Nonlinear ground state representations and sharp Hardy inequalities. (English) Zbl 1189.26031
The authors determine the optimal constant \(\mathcal C_{N,s,p}\) in the fractional Hardy inequality
\[ \iint_{\mathbb R^N\times\mathbb R^N} \frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dx\,dy \geq{\mathcal C}_{N,s,p} \int_{\mathbb R^N} \frac{|u(x)|^p}{|x|^{ps}}\, dx \tag{1} \]
(\(N\in\mathbb N\), \(0<s<1\)), where \(u\) belongs to the homogeneous Sobolev space \(\dot{W}^s_p(\mathbb R^n)\) in the case \(1\leq p\leq N/s\), or it belongs to \(\dot{W}^s_p(\mathbb R^N\setminus\{0\})\) when \(p>N/s\) (these spaces are defined as completions of \(C^\infty_0(\mathbb R^N)\) or \(C^\infty_0(\mathbb R^N\setminus\{0\})\), with regards to the left-hand side of (1)), respectively. To do so, they develop a nonlinear and non-local version of the ground state representation which even yields a reminder term in inequality (1).
The result enables to extend results from recent papers of J. Bourgain, H. Brézis and P. Mironescu [“Another look at Sobolev spaces”, Optimal control and partial differential equations. In honour of Professor Alain Bensoussan’s 60th birthday. Proceedings of the conference, Paris, France, December 4, 2000. Amsterdam: IOS Press; Tokyo: Ohmsha. 439–455 (2001; Zbl 1103.46310); J. Anal. Math. 87, 77–101 (2002; Zbl 1029.46030)] and of V. Maz’ya and T. Shaposhnikova [J. Math. Pures Appl. (9) 81, No. 9, 877–884 (2002; Zbl 1036.46026)].
Furthermore, from the sharp Hardy inequality the authors deduce the sharp constant in the Sobolev-type embedding of \(\dot{W}^s_p(\mathbb R^n)\) into the Lorentz space \(L_{p^*,p}(\mathbb R^n)\), \(1\leq p<N/s\), where \(p^*:=Np/(N-ps)\), and they characterize all optimizers.
As an appendix, the cases of equality in the rearrangement inequality in fractional Sobolev spaces become characterized.
\[ \iint_{\mathbb R^N\times\mathbb R^N} \frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dx\,dy \geq{\mathcal C}_{N,s,p} \int_{\mathbb R^N} \frac{|u(x)|^p}{|x|^{ps}}\, dx \tag{1} \]
(\(N\in\mathbb N\), \(0<s<1\)), where \(u\) belongs to the homogeneous Sobolev space \(\dot{W}^s_p(\mathbb R^n)\) in the case \(1\leq p\leq N/s\), or it belongs to \(\dot{W}^s_p(\mathbb R^N\setminus\{0\})\) when \(p>N/s\) (these spaces are defined as completions of \(C^\infty_0(\mathbb R^N)\) or \(C^\infty_0(\mathbb R^N\setminus\{0\})\), with regards to the left-hand side of (1)), respectively. To do so, they develop a nonlinear and non-local version of the ground state representation which even yields a reminder term in inequality (1).
The result enables to extend results from recent papers of J. Bourgain, H. Brézis and P. Mironescu [“Another look at Sobolev spaces”, Optimal control and partial differential equations. In honour of Professor Alain Bensoussan’s 60th birthday. Proceedings of the conference, Paris, France, December 4, 2000. Amsterdam: IOS Press; Tokyo: Ohmsha. 439–455 (2001; Zbl 1103.46310); J. Anal. Math. 87, 77–101 (2002; Zbl 1029.46030)] and of V. Maz’ya and T. Shaposhnikova [J. Math. Pures Appl. (9) 81, No. 9, 877–884 (2002; Zbl 1036.46026)].
Furthermore, from the sharp Hardy inequality the authors deduce the sharp constant in the Sobolev-type embedding of \(\dot{W}^s_p(\mathbb R^n)\) into the Lorentz space \(L_{p^*,p}(\mathbb R^n)\), \(1\leq p<N/s\), where \(p^*:=Np/(N-ps)\), and they characterize all optimizers.
As an appendix, the cases of equality in the rearrangement inequality in fractional Sobolev spaces become characterized.
Reviewer: Petr Gurka (Praha)
MSC:
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
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