Geodesic
Uniform boundedness in $L^p$ $(p=p(x))$ of some families of convolution operators
Matematičeskie zametki, Tome 59 (1996) no. 2, pp. 291-302.

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Suppose that a measurable $2\pi$-periodic essentially bounded function (the kernel) $k_\lambda=k_\lambda(x)$ is given for any real $\lambda\ge1$. We consider the following linear convolution operator in $L_p$: $$ \mathscr K_\lambda=\mathscr K_\lambda f =(\mathscr K_\lambda f)(x)=\int_{-\pi}^\pi f(t)k_\lambda(t-x)\,dt. $$ Uniform boundedness of the family of operators $\{\mathscr K_\lambda\}_{\lambda\ge1}$ is studied. Conditions on the variable exponent $p=p(x)$ and on the kernel $k_\lambda$, that ensure the uniform boundedness of the operator family $\{\mathscr K_\lambda\}_{\lambda\ge1}$ in $L_p$ are obtained. The condition on the exponent $p=p(x)$ is given in its final form. 
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     author = {I. I. Sharapudinov},
     title = {Uniform boundedness in $L^p$ $(p=p(x))$ of some families of convolution operators},
     journal = {Matemati\v{c}eskie zametki},
     pages = {291--302},
     publisher = {mathdoc},
     volume = {59},
     number = {2},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_59_2_a14/}
}
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I. I. Sharapudinov. Uniform boundedness in $L^p$ $(p=p(x))$ of some families of convolution operators. Matematičeskie zametki, Tome 59 (1996) no. 2, pp. 291-302. http://geodesic.mathdoc.fr/item/MZM_1996_59_2_a14/

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