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An asymmetric affine Pólya-Szegő principle. (English) Zbl 1241.26014
The classical Pólya-Szegő 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 Pólya-Szegő 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 \(\mathbb R^n\). A. Cianchi, E. Lutwak, D. Yang and G. Zhang [Calc. Var. Partial Differ. Equ. 36, No. 3, 419–436 (2009; Zbl 1202.26029)] proved the affine Pólya-Szegő principle \(\mathcal E_p(f^{\ast})\leq\mathcal E_p(f)\). E. Lutwak, D. Yang and 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.

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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