Embeddings of fractional Sobolev spaces and estimates of Fourier transforms

V. I. Kolyada

Sbornik. Mathematics, Tome 192 (2001) no. 7, pp. 979-1000

Résumé

Fractional anisotropic Sobolev–Liouville spaces $L_p^{r_1,\dots,r_n}(\mathbb R^n)$ are investigated for $1\leqslant p\infty$ and positive $r_k$. For functions in these spaces estimates of norms in modified spaces of Lorentz and Besov kinds, defined in terms of iterative rearrangements, are established. These estimates are used to prove inequalities for the Fourier transforms of functions in $L_1^{r_1,\dots,r_n}$. This paper continues works of the author in which similar issues have been discussed for integer $r_k$. The methods used in the paper are based on estimates of iterative rearrangements. This approach enables one to simplify proofs and at the same time to obtain stronger results. In particular, the analysis of the limit case $p=1$ becomes much easier.