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

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.