The well-known characterization of Hardy spaces H_p(𝔻), 0 < p < ∞, in terms of the Littlewood-Paley g-function S f (ζ) = (∫_0^1 |f' (r ζ)|^2 (1 - r) dr)^1/2∈L_p is generalized to Hardy-type spaces X_A corresponding to quasi-Banach lattices X on the unit circle 𝕋 under the assumption that the Hardy-Littlewood maximal operator M is bounded in (X^δ)' with some δ> 0. As an application to composition operators C_φ, we derive an exact criterion for the boundedness and compactness of C_φ: ℬ^ω→ X_A, where ℬ^ω= {f |sup |f'|/ω< ∞} is the weighted Bloch space with a log-convex radial weight ω, generalizing recent results in the one-dimensional setting.

Let d≥ 1 and φ: B_d→𝔻 be a holomorphic function, where B_d denotes the open unit ball of ℂ^d and 𝔻 = B_1. Let b: 𝔻→𝔻 be a holomorphic function and ℋ(b) denote the corresponding de Branges-Rovnyak space. We show that compactness of the composition operator C_φ from ℋ(b) to the Hardy space H^2(B_d) is related to natural restrictions on the Nevanlinna counting functions of the slice-functions φ_ζ, ζ∈∂ B_d.

Let n≥ 1 and φ : 𝔻^n→𝔻 be a holomorphic function, where 𝔻 denotes the open unit disk of ℂ . Let Θ : 𝔻→𝔻 be an inner function and K^p_Θ , p>0 , denote the corresponding model space. We obtain characterizations of the compact composition operators C_φ : K^2_Θ→ H^2(𝔻^n) , n≥ 1 , and C_φ : K^p_Θ→ H^p(𝔻^n) , 1

We prove analogs of Peyrie`re's mutual singularity theorem for standard and generalized Riesz products on the unit sphere of C n , n >= 2. As a corollary, we obtain an analog of Zygmund's dichotomy for the Riesz products under consideration.

Let $\varphi: B_d\to\mathbb{D}$, $d\ge 1$, be a holomorphic function, where $B_d$ denotes the open unit ball of $\mathbb{C}^d$ and $\mathbb{D}= B_1$. Let $\Theta: \mathbb{D} \to \mathbb{D}$ be an inner function and let $K^p_\Theta$ denote the corresponding model space. For $p>1$, we characterize the compact composition operators $C_\varphi: K^p_\Theta \to H^p(B_d)$, where $H^p(B_d)$ denotes the Hardy space.

Let RBMO(μ) = RBMO(ℝ^m, μ) denote the regular BMO space introduced by X. Tolsa for an n-dimensional finite positive measure on ℝ^m, 0<n ≤ m. We characterize the bounded Calderón-Zygmund operators T:RBMO(μ) →RBMO(μ) in terms of the function T1.

Let X be a quasi-Banach space of analytic functions in the unit disc and let q>0. A finite positive Borel measure μ in the closed unit disc 𝔻 is called a q-reverse Carleson measure for X if and only if there exists a constant C>0 such that f_X≤ C f_L^q(𝔻,dμ) for all f∈ X∩ C(𝔻). We fully characterize the q-reverse Carleson measures with all q>0 for Hardy spaces H^p(𝔻) with all 0<p≤∞, for the space BMOA(𝔻) and for the Bloch space. In addition, we describe q-reverse Carleson measures for the holomorphic Triebel–Lizorkin spaces HF_0^q,r and the holomorphic Besov spaces HB_0^q,r. Related results are obtained for the Hardy spaces and certain holomorphic Triebel–Lizorkin spaces in the unit ball of ℂ^d.

Пусть $I$ - внутренняя функция в области $\mathcal{D}=B_{n_1}\times B_{n_2}\times…\times B_{n_k}$, где $B_n$ - открытый единичный шар из $\mathbb{C}^n$, $n\geqslant 1$. В работе построены доминантные множества для пространства $H^2 \ominus I H^2$, где $H^2=H^2(\mathcal{D})$ - стандартное пространство Харди. Библиография: 12 названий.

Let 𝔻 denote the open unit disk and 𝕋=∂𝔻 . We characterize the reverse Carleson measures for the Hardy space H^p(𝔻^n) , 10 . Also, we prove that the reverse Carleson inequality for a pluriharmonic measure μ defined on the torus 𝕋^n is equivalent to the uniform reverse inequalities for almost all slice-measures μ _ζ , ζ∈𝕋^n .

We estimate the energy and Hausdorff dimensions of the Riesz products on the unit sphere of ℂ^n, n≥ 2. Also, we obtain similar results for the pluriharmonic measures on the torus.

Let φ _j , j=1,2, … , N , be holomorphic self-maps of the unit disk 𝔻 of ℂ . We prove that the compactness of a linear combination of the composition operators C_φ _j: f↦ f∘φ _j on the Hardy space H^p(𝔻) does not depend on p for 0

Let $B_n$ denote the unit ball of $\mathbb{C}^n$, $n\ge 1$, and let $\mathcal{D}$ denote a finite product of $B_{n_j}$, $j\ge 1$. Given a non-constant holomorphic function $b: \mathcal{D} \to B_1$, we study the corresponding family $\sigma_\alpha[b]$, $\alpha\in\partial B_1$, of Clark measures on the distinguished boundary $\partial\mathcal{D}$. We construct a natural unitary operator from the de Branges-Rovnyak space $\mathcal{H}(b)$ onto the Hardy space $H^2(\sigma_\alpha)$. As an application, for $\mathcal{D}= B_n$ and an inner function $I: B_n \to B_1$, we show that the property $\sigma_1[I]\ll\sigma_1[b]$ is directly related to the membership of an appropriate explicit function in $\mathcal{H}(b)$.

Let I be an inner function in 𝒟 = B_n_1× B_n_2⋯× B_n_k, where B_n denotes the open unit ball of ℂ^n, n≥ 1. We construct dominant sets for the space H^2 ⊖ I H^2, where H^2 = H^2(𝒟) denotes the standard Hardy space.

Let $H^p=H^p(B_d)$ denote the Hardy space in the open unit ball $B_d$ of $\mathbb{C}^d$, $d\ge 1$. We characterize the reverse Carleson measures for $H^p$, $10$. Given a non-inner holomorphic function $b: B_d \to B_1$, we obtain properties of the reverse Carleson measures for the de Branges-Rovnyak space $\mathcal{H}(b)$.

Let μ \mu be an n n -dimensional finite positive measure on R m \mathbb {R}^m . We obtain a T 1 T1 condition sufficient for the boundedness of Calderón–Zygmund operators on R B M O ( μ ) \mathrm {RBMO}(\mu ) , the regular BMO space of Tolsa [Math. Ann. 319 (2001), pp. 89–149].

Пусть $D\subset \mathbb{R}^d$ - ограниченная липшицева область, $\omega$ - модуль непрерывности высокого порядка и пусть $T$ - сверточный оператор Кальдерона-Зигмунда. В работе дано описание усеченных операторов $T_D$, которые ограничены на пространстве Зигмунда $\mathcal{C}_{\omega}(D)$. Полученное описание основано на свойствах функций $T_D P$ для подходящих многочленов $P$, суженных на область $D$. Библиография: 16 названий.

Let \(\mathbb {D}\) denote the unit disc of \(\mathbb {C}\) and let \(\Omega \) denote the unit ball \(B_n\) of \(\mathbb {C}^n\) or the unit polydisc \(\mathbb {D}^n\), \(n\ge 2\). Given a non-constant holomorphic function \(b: \Omega \rightarrow \mathbb {D}\), we study the corresponding family \(\sigma _\alpha [b]\), \(\alpha \in \partial \mathbb {D}\), of Clark measures on \(\partial \Omega \). For \(\Omega = B_n\) and an inner function \(I: B_n \rightarrow \mathbb {D}\), we show that the property \(\sigma _1[I]\ll \sigma _1[b]\) is directly related to the membership of an appropriate function in the de Branges–Rovnyak space \(\mathcal {H}(b)\).

Let $\mathbb{D}$ denote the unit disc of $\mathbb{C}$ and let $\mathbb{T}= \partial\mathbb{D}$. Given a holomorphic function $\varphi: \mathbb{D}^n \to \mathbb{D}$, $n\ge 2$, we study the corresponding family $\sigma_\alpha[\varphi]$, $\alpha\in\mathbb{T}$, of Clark measures on the torus $\mathbb{T}^n$. If $\varphi$ is an inner function, then we introduce and investigate related isometric operators $T_\alpha$ mapping analogs of model spaces into $L^2(\sigma_\alpha)$, $\alpha\in\mathbb{T}$.

Let Bd denote the unit ball of Cd, d≥1. Given a holomorphic function φ:Bd→B1, we study the corresponding family σα[φ], α∈∂B1, of Clark measures on the unit sphere ∂Bd. If φ is an inner function, then we introduce and investigate related unitary operators Uα mapping analogs of model spaces onto L2(σα), α∈∂B1. In particular, we explicitly characterize the set of Uα⁎f such that fσα is a pluriharmonic measure. Also, for an arbitrary holomorphic φ:Bd→B1, we use the family σα[φ] to compute the essential norm of the composition operator Cφ:H2(B1)→H2(Bd).

Let w and v be arbitrary radial weights on the unit disk $${\mathbb {D}}$$ . We characterize those univalent symbols $$g\in Hol({\mathbb {D}})$$ for which the Volterra operator $$T_g$$ maps boundedly the growth space $${\mathcal {A}}^w({\mathbb {D}})$$ into $${\mathcal {A}}^v({\mathbb {D}})$$ .