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Let a → : = (a 1 , ... , a n) ∈ [ 1 , ∞) n , p ∈ (0 , 1) , and α:= 1/p − 1. For any x ∈ ℝn and t ∈ [0, ∞), let Φ p (x , t) : = { t 1 + (t [ x ] a → ν) 1 − p if ν α ∉ ℕ , t 1 + (t [ x ] a → ν) 1 − p [ log (e + ∣ x ∣ a →) ] p if ν α ∈ ℕ , where [ ⋅ ] a → := 1 + ∣ ⋅ ∣ a → , ∣ ⋅ ∣ a → denotes the anisotropic quasi-homogeneous norm with respect to a → , and ν:= a1 +...+ an. Let H a → p (ℝ n) , ℒ α a → (ℝ n) , and H a → Φ p (ℝ n) be, respectively, the anisotropic Hardy space, the anisotropic Campanato space, and the anisotropic Musielak-Orlicz Hardy space associated with Φp on ℝn. In this article, via first establishing the wavelet characterization of anisotropic Campanato spaces, we prove that for any f ∈ H a → p (ℝ n) and g ∈ ℒ α a → (ℝ n) , the product of f and g can be decomposed into S(f, g) + T(f, g) in the sense of tempered distributions, where S is a bilinear operator bounded from H a → p (ℝ n) × ℒ α a → (ℝ n) to L1(ℝn) and T is a bilinear operator bounded from H a → p (ℝ n) × ℒ α a → (ℝ n) to H a → Φ p (ℝ n) . Moreover, this bilinear decomposition is sharp in the dual sense that any Y ⊂ H a → Φ p (ℝ n) that fits into the above bilinear decomposition should satisfy (L 1 ( R n) + Y) ∗ = (L 1 ( R n) + H a → Φ p ( R n)) ∗ . As applications, for any non-constant b ∈ ℒ α a → (ℝ n) and any sublinear operator T satisfying some mild bounded assumptions, we find the largest subspace of H a → p (ℝ n) , denoted by H a → , b p (ℝ n) , such that the commutator [b, T] is bounded from H a → , b p (ℝ n) to L1(ℝn). In addition, when T is an anisotropic Calderón-Zygmund operator, the boundedness of [b, T] from H a → , b p (ℝ n) to L1(ℝn)(or to H a → 1 (ℝ n) ) is also presented. The key of their proofs is the wavelet characterization of function spaces under consideration. [ABSTRACT FROM AUTHOR]

As a continuation of the study of polynomial approximation and composition operators on Dirichlet spaces of unit disk, which has settled a problem posed by Cima in 1976, the present paper aims to consider the case of the unbounded domains, such as the half-plane. Specifically, we may obtain the rational approximations in the Dirichlet spaces and characterize the composition operators which have a dense range on the Dirichlet spaces over the half-plane. Moreover, this paper also considers the relationship between the Dirichlet spaces and Hardy spaces on half-plane. [ABSTRACT FROM AUTHOR]

In this article, we establish a transference between the n-dimensional Euclidean space R n and the n-torus T n about the H p - L p , ∞ boundedness of maximal multipliers. As an application, we obtain that the maximal oscillatory integral S α , β ∗ is bounded from H p R n to L p , ∞ R n under the sharp relation among α , β and p. [ABSTRACT FROM AUTHOR]

In this paper, the authors completely characterize when two dual truncated Toeplitz operators are essentially commuting and when the semicommutator of two dual truncated Toeplitz operators is compact. Their main idea is to study dual truncated Toeplitz operators via Hankel operators, Toeplitz operators and function algebras. [ABSTRACT FROM AUTHOR]

Suppose $ \alpha >-1 $ α > − 1 and $ 1\leq p \leq \infty $ 1 ≤ p ≤ ∞. Let $ f=\mathcal {P}_{\alpha }[F] $ f = P α [ F ] be an α-harmonic mapping on $ {\mathbb D} $ D with the boundary F being absolute continuous and $ \dot {F}\in L^p(0,2\pi) $ F ˙ ∈ L p (0 , 2 π) , where $ \dot {F}({\rm e}^{i\theta }):=\frac {{\rm d}}{{\rm d}\theta }F({\rm e}^{i\theta }) $ F ˙ (e iθ) := d d θ F (e iθ). In this paper, we investigate the membership of $ f_z $ f z and $ f_{\bar {z}} $ f z ¯ in the space $ \mathcal {H}_{\mathcal {G}}^{p}(\mathbb {D}) $ H G p (D) , the generalized Hardy space. We prove, if $ \alpha >0 $ α > 0 , then both $ f_z $ f z and $ f_{\bar {z}} $ f z ¯ are in $ \mathcal {H}_{\mathcal {G}}^{p}(\mathbb {D}) $ H G p (D). If $ \alpha α < 0 , then $ f_z $ f z and $ f_{\bar {z}}\in \mathcal {H}_{\mathcal {G}}^{p}(\mathbb {D}) $ f z ¯ ∈ H G p (D) if and only if f is analytic. Finally, we investigate a Schwarz Lemma for α-harmonic functions. [ABSTRACT FROM AUTHOR]

We define a weighted maximal operator ...p* and prove that it is bounded from Hp(G) to Lp(G), for p ∈ (0,1). [ABSTRACT FROM AUTHOR]

In the view of existing works on operators (namely, λ- Hankel and λ- Slant Hankel operators etc.) done by Avendano, Datt and Aggrawal, we take up an operator equation ... and solve it for λ, μ ∈ C. Its solution yields an important class of slant Hankel operators, (namely, (λ, μ)-slant Hankel operators). After that, we provide certain basic properties of (λ, μ)- slant Hankel operator on L²(T) of unit circle T. Finally, we come up with a results for (λ, μ)- slant Hankel operator, which is analogous to classical Kronecker theorem for Hankel matrix. [ABSTRACT FROM AUTHOR]

In this article, we will give an affirmative answer to Gilbert’s conjecture on Hardy spaces of Clifford analytic functions in upper half-space of R 8 . It is based on an explicit construction of Clifford algebra C l 8 and Spinor space R 8 by octonion algebra. Furthermore, it provides an associative approach to the theory of octonionic analytic functions. Additionally, certain classical results about octonionic analytic functions have been reformulated, and a related subject has been treated in Octonionic Hardy space in upper half-space. [ABSTRACT FROM AUTHOR]

We begin a study of Schur analysis when the variable is now a matrix rather than a complex number. We define the corresponding Hardy space, Schur multipliers and their realizations, and interpolation. Possible applications of the present work include matrices of quaternions, matrices of split quaternions, and other algebras of hypercomplex numbers. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new approach to the classical Hardy spaces on Euclidean space. The new ideas of this paper are (i) introduce new test functions and distributions; (ii) we use the classical Calderón reproducing formula on L 2 (R d) only; (iii) the Hardy space is defined by the collections of all new distributions with the classical wavelet-type representation and the norms of Hardy space are defined as the norms of the classical atomic Hardy spaces. [ABSTRACT FROM AUTHOR]

The invariant subspace problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota) the ISP can be solved by proving that every minimal invariant subspace of a universal operator is one dimensional. In this work, we obtain conditions for T φ ∗ | M to have a non-trivial subspace where M ⊂ H 2 (D 2) is an invariant subspace of the Toeplitz operator T φ ∗ on the Hardy space over the bidisk H 2 (D 2) induced by the symbol φ ∈ H ∞ (D) . We then use this fact to obtain sufficient conditions for the ISP to be true. [ABSTRACT FROM AUTHOR]

Let ℒ 2 = (- Δ) 2 + V 2 be the Schrödinger-type operator on ℝ n ( n ≥ 5 ), let H ℒ 2 1 ⁢ (ℝ n) be the Hardy space related to ℒ 2 , and let BMO θ ⁢ (ρ) be the BMO-type space introduced by Bongioanni, Harboure and Salinas. In this paper, we investigate the boundedness of commutator [ b , T α , β , j ] , which is generated by the Riesz transform T α , β , j = V 2 ⁢ α ⁢ ∇ j ⁡ ℒ 2 - β , j = 1 , 2 , 3 , and b ∈ BMO θ ⁢ (ρ) . Here, 0 < α ≤ 1 - j 4 , j 4 < β ≤ 1 , β - α = j 4 , and the nonnegative potential V belongs to both the reverse Hölder class RH s with s ≥ n 2 and the Gaussian class associated with (- Δ) 2 . The L p boundedness of [ b , T α , β , j ] is obtained, and it is also shown that [ b , T α , β , j ] is bounded from H ℒ 2 1 ⁢ (ℝ n) to weak L 1 ⁢ (ℝ n) . [ABSTRACT FROM AUTHOR]

In this paper, we provide sufficient conditions for the functions ψ and ϕ to be the approximate duals in the Hardy space H p (R) for all 0 < p ≤ 1 . Based on these conditions, we obtain the wavelet series expansion in the Hardy space H p (R) with the approximate duals. The important properties of our approach include the following: (i) our results work for any 0 < p ≤ 1 ; (ii) we do not assume that the functions ψ and ϕ are exact duals; (iii) we provide a tractable bound for the operator norm of the associated wavelet frame operator so that it is possible to check the suitability of the functions ψ and ϕ . [ABSTRACT FROM AUTHOR]

In this paper we initiate the study of the forward and backward shifts on the discrete generalized Hardy space of a tree and the discrete generalized little Hardy space of a tree. In particular, we investigate when these shifts are bounded, find the norm of the shifts if they are bounded, characterize the trees in which they are an isometry, compute the spectrum in some concrete examples, and completely determine when they are hypercyclic. [ABSTRACT FROM AUTHOR]

We prove that a generic function from ⋂ p < 1 H p has unbounded Taylor coefficients, this applies also to the Taylor coefficients of its derivatives. Results of similar nature are valid when the space ⋂ p < 1 H p is replaced by H p ( 0 < p < 1 ) and by localized versions of such spaces. Moreover, we prove that a generic function from A (D) has Taylor coefficients outside of ℓ 1 , this applies also to the Taylor coefficients of its derivatives. Lastly, we prove that a generic function from ⋂ p < 1 h p has a harmonic conjugate that does not belong to any h q (q > 0) . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Hyers–Ulam stability of the coefficient multipliers on the Hardy space H 2 and the Bergman space A 2 , meanwhile, we also investigate the Hyers–Ulam stability of the coefficient multipliers between the Bergman space A 2 and the Hardy space H 2 . We give the necessary and sufficient condition for the coefficient multipliers to have the Hyers–Ulam stability on the Hardy space H 2 , on the Bergman space A 2 and between the Bergman space A 2 and the Hardy space H 2 , respectively. We also show that the best constant of Hyers–Ulam stability exists under different circumstances. Some results generalized the results of MacGregor and Zhu when p = 2 in MacGregor and Zhu article (Mathematika 42:413–426, 1995). Moreover, some illustrative examples are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we completely characterize the boundedness theory of anisotropic area operators from anisotropic mixed‐norm Hardy spaces Ha⃗p⃗(Rn)$H_{\vec{\mathbf {a}}}^{\vec{p}}(\mathbb {R}^{n})$ into mixed‐norm Lebesgue spaces Lp⃗(Rn)$L^{\vec{p}}(\mathbb {R}^{n})$ in terms of Carleson measures. [ABSTRACT FROM AUTHOR]

Let with , and define and denote the function family by . Let be a subset of (or more generally an ordered index set), and suppose that there exists a constant such that for all . Then (i) There exists a constant such that for all , . (ii) The λ-jump operator satisfies uniformly in for some constant . [ABSTRACT FROM AUTHOR]

In this paper we consider various subclasses of normalized, analytic functions defined in the open unit disk A = {z = C : z < 1} in the complex plane C and study the Hardy space of the functions in these subclasses. This study provides an analysis of the growth of these functions near the boundary of the open unit disk and the Taylor's coefficients of them. The study is carried out using the methods of integral means and subordination of analytic functions. Determination of explicit indices of the Hardy space and order of the growth rate of the Taylor coefficient of these functions are important results here. The novelty of the work here is an attempt to extend the study of the above mentioned features for functions in standard subclasses of analytic univalent functions which were not considered by researchers in the past. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 , s ∈ (0 , 1) , and Ω ⊂ R n be a bounded C 2 domain. The aim of this paper is to study the global endpoint regularity estimates for the fractional Dirichlet problem (- Δ) s u = f in Ω , u = 0 in R n \ Ω. More precisely, the authors prove the optimal global BMO regularity estimates (- Δ) s 2 u BMO (R n) ≤ C ‖ f ‖ L q (Ω) and ∇ s u BMO (R n) ≤ C ‖ f ‖ L q (Ω) for some q ∈ (n s , ∞) , and the global regularity estimates (- Δ) s 2 u L p (R n) ≤ C ‖ f ‖ H q (Ω) and ∇ s u L p (R n) ≤ C ‖ f ‖ H q (Ω) for any given q ∈ (n n + s , 1 ] and p ∈ [ 1 , qn n - q s) . Here, ∇ s denotes the Riesz gradient of order s and H q (Ω) denotes the Hardy space on Ω . Moreover, the authors also obtain the global regularity estimates (- Δ) t 2 u L p (R n) ≤ C ‖ f ‖ H q (Ω) and ∇ t u L p (R n) ≤ C ‖ f ‖ H q (Ω) for any given t ∈ (s , min { 1 , s + s n }) , q ∈ (n n + s - n (t - s) , 1 ] , and p ∈ [ 1 , qn n - q s + q n (t - s)) . The global regularity estimates given in this paper are further devolvement for the corresponding results in the scale of Lebesgue spaces, established by B. Abdellaoui, A. J. Fernández, T. Leonori, and A. Younes [arXiv: 2107.06535], in the endpoint case. [ABSTRACT FROM AUTHOR]

For a submodule M in Hardy module H 2 (D n) on the unit polydisc in C n , we define the n - 1 tuple of fringe operators F = (F 1 , F 2 , … , F n - 1) and the n tuple of restriction operators R = (R z 1 , R z 2 , … , R z n) with respect to M . In this paper, for the case n = 3 , it is shown that the fringe operators F are Fredholm if and only if the tuple R - λ is Fredholm, where λ ∈ D 3 , and moreover i n d (F) = - i n d (R - λ) , which answer a question of Yang (Proc Am Math Soc 131 (2):533–541, 2003) partly and generalize a result of Luo et al. (J Math Anal Appl 465(1):531–546, 2018) in the case n = 2 . Finally, we also discuss the difference quotient operators in H 2 (D n) , and apply them to explore the relationship between the fringe operators and compression operators on quotient module. [ABSTRACT FROM AUTHOR]

In this article we obtain an explicit formula for the Hilbert transform of log | f | , for the function f in Nevanlinna class having continuous extension to the real line. This family is the largest possible for which such a formula for the Hilbert transform of log | f | , can be obtained. The formula is very general and implies several previously known results. [ABSTRACT FROM AUTHOR]

The composition operator C ϕ a f = f ∘ ϕ a on the Hardy–Hilbert space H 2 (D) with affine symbol ϕ a (z) = a z + 1 - a and 0 < a < 1 has the property that the Invariant Subspace Problem for complex separable Hilbert spaces holds if and only if every minimal invariant subspace for C ϕ a is one-dimensional. These minimal invariant subspaces are always singly-generated K f : = span { f , C ϕ a f , C ϕ a 2 f , … } ¯ for some f ∈ H 2 (D) . In this article we characterize the minimal K f when f has a nonzero limit at the point 1 or if its derivative f ′ is bounded near 1. We also consider the role of the zero set of f in determining K f . Finally we prove a result linking universality in the sense of Rota with cyclicity. [ABSTRACT FROM AUTHOR]

A version of Littlewood–Paley–Rubio de Francia inequality for bounded multi‐parameter Vilenkin systems is proved: For any family of disjoint sets Ik=Ik1×⋯×IkD⊆Z+D$I_k = I_k^1 \times \cdots \times I_k^D \subseteq {\mathbb {Z}_+^D}$ such that Ikd$I_k^d$ are intervals in Z+$\mathbb {Z}_+$ and a family of functions fk$f_k$ with Vilenkin–Fourier spectrum inside Ik$I_k$ the following holds: ∑kfkLp≤C∑kfk21/2Lp,1

The main purpose of this paper is to investigate characterizations of composition operators on Bloch and Hardy type spaces. Initially, we use general doubling weights to study the composition operators from harmonic Bloch type spaces on the unit disc D to pluriharmonic Hardy spaces on the Euclidean unit ball B n . Furthermore, we develop some new methods to study the composition operators from harmonic Bloch type spaces on D to pluriharmonic Bloch type spaces on D . Additionally, some application to new characterizations of the composition operators between pluriharmonic Lipschitz type spaces to be bounded or compact will be presented. The obtained results of this paper provide the improvements and extensions of the corresponding known results. [ABSTRACT FROM AUTHOR]

Let f be an integrable function which has integral 0 on R n . What is the largest condition on | f | that guarantees that f is in the Hardy space H 1 (R n) ? When f is compactly supported, it is well-known that the largest condition on | f | is the fact that | f | ∈ L log L (R n) . We consider the same kind of problem here, but without any condition on the support. We do so for H 1 (R n) , as well as for the Hardy space H log (R n) which appears in the study of pointwise products of functions in H 1 (R n) and in its dual BMO. [ABSTRACT FROM AUTHOR]

Let A be an expansive dilation on Rn and let (·) L(Rn). Also let p(·) : Rn → (0, ) be a variable exponent function satisfying the globally log-H¨older continuous condition. In this paper, the authors first introduce the variable anisotropic Herz-Morrey-Hardy spaces HM and HMK nd then establish their atomic decompositions. As applications, the authors obtain the boundedness of some sublinear operators from HM [ABSTRACT FROM AUTHOR]

We define conic reductions X ν red for torus actions on the boundary X of a strictly pseudo-convex domain and for a given weight ν labeling a unitary irreducible representation. There is a natural residual circle action on X ν red . We have two natural decompositions of the corresponding Hardy spaces H(X) and H (X ν red) . The first one is given by the ladder of isotypes H (X) k ν , k ∈ Z ; the second one is given by the k-th Fourier components H (X ν red) k induced by the residual circle action. The aim of this paper is to prove that they are isomorphic for k sufficiently large. The result is given for spaces of (0, q)-forms with L 2 -coefficient when X is a CR manifold with non-degenerate Levi form. [ABSTRACT FROM AUTHOR]

Let be an inner function in the domain , where is the open unit ball in , . We construct dominant sets for the space , where is the standard Hardy space. [ABSTRACT FROM AUTHOR]

The field of geometric function theory has thoroughly investigated starlike functions concerning symmetric points. The main objective of this work is to derive certain geometric properties, such as the starlikeness of order δ , convexity of order δ , k-starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, and pre-starlikeness for the Galué type Struve function (GTSF). Furthermore, the conditions for GTSF belonging to the Hardy space are also derived. The results obtained in this work generalize several results available in the literature. [ABSTRACT FROM AUTHOR]

In various applications, the adoption of optimal energy matching pursuit with dictionary elements is common. When the dictionary elements are indexed by parameters within a bounded region, exhaustion-type algorithms can be employed. This article aims to investigate a process that converts the optimal parameter selection in unbounded regions to a bounded and closed (compact) sub-domain. Such a process provides accessibility for energy matching pursuit in a wide range of applications. The paper initially focuses on the open unit disc and the upper-half complex plane, introducing adaptive Fourier decomposition as the underlying methodology. It then extends this concept to general Hilbert spaces with a dictionary and Bochner spaces for random signals. Computational examples are included to illustrate the concepts presented. [ABSTRACT FROM AUTHOR]

Consider the multiplication operator MB in $L^2(\mathbb{T})$ , where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of MB in $L^2(\mathbb{T})$. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(\mathbb{T})$. Moreover, we analyse their properties while keeping the whole Hardy space, model space and Beurling-type subspaces invariant. Furthermore, we extended our study concerning conjugations in the case of finite Blaschke products. [ABSTRACT FROM AUTHOR]

We use the notion of radial differential operator of order t > 0 to introduce the weighted composition–differentiation operator E ψ , φ t (f) = ψ · (R t f) ∘ φ on the Hardy and Bergman spaces of the unit ball and the polydisk in C n . We obtain necessary and sufficient conditions on the functions φ and ψ to ensure that the operator E ψ , φ t is Hilbert–Schmidt. [ABSTRACT FROM AUTHOR]

Hardy's inequality on H p spaces, p ∈ (0 , 1 ] , in the context of orthogonal expansions is investigated for general bases on a wide class of domains in R d with Lebesgue measure. The obtained result is applied to various Hermite, Laguerre, and Jacobi expansions. For that purpose some delicate estimates of the higher order derivatives for the underlying functions and of the associated heat or Poison kernels are proved. Moreover, sharpness of studied Hardy's inequalities is justified by a construction of an explicit counterexample, which is adjusted to all considered settings. [ABSTRACT FROM AUTHOR]

We discuss de Branges–Rovnyak spaces ℋ ⁢ (b) generated by nonextreme and rational functions b and local Dirichlet spaces of order m introduced in [S. Luo, C. Gu and S. Richter, Higher order local Dirichlet integrals and de Branges–Rovnyak spaces, Adv. Math. 385 2021, Paper No. 107748]. In that paper, the authors characterized nonextreme b for which the operator Y = S | ℋ ⁢ (b) , the restriction of the shift operator S on H 2 to ℋ ⁢ (b) , is a strict 2 ⁢ m -isometry and proved that such spaces ℋ ⁢ (b) are equal to local Dirichlet spaces of order m. Here we give a characterization of local Dirichlet spaces of order m in terms of the m-th derivatives that is a generalization of a known result on local Dirichlet spaces. We also find explicit formulas for b in the case when ℋ ⁢ (b) coincides with local Dirichlet space of order m with equality of norms. Finally, we prove a property of wandering vectors of Y analogous to the property of wandering vectors of the restriction of S to harmonically weighted Dirichlet spaces obtained in [D. Sarason, Harmonically weighted Dirichlet spaces associated with finitely atomic measures, Integral Equations Operator Theory 31 1998, 2, 186–213]. [ABSTRACT FROM AUTHOR]

Let W be a finite reflection group associated with a root system R in ℝ d . Let C + denote a positive Weyl chamber distinguished by a choice of R + , a set of positive roots. We define and investigate Hardy and BMO spaces on C + in the framework of boundary conditions given by a homomorphism η ∈ Hom ⁡ (W , ℤ ^ 2) which attaches the ± signs to the facets of C + . Specialized to orthogonal root systems, atomic decompositions in H η 1 and h η 1 are obtained and the duality problem is also treated. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the complex symmetric Toeplitz operator on the Hardy space via a kind of canonical conjugation on H² introduced by M.S. Ferreira in [1]. In model space equipped with a conjugation, we prove some results for truncated toeplitz operators in relationship with complex symmetry according to [2]. [ABSTRACT FROM AUTHOR]

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.

We establish a factorization theorem for the Hardy space in terms of a multilinear fractional integral operator. As an application, a characterization of Lipschitz spaces via the boundedness of commutators of the multilinear fractional integral operator is obtained. [ABSTRACT FROM AUTHOR]

This paper defines local weighted Hardy spaces with variable exponents. Local Hardy spaces permit atomic decomposition, which is one of the main themes in this paper. A consequence is that the atomic decomposition is obtained for the functions in the Lebesgue spaces with exponentially decaying exponent. As applications, we obtain the boundedness of singular integral operators, the Littlewood–Paley characterization, and wavelet decomposition. [ABSTRACT FROM AUTHOR]

Let μ be a finite positive Borel measure on [0, 1) and let H (D) be the space of all analytic function in the unit disc D . The Cesàro-like operator C μ is defined in H (D) as follows: If f ∈ H (D) , f (z) = ∑ n = 0 ∞ a n z n (z ∈ D) , then C μ (f) (z) = ∑ n = 0 ∞ μ n ∑ k = 0 n a k z n , z ∈ D , where for n ≥ 0 , μ n denotes the n-th moment of the measure μ , that is, μ n = ∫ 0 1 t n d μ (t) . For s > 1 , let X be a Banach subspace of H (D) lying between the mean Lipschtz space Λ 1 s s and the Bloch space B . In this paper we characterize the measures μ as above for which C μ is bounded (compact) from X into any of the Hardy spaces H p (1 ≤ p ≤ ∞) . [ABSTRACT FROM AUTHOR]