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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.

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.

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 $n\times n,\ n\ge 2$.

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.

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.

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

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.

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 ...

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.

The boundedness of singular and fractional integral operator on Lebesgue and Hardy spaces have been well studied. The theory of Herz space and Herz type Hardy space, as a local version of Lebesgue and Hardy space, have been developed. The main purpose of this paper is to establish the endpoint continuity properties of some multilinear operators related to certain non-convolution type fractional singular integral operators on Herz and Herz type Hardy spaces and the endpoint estimates for the multilinear operators on Herz and Herz type Hardy spaces are obtained.

Let p(·) : ℝ n → (0, ∞] be a variable exponent function satisfying the globally log-Hölder continuous and let Θ be a continuous multi-level ellipsoid cover of ℝ n introduced by Dekel et al. [12]. In this article, we introduce highly geometric Hardy spaces Hp (·)(Θ) via the radial grand maximal function and then obtain its atomic decomposition, which generalizes that of Hardy spaces Hp (Θ) on ℝ n with pointwise variable anisotropy of Dekel et al. [16] and variable anisotropic Hardy spaces of Liu et al. [24]. As an application, we establish the boundedness of variable anisotropic singular integral operators from Hp (·)(Θ) to Lp (·)(ℝ n ) in general and from Hp (·)(Θ) to itself under the moment condition, which generalizes the previous work of Bownik et al. [6] on Hp (Θ).

We study infinite-time admissibility and infinite-time exact observability of Volterra systems in Hilbert spaces using a Hardy space approach, with an emphasis on the former. The problem is reduced...

Let $${H^2}\left( {{\mathbb{D}^2}} \right)$$ be the Hardy space over the bidisk $${\mathbb{D}^2}$$ , 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 $${N_\varphi } = {H^2}\left( {{\mathbb{D}^2}} \right)\Theta {M_\varphi }$$ . 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].

In this work, new characterizations of Herz and Herz-type Hardy spaces associated with the Dunkl operator on the real line are introduced.

Given an inner function θ in the upper half-plane, consider the subspace H2 ⊖ θH2 of the Hardy space H2. For a finite collection Λ of points on the complex plane, the subspace of functions from Kθ that vanish on Λ can be represented in the form g ∙ Kω, where ω is an inner function and g is an isometric multiplier on Kω. We obtain a description of the functions ω and g in terms of θ and Λ.

Let I ⊂ R I\subset \mathbb {R} be an interval and let f f , φ \varphi be arbitrary elements of H 1 ( I ) H^1(I) and B M O ( I ) BMO(I) , respectively, with ∫ I φ = 0 \int _I\varphi =0 . The paper contains the proof of the estimate ∫ I f φ ≤ 2 ‖ f ‖ H 1 ( I ) ‖ φ ‖ B M O ( I ) , \begin{equation*} \int _I f\varphi \leq \sqrt {2}\|f\|_{H^1(I)}\|\varphi \|_{BMO(I)}, \end{equation*} and it is shown that 2 \sqrt {2} cannot be replaced by a smaller universal constant. The argument rests on the existence of a special function enjoying appropriate size and concavity requirements.

Let $$\widetilde{T}_{\sigma }$$ be a bilinear pseudo-differential operator and $$\vec {\omega }=(\omega _{1},\omega _{2})\in A_{\vec {P}}$$ , where $$\vec {P}=(p_{1},p_{2})$$ , $$p_{i}\in [1,\infty )$$ with $$i=1,\ 2$$ , and $$A_{\vec {P}}$$ denotes the multiple weight class. In this paper, we will prove that $$\widetilde{T}_{\sigma }$$ is bounded from the product of weighted Lebesgue spaces $$L^{p_{1}}_{\omega _{1}}(\mathbb {R}^{n})\times L^{p_{2}}_{\omega _{2}}(\mathbb {R}^{n})$$ into the $$L^{p}_{\vec {\nu }_{\omega }}(\mathbb {R}^{n})$$ , where $$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$$ and $$\vec {\nu }_{\omega }(x)=\mathop {\prod }\nolimits ^{2}_{i=1}[\omega _{i}(x)]^{\frac{p}{p_{i}}}$$ , and bounded from the weighted Morrey spaces $$L^{p_{1},\kappa ,\rho }_{\omega _{1}}(\mathbb {R}^{n})\times L^{p_{2},\kappa ,\rho }_{\omega _{2}}(\mathbb {R}^{n})$$ into the $$L^{p,\kappa ,\rho }_{\vec {\nu }_{\omega }}(\mathbb {R}^{n})$$ . Furthermore, the endpoint estimate for $$\widetilde{T}_{\sigma }$$ on the weighted Hardy space $$H^{1}_{\omega }(\mathbb {R}^{n})$$ is also obtained.

In this paper, the authors completely characterize the finite rank commutator and semi-commutator of two monomial Toeplitz operators on the pluriharmonic Hardy space of the torus or the unit sphere. As a consequence, many non-trivial examples of (semi-)commuting Toeplitz operators on the pluriharmonic Hardy spaces are given.

This article is the second part of two works by the same authors on the same topic. Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors introd...

In this paper we study weighted composition operators W g , ψ commuting with self-adjoint weighted composition operators W f , ϕ on the Hardy space H 2 . In particular, we investigate which combina...

We present an effective numerical method for the recovery of biharmonic functions in a disk from continuous boundary values of these functions and of their normal derivatives using wavelets that are harmonic in the disk and interpolating on its boundary on dyadic rational grids. The expansions of solutions of boundary value problems into cumbersome interpolation series in the wavelet basis are replaced by sequences of their partial sums that are compactly presentable in the subspace bases of the corresponding multiresolution analysis (MRA) of Hardy spaces $$h_{\infty}(K)$$ of functions harmonic in the disk. Effective estimates are obtained for the approximation of solutions by partial sums of any order in terms of the best approximation of the boundary functions by trigonometric polynomials of a slightly smaller order. As a result, to provide the required accuracy of the representation of the unknown biharmonic functions, one can choose in advance the scaling parameter of the corresponding MRA subspace such that the interpolation projection to this space defines a simple analytic representation of the corresponding partial sums of interpolation series in terms of appropriate compressions and shifts of the scaling functions, skipping complicated iterative procedures for the numerical construction of the coefficients of expansion of the boundary functions into series in interpolating wavelets. We write solutions using interpolating and interpolating-orthogonal wavelets based on modified Meyer wavelets; the latter are convenient to apply if the boundary values of the boundary value problem are given approximately, for example, are found experimentally. In this case, one can employ the usual, well-known procedures of discrete orthogonal wavelet transformations for the analysis and refinement (correction) of the boundary values.

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.

The aim of this paper is to establish the boundednes of the commutator $[b,T_{\alpha }]$ [ b , T α ] generated by θ-type generalized fractional integral $T_{\alpha }$ T α and $b\in \widetilde{\mathrm{RBMO}}(\mu )$ b ∈ RBMO ˜ ( μ ) over a non-homogeneous metric measure space. Under the assumption that the dominating function λ satisfies the ϵ-weak reverse doubling condition, the author proves that the commutator $[b,T_{\alpha }]$ [ b , T α ] is bounded from the Lebesgue space $L^{p}(\mu )$ L p ( μ ) into the space $L^{q}(\mu )$ L q ( μ ) for $\frac{1}{q}=\frac{1}{p}-\alpha $ 1 q = 1 p − α and $\alpha \in (0,1)$ α ∈ ( 0 , 1 ) , and bounded from the atomic Hardy space $\widetilde{H}^{1}_{b}(\mu )$ H ˜ b 1 ( μ ) with discrete coefficient into the space $L^{\frac{1}{1-\alpha },\infty }(\mu )$ L 1 1 − α , ∞ ( μ ) . Furthermore, the boundedness of the commutator $[b,T_{\alpha }]$ [ b , T α ] on a generalized Morrey space and a Morrey space is also got.

In the present paper, we prove the boundedness of Dunkl-Hausdorff operators in space $$L^p_{\alpha }({\mathbb {R}})$$ and in the Hardy space $$H^{1}_{\alpha }({\mathbb {R}})$$ associated with the Dunkl operators, investigate continuous Dunkl wavelet transformation, and obtain some useful results. The relation between Dunkl wavelet transformation and Dunkl-Hausdorff operator is also established.

We define a scale of Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$, $p\in[1,\infty]$, that are invariant under suitable Fourier integral operators of order zero. This builds on work by Smith for $p=1$. We also introduce a notion of off-singularity decay for kernels on the cosphere bundle of $\mathbb{R}^{n}$, and we combine this with wave packet transforms and tent spaces over the cosphere bundle to develop a full Hardy space theory for oscillatory integral operators. In the process we extend the known results about $L^{p}$-boundedness of Fourier integral operators, from local boundedness to global boundedness for a larger class of symbols., Comment: 59 pages. Final version before publication

After the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂ ¯ {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are L p ⁢ ( ℝ n ) {L^{p}({\mathbb{R}^{n}})} bounded for 1 < p < ∞ {1 , but only bounded on local Hardy spaces h p ⁢ ( ℝ n ) {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for 0 < p ≤ 1 {0 . Though much work has been done on the L p ⁢ ( ℝ n 1 × ℝ n 2 ) {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 1 < p < ∞ {1 and Hardy H p ⁢ ( ℝ n 1 × ℝ n 2 ) {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 0 < p ≤ 1 {0 for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of 0 < p ≤ 1 {0 . The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces h p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for 0 < p ≤ 1 {0 recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

Let φ \varphi be a nonconstant analytic self-map of the open unit disk in C \mathbb {C} , with ‖ φ ‖ ∞ > 1 \|\varphi \|_{\infty }>1 . Consider the operator D φ D_{\varphi } , acting on the Hardy space H 2 H^{2} , given by differentiation followed by composition with φ \varphi . We obtain results relating to the adjoint, norm, and spectrum of such an operator.

We introduce a decreasing one-parameter family $\mathfrak{X}^{\gamma}(M)$, $\gamma>0$, of Banach subspaces of the Hardy-Goldberg space $\mathfrak{h}^1(M)$ on certain nondoubling Riemannian manifolds with bounded geometry and we investigate their properties. In particular, we prove that $\mathfrak{X}^{1/2}(M)$ agrees with the space of all functions in $\mathfrak{h}^1(M)$ whose Riesz transform is in $L^1(M)$, and we obtain the surprising result that this space does not admit an atomic decomposition., Comment: 21 pages

Let φ \varphi be an analytic self-map of the open unit disk D \mathbb {D} . We study the complex symmetry of composition operators C φ C_\varphi on weighted Hardy spaces induced by a bounded sequence. For any analytic self-map of D \mathbb {D} that is not an elliptic automorphism, we establish that if C φ C_{\varphi } is complex symmetric, then either φ ( 0 ) = 0 \varphi (0)=0 or φ \varphi is linear. In the case of weighted Bergman spaces A α 2 A^{2}_{\alpha } , we find the non-automorphic linear fractional symbols φ \varphi such that C φ C_{\varphi } is complex symmetric.

A basic building block in Classical Potential Theory is the fundamental solution of the Laplace equation in ${\mathbb R}^d$ (Newtonian kernel). The main goal of this article is to study the rates of nonlinear $n$-term approximation of harmonic functions on the unit ball $B^d$ from shifts of the Newtonian kernel with poles outside $\overline{B^d}$ in the harmonic Hardy spaces. Optimal rates of approximation are obtained in terms of harmonic Besov spaces. The main vehicle in establishing these results is the construction of highly localized frames for Besov and Triebel-Lizorkin spaces on the sphere whose elements are linear combinations of a fixed number of shifts of the Newtonian kernel.

The main purpose of this paper is to establish, using the bi-parameter Littlewood–Paley–Stein theory (in particular, the bi-parameter Littlewood–Paley–Stein square functions), a Calderón–Torchinsky type theorem for the following Fourier multipliers on anisotropic product Hardy spaces H p ⁢ ( ℝ n 1 × ℝ n 2 ; A → ) {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}};\vec{A})} ( 0 < p ≤ 1 {0 ): T m ⁢ f ⁢ ( x , y ) = ∫ ℝ n 1 × ℝ n 2 m ⁢ ( ξ , η ) ⁢ f ^ ⁢ ( ξ , η ) ⁢ e 2 ⁢ π ⁢ i ⁢ ( x ⋅ ξ + y ⋅ η ) ⁢ d ⁢ ξ ⁢ d ⁢ η . T_{m}f(x,y)=\int_{\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}}m(\xi,\eta)\hat{f% }(\xi,\eta)e^{2\pi i(x\cdot\xi+y\cdot\eta)}\mathop{}\!d\xi\mathop{}\!d\eta. Our main theorem is the following: Assume that m ⁢ ( ξ , η ) {m(\xi,\eta)} is a function on ℝ n 1 × ℝ n 2 {\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}} satisfying sup j , k ∈ ℤ ⁡ ∥ m j , k ∥ W ( s 1 , s 2 ) ⁢ ( A → ) < ∞ \sup_{j,k\in\mathbb{Z}}\lVert m_{j,k}\rVert_{W^{(s_{1},s_{2})}(\vec{A})} with s 1 > ζ 1 , - - 1 ⁢ ( 1 p - 1 2 ) {s_{1}>\zeta_{1,-}^{-1}(\frac{1}{p}-\frac{1}{2})} , s 2 > ζ 2 , - - 1 ⁢ ( 1 p - 1 2 ) {s_{2}>\zeta_{2,-}^{-1}(\frac{1}{p}-\frac{1}{2})} , where ζ 1 , - {\zeta_{1,-}} and ζ 2 , - {\zeta_{2,-}} depend only on the eigenvalues and are defined in the first section. Then T m {T_{m}} is bounded from H p ⁢ ( ℝ n 1 × ℝ n 2 ; A → ) {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}};\vec{A})} to H p ⁢ ( ℝ n 1 × ℝ n 2 ; A → ) {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}};\vec{A})} for all 0 < p ≤ 1 {0 and ∥ T m ∥ H p ⁢ ( A → ) → H p ⁢ ( A → ) ≤ C A → , s 1 , s 2 , p ⁢ sup j , k ∈ ℤ ⁡ ∥ m j , k ∥ W ( s 1 , s 2 ) ⁢ ( A → ) , \lVert T_{m}\rVert_{H^{p}(\vec{A})\to H^{p}(\vec{A})}\leq C_{{\vec{A},s_{1},s_% {2},p}}\sup_{j,k\in\mathbb{Z}}\lVert m_{j,k}\rVert_{W^{(s_{1},s_{2})}(\vec{A})}, where W ( s 1 , s 2 ) ⁢ ( A → ) {W^{(s_{1},s_{2})}(\vec{A})} is a bi-parameter anisotropic Sobolev space on ℝ n 1 × ℝ n 2 {\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}} with C A → , s 1 , s 2 , p {C_{{\vec{A},s_{1},s_{2},p}}} is a positive constant that depends on A → , s 1 , s 2 , p {\vec{A},s_{1},s_{2},p} . Here we use the notations m j , k ⁢ ( ξ , η ) = m ⁢ ( A 1 ∗ j ⁢ ξ , A 2 ∗ k ⁢ η ) ⁢ φ ( 1 ) ⁢ ( ξ ) ⁢ φ ( 2 ) ⁢ ( η ) {m_{j,k}(\xi,\eta)=m(A_{1}^{\ast j}\xi,A_{2}^{\ast k}\eta)\varphi^{(1)}(\xi)% \varphi^{(2)}(\eta)} , where φ ( 1 ) ⁢ ( ξ ) {\varphi^{(1)}(\xi)} is a suitable cut-off function on ℝ n 1 {\mathbb{R}^{n_{1}}} and φ ( 2 ) ⁢ ( η ) {\varphi^{(2)}(\eta)} is a suitable cut-off function on ℝ n 2 {\mathbb{R}^{n_{2}}} , respectively.

We study the influence of Szegő projector on the \begin{document}$ L^2- $\end{document} critical non linear focusing Schrodinger equation, leading to the quintic focusing NLS–Szegő equation on the line \begin{document}$ \begin{equation*} i\partial_t u + \partial_x^2 u + \Pi(|u|^4 u) = 0, \quad (t, x)\in \mathbb{R}\times \mathbb{R}, \qquad u(0, \cdot) = u_0. \end{equation*} $\end{document} It has no Galilean invariance but the momentum \begin{document}$ P(u) = \langle -i\partial_x u, u\rangle_{L^2} $\end{document} becomes the \begin{document}$ \dot{H}^{\frac{1}{2}}- $\end{document} norm. Thus this equation is globally well-posed in \begin{document}$ H^1_+ = \Pi(H^1(\mathbb{R})) $\end{document} , for every initial datum \begin{document}$ u_0 $\end{document} . The solution \begin{document}$ L^2- $\end{document} scatters both forward and backward in time if \begin{document}$ u_0 $\end{document} has sufficiently small mass. By using the concentration–compactness principle, we prove the orbital stability of some weak type of the traveling wave : \begin{document}$ u_{\omega, c}(t, x) = e^{i\omega t}Q(x+ct) $\end{document} , for some \begin{document}$ \omega, c>0 $\end{document} , where \begin{document}$ Q $\end{document} is a ground state associated to Gagliardo–Nirenberg type functional \begin{document}$ \begin{equation*} I^{(\gamma)}(f) = \frac{\|\partial_x f\|_{L^2}^2\|f\|_{L^2}^{4}+\gamma \langle -i \partial_x f , f\rangle_{L^2}^2 \|f\|_{L^2}^2}{\|f\|_{L^{6}}^{6}}, \qquad \forall f\in H^1_+ \backslash \{0\}, \end{equation*} $\end{document} for some \begin{document}$ \gamma\geq 0 $\end{document} . Its Euler–Lagrange equation is a non local elliptic equation. The ground states are completely classified in the case \begin{document}$ \gamma = 2 $\end{document} , leading to the actual orbital stability for appropriate traveling waves. As a consequence, the scattering mass threshold of the focusing quintic NLS–Szegő equation is strictly below the mass of ground state associated to the functional \begin{document}$ I^{(0)} $\end{document} , unlike the recent result by Dodson [ 8 ] on the usual quintic focusing non linear Schrodinger equation.

Let $\vec{p}\in(0,\infty)^n$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. In this article, via the non-tangential grand maximal function, the authors first introduce the anisotropic mixed-norm Hardy spaces $H_A^{\vec{p}}(\mathbb{R}^n)$ associated with $A$ and then establish their radial or non-tangential maximal function characterizations. Moreover, the authors characterize $H_A^{\vec{p}}(\mathbb{R}^n)$, respectively, by means of atoms, finite atoms, Lusin area functions, Littlewood-Paley $g$-functions or $g_{\lambda}^\ast$-functions via first establishing an anisotropic Fefferman-Stein vector-valued inequality on the mixed-norm Lebesgue space $L^{\vec{p}}(\mathbb{R}^n)$. In addition, the authors also obtain the duality between $H_A^{\vec{p}}(\mathbb{R}^n)$ and the anisotropic mixed-norm Campanato spaces. As applications, the authors establish a criterion on the boundedness of sublinear operators from $H_A^{\vec{p}}(\mathbb{R}^n)$ into a quasi-Banach space. Applying this criterion, the authors then obtain the boundedness of anisotropic convolutional $\delta$-type and non-convolutional $\beta$-order Calder\'{o}n-Zygmund operators from $H_A^{\vec{p}}(\mathbb{R}^n)$ to itself [or to $L^{\vec{p}}(\mathbb{R}^n)$]. As a corollary, the boundedness of anisotropic convolutional $\delta$-type Calder\'on-Zygmund operators on the mixed-norm Lebesgue space $L^{\vec{p}}(\mathbb{R}^n)$ with $\vec{p}\in(1,\infty)^n$ is also presented., Comment: 52 pages, Submitted. arXiv admin note: text overlap with arXiv:1801.06251, arXiv:1908.03291