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27 pages, 910 KiB

Open AccessArticle

Mixed-Norm Amalgam Spaces and Their Predual

by Houkun Zhang and Jiang Zhou

Symmetry 2022, 14(1), 74; https://doi.org/10.3390/sym14010074 - 4 Jan 2022

Cited by 5 | Viewed by 1903

In this paper, we introduce mixed-norm amalgam spaces

{(L^{\vec{p}}, L^{\vec{s}})}^{α} (R^{n})

and prove the boundedness of maximal function. Then, the dilation argument obtains the necessary and sufficient conditions of fractional integral operators’ [...] Read more.

In this paper, we introduce mixed-norm amalgam spaces

{(L^{\vec{p}}, L^{\vec{s}})}^{α} (R^{n})

and prove the boundedness of maximal function. Then, the dilation argument obtains the necessary and sufficient conditions of fractional integral operators’ boundedness. Furthermore, the strong estimates of linear commutators

[b, I_{γ}]

generated by

b \in B M O (R^{n})

and

I_{γ}

on mixed-norm amalgam spaces

{(L^{\vec{p}}, L^{\vec{s}})}^{α} (R^{n})

are established as well. In order to obtain the necessary conditions of fractional integral commutators’ boundedness, we introduce mixed-norm Wiener amalgam spaces

(L^{\vec{p}}, L^{\vec{s}}) (R^{n})

. We obtain the necessary and sufficient conditions of fractional integral commutators’ boundedness by the duality theory. The necessary conditions of fractional integral commutators’ boundedness are a new result even for the classical amalgam spaces. By the equivalent norm and the operators

S t_{r}^{(p)} (f) (x)

, we study the duality theory of mixed-norm amalgam spaces, which makes our proof easier. In particular, note that predual of the primal space is not obtained and the predual of the equivalent space does not mean the predual of the primal space. Full article

14 pages, 289 KiB

Open AccessArticle

On Generalizations of Sampling Theorem and Stability Theorem in Shift-Invariant Subspaces of Lebesgue and Wiener Amalgam Spaces with Mixed-Norms

by Junjian Zhao, Marko Kostić and Wei-Shih Du

Symmetry 2021, 13(2), 331; https://doi.org/10.3390/sym13020331 - 18 Feb 2021

Cited by 8 | Viewed by 1938

Abstract

In this paper, we establish generalized sampling theorems, generalized stability theorems and new inequalities in the setting of shift-invariant subspaces of Lebesgue and Wiener amalgam spaces with mixed-norms. A convergence theorem of general iteration algorithms for sampling in some shift-invariant subspaces of [...] Read more.

L_{\vec{p}} (R^{d})

are also given. Full article

(This article belongs to the Special Issue Recent Advances on Nonlinear Analysis and Optimization in Symmetry and Asymmetry)

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