Sharp estimates of the Sobolev norm of u times the gradient of v

[摘要] We give a new proof of the following inequality. In any dimension n greater than or equal to 2 and for 1 < p < n let s = (n + p)/2p. Then \\u del nu\\(Lp(Rn)) less than or equal to C\\u\\(Lp,s(Rn))\\nu\\(Lp,s(Rn)), where L(p,s)(R(n)) denotes the usual Sobolev space and del nu denotes the gradient of nu. The choice of s is optimal, as is the requirement that n > p. In addition, some Sobolev norms of u del nu can be estimated. (C) 1997 Academic Press.

[发布日期] 1997-01-15