Boundedness for pseudodifferential operators on multivariate α-modulation spaces

The α-modulation spaces M ^s,α _p,q (R ^d ), α∈[0,1], form a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that a pseudodifferential operator σ(x,D) with symbol in the Hörmander class S ^b _ρ,0 extends to a bounded operator σ(x,D):M ^s,α _p,q (R ^d )→M ^s-b,α _p,q (R ^d ) provided 0≤α≤ρ≤1, and 1<p,q<∞. The result extends the well-known result that pseudodifferential operators with symbol in the class S ^b _1,0 maps the Besov space B ^s _p,q (R ^d ) into B ^s-b _p,q (R ^d ).     

Banach frames for multivariate α-modulation spaces

The α-modulation spaces , α∈[0,1], form a family of spaces that include the Besov and modulation spaces as special cases. This paper is concerned with construction of Banach frames for α-modulation spaces in the multivariate setting. The frames constructed are unions of independent Riesz sequences based on tensor products of univariate brushlet functions, which simplifies the analysis of the full frame. We show that the multivariate α-modulation spaces can be completely characterized by the Banach frames constructed.     

Boundedness of fractional integral operators on α-modulation spaces

In this paper, we obtain the boundedness of the fractional integral operators, the bilinear fractional integral operators and the bilinear Hilbert transform on α-modulation spaces.     

Product (α1, α2)-modulation spaces

We study fundamental properties of product (α₁, α₂)-modulation spaces built by (α₁, α₂)-coverings of ℝⁿ¹ × ℝⁿ². Precisely we prove embedding theorems between these spaces with different parameters and other classical spaces. Furthermore, we specify their duals. The characterization of product modulation spaces via the short time Fourier transform is also obtained. Families of tight frames are constructed and discrete representations in terms of corresponding sequence spaces are derived. Fourier multipliers are studied and as applications we extract lifting properties and the identification of our spaces with (fractional) Sobolev spaces with mixed smoothness.

     

Full characterization of the embedding relations between α-modulation spaces

In this paper, we consider the embedding relations between any two a-modulation spaces. Based on an observation that the a-modulation space with smaller a can be regarded as a corresponding decomposition space associated with a-covering for larger a, we give a complete characterization of the Fourier multipliers between a-modulation spaces with different a. Then we establish a full version of optimal embedding relations between a-modulation spaces.     

Trace ideals for pseudo-differential operators and their commutators with symbols in α-modulation spaces

That symbols in the modulation space M ^1,1 generate pseudo-differential operators of the trace class was first stated by Feichtinger and proved by Gröchenig in [13]. In this paper, we show that the same is true if we replace M ^1,1 by the more general α-modulation spaces, which include modulation spaces (α = 0) and Besov spaces (α = 1) as special cases. The result with α = 0 corresponds to that of Gröchenig, and the one with α = 1 is a new result which states the trace property of the operators with symbols in the Besov space. As an application, we discuss the trace property of the commutator [α (X, D), a], where; a(χ) is a Lipschitz function and σ(χ, ξ) belongs to an α-modulation space.     

Phillips’ lemma for L-embedded Banach spaces

In this note the following version of Phillips’ lemma is proved. The L-projection of an L-embedded space—that is of a Banach space which is complemented in its bidual such that the norm between the two complementary subspaces is additive—is weak*-weakly sequentially continuous.     

Weakly relaxed α-pseudomonotonicity and equilibrium problem in Banach spaces

In this paper, we introduce the weakly relaxed ??-pseudomonotone mapping which generalizes the class of pseudomonotone mappings. Some examples are given which show that the weakly relaxed ??-pseudomonotonicity is a proper generalization of pseudomonotonicity for both scalar and vector valued bi-functions. Using the KKM technique, we establish the existence of solutions of equilibrium problems and vector equilibrium problems with weakly relaxed ??-pseudomonotonicity in the reflexive Banach spaces. The present work extends some corresponding results of the variational-like inequalities (Fang and Huang in J. Optim. Theory Appl. 118:327?C338, 2003; Lee and Lee in J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 11: 231?C242, 2004) to equilibrium problems.     

A Banach–Zarecki Theorem for functions with values in Banach spaces

A Banach–Zarecki Theorem for a Banach space-valued function  \(F : [0,1] \rightarrow X\) with compact range is presented. We define the strong absolute continuity ( \(sAC_{||.||_{F}}\) ) and the bounded variation ( \(BV_{||.||_{F}}\) ) of \(F\) with respect to the Minkowski functional \(||.||_{F}\) associated to the closed absolutely convex hull \(C_{F}\) of \(F([0,1])\) . It is proved that \(F\) is \(sAC_{||.||_{F}}\) if and only if \(F\) is \(BV_{||.||_{F}}\) , weak continuous on \([0,1]\) and satisfies the weak property \((N)\) .     

“Compactly” supported frames for spaces of distributions on the ball

Frames are constructed on the unit ball B ^d in ${\mathbb{R}^d}$ consisting of smooth functions with small shrinking supports. The new frames are designed so that they can be used for decomposition of weighted Triebel–Lizorkin and Besov spaces on B ^d with weight ${w_\mu(x):=(1-|x|^2)^{\mu-1/2}, \mu}$ half integer,?μ?≥ 0.     

Hardy spaces for α-harmonic functions in regular domains

We investigate Hardy spaces H ^p for singular α-harmonic functions in bounded domains with regular boundaries. We show the correspondence between these spaces and suitable L ^p spaces and measure spaces.     

Korpelevich’s method for variational inequality problems in Banach spaces

We propose a variant of Korpelevich’s method for solving variational inequality problems with operators in Banach spaces. A full convergence analysis of the method is presented under reasonable assumptions on the problem data.     

Hölder metric subregularity for multifunctions in type Banach spaces

Using the Borwein–Preiss variational principle and in terms of the proximal coderivative, we provide a new type of sufficient conditions for the Hölder metric subregularity and Hölder error bounds in a class of smooth Banach spaces. As an application, new characterizations for the tilt stability of Hölder minimizers are established.     

Convergence theorems for λ-strict pseudo-contractions in q-uniformly smooth Banach spaces

In this paper, we continue to discuss the properties of iterates generated by a strict pseudo- contraction or a finite family of strict pseudo-contractions in a real q-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [Marino, G., Xu, H. K.： Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl., 324, 336-349 （2007）]. In order to get a strong convergence theorem, we modify the normal Mann＇s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C. This result extends a recent result of Kim and Xu [Kim, T. H., Xu, H. K.： Strong convergence of modified Mann iterations. Nonl. Anal., 61, 51- 60 （2005）] both from nonexpansive mappings to λ-strict pseudo-contractions and from Hilbert spaces to q-uniformly smooth Banach spaces.     

Moreau’s decomposition in Banach spaces

Moreau’s decomposition is a powerful nonlinear hilbertian analysis tool that has been used in various areas of optimization and applied mathematics. In this paper, it is extended to reflexive Banach spaces and in the context of generalized proximity measures. This extension unifies and significantly improves upon existing results.     

Necessary conditions for Gabor frames

The goal of this paper is to establish a set of necessary conditions for Gabor frames. These conditions are also sufficient for tight frames.