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In this paper, we characterize weighted composition operators that preserve frames on the weighted Hardy spaces in the unit disk. In particular, we obtain the symbol properties of the bounded invertible weighted composition operators. Moreover, we establish the equivalence between bounded invertible operators and frame-preserving operators. Furthermore, we show that weighted composition operator preserves frames if and only if it preserves the Riesz bases property. Additionally, we investigate the weighted composition operators that preserve tight or normalized tight frames on the Dirichlet space. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete real-variable theory of local variable Hardy spaces. First, we present various real-variable characterization in terms of several local maximal functions. Next, the new atomic and the finite atomic decomposition for the local variable Hardy spaces are established. As an application, we also introduce the local variable Campanato space which is showed to be the dual space of the local variable Hardy spaces. Analogous to the homogeneous case, some equivalent definitions of the dual of local variable Hardy spaces are also considered. Finally, we show the boundedness of inhomogeneous Calderón-Zygmund singular integrals and local fractional integrals on local variable Hardy spaces and their duals. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the weak Lipschitz spaces WLipq,α, 1 < q < ∞, 0 < α < 1 which are the analog of weak Lebesgue spaces Lq,∞ in the setting of Lipschitz space. We obtain the equivalence between the norm and . As an application, we show that the commutator Mβb is bounded from Lp to Lq,∞ for some p ∈ (1, ∞) and if and only if b is in Lipα. We also introduce the weak central bounded mean oscillation space WCBMOq,α and give a characterization of WCBMOq,α via the boundedness of the commutators of Hardy type operators. [ABSTRACT FROM AUTHOR]

Let H2(ⅅ2) be the Hardy space over the bidisk ⅅ2, and let Mψ,ϕ = [(ψ(z) − ϕ(w))2] be the submodule generated by (ψ(z) − ϕ(w))2, where ψ(z) and ϕ(w) are nonconstant inner functions. The related quotient module is denoted by Nψ,ϕ = H2(ⅅ2) ⊖ Mψ,ϕ. In this paper, we give a complete characterization for the essential normality of Nψ,ϕ. In particular, if ψ(z)= z, we simply write Mψ,ϕ and Nψ,ϕ as Mϕ and Nϕ respectively. This paper also studies compactness of evaluation operators L(0)∣nϕ and R(0)ϕnϕ, essential spectrum of compression operator Sz on Nϕ, essential normality of compression operators Sz and Sw on Nϕ. [ABSTRACT FROM AUTHOR]

It is known that the Invariant Subspace Problem for Hilbert spaces is equivalent to the statement that all minimal non-trivial invariant subspaces for a universal operator are one dimensional. In this paper, we characterize all linear fractional composition operators and their adjoints that have universal translates on the space S 2 (D) . Moreover, we characterize all adjoints of linear fractional composition operators that have universal translates on the Hardy space H 2 (D) . In addition, we consider the minimal invariant subspaces of the composition operator C φ a on S 2 (D) , where ϕa(z) = az + 1 − a, a ∈ (0, 1). Finally, some relationships between complex symmetry and universality for bounded linear operators and commuting pairs of operators on a complex separable, infinite dimensional Hilbert space are explored. [ABSTRACT FROM AUTHOR]

In this paper, we completely characterize the positive Borel measures μ on the unit ball B n such that the differential type operator ℛ m of order m ∈ ℕ is bounded from Hardy type tent space ℋ T q , α p (B n) into Ls(μ) for full range of p, q, s, α. Subsequently, the corresponding compact description of differential type operator ℛ m is also characterized. As an application, we obtain the boundedness and compactness of integration operator Jg from ℋ T q , α p (B n) to ℋ T s , β t (B n) , and the methods used here are adaptable to the Hardy spaces. [ABSTRACT FROM AUTHOR]

Let H 2 (D 2) be the Hardy space over the bidisk D 2 , and let Mφ = [(z − φ(w))2] be the submodule generated by (z − φ(w))2, where φ(w) is a function in H∞(w). The related quotient module is denoted by N φ = H 2 (D 2) Θ M φ . In the present paper, we study the Fredholmness of compression operators Sz, Sw on Nφ. When φ(w) is a nonconstant inner function, we prove that the Beurling type theorem holds for the fringe operator Fw on [(z − w)2] ⊖ z[(z − w)2] and the Beurling type theorem holds for the fringe operator Fz on Mφ ⊖ wMφ if φ(0) = 0. Lastly, we study some properties of Fw on [(z − w2)2] ⊖ z[(z − w2)2]. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for −1/2 ≤ β ≤ 0, suppose M is an invariant subspaces of the Hardy-Sobolev spaces H (D) for T , then M ⊖ zM is a generating wandering subspace of M, that is, $$M = {\left[ {M \ominus zM} \right]_{T_z^\beta }}$$ . Moreover, any non-trivial invariant subspace M of H (D) is also generated by the quasi-wandering subspace P T M , that is, $$M = {\left[ {{P_M}T_z^\beta {M^ \bot }} \right]_{T_z^\beta }}$$ . [ABSTRACT FROM AUTHOR]

In this paper, we generalize the concept of asymptotic Hankel operators on H 2 (D) to the Hardy space H 2 (D n) (over polydisk) in terms of asymptotic Hankel and partial asymptotic Hankel operators and investigate some properties in case of its weak and strong convergence. Meanwhile, we introduce ith-partial Hankel operators on H 2 (D n) and obtain a characterization of its compactness for n > 1. Our main results include the containment of Toeplitz algebra in the collection of all strong partial asymptotic Hankel operators on H 2 (D n) . It is also shown that a Toeplitz operator with symbol ϕ is asymptotic Hankel if and only if ϕ is holomorphic function in L ∞ (T n) . [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish, using the Littlewood-Paley-Stein theory (in particular, the Littlewood-Paley-Stein square functions), a Calderón-Torchinsky type theorem for the following Fourier multipliers on anisotropic Hardy spaces Hp (ℝn; A) associated with expensive dilation A: T m f (x) = ∫ ℝ n m (ξ) f ^ (ξ) e 2 π i x ⋅ ξ d ξ . Our main Theorem is the following: Assume that m(ξ) is a function on ℝn satisfying sup j ∈ ℤ ‖ m j ‖ W s (A ∗) < ∞ with s > ζ − − 1 (1 p − 1 2) Then Tm is bounded from Hp(ℝn; A) to Hp(ℝn; A) for all 0 < p ≤ 1 and ‖ T m ‖ H A p → H A p ≲ sup j ∈ ℤ ‖ m j ‖ W s (A ∗) , where A* denotes the transpose of A. Here we haveusedthe notations mj (ξ)= m(A*jξ)φ(ξ)and is a suitable cut-off function on ℝn, and Ws(A*) is an anisotropic Sobolev space associated with expansive dilation A* on ℝn. [ABSTRACT FROM AUTHOR]

Usually, the condition that T is bounded on L2(ℝn) is assumed to prove the boundedness of an operator T on a Hardy space. With this assumption, one only needs to prove the uniformly boundness of T on atoms, since T(f)= ∑i λiT(ai), provided that f = ∑i λiai in L2 (ℝn), where ai is an L2 atom of this Hardy space. So far, the L2 atomic decomposition of local Hardy spaces hp(ℝn), 0 > p ≤ 1, hasn't been established. In this paper, we will solve this problem, and also show that hp(ℝn) can also be characterized by discrete Littlewood-Paley functions. [ABSTRACT FROM AUTHOR]

A model space is a subspace of the Hardy space which is invariant under the backward shift, and a truncated Toeplitz operator is the compression of a Toeplitz operator on some model space. In this paper we prove a necessary and sufficient condition for the commutator of two truncated Toeplitz operators on a model space to be compact. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish the Hörmander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy spaces for k ≥ 3 using the multi-parameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k = 3, and the method works for all the cases k ≥ 3: where $$x = \left( {{x_1},{x_2},{x_3}} \right) \in {\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}$$ and $$\xi = \left( {{\xi _1},{\xi _2},{\xi _3}} \right) \in {\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}$$ . One of our main results is the following: Assume that m( ξ) is a function on $${\mathbb{R}^{{n_1} + {n_2} + {n_3}}}$$ satisfying $$\mathop {\sup }\limits_{j,k,l \in \mathbb{Z}} {\left\| {{m_{j,k,l}}} \right\|_{{W^{\left( {{s_1},{s_2},{s_3}} \right)}}}} < \infty $$ with $$s_{i} > n_{i}(\frac{1}{p}-\frac{1}{2})$$ for 1 ≤ i ≤ 3. Then T is bounded from $${H^p}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}} \right)$$ to $${H^p}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}} \right)$$ for all 0 < p ≤ 1 and $$\left\| {{T_m}} \right\|{}_{{H^P} \to {H^P}} \lesssim \mathop {\sup }\limits_{j,k,l \in \mathbb{Z}} {\left\| {{m_{j,k,l}}} \right\|_{{W^{\left( {{s_1},{s_{2,}}{s_3}} \right)}}}}$$ . Moreover, the smoothness assumption on s for 1 ≤ i ≤ 3 is optimal. Here we have used the notations m ( ξ) = m(2 ξ , 2 ξ , 2 ξ )Ψ( ξ )Ψ( ξ )Ψ( ξ ) and Ψ( ξ ) is a suitable cut-off function on $${\mathbb{R}^{{n_i}}}$$ for 1 ≤ i ≤ 3, and $${W^{\left( {{s_1},{s_2},{s_3}} \right)}}$$ is a three-parameter Sobolev space on $${\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}$$ . Because the Fefferman criterion breaks down in three parameters or more, we consider the L boundedness of the Littlewood-Paley square function of T f to establish its boundedness on the multi-parameter Hardy spaces. [ABSTRACT FROM AUTHOR]

In this paper, we give counterexamples to show that variation and oscillation operators related to rough truncations of the Hilbert transform are not bounded from H to L , and prove variation, oscillation and λ-jump operators related to smooth truncations are bounded from H to L . [ABSTRACT FROM AUTHOR]

Let ( X, d, μ) be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Under the weak reverse doubling condition, the authors prove that the generalized homogeneous Littlewood-Paley g-function ġ ( r ∈ [2,∞)) is bounded from Hardy space H ( μ) into L ( μ). Moreover, the authors show that, if f ∈ RBMO( μ), then [ ġ ( f)] is either infinite everywhere or finite almost everywhere, and in the latter case, [ ġ ( f)] belongs to RBLO( μ) with the norm no more than | f| multiplied by a positive constant which is independent of f. As a corollary, the authors obtain the boundedness of ġ from RBMO( μ) into RBLO( μ). The vector valued Calderón-Zygmund theory over ( X, d, μ) is also established with details in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce $${L^{{\sigma _1}}}{\left( {\log L} \right)^{{\sigma _2}}}$$ conditions satisfied by the variable kernels Ω( x, z) for 0 ≤ σ ≤ 1 and σ ≥ 0. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators T, fractional integrals T and parametric Marcinkiewicz integrals μ with variable kernels on the Hardy spaces H(R) and weak Hardy spaces WH(R). Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy-Lorentz spaces H(R) for all p < q < ∞. [ABSTRACT FROM AUTHOR]

This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group H. The sharp bounds for the strong type ( p, p) (1 ≤ p ≤ ∞) estimates of n-dimensional Hausdorff operators on H are obtained. The sharp bounds for strong ( p, p) estimates are further extended to multilinear cases. As an application, we derive the sharp constant for the multilinear Hardy operator on H. The weak type ( p, p) (1 ≤ p≤∞) estimates are also obtained. Keywords Hausdorff operator, Heisenberg group, multilinear, sharp estimate [ABSTRACT FROM AUTHOR]

In this paper, we show that two Toeplitz operators T and T on the Hardy space of the polydisk can commute if and only if the Berezin transform of the commutator [ T, T] is n-harmonic. [ABSTRACT FROM AUTHOR]

We first study the Volterra operator V acting on spaces of Dirichlet series. We prove that V is bounded on the Hardy space ℋ 0 p for any 0 < p ≤ ∞, and is compact on ℋ 0 p for 1

g. We prove that if Tg is bounded on the Hardy space ℋ p , then it is bounded on the Bergman space ℋ w p . [ABSTRACT FROM AUTHOR]

Let (χ, ρ, μ) be a space of homogeneous type in the sense of Coifman and Weiss, and Y(χ) a ball quasi-Banach function space on χ, which supports both a Fefferman-Stein vector-valued maximal inequality and the boundedness of the powered Hardy-Littlewood maximal operator on its associate space. The authors first introduce the Hardy space HY(χ) associated with Y(χ), via the Lusin-area function, and then establish its various equivalent characterizations, respectively, in terms of atoms, molecules, and Littlewood—Paley g-functions and gλ*-functions. As an application, the authors obtain the boundedness of Calderón—Zygmund operators from HY (χ) to Y(χ), or to HY(χ) via first establishing a boundedness criterion of linear operators on HY(χ). All these results have a wide range of generality and, particularly, even when they are applied to variable Hardy spaces, the obtained results are also new. The major novelties of this article exist in that, to escape the reverse doubling condition of μ and the triangle inequality of ρ, the authors subtly use the wavelet reproducing formula, originally establish an admissible molecular characterization of HY(χ), and fully apply the geometrical properties of χ expressed by dyadic reference points or dyadic cubes. [ABSTRACT FROM AUTHOR]

Let T be a strongly singular Calderón-Zygmund operator and b ∈ Lloc(ℝn). This article finds out a class of non-trivial subspaces BMOω,p,n(ℝn)of BMO(ℝn) for certain ω ∈ A1, 0 < p ≤ 1 and 1 < u ≤ ∞, such that the commutator [b, T] is bounded from weighted Hardy space H ω p (ℝ n) to weighted Lebesgue space L ω p (ℝ n) if b ∈ BMOω,p,∞(ℝn), and is bounded from weighted Hardy space H ω p (ℝ n) to itself if T*1 = 0 and b ∈ BMOω,p,u(ℝn)for 1 < u < 2. [ABSTRACT FROM AUTHOR]

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]

On the Dirichlet space of the unit ball, we study some algebraic properties of Toeplitz operators. We give a relation between Toeplitz operators on the Dirichlet space and their analogues defined on the Hardy space. Based on this, we characterize when finite sums of products of Toeplitz operators are of finite rank. Also, we give a necessary and sufficient condition for the commutator and semi-commutator of two Toeplitz operators being zero. [ABSTRACT FROM AUTHOR]

Let A be an expansive dilation on ℝ and φ : ℝ × [0, ∞) → [0, ∞) an anisotropic Musielak-Orlicz function. Let H (ℝ) be the anisotropic Hardy space of Musielak-Orlicz type defined via the grand maximal function. In this article, the authors establish its molecular characterization via the atomic characterization of H (ℝ). The molecules introduced in this article have the vanishing moments up to order s and the range of s in the isotropic case (namely, A := 2 I ) coincides with the range of well-known classical molecules and, moreover, even for the isotropic Hardy space H(ℝ) with p ∈ (0, 1] (in this case, A := 2 I , φ( x, t) := t for all x ∈ ℝ and t ∈ [0, ∞)), this molecular characterization is also new. As an application, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators from H (ℝ) to L (ℝ) or from H (ℝ) to itself. [ABSTRACT FROM AUTHOR]