SHARP AFFINE Lp SOBOLEV INEQUALITIES

In this paper we prove a sharp affine Lp Sobolev inequality for functions on R. The new inequality is significantly stronger than (and directly implies) the classical sharp Lp Sobolev inequality of Aubin [A2] and Talenti [T], even though it uses only the vector space structure and standard Lebesgue measure on R. For the new inequality, no inner product, norm, or conformal structure is needed at all. In other words, the inequality is invariant under all affine transformations of R. That such an inequality exists is surprising because the classical sharp Lp Sobolev inequality relies strongly on the Euclidean geometric structure of R, especially on the isoperimetric inequality. Zhang [Z] formulated and proved the sharp affine L1 Sobolev inequality and established its equivalence to an L1 affine isoperimetric inequality that is also proved in [Z]. He also showed that the affine L1 Sobolev inequality is stronger than the classical L1 Sobolev inequality. The L1 Sobolev inequality is known to be equivalent to the isoperimetric inequality (see, for example, [F], [FF], [M], [BZ], [O], and [SY]). The geometry behind the sharp Lp Sobolev inequality is also the isoperimetric inequality. For the affine Sobolev inequalities the situation is quite different. The geometric inequality and the critical tools used to establish the affine L1 Sobolev inequality are not strong enough to enable us to establish the affine Lp Sobolev inequality for p > 1. A new geometric inequality and new tools are needed. The inequality needed is an affine Lp affine isoperimetric inequality recently established by the authors in [LYZ1] (see Campi and Gronchi [CG] for a recent alternate approach). We will also need the solution of an Lp extension of the classical Minkowski problem obtained in [L2]. It is crucial to observe that while the geometric core of the classical Lp Sobolev inequality (i.e., the isoperimetric inequality) is the same for all p, the geometric inequality (i.e., the affine Lp isoperimetric inequality) behind the new affine Lp Sobolev inequality is different for different p. Let R denote n–dimensional Euclidean space; throughout we will assume that n ≥ 2. Let H(R) denote the usual Sobolev space of real-valued functions of R with Lp partial derivatives.