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For s ∈ ℝ , denote by P k s f the "projection" of a function f in D ℝ d into the eigenspaces of the Dunkl Laplacian Δ k corresponding to the eigenvalue − s 2. The parameter k comes from Dunkl's theory of differential-difference operators. We shall characterize the range of P k s on the space of functions f ∈ D ℝ d supported inside the closed ball B O , R ¯. As an application, we provide a spectral version of the Paley-Wiener theorem for the Dunkl transform. [ABSTRACT FROM AUTHOR]

We prove the condition “c is neither 0 nor a negative integer” can be dropped on the boundedness of a class of integral operators Sa,b,c on Lp space, which improves the result by Krues and Zhu. Besides, the exact norm of Sa,b,c on Lp space is also obtained under the assumption c=n+1+a+b. [ABSTRACT FROM AUTHOR]

We study the problem of isometric extension on a sphere of the space sn(H). We give an affirmative answer to Tingley's problem in the space sn(H). [ABSTRACT FROM AUTHOR]

Let Tσ be the multilinear Fourier multiplier operator associated with multiplier σ satisfying the Sobolev regularity that sup ...(ℝmn < ∞ for some sκ ∈ (n/2,n] (κ = 1,...,m). The authors prove that if b1...,bm,...,∈ BMO(ℝn and ... then the commutator TσΣb is bounded from LP1(ℝn, w1) x... x LPm(ℝn, wm) to LP(ℝn, ...). Moreover, the authors also prove that if b1,...,bm ∈ VMO(ℝn) and ... ∈...then the commutator TπΣb, is compact operator from LP1 (ℝn, w1) x... x LPm(ℝn,wm) to LP(ℝn,...). [ABSTRACT FROM AUTHOR]

We give a necessary and sufficient condition on a function h : ℝ → ℝ under which the nonlinear composition operator H, associated with the function h, Hu(t) = h(u(t)), acts in the space κΦBV[a, b] and satisfies a local Lipschitz condition. [ABSTRACT FROM AUTHOR]

We study the boundedness of Φ-admissible sublinear singular operators on Orlicz-Morrey spaces MΦ, φ(ℝn). These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator and Calderón-Zygmund singular integral operator. [ABSTRACT FROM AUTHOR]

The boundedness and compactness of generalized composition operators on Zygmund-Orlicz type spaces and Bloch-Orlicz type spaces are established in this paper. [ABSTRACT FROM AUTHOR]

The oscillatory hyper-Hilbert transform along curves is of the following form: Hn,α,βf(x) = ∫ 0¹f(x - ⌈(t))e it-β t-1-α dt, where α ≥ 0, β ≥ 0, and ⌈(t) = {tP1, tP2,..., tPn). The study on this operator is motivated by the hyper-Hilbert transform and the strongly singular integráis. The LP bounds for Hn,α,β have been given by Chen et al. (2008 and 2010). In this paper, for some α, β, and p, the boundedness of Hn, α, β on Sobolev spaces LsP(ℝn) and the boundedness of this operator from Ls²(ℝn) to L²(ℝn) are obtained. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to give the boundedness of the Marcinkiewicz integralµΩ on Herz spaces ... and ..., where the two main indices are variable. Meanwhile, we consider the boundedness of the higher order commutator µΩ,bm generated by µΩ and function b in BMO(ℝn) on these spaces. [ABSTRACT FROM AUTHOR]

We define and investigate generalized local Morrey spaces and generalized local Campanato spaces, within a context of a general quasimetric measure space. The locality is manifested here by a restriction to a subfamily of involved balls. The structural properties of these spaces and the maximal operators associated to them are studied. In numerous remarks, we relate the developed theory, mostly in the "global" case, to the cases existing in the literature. We also suggest a coherent theory of generalized Morrey and Campanato spaces on open proper subsets of ℝn. [ABSTRACT FROM AUTHOR]

Let KrM' (R) be the generalized tempered distributions of eM(x) -growth with restricted order r ϵ N0, where the function M (x) grows faster than any linear functions as ∣x∣ → ∞. We show the convergence of multiresolution expansions of KrM' (R) in the test function space KrM (R) of KrM' (R). In addition, we show that the kernel of an integral operator K : KrM' (R) → KrM' (R) provides approximation order in KrM' (R) in the context of shift-invariant spaces. [ABSTRACT FROM AUTHOR]

Let f be an analytic function in the unit disc D. The Volterra integral operator If is defined as follows: If(h)(z) = ∫z0 f (w)h' (w)dw, h ∈ H (D), z ∈ D. In this paper, we compute the norm of If on some analytic function spaces. [ABSTRACT FROM AUTHOR]

A function space, Lπ,8) (O), 0 = π < 8, is defined. It is proved that Lπ,8) (O) is a Banach space which is a generalization of exponential class. An alternative definition of Lπ,8) (O) space is given. As an application, we obtain weak monotonicity property for very weak solutions of A-harmonic equation with variable coefficients under some suitable conditions related to Lπ,8) (O), which provides a generalization of a known result due toMoscariello. A weighted space Lπ,8) L (O) is also defined, and the boundedness for the Hardy-Littlewood maximal operatorMw and a Calderón-Zygmund operator T with respect to Lπ,8) (O) is obtained. [ABSTRACT FROM AUTHOR]

The authors consider the generalized commutator of fractional Hardy operator with a rough kernel as follows: ... where Ω ∊ ... with m ∊ Z+. The authors prove that Hm∊,A,β is bounded on Herz type space and λ-Central Morrey space with m ≥ 1, which is an open problem for m > 2. ∊ [ABSTRACT FROM AUTHOR]

In many previous papers, an integral transform ...γ,ß was just considered as a transform on appropriate function spaces. In this paper we deal with the integral transform as an operator on a function space. We then apply various operator theories to ...γ,ß. Finally we give an application for the spectral representation of a self-adjoint operator which plays a key role in quantum mechanics. [ABSTRACT FROM AUTHOR]

The existence of the second (according to the module) eigenvalue λ2 of a completely continuous nonnegative operator A is proved under the conditions that A acts in the space Lp(Ω) or C(Ω) and its exterior square A?A is also nonnegative. For the case when the operators A and A ∧ A are indecomposable, the simplicity of the first and second eigenvalues is proved, and the interrelation between the indices of imprimitivity of A and A ∧ A is examined. For the case when A and A ∧ A are primitive, the difference (according to the module) of λ1 and λ2 from each other and from other eigenvalues is proved. [ABSTRACT FROM AUTHOR]