Invertibility of Convolution Operator in $L_{p}$ Spaces in Applications to $R$-semigroups

Sufficient conditions for existence of an inverse operator to convolution operator in $L_ {p}(\mathbb{R}),\ p\in(2,\infty)$, spaces and sufficient conditions for density of the convolution operator range are obtained. These convolution operator properties considered in applications to a wide class of regularizing semigroups — to $R$-semigroups.