Uniform boundedness in $L^p$ $(p=p(x))$ of some families of convolution operators

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.