On Caffarelli-Kohn-Nirenberg Inequalities for Block-Radial Functions

Abstract

The paper provides weighted Sobolev inequalities of the Caffarelli-Kohn-Nirenberg type for functions with multi-radial symmetry. An elementary example of such inequality is the following inequality of Hardy type for functions u = u(r 1

Pointwise Estimates for Block-Radial Functions of Sobolev Classes

“…For the case s = 1 the estimate (2.9) for a subspace ofḢ 1, p γ (R N ) was given previously as Corollary 1 in [17]. Part (ii) of the corollary extends it to the wholė…”

“…The main technical tool used in the paper is the method of atomic decompositions. Strauss type inequalities for Sobolev spaces of integer smoothness can be also proved by more elementary methods, and, as mentioned above, such inequalities were obtained by authors in [17] s = 1. However, the inequality that could be obtained by a simpler argument, (2.9), is less sharp than (2.8), and is verified for, generally, a more narrow class of functions.…”

“…A preliminary and more coarse pointwise estimate for multiradial functions iṅ H 1, p (R N ) with the right hand side as in (2.9) below has been previously given by the authors in Corollary 1, [17], for a subspace ofḢ 1, p (R N ). For a large range of parameters this subspace is strictly smaller thanḢ 1, p (R N ), and perhaps more significantly, for a large range of parameters inequality (2.9) is not an optimal estimate.…”

“…The authors would also like to bring it to the attention of the reader that Proposition 1 of [17] that alleges that the subspace of the functions considered in [17] cannot be generally enlarged toḢ 1, p γ (R N ), contains a computational error (a request for publication an erratum has been made) and should be ignored.…”

“…However, the inequality that could be obtained by a simpler argument, (2.9), is less sharp than (2.8), and is verified for, generally, a more narrow class of functions. The main objective of using the approach of this paper was to refine the estimates of [17] for the classical Sobolev spaces, but as it happened, generalization to Sobolev spaces of the fractional did not complicate the proofs.…”