Calderón–Zygmund Operators in the Bessel Setting for All Possible Type Indices

We show that many harmonic analysis operators in the Bessel setting,including maximal operators,Littlewood–Paley–Stein type square functions,multipliers of Laplace or Laplace–Stieltjes transform type and Riesz transforms are,or can be viewed as,Calderón–Zygmund operators for all possible values of type parameter λ in this context.This extends results existing in the literature,but being justified only for a restricted range of λ.     

Atomic Decompositions of Triebel-Lizorkin Spaces with Local Weights and Applications

In this paper,the authors characterize the inhomogeneous Triebel-Lizorkin spaces Fp,q s,w(Rn)with local weight w by using the Lusin-area functions for the full ranges of the indices,and then establish their atomic decompositions for s ∈ R,p ∈(0,1] and q ∈ [p,∞).The novelty is that the weight w here satisfies the classical Muckenhoupt condition only on balls with their radii in(0,1].Finite atomic decompositions for smooth functions in Fp,q s,w(Rn)are also obtained,which further implies that a(sub)linear operator that maps smooth atoms of Fp,q s,w(Rn)uniformly into a bounded set of a(quasi-)Banach space is extended to a bounded operator on the whole Fp,q s,w(Rn).As an application,the boundedness of the local Riesz operator on the space Fp,q s,w(Rn)is obtained.