General logarithmic Sobolev inequalities and Orlicz imbeddings

We obtain, for a large class of measures μ, general inequalities of the form $\text{∫R}{}^{\mathrm{n}}\text{|u|}{}^{\mathrm{p}}\text{A(}\text{log}{}^1\text{|u|) dμ ? K(6u : W}{}^{\mathrm{m,p}}\text{(R}{}^{\mathrm{n}}\text{,dμ)6}{}^{\mathrm{p}}\text{+ 6 u 6}{}^{\mathrm{p}}\text{A(}\text{log}{}^1\text{6 u 6))}$, where $\text{6u6 = 6 u: L}{}^{\mathrm{p}}\text{(R}{}^{\mathrm{n}}\text{,dμ)6}{}^{\mathrm{p}}\text{,}\text{log}{}^1\text{t =}\text{max}\text{{1,}\text{log}\text{t}}$, and the function A depends in an appropriate way on μ. Our results extend similar results obtained by Rosen for the case p = 2, A(t) = t^s. We also investigate some implications of these inequalities for the imbedding of Sobolev spaces into Orlicz spaces.