User nagaraj - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T17:34:14Z http://mathoverflow.net/feeds/user/25035 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81034/compact-embeddings-of-sobolev-spaces-a-counterexample-showing-the-rellich-kondra/101908#101908 Answer by Nagaraj for Compact Embeddings of Sobolev Spaces: A Counterexample Showing The Rellich-Kondrachov Theorem Is Sharp Nagaraj 2012-07-11T01:59:02Z 2012-07-11T01:59:02Z <p>This question has cropped up in a work of mine.When the p* norm is computed without using the gradient value of the sequence,the Gagliardo-Nirenberg <code>default condition' 1-n/p + n/p* = 0 appears to impose a restriction on the sequence index epsilon,since the sequence index disappears.This has the</code>effect' that the W^(1,p) bound condition the sequence has to compulsorily verify is ignored and this sounds illogical,specially when the Rellich-Kondrachov theorem tacitly makes use of such a bound in its proof.On the other hand,since the gradient values of the sequence have to be used in integration to verify the W^(1,p) bound and keeping in mind that there is a formula that expresses a C^1 function of compact support in terms of its gradient(this by integration by parts of the fundamental solution of the Laplacian),it is possible to obtain estimates for the p* norm in terms of the L^infinity norm of the defining test function u and what is more,such an estimate involves positive powers of the sequence index epsilon even though the Gagliardo-Nirenberg default condition is used,showing that the sequence actually converges to zero in p* norm! The actual meaning therefore is that the p* norm computations,one without using the gradient of the sequence and the other using the gradient and its implied value in integration,are different and if forcibly compared,leads to the contradiction that u vanishes identically.We have worked out the proof that relies on nontrivial facts such as strong bounds for the Hardy-Littlewood maximal function. It is to be noted further that the condition q &lt; p* is only a sufficient condition to establish the Rellich-Kondrachov theorem by interpolation and there is nothing to support that the theorem can not be established by a procedure that does not require interpolation. In summary,the counter-example seems vacuous,showing that compactness at p* may still be an open problem!</p>