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For a fixed analytic function g on the unit disc D , we consider the analytic paraproducts induced by g, which are defined by T g f ( z ) = ∫ 0 z f ( ζ ) g ′ ( ζ ) d ζ , S g f ( z ) = ∫ 0 z f ′ ( ζ ) g ( ζ ) d ζ , and M g f ( z ) = f ( z ) g ( z ) . The boundedness of these operators on various spaces of analytic functions on D is well understood. The original motivation for this work is to understand the boundedness of compositions of two of these operators, for example T g 2 , T g S g , M g T g , etc. Our methods yield a characterization of the boundedness of a large class of operators contained in the algebra generated by these analytic paraproducts acting on the classical weighted Bergman and Hardy spaces in terms of the symbol g. In some cases it turns out that this property is not affected by cancellation, while in others it requires stronger and more subtle restrictions on the oscillation of the symbol g than the case of a single paraproduct.

This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.

Property $$(UW {\scriptstyle \Pi })$$ for a bounded linear operator $$T\in L(X)$$ on a Banach space X means that the points $$\lambda $$ of the approximate point spectrum for which $$\lambda I-T$$ is upper semi-Weyl are exactly the poles of the resolvent of T. In this paper we give some new results on this property and relate it with some other variants of Browder’s theorem, as property (gaz) and property $$(Z{\scriptstyle \Pi _a})$$ . In the last part the theory is applied to Toeplitz operators $$T_\phi $$ with symbol $$\phi $$ on Hardy spaces.

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 $\mathcal{H}(b)$.

In this paper we first prove the existence of the n-best approximation in terms of the parameterized Szego kernels in the stochastic complex Hardy space of the unit disc. It is a generalization to random signals of the corresponding result for the Hardy space of the disc, and has applications in signal analysis. The result may be generalized to the randomized polydisc case with potential applications in image processing. A practical algorithm for the approximation is proposed.

This paper concerns the reconstruction of a function f in the Hardy space of the unit disc $$\mathbb{D}$$ by using a sample value f (a) and certain n-intensity measurements $$\left| {\left\langle {f,{E_{{a_1} \cdots {a_n}}}} \right\rangle} \right|$$ , where a1, ⋯, $${a_n} \in \mathbb{D}$$ , and $${E_{{a_1} \cdots {a_n}}}$$ is the n-th term of the Gram-Schmidt orthogonalization of the Szego kernels $${k_{{a_1}}}, \cdots ,{k_{{a_n}}}$$ , or their multiple forms. Three schemes are presented. The first two schemes each directly obtain all the function values f (z). In the first one we use Nevanlinna’s inner and outer function factorization which merely requires the 1-intensity measurements equivalent to know the modulus |f (z)|. In the second scheme we do not use deep complex analysis, but require some 2- and 3-intensity measurements. The third scheme, as an application of AFD, gives sparse representation of f (z) converging quickly in the energy sense, depending on consecutively selected maximal n-intensity measurements $$\left| {\left\langle {f,{E_{{a_1} \cdots {a_n}}}} \right\rangle} \right|$$ .

For any 1 ≤ p ≤ ∞, let Sp (ⅅ) be the space of holomorphic functions ƒ on ⅅ such that ƒ′ belongs to the Hardy space Hp(ⅅ), with the norm ∥ƒ∥∑ = ∥f∥∞+∥ƒ′∥p. We prove that every approximate local isometry of Sp(ⅅ) is a surjective isometry and that every approximate 2-local isometry of Sp(ⅅ) is a surjective linear isometry. As a consequence, we deduce that the sets of isometric re ections and generalized bi-circular projections on Sp(ⅅ) are also topologically re exive and 2-topologically reflexive.

In this paper, we find a necessary and sufficient condition for the modulus of continuity for which subsequences of Fejér means with respect to Vilenkin systems are bounded from the Hardy space H p {H_{p}} to the Lebesgue space L p {L_{p}} for all 0 < p < 1 2 {0 .

We will establish the boundedness of the Fourier multiplier operator T m ⁢ f T_{m}f on multi-parameter Hardy spaces H p ⁢ ( R n 1 × ⋯ × R n r ) H^{p}(\mathbb{R}^{n_{1}}\times\cdots\times\mathbb{R}^{n_{r}}) ( 0 < p ≤ 1 0 ) when the multiplier 𝑚 is of optimal smoothness in multi-parameter Besov spaces B 2 , q ( s 1 , … , s r ) ⁢ ( R n 1 × ⋯ × R n r ) B^{{(s_{1},\ldots,s_{r})}}_{2,q}(\mathbb{R}^{n_{1}}\times\cdots\times\mathbb{R}^{n_{r}}) , where T m ⁢ f ⁢ ( x ) = ∫ R n 1 × ⋯ × R n r m ⁢ ( ξ ) ⁢ f ^ ⁢ ( ξ ) ⁢ e 2 ⁢ π ⁢ i ⁢ x ⋅ ξ ⁢ d ξ T_{m}f(x)=\int_{\mathbb{R}^{n_{1}}\times\cdots\times\mathbb{R}^{n_{r}}}m(\xi)\hat{f}(\xi)e^{2\pi ix\cdot\xi}\,d\xi for x ∈ R n 1 × ⋯ × R n r x\in{\mathbb{R}^{n_{1}}\times\cdots\times\mathbb{R}^{n_{r}}} . We will show ∥ T m ∥ H p → H p ≲ sup j 1 , … , j r ∈ Z ⁡ ∥ m j 1 , … , j r ∥ B 2 , q ( s 1 , … , s r ) , \lVert T_{m}\rVert_{H^{p}\to H^{p}}\lesssim\sup_{j_{1},\ldots,j_{r}\in\mathbb{Z}}\lVert m_{j_{1},\ldots,j_{r}}\rVert_{B^{{(s_{1},\ldots,s_{r})}}_{2,q}}, where 0 < q < ∞ 0 and s i > n i ⁢ ( 1 p - 1 2 ) s_{i}>n_{i}\bigl{(}\frac{1}{p}-\frac{1}{2}\bigr{)} . Here we have used the notation m j 1 , … , j r ⁢ ( ξ ) = m ⁢ ( 2 j 1 ⁢ ξ 1 , … , 2 j r ⁢ ξ r ) ⁢ ψ ( 1 ) ⁢ ( ξ 1 ) ⁢ ⋯ ⁢ ψ ( r ) ⁢ ( ξ r ) , m_{j_{1},\ldots,j_{r}}(\xi)=m(2^{j_{1}}\xi_{1},\ldots,2^{j_{r}}\xi_{r})\psi^{(1)}(\xi_{1})\cdots\psi^{(r)}(\xi_{r}), and ψ ( i ) ⁢ ( ξ i ) \psi^{(i)}(\xi_{i}) is a suitable cut-off function on R n i \mathbb{R}^{n_{i}} for 1 ≤ i ≤ r 1\leq i\leq r . This multi-parameter Hörmander multiplier theorem is in the spirit of the earlier work of Baernstein and Sawyer in the one-parameter setting and sharpens our recent result of Hörmander multiplier theorem in the bi-parameter case which was established using R. Fefferman’s boundedness criterion. Because R. Fefferman’s boundedness criterion fails in the cases of three or more parameters, it is substantially more difficult to establish such Hörmander multiplier theorems in three or more parameters than in the bi-parameter case. To assume only the optimal smoothness on the multipliers, delicate and hard analysis on the sharp estimates of the square functions on arbitrary atoms are required. Our main theorems give the boundedness on the multi-parameter Hardy spaces under the smoothness assumption of the multipliers in multi-parameter Besov spaces and show the regularity conditions to be sharp.

Weighted counterparts of the one-sided Littlewood–Paley inequalities for arbitrary rectangles in R 2 \mathbb {R}^2 are proved. For a partition I \mathcal {I} of the plane R 2 \mathbb {R}^2 into rectangles with sides parallel to coordinate axes and a weight w ( ⋅ , ⋅ ) w(\,\cdot \,, \,\cdot \,) satisfying the two-parameter Muckenhoupt condition A p / 2 A_{p/2} for 2 > p > ∞ 2 > p > \infty , the following inequality holds: c p , w ‖ { M I f } I ∈ I ‖ L w p ( l 2 ) ≤ ‖ f ‖ L w p , \begin{equation*} c_{p, w}\lVert \{M_I f\}_{I \in \mathcal {I}} \rVert _{L^p_w(l^2)} \leq \lVert f \rVert _{L_w^p} , \end{equation*} where the symbols M I f ^ = f ^ χ I \widehat {M_I f} = \widehat {f} \chi _{I} denote the corresponding Fourier multipliers. For I \mathcal {I} as above, p p in the range 0 > p > 2 0 > p > 2 , and weights w w satisfying a dual condition α r ( p ) \alpha _{r(p)} , the following inequality holds ‖ ∑ I ∈ I f I ‖ L w p ≤ C p , w ‖ { f I } I ∈ I ‖ L w p ( l 2 ) , where supp ⁡ f I ^ ⊆ I for I ∈ I . \begin{equation*} \Big \|{\sum }_{I \in \mathcal {I}} f_I\Big \|_{L^p_w} \leq C_{p, w} \big \| \left \{ f_I \right \}_{I \in \mathcal {I}} \big \|_{L^p_w(l^2)} , \text { where } \operatorname {supp}{\widehat {f_I}} \subseteq I \text { for } I \in \mathcal {I}. \end{equation*} The proof is based on the theory of two-parameter singular integral operators on Hardy spaces developed by R. Fefferman and some of its more recent weighted generalizations. The former and the latter inequalities are extensions to the weighted setting, respectively, for Journe’s result of 1985 and Osipov’s result of 2010.

In this paper, we extend the classical Bohr's inequality to the setting of the non-commutative Hardy space $H^1$ associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr's inequality for operators in the von Neumann-Schatten class $\cl C_1$ and square matrices of any finite order. Interestingly, we establish that the optimal bound for $r$ in the above mentioned Bohr's inequality concerning von Neumann-Shcatten class is 1/3 whereas it is 1/2 in the case of $2\times 2$ matrices and reduces to $\sqrt{2}-1$ for the case of $3\times 3$ matrices. We also obtain a generalization of our above-mentioned Bohr's inequality for finite matrices where we show that the optimal bound for $r$, unlike above, remains 1/3 for every fixed order $n\times n,\ n\ge 2$.

In this paper, we study the spectrum and the invariant subspace of compressed shifts on the Beurling type quotient module $${{\cal K}_\theta}$$ of Hardy space H2(ⅅ2). Suppose θ is a rational inner function with degree (1, 1). Firstly, we characterize the spectrum of $${S_{{z_1}}}$$ on $${{\cal K}_\theta}$$ ; secondly, we characterize the Agler type subspace of the compressed shift.

We show that any weakly separated Bessel system of model spaces in the Hardy space on the unit disc is a Riesz system and we highlight some applications to interpolating sequences of matrices. This will be done without using the recent solution of the Feichtinger conjecture, whose natural generalization to multidimensional model subspaces of ${\mathrm {H}}^2$ turns out to be false.

Let $$D\ne \mathbb {C}$$ be a simply connected domain and f be a Riemann mapping from $$\mathbb {D}$$ onto D. The Hardy number of D is the supremum of all p for which f belongs in the Hardy space $${H^p}\left( \mathbb {D} \right) $$ . A comb domain is a domain whose complement is the union of an infinite number of vertical rays symmetric with respect to the real axis. In this paper we prove that, for $$p>0$$ , there is a comb domain with Hardy number equal to p if and only if $$p\in [1,+\infty ]$$ . It is known that the Hardy number is related to the moments of the exit time of Brownian motion from the domain. In fact, Burkholder proved that the Hardy number of a simply connected domain is twice the supremum of all $$p>0$$ for which the p-th moment of the exit time of Brownian motion is finite. Therefore, our result implies that given $$ p < q$$ there exists a comb domain with finite p-th moment but infinite q-th moment if and only if $$q\ge 1/2$$ . This answers a question posed by Boudabra and Markowsky.

In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna-Pick spaces $\mathcal H$, we characterize all multipliers of the weak product space $\mathcal H \odot \mathcal H$. In particular, we show that if $\mathcal H$ has the so-called column-row property, then the multipliers of $\mathcal H$ and of $\mathcal H \odot \mathcal H$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball. As a key device, we exhibit a natural operator space structure on $\mathcal H \odot \mathcal H$, which enables the use of dilations of completely bounded maps., Comment: 21 pages

We obtain a noncommutative multivariable analogue of Louhichi and Olofsson characterization of Toeplitz operators with harmonic symbols on the weighted Bergman space $A_m({\bf D})$, as well as Eschmeier and Langendorfer extension to the unit ball of ${\bf C}^n$. All our results are proved in the more general setting of noncommutative poly-hyperballs ${\bf D_n^m}(H)$, ${\bf n,m}\in {\bf N}^k$, and are used to characterize the bounded free $k$-pluriharmonic functions with operator coefficients on poly-hyperballs and to solve the associated Dirichlet extension problem. In particular, the results hold for the reproducing kernel Hilbert space with kernel $$ \kappa_{\bf m}(z,w):=\prod_{i=1}^k \frac{1}{(1-\bar z_i w_i)^{m_i}},\qquad z,w\in {\bf D}^k, $$ where $m_i\geq 1$. This includes the Hardy space, the Bergman space, and the weighted Bergman space over the polydisk.

In [3], Chen, Deng, Ding and Fan proved that the fractional power dissipative operator is bounded on Lebesgue spaces Lp(ℝn), Hardy spaces Hp(ℝn) and general mixed norm spaces, which implies almost everywhere convergence of such operator. In this paper, we study the rate of convergence on fractional power dissipative operator on some sobolev type spaces.

We introduce the notion of quasi-inner function and show that the product u = ρ ∞ ∏ ρ v of m + 1 ratios of local L-factors ρ v ( z ) = γ v ( z ) / γ v ( 1 − z ) over a finite set F of places of Q inclusive of the archimedean place is quasi-inner on the left of the critical line ℜ ( z ) = 1 2 in the following sense. The off diagonal part u 21 of the matrix of the multiplication by u in the orthogonal decomposition of the Hilbert space L 2 of square integrable functions on the critical line into the Hardy space H 2 and its orthogonal complement is a compact operator. When interpreted on the unit disk, the quasi-inner condition means that the associated Haenkel matrix is compact. We show that none of the individual non-archimedean ratios ρ v is quasi-inner and, in order to prove our main result we use Gauss multiplication theorem to factor the archimedean ratio ρ ∞ into a product of m quasi-inner functions whose product with each ρ v retains the property to be quasi-inner. Finally we prove that Sonin's space is simply the kernel of the diagonal part u 22 for the quasi-inner function u = ρ ∞ , and when u ( F ) = ∏ v ∈ F ρ v the kernels of the u ( F ) 22 form an inductive system of infinite dimensional spaces which are the semi-local analogues of (classical) Sonin's spaces.

In the present paper, we obtain some new results, and we generalize some known results for the Hausdorff operators. We have studied the generalized Hausdorff operators H α , φ on the Dunkl-type homogeneous weighted Herz spaces K ̇ α , q β , p ℝ and Dunkl Herz-type Hardy spaces H K ̇ α , q β , p , N ℝ . We have determined simple sufficient conditions for these operators to be bounded on these spaces. As applications, we provide necessary and sufficient conditions for generalized Cesàro operator to be bounded on K ̇ α , q β , p ℝ and Hardy inequality for K ̇ α , q β , p ℝ .

In this paper, we study unbounded complex symmetric Toeplitz operators on the Hardy space $$H^{2}(\mathbb{D})$$ and the Fock space ℱ2. The technique used to investigate the complex symmetry of unbounded Toeplitz operators is different from that used to investigate the complex symmetry of bounded Toeplitz operators.

We study Toeplitz operators from Hardy spaces to weighted Bergman spaces in the unit ball of Cn. Toeplitz operators are closely related to many classical mappings, such as composition operators, the Volterra type integration operators and Carleson embeddings. We characterize the boundedness and compactness of Toeplitz operators from Hardy spaces Hp to weighted Bergman spaces A for the different values of p and q in the unit ball.

We study idempotent, model, and Toeplitz operators that attain the norm. Notably, we prove that if $\mathcal{Q}$ is a backward shift invariant subspace of the Hardy space $H^2(\mathbb{D})$, then the model operator $S_{\mathcal{Q}}$ attains its norm. Here $S_{\mathcal{Q}} = P_{\mathcal{Q}}M_z|_{\mathcal{Q}}$, the compression of the shift $M_z$ on the Hardy space $H^2(\mathbb{D})$ to $\mathcal{Q}$., 15 pages.To appear in Linear Algebra and Its Applications

In this paper, we study weighted composition operators on the Fock space $${{\cal F}^2}$$ . We prove that each bounded composition operator on $${{\cal F}^2}$$ is complex symmetric. This is in sharp contrast with the phenomenon on the Hardy space $${H^2}(\mathbb{D})$$ . We characterize Hermitian weighted composition operators and algebraic weighted composition operators with degree less than or equal to two on $${{\cal F}^2}$$ . In addition, we investigate cyclicity and hypercyclicity of complex symmetric weighted composition operators. We also characterize those weighted composition operators that preserve frames, tight frames or normalized tight frames on $${{\cal F}^2}$$ . Finally, we study mean ergodicity and uniformly mean ergodicity of weighted composition operators.

Let X be a ball quasi-Banach function space on Rn and HX(Rn) the associated Hardy space. In this article, under the assumptions that the Hardy–Littlewood maximal operator satisfies some Fefferman–S...

The main goal of this paper is to investigate the boundedness and essential norm of a class of product-type operators T u , v , φ m , m ∈ ℕ from Hardy spaces into n th weighted-type spaces. As a corollary, we obtain some equivalent conditions for compactness of such operators.

We provide a constructive proof of H1 (ℝd) (the classical Hardy space) factorization in terms of fractional commutators in Lorentz spaces. As a direct application, we obtain a characterization of functions in BMO space. Furthermore, we also obtain a Lorentz compactness characterization of fractional commutators.

In this paper we consider composition operators on Hardy-Sobolev spaces in connections with $BMO$-quasiconformal mappings. Using the duality of Hardy spaces and $BMO$-spaces we prove that $BMO$-quasiconformal mappings generate bounded composition operators from Hardy-Sobolev spaces to Sobolev spaces., Comment: 12 pages

We study the problem of how extremal functions for linear functionals over a Bergman space are influenced by the properties of the functions generating these functionals. For different classes of generating functions, we obtain a sufficiently exact description of qualitative properties of the corresponding extremal functions. The method developed here can be used to study similar problems in Hardy spaces.

In this paper, we study the boundedness and compactness of the Volterra integral operators from Dirichlet type spaces $${\mathcal {D}}^p_{p-2+s}$$ into Hardy spaces $${\mathcal {H}}^q$$ when $$1

A rational function belongs to the Hardy space, H 2 H^2 , of square-summable power series if and only if it is bounded in the complex unit disk. Any such rational function is necessarily analytic in a disk of radius greater than one. The inner-outer factorization of a rational function r ∈ H 2 \mathfrak {r} \in H^2 is particularly simple: The inner factor of r \mathfrak {r} is a (finite) Blaschke product and (hence) both the inner and outer factors are again rational. We extend these and other basic facts on rational functions in H 2 H^2 to the full Fock space over C d \mathbb {C} ^d , identified as the non-commutative (NC) Hardy space of square-summable power series in several NC variables. In particular, we characterize when an NC rational function belongs to the Fock space, we prove analogues of classical results for inner-outer factorizations of NC rational functions and NC polynomials, and we obtain spectral results for NC rational multipliers.

We give conditions for boundedness of Hausdorff operators on real Hardy spaces H1 over homogeneous spaces of locally compact groups with local doubling property. Special cases of the hyperbolic plane and 2-sphere are considered. In this cases Hausdorff operators look as orbital like integrals.

in this paper, we investigated Toeplitz like operators on vector valued Hardy spaces. Toeplitz like operators on 2-nuclear tensor product of Hardy spaces are then constructed and described using the theory of p-nuclear tensor product of Banach spaces, and their basic algebraic properties and spectrum are analyzed.

Introduction. Quoting from a well-known American mathematician Lipman Bers [1] “It would be tempting to rewrite history and to claim that quasiconformal transformations have been discovered in connection with gas-dynamical problems. As a matter of fact, however, the concept of quasiconformality was arrived at by Grotzsch [2] and Ahlfors [3] from the point of view of function theory”. The present work is devoted to the theory of analytic solutions of the Beltrami equation which directly related to the quasi-conformal mappings. The function is, in general, assumed to be measurable with almost everywhere in the domain under consideration. Solutions of equation (1) are often referred to as analytic functions in the literature. Research methods.

Let $$f = P[F]$$ denote the Poisson integral of F in the unit disk $${\mathbb {D}}$$ with F being absolutely continuous in the unit circle $${\mathbb {T}}$$ and $${\dot{F}}\in L^{p}({\mathbb {T}})$$ , where $${\dot{F}}(e^{it})=\frac{d}{dt} F(e^{it})$$ and $$p\ge 1$$ . Recently, the author in Zhu (J Geom Anal, 2020) proved that (1) if f is a harmonic mapping and $$1\le p< 2$$ , then $$f_{z}$$ and $$\overline{f_{{\overline{z}}}}\in \mathcal {B}^{p}({\mathbb {D}}),$$ the classical Bergman spaces of $${\mathbb {D}}$$ [12, Theorem 1.2]; (2) if f is a harmonic quasiregular mapping and $$1\le p\le \infty $$ , then $$f_{z},$$ $$\overline{f_{{\overline{z}}}}\in \mathcal {H}^{p}({\mathbb {D}}),$$ the classical Hardy spaces of $${\mathbb {D}}$$ [12, Theorem 1.3]. These are the main results in Zhu (J Geom Anal, 2020). The purpose of this paper is to generalize these two results. First, we prove that, under the same assumptions, [12, Theorem 1.2] is true when $$1\le p< \infty $$ . Also, we show that [12, Theorem 1.2] is not true when $$p=\infty $$ . Second, we demonstrate that [12, Theorem 1.3] still holds true when the assumption f being a harmonic quasiregular mapping is replaced by the weaker one f being a harmonic elliptic mapping.

Let $$(\Omega ,{\mathcal {F}},{\mathbb {P}})$$ be a probability space, $$\varphi :\ \Omega \times [0,\infty )\rightarrow [0,\infty )$$ a Musielak–Orlicz function, and $$q\in (0,\infty ]$$ . In this article, the authors introduce five martingale Musielak–Orlicz–Lorentz Hardy spaces and prove that these new spaces have some important features such as atomic characterizations, the boundedness of $$\sigma $$ -sublinear operators, and martingale inequalities. This new scale of martingale Hardy spaces requires the introduction of the Musielak–Orlicz–Lorentz space $$L^{\varphi ,q}(\Omega )$$ . In particular, the authors show that this Lorentz type space has some fundamental properties including the completeness, the convergence, real interpolations, and the Fefferman–Stein vector-valued inequality for the Doob maximal operator. As applications, the authors prove that the maximal Fejer operator is bounded from the martingale Musielak–Orlicz–Lorentz Hardy space $$H_{\varphi ,q}[0,1)$$ to $$L^{\varphi ,q}[0,1)$$ , which further implies some convergence results of the Fejer means. Moreover, all the above results are new even for Musielak–Orlicz functions with particular structure such as weight, weight Orlicz, and double-phase growth. The main approach used in this article can be viewed as a combination of the stopping time argument in probability theory and the real-variable technique of function spaces in harmonic analysis.

In this paper, we give optimal embedding relations between local Hardy spaces and α-modulation spaces. By a different approach, we extend the main results obtained by Kobayashi, Miyachi and Tomita in [Studia Math., 192 (2009), 79–96].

The purpose of this paper is to introduce a new operator on Hardy space ℍ2 called the F-composition operator which is generalization of composition operator. Some basic properties of this operator ...

In this paper, we are concerned with Toeplitz operators whose symbols are complex Borel measures. When a complex Borel measure μ on the unit circle is given, we give a formal definition of a Toeplitz operator T μ with symbol μ , as an unbounded linear operator on the Hardy space. We then study various properties of T μ . Among them, there is a theorem that the domain of T μ is represented by a trichotomy. Also, it was shown that if the domain of T μ contains at least one polynomial, then T μ is densely defined. In addition, we give evidence for the conjecture that T μ with a singular measure μ reduces to a trivial linear operator.

Let (X,d,μ) be a space of homogeneous type, with upper dimension μ, in the sense of R. R. Coifman and G. Weiss. Let η be the Holder regularity index of wavelets constructed by P. Auscher and T. Hytonen. In this article, the authors introduce the local Hardy space h∗,p(X) via local grand maximal functions and also characterize h∗,p(X) via local radial maximal functions, local non‐tangential maximal functions, local atoms and local Littlewood–Paley functions. Furthermore, the authors establish the relationship between the global and the local Hardy spaces. Finally, the authors also obtain the finite atomic characterizations of h∗,p(X). As an application, the authors give the dual spaces of h∗,p(X) when p∈(ω/(ω+η),1), which further completes the result of G. Dafni and H. Yue on the dual space of h∗,1(X). This article also answers the question of R. R. Coifman and G. Weiss on the nonnecessity of any additional geometric assumptions except the doubling condition for the radial maximal function characterization of Hcw1(X) when μ(X)

We consider weighted norm inequalities for multilinear multipliers whose symbols satisfy a product-type Hörmander condition. Our approach is to consider a more general family of multilinear singular integral operators associated to a family of smooth kernels that satisfy an L r L^r -Schwartz regularity condition. We give conditions for these operators to satisfy weighted Hardy space estimates and derive our results for multipliers as a special case. As an additional application, we use Rubio de Francia extrapolation to prove multilinear estimates on the variable exponent Hardy spaces.

Let $$\overrightarrow a \,: = \,\left( {{a_1}, \ldots,{a_n}} \right) \in {\left[ {1,\infty } \right)^n},\,\overrightarrow {p\,}: = \left( {{p_1}, \ldots,{p_n}} \right) \in {\left( {0,1} \right]^n},H_{\overrightarrow a }^{\overrightarrow p }\left( {{\mathbb{R}^n}} \right)$$ be the anisotropic mixed-norm Hardy space associated with $$\overrightarrow a $$ defined via the radial maximal function, and let f belong to the Hardy space $$H_{\overrightarrow a }^{\overrightarrow p }\left( {{\mathbb{R}^n}} \right)$$ . In this article, we show that the Fourier transform $$\widehat{f}$$ coincides with a continuous function g on ℝn in the sense of tempered distributions and, moreover, this continuous function g, multiplied by a step function associated with $$\overrightarrow a $$ , can be pointwisely controlled by a constant multiple of the Hardy space norm of f. These proofs are achieved via the known atomic characterization of $$H_{\overrightarrow a }^{\overrightarrow p }\left( {{\mathbb{R}^n}} \right)$$ and the establishment of two uniform estimates on anisotropic mixed-norm atoms. As applications, we also conclude a higher order convergence of the continuous function g at the origin. Finally, a variant of the Hardy-Littlewood inequality in the anisotropic mixed-norm Hardy space setting is also obtained. All these results are a natural generalization of the well-known corresponding conclusions of the classical Hardy spaces Hp(ℝn) with p ∈ (0, 1], and are even new for isotropic mixed-norm Hardy spaces on ∈n.

We obtain a new version of Hardy theorem about power series of several variables reciprocal to the power series with positive coefficients. We prove that if the sequence {as} = as1,s2,...,sn, ||s|| ≥ K satisfies condition of logarithmically convexity and the first coefficient a0 is sufficiently large then reciprocal power series has only negative coefficients {bs} = bs1,s2,...,sn, except b0,0,...,0 for any K. The classical Hardy theorem corresponds to the case K = 0, n = 1. Such results are useful in Nevanlinna - Pick theory. For example, if function k(x, y) can be represented as power series Σn≥0 an(x-y)n, an > 0, and reciprocal function 1/k(x,y) can be represented as power series Σn≥0 bn(x-y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna-Pick property. The reproducing kernel 1/1-x-y of the classical Hardy space H2(D) is a prime example for our theorems.