Hardy-Sobolev Spaces Research Articles

Extrapolation of solvability of the regularity and the Poisson regularity problems in rough domains

Let Ω⊂Rn+1, n≥2, be an open set satisfying the corkscrew condition with n-Ahlfors regular boundary ∂Ω, but without any connectivity assumption. We study the connection between solvability of the regularity problem for divergence form elliptic operators with boundary data in the Hajłasz-Sobolev space M1,1(∂Ω) and the weak-A∞ property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in M1,1(∂Ω) is equivalent to the solvability of the regularity problem in M1,p(∂Ω) for some p>1. We also prove analogous extrapolation results for the Poisson regularity problem defined on tent spaces. Moreover, under the hypothesis that ∂Ω supports a weak (1,1)-Poincaré inequality, we show that the solvability of the regularity problem in the Hajłasz-Sobolev space M1,1(∂Ω) is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives.

Carleson measures for Hardy-Sobolev spaces in the Siegel upper half-space

We give a capacitary type characterization of Carleson measures for a class of Hardy-Sobolev spaces (also known as weighted Dirichlet spaces) on the Siegel upper half-space, introduced by Arcozzi et al. in [7]. This answers in part a question raised by the same authors.

The Calogero–Moser derivative nonlinear Schrödinger equation

AbstractWe study the Calogero–Moser derivative nonlinear Schrödinger NLS equation posed on the Hardy–Sobolev space with suitable . By using a Lax pair structure for this ‐critical equation, we prove global well‐posedness for and initial data with sub‐critical or critical ‐mass . Moreover, we prove uniqueness of ground states and also classify all traveling solitary waves. Finally, we study in detail the class of multi‐soliton solutions and we prove that they exhibit energy cascades in the following strong sense such that as for every .

Frequency analysis with multiple kernels and complete dictionary

In signal analysis, among the effort of seeking for efficient representations of a signal into the basic ones of meaningful frequencies, to extract principal frequency components, consecutively one after another or n at one time, is a fundamental strategy. For this goal, we define the concept of mean-frequency and develop the related frequency decomposition with the complete Szegö kernel dictionary, the latter consisting of the multiple kernels, being defined as the parameter-derivatives of the Szegö kernels. Several major energy matching pursuit type sparse representations, including greedy algorithm (GA), orthogonal greedy algorithm (OGA), adaptive Fourier decomposition (AFD), pre-orthogonal adaptive Fourier decomposition (POAFD), n-Best approximation, and unwinding Blaschke expansion, are analyzed and compared. Of which an order in reconstruction efficiency between the mentioned algorithms is given based on the detailed study of their respective remainders. The study spells out the natural connections between the multiple kernels and the related Laguerre system, and in particular, shows that both, like the Fourier series, extract out the O(n−σ) order convergence rate from the functions in the Hardy-Sobolev space of order σ>0. The existence of the n-Best approximation with the complete Szegö dictionary is proved and the related algorithm aspects are discussed. The included experiments form a significant integration part of the study, for they not only illustrate the theoretical results but also provide cross comparison between various ways of combination between the matching pursuit algorithms and the dictionaries in use. Experiments show that the complete dictionary remarkably improves approximation efficiency.

Fredholm composition operators on Hardy-Sobolev spaces with bounded reproducing kernel

Fredholm composition operators on Hardy-Sobolev spaces with bounded reproducing kernel

Almost Everywhere Convergence of Bochner–Riesz Means on Hardy–Sobolev Spaces

We investigate the convergence of Bochner–Riesz means on Hardy–Sobolev spaces. The relation between the smoothness imposed on functions and the rate of almost everywhere convergence of the generalized Bochner–Riesz means is given.

On Korn’s First Inequality in a Hardy-Sobolev Space

AbstractKorn’s first inequality states that there exists a constant such that the ${\mathcal {L}}^{2}$ L 2 -norm of the infinitesimal displacement gradient is bounded above by this constant times the ${\mathcal {L}}^{2}$ L 2 -norm of the infinitesimal strain, i.e., the symmetric part of the gradient, for all infinitesimal displacements that are equal to zero on the boundary of a body ℬ. This inequality is known to hold when the ${\mathcal {L}}^{2}$ L 2 -norm is replaced by the ${\mathcal {L}}^{p}$ L p -norm for any $p\in (1,\infty )$ p ∈ ( 1 , ∞ ) . However, if $p=1$ p = 1 or $p=\infty $ p = ∞ the resulting inequality is false. It was previously shown that if one replaces the ${\mathcal {L}}^{\infty}$ L ∞ -norm by the $\operatorname{BMO}$ BMO -seminorm (Bounded Mean Oscillation) then one maintains Korn’s inequality. (Recall that ${\mathcal {L}}^{\infty}({\mathcal {B}})\subset \operatorname{BMO}({\mathcal {B}}) \subset {\mathcal {L}}^{p}({\mathcal {B}})\subset {\mathcal {L}}^{1}({ \mathcal {B}})$ L ∞ ( B ) ⊂ BMO ( B ) ⊂ L p ( B ) ⊂ L 1 ( B ) , $1< p<\infty $ 1 < p < ∞ .) In this manuscript it is shown that Korn’s inequality is also maintained if one replaces the ${\mathcal {L}}^{1}$ L 1 -norm by the norm in the Hardy space ${\mathcal {H}}^{1}$ H 1 , the predual of $\operatorname{BMO}$ BMO . One caveat: the results herein are only applicable to the pure-displacement problem with the displacement equal to zero on the entire boundary of ℬ.

Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels

<abstract><p>For any real $ \beta $ let $ H^2_\beta $ be the Hardy-Sobolev space on the unit disc $ {\mathbb D} $. $ H^2_\beta $ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $ \beta > 1/2 $. In this paper, we prove that $ C_{\varphi } $ has dense range in $ H_{\beta }^{2} $ if and only if the polynomials are dense in a certain Dirichlet space of the domain $ \varphi({\mathbb D}) $ for $ 1/2 < \beta < 1 $. It follows that if the range of $ C_{\varphi } $ is dense in $ H_{\beta }^{2} $, then $ \varphi $ is a weak-star generator of $ H^{\infty} $, although the conclusion is false for the classical Dirichlet space $ \mathfrak{D} $. Moreover, we study the relation between the density of the range of $ C_{\varphi } $ and the cyclic vector of the multiplier $ M_{\varphi}^{\beta}. $</p></abstract>

Fractional Leibniz Rules Associated to Bilinear Hermite Pseudo-Multipliers

Abstract We obtain a fractional Leibniz rule associated to bilinear Hermite pseudo-multipliers in the context of Hermite Besov and Hermite Triebel–Lizorkin spaces. As a byproduct, we show that the classes of bounded functions in these spaces (which include Hermite Sobolev and Hermite Hardy–Sobolev spaces) are algebras under pointwise multiplication. To obtain these results we develop appropriate decompositions for bilinear pseudo-multipliers and molecular estimates for certain families of functions in the Hermite setting.

Исследование ограниченности теплицевых операторов в пространствах Харди-Соболева в единичном шаре из CN

In this paper, we describe those symbols on a unit sphere, for which the corresponding multiple Toeplitz operator is bounded in the Hardy-Sobolev space in a unit ball of a complex space.

Sums of Dual Toeplitz Products on the Orthogonal Complements of the Hardy–Sobolev Spaces

Sums of Dual Toeplitz Products on the Orthogonal Complements of the Hardy–Sobolev Spaces

Composition operators on Hardy-Sobolev spaces and BMO-quasiconformal mappings

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.

Correction to: Embedding theorems for Sobolev and Hardy–Sobolev spaces and estimates of Fourier transforms

We detected some errors in the formulation of Theorem 6.3 and in the proof of Theorem 6.5 in our paper [4].

Composition operators on Hardy-Sobolev spaces and BMO-quasiconformal mappings

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.

Compact Toeplitz operators products on Hardy–Sobolev spaces over the unit polydisk

We study the compactness of finite sums of products of two Toeplitz operators on Hardy–Sobolev spaces over the unit polydisk Hβ2(𝔻n). We calculate the essential norm of these operators and answer the question of when a Toeplitz operator on Hβ2(𝔻n) is Fredholm.

Regularities of Time-Fractional Derivatives of Semigroups Related to Schrodinger Operators with Application to Hardy-Sobolev Spaces on Heisenberg Groups

In this paper, assume that L=−Δℍn+V is a Schrödinger operator on the Heisenberg group ℍn, where the nonnegative potential V belongs to the reverse Hölder class BQ/2. By the aid of the subordinate formula, we investigate the regularity properties of the time-fractional derivatives of semigroups e−tLt>0 and e−tLt>0, respectively. As applications, using fractional square functions, we characterize the Hardy-Sobolev type space HL1,αℍn associated with L. Moreover, the fractional square function characterizations indicate an equivalence relation of two classes of Hardy-Sobolev spaces related with L.

Blaschke Decompositions on Weighted Hardy Spaces

Recently, several authors have considered a nonlinear analogue of Fourier series in signal analysis, referred to as either the unwinding series or adaptive Fourier decomposition. In these processes, a signal is represented as the real component of the boundary value of an analytic function $$F: \partial {\mathbb {D}}\rightarrow {\mathbb {C}}$$ and by performing an iterative method to obtain a sequence of Blaschke decompositions, the signal can be efficiently approximated using only a few terms. To better understand the convergence of these methods, the study of Blaschke decompositions on weighted Hardy spaces was initiated by Coifman and Steinerberger, under the assumption that the complex valued function F has an analytic extension to $${\mathbb {D}}_{1+\epsilon }$$ for some $$\epsilon >0$$ . This provided bounds on weighted Hardy norms involving a single zero, $$\alpha \in {\mathbb {D}}$$ , of F and its Blaschke decomposition. That work also noted that in many specific examples, the unwinding series of F converges at an exponential rate to F, which when coupled with an efficient algorithm to compute a Blaschke decomposition, has led to the hope that this process will provide a new and efficient way to approximate signals. In this work, we continue the study of Blaschke decompositions on weighted Hardy Spaces for functions in the larger space $${\mathcal {H}}^2({\mathbb {D}})$$ under the assumption that the function has finitely many roots in $${\mathbb {D}}$$ . This is meaningful, since there are many functions that meet this criterion but do not extend analytically to $${\mathbb {D}}_{1+\epsilon }$$ for any $$\epsilon >0$$ , for example $$F(z)=\log (1-z)$$ . By studying the growth rate of the weights, we improve the bounds provided by Coifman and Steinerberger to obtain new estimates containing all roots of F in $${\mathbb {D}}$$ . This provides us with new insights into Blaschke decompositions on classical function spaces including the Hardy–Sobolev spaces and weighted Bergman spaces, which correspond to making specific choices for the aforementioned weights. Further, we state a sufficient condition on the weights for our improved bounds to hold for any function in the Hardy space, $${\mathcal {H}}^2({\mathbb {D}})$$ , in particular functions with an infinite number of roots in $${\mathbb {D}}$$ . These results may help to better explain why the exponential convergence of the unwinding series is seen in many numerical examples.

On the Continuity and Compactness of Pseudodifferential Operators on Localizable Hardy Spaces

In this paper we establish Sobolev type compact embedding theorems for Hormander classes of pseudodifferential operators $OpS^{-\alpha }_{1,\delta }$ on localizable Hardy space. Our work include new optimal boundedness results. As application, we obtain compact embeddings for compactly supported distributions with respect to the space variables in the nonhomogeneous localizable Hardy-Sobolev spaces.

The characterizations of Hardy–Sobolev spaces by fractional square functions related to Schrödinger operators

The characterizations of Hardy–Sobolev spaces by fractional square functions related to Schrödinger operators

Hankel operators with pluriharmonic symbols on Hardy-Sobolev spaces

ABSTRACT In this paper, we characterize the kernels and finite rank properties of the Hankel operators and small Hankel operators with pluriharmonic symbols on Hardy-Sobolev spaces and pluriharmonic Hardy-Sobolev spaces . For , the result was given by Y. J. Lee.