The Caffarelli–Kohn–Nirenberg inequalities for radial functions

We establish the full range of the Caffarelli–Kohn–Nirenberg inequalities for radial functions in the Sobolev and the fractional Sobolev spaces of order $0 < s \le 1$. In particular, we show that the range of the parameters for radial functions is strictly larger than the one without symmetric assumption. Previous known results reveal only some special ranges of parameters even in the case $s=1$. The known proofs used the Riesz potential and inequalities for fractional integrations. Our proof is new, elementary, and is based on one-dimensional case. Applications on compact embeddings are also mentioned.