Sharp Sobolev embeddings and related Hardy inequalities
The paper surveys some results on sharp Sobolev embeddings and corresponding Hardy-type inequalities in the critical case. More precisely, let \(1<p<\infty \), \(s=n/p\), and let \(A_p\) stand either for the fractional Sobolev space \(H^s_p(\mathbb R^n)\) or the Besov space \(B^s_{p,p}(\mathbb R^n)\). Suppose that \(\kappa \: (0,1] \rightarrow \mathbb R\) is positive, continuous and decreasing. The main result of the paper states that the following statements are equivalent: NEWLINENEWLINENEWLINE(i) \(\int ^1_0 (\frac {\kappa (t)f^*(t)}{1+|\log t|})^p \frac {dt}{t} \lesssim \|f|A_p\|^p \quad \text{for all} \;f\in A_p\); NEWLINENEWLINENEWLINE(ii) \(\int _{|x|<1} (\frac {\kappa (|x|)f(x)}{1+|\log |x\|})^p \frac {dx}{|x|^n} \lesssim \|f|A_p \|^p \quad \text{for all} \;f\in A_p\); NEWLINENEWLINENEWLINE(iii) \(\kappa \) is bounded.NEWLINENEWLINEFor the entire collection see [Zbl 0933.00035].
scientific article