General logarithmic Sobolev inequalities and Orlicz imbeddings

AbstractWe obtain, for a large class of measures μ, general inequalities of the form ∫Rn|u|p A(log∗|u|) dμ ⩾ K(‖u : Wm,p(Rn,dμ)‖p + ‖ u ‖p A(log∗ ‖ u ‖)), where ‖u‖ = ‖ u: Lp(Rn,dμ)‖p, log∗ t = max{1, log 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) = ts. We also investigate some implications of these inequalities for the imbedding of Sobolev spaces into Orlicz spaces