On Uniform Boundedness of Some Families of Integral Convolution Operators in Weighted Variable Exponent Lebesgue Spaces

Let $k_\lambda(x)$ be a measurable essentially bounded $2\pi$-periodic function (kernel), where $\lambda\ge1$. Conditions on the weight and on the kernels $\{k_\lambda(x)\}_{\lambda\ge1}$ that provide the family of convolution operators $\{\mathcal{K}_\lambda f(x):\mathcal{K}_\lambda f(x)=\int_Ef(t)k_\lambda(t-x)\,dt\}_{\lambda\ge1}$ $(E=[-\pi,\pi])$ uniform boundedness in weighted variable exponent Lebesgue space $L^{p(x)}_{2\pi,w}$ are investigated.