A twist in sharp Sobolev inequalities with lower order remainder terms

Abstract

Abstract Let ( M , g ) {(M,g)} be a smooth compact Riemannian manifold of dimension n ≥ 3 {n\geq 3} . Let also A be a smooth symmetrical positive ( 0 , 2 ) {(0,2)} -tensor field in M. By the Sobolev embedding theorem, we can write that there exist K , B > 0 {K,B>0} such that for any u ∈ H 1 ⁢ ( M ) {u\in H^{1}(M)} , (0.1) ∥ u ∥ L 2 ⋆ 2 ≤ K ⁢ ∥ ∇ A ⁡ u ∥ L 2 2 + B ⁢ ∥ u ∥ L 1 2 \|u\|_{L^{2^{\star}}}^{2}\leq K\|\nabla_{A}u\|_{L^{2}}^{2}+B\|u\|_{L^{1}}^{2} where H 1 ⁢ ( M ) {H^{1}(M)} is the standard Sobolev space of functions in L 2 {L^{2}} with one derivative in L 2 {L^{2}} , | ∇ A ⁡ u | 2 = A ⁢ ( ∇ ⁡ u , ∇ ⁡ u ) {|\nabla_{A}u|^{2}=A(\nabla u,\nabla u)} and 2 ⋆ {2^{\star}} is the critical Sobolev exponent for H 1 {H^{1}} . We compute in this paper the value of the best possible K in (0.1) and investigate the validity of the corresponding sharp inequality.