Sobolev inequalities via Muramatu's integral formula
Abstract
For the Sobolev space Wmp(Rn) with positive integer m and 1<p<infty, sometimes replaced by 1<=p<infty, we consider the case m-n/p<0 and the case m-n/p=0, and give new proofs of the Sobolev embedding theorems by Muramatu's integral formula. When m-n/p<0, the embedding into Lq(Rn) with q satisfying m-n/p=-n/q is derived without the Hardy-Littlewood-Sobolev inequality by incorporating the method to prove it. When m-n/p=0, we prove the embedding into the BMO space or the VMO space as well as Trudinger's inequality.
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