Calderón–Zygmund operators with operator‐valued kernel on homogeneous Besov spaces

We consider generalized Calderón–Zygmund operators whose kernel takes values in the space of all continuous linear operators between two Banach spaces. In the spirit of the T (1) theorem of David and Journé we prove boundedness results for such operators on vector‐valued Besov spaces. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

双线性Fourier乘子在Triebel-Lizorkin和Besov空间的有界性

本文利用Littlewood-Paley分解,Fourier变换和逆变换等方法,研究了双线性Fourier乘子在非齐次正光滑性Triebel-Lizorkin空间和Besov空间的有界性.     

Embedding vector‐valued Besov spaces into spaces of γ ‐radonifying operators

It is shown that a Banach space E has type p if and only for some (all) d ≥ 1 the Besov space B^{(1/p – 1/2)d} _p,p (?^d ; E) embeds into the space γ (L²(?^d ), E) of γ ‐radonifying operators L²(?^d ) → E. A similar result characterizing cotype q is obtained. These results may be viewed as E ‐valued extensions of the classical Sobolev embedding theorems. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On operator-valued Fourier multipliers

We give a simpler proof of a result on operator-valued Fourier multipliers on Lp([0,2π]d;X) using an induction argument based on a known result when d=1.     

Fourier multipliers and spectral measures in Banach function spaces

It is classical that amongst all spaces L^p (G), 1 ≤ p ≤ ∞, for , or say, only L² (G) (that is, p = 2) has the property that every bounded Borel function on the dual group Γ determines a bounded Fourier multiplier operator in L² (G). Stone’s theorem asserts that there exists a regular, projection-valued measure (of operators on L² (G)), defined on the Borel sets of Γ, with Fourier-Stieltjes transform equal to the group of translation operators on L² (G); this fails for every p ≠ 2. We show that this special status of L² (G) amongst the spaces L^p (G), 1 ≤ p ≤ ∞, is actually more widespread; it continues to hold in a much larger class of Banach function spaces defined over G (relative to Haar measure).      

Besov spaces and Herz spaces on local fields

The relationship of Besov spaces and Herz spaces on local fields is given. As an application, one multiplier theorem is obtained. And the decompositional characterization of the weighted Besov spaces is established.     

Fourier multipliers on power‐weighted Hardy spaces

Using Herz spaces, we obtain a sufficient condition for a bounded measurable function on ?ⁿ to be a Fourier multiplier on H^p_α (?ⁿ ) for 0 < p < 1 and –n < α ≤ 0. Our result is sharp in a certain sense and generalizes a recent result obtained by Baernstein and Sawyer. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Commutators for Fourier multipliers on Besov Spaces

If T is any bounded linear operator on Besov spaces B_p^σ_j,q_j(Rⁿ)(j=0,1, and 0<σ₁<σ<σ₀), it is proved that the commutator [T,T_μ]=TT_μ−T_μT is bounded on B_p^σ,q(Rⁿ), if T_μ is a Fourier multiplier such that μ is any (possibly unbounded) symbol with uniformly bounded variation on dyadic coronas.     

Operator–valued Fourier multiplier theorems on Besov spaces

We investigate analytical properties of a measure geometric Laplacian which is given as the second derivative w.r.t. two atomless finite Borel measures μ and ν with compact supports supp μ ? supp ν on the real line. This class of operators includes a generalization of the well‐known Sturm‐Liouville operator as well as of the measure geometric Laplacian given by . We obtain for this differential operator under both Dirichlet and Neumann boundary conditions similar properties as known in the classical Lebesgue case for the euclidean Laplacian like Gauß‐Green‐formula, inversion formula, compactness of the resolvent and its kernel representation w.r.t. the corresponding Green function. Finally we prove nuclearity of the resolvent and give two representations of its trace.     

Fourier multipliers of classical modulation spaces

Based on the observation that translation invariant operators on modulation spaces are convolution operators we use techniques concerning pointwise multipliers for generalized Wiener amalgam spaces in order to give a complete characterization of the Fourier multipliers of modulation spaces. We deduce various applications, among them certain convolution relations between modulation spaces, as well as a short proof for a generalization of the main result of a recent paper by Bènyi et al., see [À. Bènyi, L. Grafakos, K. Gröchenig, K.A. Okoudjou, A class of Fourier multipliers for modulation spaces, Appl. Comput. Harmon. Anal. 19 (1) (2005) 131–139]. Finally, we show that any function with ([d/2]+1)-times bounded derivatives is a Fourier multiplier for all modulation spaces with p(1,∞) and q[1,∞].     

Besov type function spaces defined on metric-measure spaces

The purpose of this article is to study the Besov type function spaces for maps which are defined on abstract metric-measure spaces. We extend some of the embedding theorems of the classical Besov spaces to the setting of abstract spaces.     

Well‐posedness of fractional degenerate differential equations with finite delay on vector‐valued functional spaces

We study the well‐posedness of the fractional degenerate differential equations with finite delay on Lebesgue–Bochner spaces , periodic Besov spaces and periodic Triebel–Lizorkin spaces , where A and M are closed linear operators on a Banach space X satisfying , F is a bounded linear operator from (resp. and ) into X, where is given by when and . Using known operator‐valued Fourier multiplier theorems, we give necessary or sufficient conditions for the well‐posedness of in the above three function spaces.     

Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

We study the well‐posedness of the fractional differential equations with infinite delay on Lebesgue–Bochner spaces and Besov spaces , where A and B are closed linear operators on a Banach space X satisfying ,  and . Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of in the above vector‐valued function spaces on by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied.     

Existence and regularizing rate estimates of solutions to the 3‐D generalized micropolar system in Fourier‐Besov spaces

In this paper, we study global existence and asymptotic stability of solutions for the initial value problem of the three‐dimensional (3‐D) generalized incompressible micropolar system in Fourier‐Besov spaces. Besides, we also establish some regularizing rate estimates of the higher‐order spatial derivatives of solutions, which particularly imply the spatial analyticity and the temporal decay of global solutions.     

Unimodular Fourier multipliers for modulation spaces

We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol eiα|ξ|, where α∈[0,2], are bounded on all modulation spaces, but, in general, fail to be bounded on the usual Lp-spaces. As a consequence, the phase-space concentration of the solutions to the free Schrödinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers |ξ|^−δsin(α|ξ|) for 0?δ?α.     

Besov maximal regularity for a class of degenerate integro-differential equations with infinite delay in Banach spaces

The theory of operator-valued Fourier multipliers is used to obtain characterizations for well-posedness of a large class of degenerate integro-differential equations of second order in time in Banach spaces. Specifically, we treat the case of vector-valued Besov spaces on the real line. It is important to note that in particular, the results are applicable to the more familiar scale of vector-valued Hölder spaces. The equations under consideration are important in several applied problems in physics and material science, in particular for phenomena where memory effects are important. Several models in the area of viscoelasticity, including heat conduction and wave propagation correspond to the general class of integro-differential equations considered here. The importance of the results is that they can be used to treat nonlinear equations.     

Bilinear multipliers on Lorentz spaces

We give one sufficient and two necessary conditions for boundedness between Lebesgue or Lorentz spaces of several classes of bilinear multiplier operators closely connected with the bilinear Hilbert transform. The author has been partially supported by grants DGESIC PB98-1246 and BMF 2002-04013.