an:06008816
Zbl 1241.26014
Haberl, Christoph; Schuster, Franz E.; Xiao, Jie
An asymmetric affine P??lya-Szeg?? principle
EN
Math. Ann. 352, No. 3, 517-542 (2012).
00295412
2012
j
26D10 46E35
P??lya-Szeg?? principle; symmetric rearrangement; asymmetric affine energy; Sobolev inequality; logarithmic Sobolev inequality; Nash inequality; Moser-Trudinger inequality; Morrey-Sobolev inequality; Gagliardo-Nirenberg inequality
The classical P??lya-Szeg?? principle states that the symmetric rearrangement \(f^{\ast}\) of a function \(f\in W^{1,p}(\mathbb R^n)\), \(p\geq1\), remains in \(W^{1,p}(\mathbb R^n)\) and, moreover, \(\|\nabla f^{\ast}\|_p\leq\|\nabla f\|_p\). In the affine P??lya-Szeg?? inequality the \(L^p\) norm of the gradient \(\|\nabla f\|_p\) is replaced by the \(L^p\) affine energy
\[
\mathcal E_p(f)=c_{n,p}\bigl(\int_{S^{n-1}}\| D_uf\|_p^{-n}du\bigr)^{-1/n},
\]
where \(D_u f\) is the directional derivative of \(f\) in the direction \(u\) and the constant \(c_{n,p}\) is such that
\[
\mathcal E_p(f^{\ast})=\|\nabla f^{\ast}\|_p
\]
for \(f\in W^{1,p}(\mathbb R^n)\). Note that, unlike \(\|\nabla f\|_p\), \(\mathcal E_p(f)\) is invariant under volume preserving affine transformations on \(\mathbb R^n\). \textit{A. Cianchi}, \textit{E. Lutwak}, \textit{D. Yang} and \textit{G. Zhang} [Calc. Var. Partial Differ. Equ. 36, No. 3, 419--436 (2009; Zbl 1202.26029)] proved the affine P??lya-Szeg?? principle \(\mathcal E_p(f^{\ast})\leq\mathcal E_p(f)\). \textit{E. Lutwak}, \textit{D. Yang} and \textit{G. Zhang} [J. Differ. Geom. 62, No. 1, 17--38 (2002; Zbl 1073.46027)] proved that
\[
\mathcal E_p(f)\leq\|\nabla f\|_p
\]
thus showing that the affine inequality is stronger than the original Euclidean one.
The authors introduce the asymmetric \(L^p\) affine energy
\[
\mathcal E_p^+(f)=2^{1/p}c_{n,p}\bigl(\int_{S^{n-1}}\| D_u^+f\|_p^{-n}du\bigr)^{-1/n},
\]
where \(D_u^+ f=\max\{D_uf,0\}\). The asymmetric affine energy \(\mathcal E_p^+(f)\) is again invariant under volume preserving affine transformations on \(\mathbb R^n\), but it differs from the (symmetric) affine energy \(\mathcal E_p(f)\) by the fact that the odd parts of the directional derivative do not vanish. The authors prove that for \(p\geq1\) and \(f\in W^{1,p}(\mathbb R^n)\) also \(f^\ast\) belongs to \(W^{1,p}(\mathbb R^n)\) and the estimate \(\mathcal E_p^+(f^\ast)\leq\mathcal E_p^+(f)\) holds. Using this result they derive new sharp asymmetric affine versions of various inequalities like the classical Sobolev and logarithmic Sobolev inequalities, the Nash inequality, the Moser-Trudinger inequality, the Morrey-Sobolev inequality, and the Gagliardo-Nirenberg inequality.
Ji????\ R??kosn??k (Praha)
Zbl 1202.26029; Zbl 1073.46027