mathcal H}_\mu $ from $ {\mathcal{B}}^\alpha $ to $ {\mathcal{B}} $ and BMOA.

By \(BMO_{\text {o}}(\mathbb {R})\) we denote the space consisting of all those odd and bounded mean oscillation functions on \(\mathbb {R}\). In this paper we characterize the functions in \(BMO_{\text {o}}(\mathbb {R})\) with bounded support as those ones that can be written as a sum of a bounded function on \((0,\infty )\) plus the balayage of a Carleson measure on \((0,\infty )\times (0,\infty )\) with respect to the Poisson semigroup associated with the Bessel operator $$\begin{aligned} B_\lambda :=-x^{-\lambda }\frac{d}{dx}x^{2\lambda }\frac{d}{dx}x^{-\lambda },\quad \lambda >0. \end{aligned}$$ This result can be seen as an extension to Bessel setting of a classical result due to Carleson.

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L=-\Delta+|x|^{2}$\end{document}L=−Δ+|x|2 be a Hermite operator, where Δ is the Laplacian on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{d}$\end{document}Rd. In this paper we define a new version of Carleson measure associated with Hermite operator, which is adapted to the operator L. Then, we will use it to characterize the dual spaces and predual spaces of the Hardy spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{L}^{p}(\mathbb {R}^{d})$\end{document}HLp(Rd) associated with L.

Let $\mathcal{L}$ be a Schr\"odinger operator of the form $\mathcal{L} = -\Delta+V$ acting on $L^2(\mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse H\"older class $B_q$ for some $q\geq n.$ Let $L^{p,\lambda}(\mathbb{R}^{n})$, $0\le \lambda, Comment: Minor corrections. To be appeared in Taiwanese Journal of Mathematics. arXiv admin note: substantial text overlap with arXiv:1710.01160

Let L be a Schr\"odinger operator of the form L=-\Delta+V acting on L^2(\mathbb R^n) where the nonnegative potential V belongs to the reverse H\"older class B_q for some q>= n. Let BMO denote the BMO space associated to the Schr\"odinger operator L. In this article we will show that a function f in BMO_L is the trace of the solution of u_t+L u=0, u(x,0)= f(x), where u satisfies a Carleson-type condition. Conversely, this Carleson condition characterizes all the L-carolic functions whose traces belong to the space BMO_L. This result extends the analogous characterization founded by Fabes and Neri for the classical BMO space of John and Nirenberg., Comment: 22 pages

In 1985, Coifman, Meyer, and Stein gave the duality of the tent spaces; that is, $(T_{q}^{p}(\mathbb{R}^{n+1}_{+}))^{\ast}=T_{q'}^{p'}(\mathbb{R}^{n+1}_{+})$ for $1\lt p,q\lt \infty$ , and $(T_{\infty}^{1}(\mathbb{R}^{n+1}_{+}))^{\ast}=\mathscr{C}(\mathbb{R}^{n+1}_{+})$ , $(T_{q}^{1}(\mathbb{R}^{n+1}_{+}))^{\ast}=T_{q'}^{\infty}(\mathbb{R}^{n+1}_{+})$ for $1\lt q\lt \infty$ , where $\mathscr{C}(\mathbb{R}^{n+1}_{+})$ denotes the Carleson measure space on $\mathbb{R}^{n+1}_{+}$ . We prove that $(\mathscr{C}_{v}(\mathbb{R}^{n+1}_{+}))^{\ast}=T_{\infty}^{1}(\mathbb{R}^{n+1}_{+})$ , which we introduced recently, where $\mathscr{C}_{v}(\mathbb{R}^{n+1}_{+})$ is the vanishing Carleson measure space on $\mathbb{R}^{n+1}_{+}$ . We also give the characterizations of $T_{q}^{\infty}(\mathbb{R}^{n+1}_{+})$ by the boundedness of the Poisson integral. As application, we give the boundedness and compactness on $L^{q}(\mathbb{R}^{n})$ of the paraproduct $\pi_{F}$ associated with the tent space $T_{q}^{\infty}(\mathbb{R}^{n+1}_{+})$ , and we extend partially an interesting result given by Coifman, Meyer, and Stein, which establishes a close connection between the tent spaces $T_{2}^{p}(\mathbb{R}^{n+1}_{+})$ $(1\le p\le\infty)$ and $L^{p}(\mathbb{R}^{n})$ , $H^{p}(\mathbb{R}^{n})$ and $\mathit{BMO}(\mathbb{R}^{n})$ spaces.

Let \(A^p_\omega \) denote the Bergman space in the unit disc \(\mathbb {D}\) of the complex plane induced by a radial weight \(\omega \) with the doubling property \(\int _{r}^1\omega (s)\,ds\le C\int _{\frac{1+r}{2}}^1\omega (s)\,ds\). The tent space \(T^q_s(\nu ,\omega )\) consists of functions such that $$\begin{aligned} \begin{aligned} \Vert f\Vert _{T^q_s(\nu ,\omega )}^q =\int _\mathbb {D}\left( \int _{\varGamma (\zeta )}|f(z)|^s\,d\nu (z)\right) ^\frac{q}{s}\omega (\zeta )\,dA(\zeta ) 0\), by considering a generalized area operator. The results are provided in terms of Carleson measures for \(A^p_\omega \).

Let μ be a positive Borel measure on the unit disc D . In this note, we show that the inclusion mapping from analytic Morrey spaces L 2 , λ ( D ) to quadratic tent-type spaces T λ ∞ ( μ ) is bounded (compact) if and only if μ is a Carleson measure (vanishing Carleson measure). As a byproduct we get a new version of well-known Carleson measure theorem of Hardy spaces. The inverse Carleson measure problem of L 2 , λ ( D ) is also studied.

In this work, we give new sufficient conditions for Littlewood–Paley–Stein square function operators and necessary and sufficient conditions for Calderon–Zygmund operators to be bounded on Hardy spaces $$H^p$$ with indices smaller than 1. New Carleson measure type conditions are defined for Littlewood–Paley–Stein operators, and the authors show that they are sufficient for the associated square function to be bounded from $$H^p$$ into $$L^p$$ . New polynomial growth $$BMO$$ conditions are also introduced for Calderon–Zygmund operators. These results are applied to prove that Bony paraproducts can be constructed such that they are bounded on Hardy spaces with exponents ranging all the way down to zero.

We apply interpolation techniques to study behaviour of the canonical inclusion maps of quasi-Banach spaces of analytic functions on the open unit disk of the plane into (quasi)-Banach function lattices on the closed or open unit disk equipped with a Borel measure. These results are applied to abstract Hardy spaces generated by symmetric spaces. We investigate relationships between boundedness or compactness of the canonical inclusion maps and generalized variants of Carleson measures and show applications to composition operators on abstract Hardy spaces. We specialize our results to Hardy–Lorentz spaces.

A general method for estimating the approximation numbers of composition operators on the Hardy space H 2, using finite-dimensional model subspaces, is studied and applied in the case when the symbol of the operator maps the unit disc to a domain whose boundary meets the unit circle at just one point. The exact rate of decay of the approximation numbers is identified when this map is sufficiently smooth at the point of tangency; it follows that a composition operator with any prescribed slow decay of its approximation numbers can be explicitly constructed. Similarly, an asymptotic expression for the approximation numbers is found when the mapping has a sharp cusp at the distinguished boundary point. Precise asymptotic estimates in the intermediate cases, including that of maps with a corner at the distinguished boundary point, are also established.

Let \(D\) denote the classical Dirichlet space of analytic functions \(f\) in the open unit disc \(\mathbb {D}\) with finite Dirichlet integral, \(\int _\mathbb {D}|f'|^2 dA < \infty \). Furthermore, let \(D \odot D\) be the space of weak products of functions in \(D\), i.e. all functions \(h\) that can be written as \(h = \sum _{i=1}^\infty f_i g_i\) for some \(f_i, g_i \in D\) with \(\sum _{i=1}^\infty \Vert f_i\Vert \Vert g_i\Vert < \infty \). The dual of \(D \odot D\) has been characterized in 2010 by Arcozzi, Rochberg, Sawyer, and Wick as the space \(\mathcal {X}(D)\) of analytic functions \(b\) on \(\mathbb {D}\) such that \(|b'|^2 dA\) is a Carleson measure for the Dirichlet space. In this paper we show that for functions \(f\) in proper weak*-closed \(M_z^*\)-invariant subspaces of \(\mathcal {X}(D)\), the functions \((zf)'\) are in the Nevanlinna class of \(\mathbb {D}\) and have meromorphic pseudocontinuations in the Nevanlinna class of the exterior disc. We then use this result to show that every nonzero \(M_z\)-invariant subspace \(\mathcal {N}\) of \(D \odot D\) has index 1, i.e. satisfies \(\dim \mathcal {N}/z\mathcal {N}=1\).

Let $\mathcal{X}$ be a space of homogenous type and $\varphi:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$ a growth function such that $\varphi(\cdot,t)$ is a Muckenhoupt weight uniformly in $t$ and $\varphi(x,\cdot)$ an Orlicz function of uniformly upper type 1 and lower type $p\in(0,1]$. In this article, the authors introduce a new Musielak-Orlicz BMO-type space $\mathrm{BMO}^{\varphi}_A(\mathcal{X})$ associated with the generalized approximation to the identity, give out its basic properties and establish its two equivalent characterizations, respectively, in terms of the spaces $\mathrm{BMO}^{\varphi}_{A,\,\mathrm{max}}(\mathcal{X})$ and $\widetilde{\mathrm{BMO}}^{\varphi}_A(\mathcal{X})$. Moreover, two variants of the John-Nirenberg inequality on $\mathrm{BMO}^{\varphi}_A(\mathcal{X})$ are obtained. As an application, the authors further prove that the space $\mathrm{BMO}^{\varphi}_{\sqrt{\Delta}}(\mathbb{R}^n)$, associated with the Poisson semigroup of the Laplace operator $\Delta$ on $\mathbb{R}^n$, coincides with the space $\mathrm{BMO}^{\varphi}(\mathbb{R}^n)$ introduced by L. D. Ky., Comment: Acta Math. Sin. (Engl. Ser.) (to appear)

In this paper we establish the characterization of the weighted BMO via two weight commutators in the settings of the Neumann Laplacian $\Delta_{N_+}$ on the upper half space $\mathbb{R}^n_+$ and the reflection Neumann Laplacian $\Delta_N$ on $\mathbb{R}^n$ with respect to the weights associated to $\Delta_{N_+}$ and $\Delta_{N}$ respectively. This in turn yields a weak factorization for the corresponding weighted Hardy spaces, where in particular, the weighted class associated to $\Delta_{N}$ is strictly larger than the Muckenhoupt weighted class and contains non-doubling weights. In our study, we also make contributions to the classical Muckenhoupt--Wheeden weighted Hardy space (BMO space respectively) by showing that it can be characterized via area function (Carleson measure respectively) involving the semigroup generated by the Laplacian on $\mathbb{R}^n$ and that the duality of these weighted Hardy and BMO spaces holds for Muckenhoupt $A^p$ weights with $p\in (1,2]$ while the previously known related results cover only $p\in (1,{n+1\over n}]$. We also point out that this two weight commutator theorem might not be true in the setting of general operators $L$, and in particular we show that it is not true when $L$ is the Dirichlet Laplacian $\Delta_{D_+}$ on $\mathbb{R}^n_+$., Comment: 44 pages, 3 figures

We study Carleson inequalities in a framework of parabolic Hardy spaces. Similar results for parabolic Bergman spaces are discussed in [NSY1] (see also [NSY2]), where $\tau$-Carleson measures play an important roll. In the present case, $T_{\tau}$-Carleson measures are useful. We give an relation between these measures.

We consider the Carleson embeddings of the classical Hardy spaces (on the disk) into a L p ($\mu$) space, where $\mu$ is a Carleson measure on the unit disk. This includes the case of composition operators. We characterize such operators which are r-summing on H p , where p \textgreater{} 1 and r $\ge$ 1. This completely extends the former results on the subject and solves a problem open since the early seventies. Mathematics Subject Classification. Primary: 47B33 -- Secondary: 28A12; 30C85; 31A15; 46E20; 46E22; 47B06

In this paper we give a Carleson measure characterization for the compact composition operators between Dirichlet type spaces. We use this characterization to show that every compact composition operator on Dirichlet type spaces is compact on the Bloch space.

For parabolic linear operators L of second order in divergence form, we prove that the solvability of initial Lp Dirichlet problems for the whole range 1 < p < ∞ is preserved under appropriate small perturbations of the coefficients of the operators involved. We also prove that if the coefficients of L satisfy a suitable controlled oscillation in the form of Carleson measure conditions, then for certain values of p > 1, the initial Lp Dirichlet problem associated with Lu = 0 over non-cylindrical domains is solvable. The results are adequate adaptations of the corresponding results for elliptic equations.

We give a connection between the Hankel matrix acting on Dirichlet spaces D α , 0 α 2 , and the Carleson measure supported on ( − 1 , 1 ) . As an application, we prove that the generalized Hilbert operators H β are always bounded on Dirichlet spaces D α for 0 α 2 and that the range ( 0 , 2 ) of α in our results is the best possible.

Let φ : R n × [ 0 , ∞ ) → [ 0 , ∞ ) be such that φ ( x , ⋅ ) is an Orlicz function and φ ( ⋅ , t ) is a Muckenhoupt A ∞ ( R n ) weight uniformly in t . In this article, the authors introduce the Musielak–Orlicz Campanato space L φ , q , s ( R n ) ; as an application, the authors prove that some of them is the dual space of the Musielak–Orlicz Hardy space H φ ( R n ) , which, in the case when q = 1 and s = 0 , was obtained by L.D. Ky [ arXiv:1105.0486 ]. The authors also establish a John–Nirenberg inequality for functions in L φ , 1 , s ( R n ) and, as an application, the authors also obtain several equivalent characterizations of L φ , q , s ( R n ) , which, in return, further induce the φ -Carleson measure characterization of L φ , 1 , s ( R n ) .

It is shown that the Laplace transform of an Lp (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 Lp inequality for the harmonic extension is given with a numeric constant.