Approximation theory

Daghestan Electronic Mathematical Reports: Issue 8 (2017)

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

UDK: 517.5

Pages: 93 - 99

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.

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