Equivalence in Operator Algebras.
One-sided versions of maximal functions for suitable defined distributions are considered. Weighted norm equivalences of these maximal functions for weights in the Sawyer's Aq+ classes are obtained.
We obtain the equivalence of the properties and (NUC) in Orlicz function spaces. This answers a question raised by Y. Cui, R. Pluciennik and T. Wang.
We consider (p,q)-multi-norms and standard t-multi-norms based on Banach spaces of the form , and resolve some question about the mutual equivalence of two such multi-norms. We introduce a new multi-norm, called the [p,q]-concave multi-norm, and relate it to the standard t-multi-norm.
We discuss the validity of the Helmholtz decomposition of the Muckenhoupt -weighted -space for any domain in , , , and Muckenhoupt -weight . Set and . Then the Helmholtz decomposition of and and the variational estimate of and are equivalent. Furthermore, we can replace and by and , respectively. The proof is based on the reflexivity and orthogonality of and and the Hahn-Banach theorem. As a corollary of our main result, we obtain the extrapolation theorem with...
The main object of this work is to describe such weight functions w(t) that for all elements the estimate is valid with a constant K(Ω), which does not depend on f and it grows to infinity when the domain Ω shrinks, i.e. deforms into a lower dimensional convex set . In one-dimensional case means that as σ → 0. It should be noted that in the framework of the signal transmission problem such estimates describe a signal’s behavior under the influence of detection and amplification. This work...
Recently, the weak Triebel-Lizorkin space was introduced by Grafakos and He, which includes the standard Triebel-Lizorkin space as a subset. The latter has a wide applications in aspects of analysis. In this paper, the authors firstly give equivalent quasi-norms of weak Triebel-Lizorkin spaces in terms of Peetre's maximal functions. As an application of those equivalent quasi-norms, an atomic decomposition of weak Triebel-Lizorkin spaces is given.
Utilizing elementary properties of convergence of numerical sequences we prove Nikodym, Banach, Orlicz-Pettis type theorems