Rearrangement-invariant norm inequalities for convolution operators

Let $k\in (L_1+L_{\infty })\left(R^n\right)$, where, as usual, $L_1\left(R^n\right)$ denotes the class of Lebesgue-integrable functions on $R^n$ and $L_{\infty }\left(R^n\right)$ denotes the class of functions on $R^n$ that are Lebesgue-measurable and bounded almost everywhere. Given $f\in (L_1\cap L_{\infty })\left(R^n\right)$, set $\left(T_{k}f\right)\left(x\right)={\int }_{R^n}k(x-y)f\left(y\right)dy,x\in R^n.$We study inequalities of the form $\rho \left(T_{k}f\right)\le C\sigma \left(f\right),$in which $C>0$ is independent of $f\in (L_1\cap L_{\infty })\left(R^n\right)$. The functionals $\rho$ and $\sigma$ are so-called rearrangement-invariant (r.i.) norms on $M_{+}\left(R^n\right)$, the class of nonnegative measurable functions on $R^n$.

Results first proved in the general context of r.i. norms are both specialized and expanded upon in the special case of Orlicz norms.