Homogeneous variable exponent Besov and Triebel–Lizorkin spaces

We introduce homogeneous Besov and Triebel–Lizorkin spaces with variable indexes. We show that their study reduces to the study of inhomogeneous variable exponent spaces and homogeneous constant exponent spaces. Corollaries include trace space characterizations and Sobolev embeddings.     

Complex interpolation of Herz‐type Triebel–Lizorkin spaces

We study complex interpolation of Herz‐type Triebel–Lizorkin spaces by using the Calderón product method. Additionally we present complex interpolation between Herz‐type Triebel–Lizorkin spaces and Triebel–Lizorkin spaces . Moreover, we apply these results to obtain the complex interpolation of Triebel–Lizorkin spaces equipped with power weights and between (or ) spaces and Herz spaces.     

Integral operators of Marcinkiewicz type on Triebel–Lizorkin spaces

A systematic treatment is given of several classes of parametric Marcinkiewicz integrals. The boundedness on Triebel–Lizorkin spaces will be presented for these operators with rough kernels in , which relates to the Grafakos–Stefanov class. Moreover, the boundedness on Besov spaces for above operators is also considered.     

Boundedness of certain commutators on Triebel–Lizorkin spaces

In this paper, two types of commutators are considered, and the boundedness of these operators on Triebel–Lizorkin spaces are discussed.     

Rough singular integrals on Triebel–Lizorkin space and Besov space

In this paper the authors prove that the homogeneous singular integral T_Ω with ΩH¹(Sⁿ⁻¹) is bounded on the Triebel–Lizorkin spaces and the Besov spaces. These results answer an open problem proposed by Chen and Zhang in [J. Chen, C. Zhang, Boundedness of rough singular integral on the Triebel–Lizorkin spaces, J. Math. Anal. Appl. 337 (2008) 1048–1052]. The same results hold also for the rough singular integral operators T_Ω,h with radial function kernels.     

Atomic decomposition for weighted Besov and Triebel‐Lizorkin spaces

Rychkov defined weighted Besov spaces and weighted Triebel‐Lizorkin spaces coming with a weight in the class $A_p^{\rm loc}$, which is even wider than the class A_p due to Muckenhoupt. In the present paper we are concerned with the atomic decomposition of these spaces. As an application, we obtain some key theorems in the theory of function spaces. Finally we conclude this paper with some helpful examples showing the difference between weighted spaces and unweighted spaces.     

Some maximal inequalities on Triebel–Lizorkin spaces for

In this work we give some maximal inequalities in Triebel–Lizorkin spaces, which are “‐variants” of Fefferman–Stein vector‐valued maximal inequality and Peetre's maximal inequality. We will give some applications of the new maximal inequalities and discuss sharpness of some results.     

Generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on domains

In this paper, we consider a non‐smooth atomic decomposition by using a smooth atomic decomposition. Applying the non‐smooth atomic decomposition, a local means characterization and a quarkonical decomposition, we obtain a pointwise multiplier and a trace operator for generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on the whole space. We also develop the theory of those spaces on domains. We consider an extension operator and a trace operator on the upper half space and on compact oriented Riemannian manifolds.     

Characterization of Triebel–Lizorkin type spaces with variable exponents via maximal functions,local means and non‐smooth atomic decompositions

In this paper we study the maximal function and local means characterizations and the non‐smooth atomic decomposition of the Triebel–Lizorkin type spaces with variable exponents . These spaces were recently introduced by Yang et al. and cover the Triebel–Lizorkin spaces with variable exponents as well as the classical Triebel–Lizorkin spaces , even the case when . Moreover, covered by this scale are also the Triebel–Lizorkin‐type spaces with constant exponents which, in turn cover the Triebel–Lizorkin–Morrey spaces. As an application we obtain a pointwise multiplier assertion for those spaces.     

Modulation spaces and pseudodifferential operators

We use methods from time-frequency analysis to study boundedness and traceclass properties of pseudodifferential operators. As natural symbol classes, we use the modulation spaces onR ^2d , which quantify the notion of the time-frequency content of a function or distribution. We show that if a symbol lies in the modulation spaceM _,1 (R ^2d ), then the corresponding pseudodifferential operator is bounded onL ²(R ^d ) and, more generally, on the modulation spacesM _p,p (R ^d ) for 1p. If lies in the modulation spaceM _2,2 ^s (R ^2d )=L _s ^/2 (R ^2d )H ^s (R ^2d ), i.e., the intersection of a weightedL ²-space and a Sobolev space, then the corresponding operator lies in a specified Schatten class. These results hold for both the Weyl and the Kohn-Nirenberg correspondences. Using recent embedding theorems of Lipschitz and Fourier spaces into modulation spaces, we show that these results improve on the classical Calderòn-Vaillancourt boundedness theorem and on Daubechies' trace-class results.     

On a generalized nonlinear heat equation in Besov and Triebel–Lizorkin spaces

The paper deals with solutions of the Cauchy problem of a nonlinear generalized heat equation in the context of Besov and Triebel–Lizorkin spaces and where and with initial data belonging to some spaces where and .     

Wavelets and local Triebel–Lizorkin spaces with the Lorentz index

In this paper, we apply wavelets to consider local norm function spaces with the Lorentz index. Triebel–Lizorkin–Lorentz spaces are based on the real interpolation of the Triebel–Lizorkin spaces. Triebel–Lizorkin–Morrey spaces are based on local norm of the Triebel–Lizorkin spaces. We give a unified depict of spaces that include these two kinds of spaces. Each index of the five index spaces represents a property of functions. We prove the wavelet characterization of the Triebel–Lizorkin–Lorentz–Morrey spaces and use such characterization to study some basic properties of these spaces.     

Sobolev‐Jawerth embedding of Triebel‐Lizorkin‐Morrey‐Lorentz spaces and fractional integral operator on Hardy type spaces

A Sobolev type embedding for Triebel‐Lizorkin‐Morrey‐Lorentz spaces is established in this paper. As an application of this result, the boundedness of the fractional integral operator on some generalizations of Hardy spaces such as Hardy‐Morrey spaces and Hardy‐Lorentz spaces are obtained.     

Time-Frequency analysis of localization operators

We study a class of pseudodifferential operators known as time-frequency localization operators, Anti-Wick operators, Gabor-Toeplitz operators or wave packets. Given a symbol a and two windows ?₁,?₂, we investigate the multilinear mapping from to the localization operator A_a^?₁,?₂ and we give sufficient and necessary conditions for A_a^?₁,?₂ to be bounded or to belong to a Schatten class. Our results are formulated in terms of time-frequency analysis, in particular we use modulation spaces as appropriate classes for symbols and windows.     

Singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces

The paper is devoted to some only recently uncovered phenomena emerging in the study of singular integral operators (SIO's) with piecewise continuous (PC) coefficients in reflexive rearrangement-invariant spaces over Carleson curves. We deal with several kinds of indices of submultiplicative functions which describe properties of spaces (Boyd and Zippin indices) and curves (spirality indices). We consider some disintegration condition which combines properties of spaces and curves, the Boyd and spirality indices.We show that the essential spectrum of SIO associated with the Riemann boundary value problem with PC coefficient arises from the essential range of the coefficient by filling in certain massive connected sets (so-called logarithmic leaves) between the endpoints of jumps.These results combined with the Allan-Douglas local principle and with the two projections theorem enable us to study the Banach algebra generated by SIO's with matrix-valued piecewise continuous coefficients. We construct a symbol calculus for this Banach algebra which provides a Fredholm criterion and gives a basis for an index formula for arbitrary SIO's from in terms of their symbols.     

Polynomial numerical indices of Banach spaces with absolute norm

We study when a Banach space with absolute norm may have polynomial numerical indices equal to one. In the real case, we show that a Banach space X with absolute norm, which has the Radon-Nikodým property or is Asplund, satisfies n⁽²⁾(X)<1 unless it is one-dimensional. In the complex case, we show that the only Banach spaces X with absolute norm and the Radon-Nikodým property which satisfy n⁽²⁾(X)=1 are the spaces . Also, the only Asplund complex space X with absolute norm which satisfies n⁽²⁾(X)=1 is c₀(Λ).     

Generalized Anti-Wick Operators with Symbols in Distributional Sobolev spaces

Generalized Anti-Wick operators are introduced as a class of pseudodifferential operators which depend on a symbol and two different window functions. Using symbols in Sobolev spaces with negative smoothness and windows in so-called modulation spaces, we derive new conditions for the boundedness on L² of such operators and for their membership in the Schatten classes. These results extend and refine results contained in [20], [10], [5], [4], and [14].     

Polynomial automorphisms and hypercyclic operators on spaces of analytic functions

We consider hypercyclic composition operators on which can be obtained from the translation operator using polynomial automorphisms of . In particular we show that if C _S is a hypercyclic operator for an affine automorphism S on , then for some polynomial automorphism Θ and vectors a and b, where I is the identity operator. Finally, we prove the hypercyclicity of “symmetric translations” on a space of symmetric analytic functions on ℓ₁. Received: 8 June 2006 Revised: 26 September 2006     

Invariants and spaces of zero solutions of linear partial differential operators

It is shown that for open convex , d > 1 and a nontrivial polynomial P the space does not have property . If P is elliptic or homogeneous, then this holds for every open Ω. For even cannot occur and if it occurs for some Ω, then P must be hypoelliptic. Received: 18 July 2005