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This article considers systems of subspaces of the Hardy space generated by the Szego kernel. The main result of the work is to establish the convergence of orthorecursive expansions with respect to the considered systems of subspaces. Note that the conditions for the convergence of orthorecursive expansions prove to be somewhat more restrictive compared to the previously obtained conditions for the convergence of order-preserving weak greedy algorithms and frame expansions.

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 $ S_j(z) = \varepsilon_j +(z-\varepsilon_j)/2 $ be an iterated function system, where $ \varepsilon_j = e^{2j\pi i/3} $ for $ j = 0, 1, 2 $. Then, there exists a uniform self-similar measure $ \mu $ supported on a compact set $ K $, which is the attractor of $ \{S_j\}_{j = 0}^2 $. The Hausdorff dimension of the attractor $ K $ is $ \alpha = \log 3/\log 2 $. Let $ F(z) = \int_{K}(z-\omega)^{-1}d\mu(\omega) $ be the Cauchy transform of $ \mu $. In this paper, we consider the Hardy space and the multiplier property of $ F $. We prove that $ F' $ belongs to $ H^p $ for $ 0 < p < 1/(2-\alpha) $ and that $ F $ is a multiplier of some class of function space.

We obtain a class of commutators of bilinear pseudo-differential operators on products of Hardy spaces by applying the accurate estimates of the Hörmander class. And we also prove another version of these types of commutators on Herz-type spaces.

In this paper a multiplier theorem in the Hardy space H^1(\mathbb {T}) associated with Jacobi expansions of exponential type is proved, that is, a bilateral sequence \left \{\lambda _n\right \}_{n=-\infty }^{\infty } is a multiplier from H^1(\mathbb {T}) into the sequence space \ell ^1(\mathbb {Z}) associated with Jacobi expansions of exponential type, if \[ \sup _N\sum _{k=1}^{\infty }\left (\sum _{kN<|j|\le (k+1)N}|\lambda _j|\right)^2<\infty.\] This is a generalization of a multiplier theorem on usual Fourier expansions in the Hardy space H^1(\mathbb {T}), and for \lambda _n=(|n|+1)^{-1}, a Hardy type inequality for Jacobi expansions is immediate which has ever been proved by Kanjin and Sato [Math. Inequal. Appl. 7 (2004), pp. 551–555]. [ABSTRACT FROM AUTHOR]

We present two alternative proofs of Mandrekar's theorem, which states that an invariant subspace of the Hardy space on the bidisc is of Beurling type precisely when the shifts satisfy a doubly commuting condition [Proc. Amer. Math. Soc. 103 (1988), pp. 145–148]. The first proof uses properties of Toeplitz operators to derive a formula for the reproducing kernel of certain shift invariant subspaces, which can then be used to characterize them. The second proof relies on the reproducing property in order to show that the reproducing kernel at the origin must generate the entire shift invariant subspace. [ABSTRACT FROM AUTHOR]

We show that the norm in the Hardy space H^p satisfies \begin{equation} \|f\|_{H^p}^p\asymp \int _0^1M_q^p(r,f')(1-r)^{p\left (1-\frac 1q\right)}\,dr+|f(0)|^p\tag {\dagger } \end{equation} for all univalent functions provided that either q\ge 2 or \frac {2p}{2+p}

Vector-valued analytic functions in Cn, which are known to have vector-valued tempered distributional boundary values, are shown to be in the Hardy space Hp,1≤p<2, if the boundary value is in the vector-valued Lp,1≤p<2, functions. The analysis of this paper extends the analysis of a previous paper that considered the cases for 2≤p≤∞. Thus, with the addition of the results of this paper, the considered problems are proved for all p,1≤p≤∞.

Abstract In this paper, we completely characterize the boundedness and compactness of the Volterra integration operators J g $J_{g}$ acting from the Hardy-type tent spaces HT q , α p ( B n ) to the Hardy spaces H t ( B n ) in the unit ball of C n for all 0 < p , q , t < ∞ $0< p,q,t − n − 1 $\alpha >-n-1$ . The duality and factorization techniques for tent spaces of sequences play an important role in the proof of the main results.

Over the past several years, optimal polynomial approximants (OPAs) have been studied in many different function spaces. In these settings, numerous papers have been devoted to studying the properties of their zeros. In this paper, we introduce the notion of optimal polynomial approximant in the space Lp, 1 ≤ p ≤ ∞. We begin our treatment by showing existence and uniqueness for 1 < p < ∞. For the extreme cases of p = 1 and p = ∞, we show that uniqueness does not necessarily hold. We continue our development by elaborating on the special case of L2. Here, we create a test to determine whether or not a given 1st degree OPA is zero-free in ̄𝔻. Afterward, we shed light on an orthogonality condition in Lp. This allows us to study OPAs in Lp with the additional tools from the L2 setting. Throughout this paper, we focus many of our discussions on the zeros of OPAs. In particular, we show that if 1 < p < ∞, f ∈ Hp, and f(0) ≠ 0, then there exists a disk, centered at the origin, in which all the associated OPAs are zero-free. Toward the end of this paper, we use the orthogonality condition to compute the coefficients of some OPAs in Lp. To inspire further research in the general theory, we pose several open questions throughout our discussions.

Let φ be an analytic self-map of the open unit disk D and let Ψ be an analytic function on D. The weighted composition operator CΨ, φ is the operator on the Hardy space H² given by CΨ, φ f = Ψ f º φ. Under some conditions on φ1 and φ2, we try to find a subset of the numerical range of CΨ1, φ1 + CΨ2, φ2 and determine when zero lies in the interior of the numerical range of CΨ1, φ1 + CΨ2, φ2. [ABSTRACT FROM AUTHOR]

We prove the weighted boundedness for the multilinear operators associated to some integral operators for the endpoint cases. The operators include Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.

The properties of solutions of the following differential equation $ f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f = F(z) $ are studied, where $ A_{j}(z) $ and $ F(z) $ are analytic in the unit disc $ \mathbb{D} = \{z:|z| < 1\} $, $ j = 0, 1, \ldots, k-1 $. First, the growth of solutions of the equation is estimated. Second, some coefficient's conditions such that the solution of the equation belong to Hardy type spaces are showed. Finally, some related question are studied in this paper.

Abstract In this paper, the boundedness of the classic Hardy operator and its adjoint on Hardy spaces is obtained. We also discuss the boundedness for the commutators generated by the classic Hardy operator and its adjoint with B M O $BMO$ and C M O ( R + ) $CMO(\mathbb{R}^{+})$ functions on Hardy spaces.

In this paper, an equivalent quasinorm for the Lipschitz space of noncommutative martingales is presented. As an application, we obtain the duality theorem between the noncommutative martingale Hardy space hpc(ℳ){h}_{p}^{c}( {\mathcal M} ) (resp. hpr(ℳ){h}_{p}^{r}( {\mathcal M} )) and the Lipschitz space λβc(ℳ){\lambda }_{\beta }^{c}( {\mathcal M} ) (resp. λβr(ℳ){\lambda }_{\beta }^{r}( {\mathcal M} )) for 0

In this paper, we deal with some geometric properties including starlikeness and convexity of order α of Jackson’s second and third q-Bessel functions which are natural extensions of classical Bessel function Jν. In additon, we determine some conditions on the parameters such that Jackson’s second and third q-Bessel functions belong to the Hardy space and to the class of bounded analytic functions.

Abstract The purpose of the paper is to study the coefficient multipliers of the Hardy spaces H p $H^{p}$ associated with Jacobi expansions of exponential type. The main results are about the boundedness from H p $H^{p}$ to ℓ q $\ell ^{q}$ of the multiplier operators in terms of Jacobi expansions of exponential type for (i) p = 1 $p=1$ , 2 ≤ q < ∞ $2\le q

Abstract Let a function b belong to the space BMOθ(ρ) $\operatorname{BMO}_{\theta }(\rho )$, which is larger than the space BMO(Rn) $\operatorname{BMO}(\mathbb{R}^{n})$, and let a nonnegative potential V belong to the reverse Hölder class RHs $\mathit{RH}_{s}$ with n/2

Abstract Recently we have introduced a product-type operator and studied it on some spaces of analytic functions on the unit disc. Here we start investigating the operator on the space of analytic functions on the upper half-plane. We characterize the boundedness and compactness of the operator between Hardy and α-Bloch spaces on the domain.

Abstract Let L=−Δ+|x|2 $L=-\Delta+|x|^{2}$ be a Hermite operator, where Δ is the Laplacian on Rd $\mathbb {R}^{d}$. 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 HLp(Rd) $H_{L}^{p}(\mathbb {R}^{d})$ associated with L.

Abstract In this paper, we obtain that C ψ , φ ∗ $C_{\psi,\varphi}^{\ast}$ is bounded below on H 2 $H^{2}$ or A α 2 $A_{\alpha}^{2}$ if and only if C ψ , φ $C_{\psi,\varphi }$ is invertible. Moreover, we investigate the Fredholm operators C ψ 1 , φ 1 C ψ 2 , φ 2 ∗ $C_{\psi_{1},\varphi_{1}}C_{\psi_{2},\varphi_{2}}^{\ast}$ and C ψ 1 , φ 1 ∗ C ψ 2 , φ 2 $C_{\psi_{1},\varphi_{1}}^{\ast}C_{\psi_{2},\varphi_{2}}$ on H 2 $H^{2}$ and A α 2 $A_{\alpha}^{2}$ .

In 2011, Dekel et al. developed highly geometric Hardy spaces Hp(Θ), for the full range 0

Let B be the unit disc in ℝ2, ℋ be the completion of C 0 ∞ (B) under the norm ∥ u ∥ H = (∫ B | ▽ u | 2 d x − ∫ B u 2 (1 − | x | 2) 2 d x) 1 2 , ∀ u ∈ C 0 ∞ (B). By the method of blow-up analysis and an argument of rearrangement with respect to the standard hyperbolic metric d v H = d x (1 − | x | 2 ) 2 , we prove that, for any fixed α , 0 ≤ α < λ p (B) = inf u ∈ H , u ≠ 0 ∥ u ∥ H 2 ∥ u ∥ p 2 , the supremum sup u ∈ H , ∥ u ∥ H = 1 ∫ B e 4 π (1 + α ∥ u ∥ p 2) u 2 d x < + ∞ , ∀ p > 1. This is an analog of early results of Lu—Yang (Discrete Contin. Dyn. Syst., 2009) and Yang (Trans. Amer. Math. Soc., 2007), and extends those of Wang—Ye (Adv. Math., 2012) and Yang—Zhu (Ann. Global Anal. Geom., 2016). [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to prove that the boundedness of the commutator Mκ,b∗$\mathcal{M}_{\kappa,b}^{*} $ generated by the Littlewood-Paley operator Mκ∗$\mathcal{M}_{\kappa}^{*} $ and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of Mκ∗$\mathcal{M}_{\kappa}^{*} $ satisfies a certain Hörmander-type condition, the authors prove that Mκ,b∗$\mathcal{M}_{\kappa,b}^{*} $ is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from the space L log L(μ) to the weak Lebesgue space L1,∞(μ), and is bounded from the atomic Hardy spaces H1(μ) to the weak Lebesgue spaces L1,∞(μ).

Liflyand and Móricz in Liflyand and Móricz [The Hausdorff operator is bounded on the real Hardy space. Proc Amer Math Soc. 1999;128:1391–1396; Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces. Acta Math Hungar. 2002;97(1–2):133–143] studied the Hausdorff operator on the real Hardy space and proved the boundedness property and the commuting relations of this operator and Hilbert transformations. The main objective of this paper is extending these results to the context of Dunkl theory. [ABSTRACT FROM AUTHOR]

Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the author develops a partial theory of paraproducts {Πj}j=13 defined via approximations of the identity with exponential decay (and integration 1), which are extensions of paraproducts defined via regular wavelets. Precisely, the author first obtains the boundedness of Π3 on Hardy spaces and then, via the methods of interpolation and the well-known T(1) theorem, establishes the endpoint estimates for {Πj}j=13. The main novelty of this paper is the application of the Abel summation formula to the establishment of some relations among the boundedness of {Πj}j=13, which has independent interests. It is also remarked that, throughout this article, μ is not assumed to satisfy the reverse doubling condition.

Abstract Let φ : R n × [ 0 , ∞ ) → [ 0 , ∞ ) $\varphi:\mathbb{R}^{n}\times[0, \infty) \to[0, \infty)$ satisfy that φ ( x , ⋅ ) $\varphi(x, \cdot)$ , for any given x ∈ R n $x\in\mathbb{R}^{n}$ , is an Orlicz function and φ ( ⋅ , t ) $\varphi(\cdot, t)$ is a Muckenhoupt A ∞ $A_{\infty}$ weight uniformly in t ∈ ( 0 , ∞ ) $t\in(0, \infty)$ . The Musielak-Orlicz Hardy space H φ ( R n ) $H^{\varphi}(\mathbb{R}^{n})$ is defined to be the set of all tempered distributions such that their grand maximal functions belong to the Musielak-Orlicz space L φ ( R n ) $L^{\varphi}(\mathbb{R}^{n})$ . In this paper, the authors establish the boundedness of Marcinkiewicz integral μ Ω $\mu _{\Omega}$ from H φ ( R n ) $H^{\varphi}(\mathbb{R}^{n})$ to L φ ( R n ) $L^{\varphi}(\mathbb{R}^{n})$ under weaker smoothness conditions assumed on Ω. This result is also new even when φ ( x , t ) : = ϕ ( t ) $\varphi(x, t):=\phi(t)$ for all ( x , t ) ∈ R n × [ 0 , ∞ ) $(x, t)\in\mathbb{R}^{n}\times[0, \infty)$ , where ϕ is an Orlicz function.

Abstract In this paper, we consider the Riesz transform of higher order associated with the harmonic oscillator L = − Δ + | x | 2 $L=-\Delta+|x|^{2}$ , where Δ is the Laplacian on R d $\mathbb {R}^{d}$ . Moreover, the boundedness of Riesz transforms of higher order associated with Hermite functions on the Hardy space is proved.

The asymptotic profile for diffusion wave terms of solutions to the compressible Navier–Stokes–Korteweg system is studied on R2. The diffusion wave with time-decay estimate was studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002), and Kobayashi and Tsuda (2018) for compressible Navier–Stokes and compressible Navier–Stokes–Korteweg systems. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space–time L2 of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by L2 on space, decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz’s energy estimate, and the Fefferman–Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation.

In this paper, we give a new proof of the Bessonov theorem [Proc. Amer. Math. Soc. 142 (2014), pp. 1301-1313] on finite rank truncated Toeplitz operators on model spaces via the product of Hankel operators on Hardy space. [ABSTRACT FROM AUTHOR]

Operators on function spaces of form Cɸf = f ∘ ɸ, where ɸ is a fixed map are called composition operators with symbol ɸ. We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.

In this paper we obtain a characterization of little Hankel operators defined on the Bergman space of the unit disk and then extend the result to vector valued Bergman spaces. We then derive from it certain asymptotic properties of little Hankel operators.