We show that, if a certain Sobolev inequality holds, then a scale-invariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1

Contents 1 Introduction 2 2 Preliminaries 2 2.1 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The doubling property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 The Poincar'e inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 The p-Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 The non-smooth case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 p-parabolicity and p-hyperbolicity 10 3.1 An inequality for supersolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Volume growth and p-parabolicity . . . . . . . . . . . . . . . . .

We consider the regularity problem for subelliptic, sphere-valued p-harmonic maps, associated with a system of Hormander vector fields in a bounded domain of IR n . We prove that for p equal to the homogeneous dimension Q, the maps in question are locally Holder continuous. Our method of proof uses an abstract lemma, which serves as a counterpart of the duality of Hardy space and BMO (even though no Hardy spaces are available in this context) and seems to be of independent interest. Its fairly simple proof, bypassing the whole burden of proof of Fefferman's duality theorem, uses just Sobolev inequality, properties of the fundamental solution of the subelliptic Laplace operator, and an abstract version of fractional integration theorem. 1 Introduction The story begins with the paper of Muller, [59], who --- for the sake of an application to nonlinear elasticity --- proved that if the Jacobian determinant J u of a Sobolev map u 2 W 1;n loc (IR n ; IR n ) is nonnegative, then it...

Abstract. We study the interpolation property of Sobolev spaces of order 1 denoted by W 1 p,V, arising from Schrödinger operators with positive potential. We show that for 1 ≤ p1 < p < p2 < q0 with p> s0, W 1 p,V is a real interpolation space between W 1 p1,V and W 1 p2,V on some classes of manifolds and Lie groups. The constants s0, q0 depend on our hypotheses.