This article is cited in 2 papers

On the uniform boundedness of the family of shifts of Steklov functions in weighted Lebesgue spaces with variable exponent

T. N. Shakh-Emirov

Daghestan Scientific Centre of Russian Academy of Sciences, Makhachkala

Abstract: The problem of the uniform boundedness of the Steklov functions shifts families of the form $ S_{\lambda,\tau}(f)=S_{\lambda}(f)(x+\tau)=\lambda\int_{x+\tau-\frac 1{2\lambda}}^{x+\tau+\frac 1{2\lambda}}f(t)dt $ was considered. It was shown that these shifts are uniformly bounded in weighted variable exponent Lebesgue spaces $L^{p(x)}_{2\pi,w}$, where $w=w(x)$ is the weight function satisfying the analogue of Muckenhoupt's condition.

Keywords: Lebesgue spaces with variable exponent, Dini – Lipschitz condition, Steklov operators.

UDC: 517.5

Received: 09.11.2017
Revised: 28.11.2017
Accepted: 29.11.2017

DOI: 10.31029/demr.8.9