Inhomogeneous Besov and Triebel-Lizorkin spaces associated with a para-accretive function and their applications

In this paper,using inhomogeneous Calderon’s reproducing formulas and the space of test functions associated with a para-accretive function,the inhomogeneous Besov and TriebelLizorkin spaces are established.As applications,pointwise multiplier theorems are also obtained.

Function spaces of Besov-type and Triebel-Lizorkin-type——a survey

This article is devoted to presenting a recapitulative introduction for the theory of Besov-type and Triebel-Lizorkin-type spaces developed in recent years.

THE BESOV AND TRIEBEL-LIZORKIN SPACES WITH HIGH ORDER ON THE LIPSCHITZ CURVES

The Besov spaces B_p^(α,4)(Γ)and Triebel-Lizorkin spaces F_p^(α,4)(Γ)with high order x∈R on a Lipschitz curve Γ are defind,when 1≤p≤∞,1≤q≤∞.To compare to the classical case.a difference characterization of such spaces in the case|x|<1 is given also.

Characterizations of realized homogeneous Besov and Triebel-Lizorkin spaces via differences

Based on the role of the polynomial functions on the homogeneous Besov spaces, on the homogeneous Triebel-Lizorkin spaces and on their realized versions, we study and obtain characterizations of these spaces via difference operators in a certain sense.

A Characterization of Besov Spaces of Para-Accretive Type and Its Application

There are two folds in this article. One fold is to characterize the Besov spaces of para-accretive type , which reduces to the classical Besov spaces when the para-accretive function is constant, by using a discrete Calderón-type reproducing formula and Plancherel-P?lya-type inequality associated to a para-accretive function b in Rn. The other is to show that a generalized singular integral operator T with extends to be bounded from for and , where ε is the regularity exponent of the kernel of T.

Singular Integrals and Weighted Triebel-Lizorkin and Besov Spaces of Arbitrary Number of Parameters

Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderon＇s identity. This is inspired by the work of discrete Littlewood- Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe＇s covering lemma in multiparameter setting.

Herz type Besov and Triebel-Lizorkin spaces with variable exponent

The Herz type Besov and Triebel-Lizorkin spaces with variable exponent are introduced. Then characterizations of these new spaces by maximal functions are given.

T1 theorem for Besov and Triebel-Lizorkin spaces

Using the discrete Calderon type reproducing formula and the PlancherelPolya characterization for the Besov and Triebel-Lizorkin spaces, the T1 theorem for the Besov and Triebel-Lizorkin spaces was proved.

Difference Characterization of Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type

In this article,the authors introduce the spaces of Lipschitz type on spaces of homogeneous type in the sense of Coifman and Weiss,and discuss their relations with Besov and Triebel-Lizorkin spaces.As an application,the authors establish the difference characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type.A major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the considered measure of the underlying spaceχvia using the geometrical property ofχexpressed by its dyadic reference points,dyadic cubes,and the(local)lower bound.Moreover,some results when p≤1 but near to 1 are new even whenχis an RD-space.

Littlewood-Paley Functions and Triebel-Lizorkin Spaces,Besov Spaces

We establish Littlewood-Paley charaterizations of Triebel-Lizorkin spaces and Besov spaces in Euclidean spaces using several square functions defined via the spherical average,the ball average,the Bochner-Riesz means and some other well-known operators.We provide a simple proof so that we are able to extend and improve many results published in recent papers.

New Characterizations of Inhomogeneous Besovand Triebel-Lizorkin Spaces over Spaces of Homogeneous Type

In this paper we use the T1 theorem to prove a new characterization with minimum regularity and cancellation conditions for inhomogeneous Besov and Triebel-Lizorkin spaces over spaces of homogeneous type. These results are new even for R^n.

A Class of Oscillatory Singular Integrals with Hardy Kernels on Triebel-Lizorkin Spaces and Besov Spaces

In this paper,the boundedness is obtained on the Triebel-Lizorkin spaces and the Besov spaces for a class of oscillatory singular integrals with Hardy kernels.

Variable 2-Microlocal Besov–Triebel–Lizorkin-Type Spaces

This article is devoted to the study of variable 2-microlocal Besov-type and Triebel- Lizorkin-type spaces. These variable function spaces are defined via a Fourier-analytical approach. The authors then characterize these spaces by means of Q-transforms, Peetre maximal functions, smooth atoms, ball means of differences and approximations by analytic functions. As applications, some re- lated Sobolev-type embeddings and trace theorems of these spaces are Mso established. Moreover, some obtained results, such as characterizations via approximations by analytic functions, are new even for the classical variable Besov and Triebel-Lizorkin spaces.

Pointwise Characterizations of Besov and Triebel–Lizorkin Spaces with Generalized Smoothness and Their Applications

In this article,the authors first establish the point wise characterizations of Besov and Triebel-Lizorkin spaces with generalized smoothness on R;via the Hajlasz gradient sequences,which serve as a way to extend these spaces to more general metric measure spaces.Moreover,on metric spaces with doubling measures,the authors further prove that the Besov and the Triebel-Lizorkin spaces with generalized smoothness defined via Hajlasz gradient sequences coincide with those defined via hyperbolic fillings.As an application,some trace theorems of these spaces on Ahlfors regular spaces are established.

Boundedness and continuity of maximal singular integrals and maximal functions on Triebel-Lizorkin spaces

In this paper,we systematically study several classes of maximal singular integrals and maximal functions with rough kernels in Fβ(S^n-1),a topic that relates to the Grafakos-Stefanov class.The boundedness and continuity of these operators on Triebel-Lizorkin spaces and Besov spaces are discussed.

Maximal regularity of second order delay equations in Banach spaces

We give necessary and sufficient conditions of Lp-maximal regularity(resp.B sp ,q-maximal regularity or F sp ,q-maximal regularity) for the second order delay equations:u″(t)=Au(t) + Gu't + F u t + f(t), t ∈ [0, 2π] with periodic boundary conditions u(0)=u(2π), u′(0)=u′(2π), where A is a closed operator in a Banach space X,F and G are delay operators on Lp([-2π, 0];X)(resp.Bsp ,q([2π, 0];X) or Fsp,q([-2π, 0;X])).

Unboundedness properties of smoothness Morrey spaces of regular distributions on domains

We study unboundedness of smoothness Morrey spaces on bounded domains ? ? R^n in terms of growth envelopes. It turns out that in this situation the growth envelope function is finite—in contrast to the results obtained by Haroske et al.(2016) for corresponding spaces defined on R^n. A similar effect was already observed by Haroske et al.(2017), where classical Morrey spaces M_(u,p)(?) were investigated. We deal with all cases where the concept is reasonable and also include the tricky limiting cases. Our results can be reformulated in terms of optimal embeddings into the scale of Lorentz spaces L_(p,q)(?).

FAST ALGORITHM FOR CALDERóN-ZYGMUND OPERATORS:CONVERGENCE SPEED AND ROUGH KERNEL

In this article, we consider a fast algorithm for first generation Calderon-Zygmund operators. First, we estimate the convergence speed of the relative approximation algorithm. Then, we establish the continuity on Besov spaces and Triebel-Lizorkin spaces for the oper- ators with rough kernel.

NONLINEAR APPROXIMATION WITH GENERAL WAVE PACKETS

We study nonlinear approximation in the Triebel-Lizorkin spaces with dictionaries formed by dilating and translating one single function g. A general Jackson inequality is derived for best m-term approximation with such dictionaries. In some special cases where g has a special structure, a complete characterization of the approximation spaces is derived.