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**Sharp affine \(L_ p\) Sobolev inequalities.**
*(English)*
Zbl 1073.46027

The authors establish a sharp affine \(L_p\) Sobolev inequality for functions on Euclidean \(n\)-space. The case \(p=1\) was formulated and proved by G.–Y. Zhang in [J. Differ. Geom. 53, No. 1, 183–202 (1999; Zbl 1040.53089)], and he established its equivalence to an \(L_1\) affine isoperimetric inequality that was also proved in the same paper.

In this paper, the authors solve the problem for all cases \(p>1\). An important qualitative step is needed, because both the geometric inequality and the critical tools used to prove the affine \(L_1\) Sobolev inequality are not strong enough to obtain the affine inequality for \(p>1\). A new inequality and new tools are needed. The affine isoperimetric inequality had been established by the present authors in [J. Differ. Geom. 56, No. 1, 111–132 (2000; Zbl 1034.52009)] (S. Campi and P. Gronchi provided an alternative approach in [Adv. Math. 167, No.1, 128-141 (2002; Zbl 1002.52005)]). Also, the solution of an \(L_p\) extension of the classical Minkowski problem is required. In order to apply this extension, it is necessary to define for functions (rather than for bodies) the notions of \(L_p\) mixed volumes; this definition is given by means of the level sets determined by the functions and then applying Sard’s theorem to prove that the boundary of the level set is a \(C^1\) submanifold with everywhere nonzero normal vector \(\nabla f\).

While the geometric core of the classical \(L_p\) Sobolev inequality is the same for all \(p\), the geometric inequality behind the new affine \(L_p\) Sobolev inequality is different for different \(p\).

Let \(H^{1,p}(\mathbb{R}^n)\) denote the usual Sobolev space of real-valued functions of \(\mathbb{R}^n\) with \(L_p\) partial derivatives. The authors associate with each function \(f\in H^{1,p}(\mathbb{R}^n)\) a Banach space; this association is affine in nature. The volume of the unit ball of this Banach space can be bounded from above by the reciprocal of the ordinary \(L_q\)-norm of \(f\) where \(\frac 1q= \frac 1p - \frac 1n\).

The inequalities obtained in this way are invariant under affine transformations while the classical \(L_p\) Sobolev inequality obtained by Aubin and Talenti is invariant only under rigid motions.

Furthermore, the affine \(L_p\) Sobolev inequality is stronger than the classical \(L_p\) Sobolev inequality and the classical inequality can be deduced from the affine inequality using Hölder inequality.

In the particular case \(p=2\), both classical and affine Sobolev inequalities are equivalent since the \(L_2\) Banach norm is Euclidean.

The authors also present an application of the affine \(L_p\) Sobolev inequality to information theory and promise further applications in a forthcoming paper.

In this paper, the authors solve the problem for all cases \(p>1\). An important qualitative step is needed, because both the geometric inequality and the critical tools used to prove the affine \(L_1\) Sobolev inequality are not strong enough to obtain the affine inequality for \(p>1\). A new inequality and new tools are needed. The affine isoperimetric inequality had been established by the present authors in [J. Differ. Geom. 56, No. 1, 111–132 (2000; Zbl 1034.52009)] (S. Campi and P. Gronchi provided an alternative approach in [Adv. Math. 167, No.1, 128-141 (2002; Zbl 1002.52005)]). Also, the solution of an \(L_p\) extension of the classical Minkowski problem is required. In order to apply this extension, it is necessary to define for functions (rather than for bodies) the notions of \(L_p\) mixed volumes; this definition is given by means of the level sets determined by the functions and then applying Sard’s theorem to prove that the boundary of the level set is a \(C^1\) submanifold with everywhere nonzero normal vector \(\nabla f\).

While the geometric core of the classical \(L_p\) Sobolev inequality is the same for all \(p\), the geometric inequality behind the new affine \(L_p\) Sobolev inequality is different for different \(p\).

Let \(H^{1,p}(\mathbb{R}^n)\) denote the usual Sobolev space of real-valued functions of \(\mathbb{R}^n\) with \(L_p\) partial derivatives. The authors associate with each function \(f\in H^{1,p}(\mathbb{R}^n)\) a Banach space; this association is affine in nature. The volume of the unit ball of this Banach space can be bounded from above by the reciprocal of the ordinary \(L_q\)-norm of \(f\) where \(\frac 1q= \frac 1p - \frac 1n\).

The inequalities obtained in this way are invariant under affine transformations while the classical \(L_p\) Sobolev inequality obtained by Aubin and Talenti is invariant only under rigid motions.

Furthermore, the affine \(L_p\) Sobolev inequality is stronger than the classical \(L_p\) Sobolev inequality and the classical inequality can be deduced from the affine inequality using Hölder inequality.

In the particular case \(p=2\), both classical and affine Sobolev inequalities are equivalent since the \(L_2\) Banach norm is Euclidean.

The authors also present an application of the affine \(L_p\) Sobolev inequality to information theory and promise further applications in a forthcoming paper.

Reviewer: Salvador Gomis (Alicante)

### MSC:

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

26D10 | Inequalities involving derivatives and differential and integral operators |

35J20 | Variational methods for second-order elliptic equations |

52A40 | Inequalities and extremum problems involving convexity in convex geometry |

52A39 | Mixed volumes and related topics in convex geometry |

52A21 | Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry) |