Duality of Besov,Triebel–Lizorkin and Herz spaces with variable exponents

Our aim is to prove duality and reflexivity of Besov spaces, Triebel–Lizorkin spaces and Herz spaces with variable exponents.     

Duality and interpolation of anisotropic Triebel–Lizorkin spaces

We study properties of anisotropic Triebel–Lizorkin spaces associated with general expansive dilations and doubling measures on using wavelet transforms. This paper is a continuation of (Bownik in J Geom Anal 2007, to appear, Trans Am Math Soc 358:1469–1510, 2006), where we generalized the isotropic methods of dyadic -transforms of Frazier and Jawerth (J Funct Anal 93:34–170, 1990) to non-isotropic settings. By working at the level of sequence spaces, we identify the duals of anisotropic Triebel–Lizorkin spaces. We also obtain several real and complex interpolation results for these spaces. The author was partially supported by the NSF grants DMS-0441817 and DMS-0653881. The author wishes to thank Michael Frazier and Dachun Yang for valuable comments and discussions on this work.     

Duality of weighted anisotropic Besov and Triebel–Lizorkin spaces

Let A be an expansive dilation on ${{\mathbb R}^n}$ and w a Muckenhoupt ${\mathcal A_\infty(A)}$ weight. In this paper, for all parameters ${\alpha\in{\mathbb R} }$ and ${p,q\in(0,\infty)}$ , the authors identify the dual spaces of weighted anisotropic Besov spaces ${\dot B^\alpha_{p,q}(A;w)}$ and Triebel?CLizorkin spaces ${\dot F^\alpha_{p,q}(A;w)}$ with some new weighted Besov-type and Triebel?CLizorkin-type spaces. The corresponding results on anisotropic Besov spaces ${\dot B^\alpha_{p,q}(A; \mu)}$ and Triebel?CLizorkin spaces ${\dot F^\alpha_{p,q}(A; \mu)}$ associated with ${\rho_A}$ -doubling measure??? are also established. All results are new even for the classical weighted Besov and Triebel?CLizorkin spaces in the isotropic setting. In particular, the authors also obtain the ${\varphi}$ -transform characterization of the dual spaces of the classical weighted Hardy spaces on ${{\mathbb R}^n}$ .     

Complex Interpolation of Weighted Besov and Lizorkin–Triebel Spaces

We study complex interpolation of weighted Besov and Lizorkin–Triebel spaces.The used weights w0,w1 are local Muckenhoupt weights in the sense of Rychkov.As a first step we calculate the Calder′on products of associated sequence spaces.Finally,as a corollary of these investigations,we obtain results on complex interpolation of radial subspaces of Besov and Lizorkin–Triebel spaces on Rd.     

Lower bound of measure and embeddings of Sobolev,Besov and Triebel–Lizorkin spaces

In this article, we study the relation between Sobolev-type embeddings for Sobolev spaces or Hajłasz–Besov spaces or Hajłasz–Triebel–Lizorkin spaces defined on a doubling and geodesic metric measure space and lower bound for measure of balls either in the whole space or in a domain inside the space.     

Boundedness of composition of operators associated with different homogeneities on weighted Besov and Triebel–Lizorkin spaces

In this paper, we introduce weighted Besov spaces and weighted Triebel–Lizorkin spaces associated with different homogeneities and prove that the composition of two Calderón–Zygmund operators is bounded on these spaces. This extends a recent result in Han et al, Revista Mat. Iber.     

Measure Density and Extension of Besov and Triebel–Lizorkin Functions

We show that a domain is an extension domain for a Haj?asz–Besov or for a Haj?asz–Triebel–Lizorkin space if and only if it satisfies a measure density condition. We use a modification of the Whitney extension where integral averages are replaced by median values, which allows us to handle also the case \(0<p<1\). The necessity of the measure density condition is derived from embedding theorems; in the case of Haj?asz–Besov spaces we apply an optimal Lorentz-type Sobolev embedding theorem which we prove using a new interpolation result. This interpolation theorem says that Haj?asz–Besov spaces are intermediate spaces between \(L^p\) and Haj?asz–Sobolev spaces. Our results are proved in the setting of a metric measure space, but most of them are new even in the Euclidean setting, for instance, we obtain a characterization of extension domains for classical Besov spaces \(B^s_{p,q}\), \(0<s<1\), \(0<p<\infty \), \(0<q\le \infty \), defined via the \(L^p\)-modulus of smoothness of a function.     

Littlewood–Paley characterization of BMO and Triebel–Lizorkin spaces

We prove a generalization of the Littlewood–Paley characterisation of the BMO space where the shifts of a Schwartz function are replaced by a family of functions with suitable conditions imposed on them. We also prove that a certain family of Triebel–Lizorkin spaces can be characterized in a similar way.     

Harmonic Besov and Triebel–Lizorkin Spaces on the Ball

Harmonic Besov and Triebel–Lizorkin spaces on the unit ball in \({\mathbb R}^d\) with full range of parameters are introduced and studied. It is shown that these spaces can be identified with respective Besov and Triebel–Lizorkin spaces of distributions on the sphere. Frames consisting of harmonic functions are also developed and frame characterization of the harmonic Besov and Triebel–Lizorkin spaces is established.     

The Boundedness of Composition Operators on Triebel–Lizorkin and Besov Spaces with Different Homogeneities

In this paper, we introduce new Triebel–Lizorkin and Besov Spaces associated with the different homogeneities of two singular integral operators. We then establish the boundedness of composition of two Calder′on–Zygmund singular integral operators with different homogeneities on these Triebel–Lizorkin and Besov spaces.     

Two Types of Rubio de Francia Operators on Triebel–Lizorkin and Besov Spaces

Journal of Fourier Analysis and Applications - We discuss generalizations of Rubio de Francia’s inequality for Triebel–Lizorkin and Besov spaces, continuing the research from Osipov (Sb...     

The Ahlfors–Beurling transform on Morrey spaces with variable exponents

We establish the vector-valued inequalities of the Ahlfors–Beurling operator on Morrey spaces with variable exponents. As consequences of these inequalities, we have the boundedness of the Ahlfors–Beurling transform on Lebesgue spaces with variable exponents and Morrey spaces. The results obtained in this paper are new in the case of Morrey spaces.     

Domains of type 1,1 operators: a case for Triebel–Lizorkin spaces

Pseudo-differential operators of type $1\text{,}1$ are proved continuous from the Triebel–Lizorkin space $F_{p\text{,}1}^d$ to $L_p$, $1?p<\infty$, when of order d, and this is, in general, the largest possible domain among the Besov and Triebel–Lizorkin spaces. Hörmander's condition on the twisted diagonal is extended to this framework, using a general support rule for Fourier transformed pseudo-differential operators. To cite this article: J. Johnsen, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Complex interpolation and Calderón product of quasi-Banach spaces

We prove that the inner complex interpolation of two quasi-Banach lattices coincides with the closure of their intersection in their Calderón product. This generalizes a classical result by V. A. Shestakov in 1974 for Banach lattices.