Hardy-Littlewood Inequality for Quasiregular Maps on Carnot Groups

A natural generalization of analytic functions to n-dimensional Euclidean space are quasiregular mappings.(See [2] and [3].) An analogue of Theorem 1.1 for quasiregular mappings in John domains in Euclidean space appeared in [4]. Recently the analytical tools used in the proof of this result have been generalized to Carnot groups. We give an account of some of these advances and obtain an analogue of Theorem 1.1 in this context. A Carnot group is a connected, simply connected, nilpotent Lie group G of topological dimG = N ≥ 2 equipped with a graded Lie algebra G = V1 ⊕ · · · ⊕ Vr so that [V1, Vi] = Vi+1 for i=1,2,...,r-1 and [V1, Vr] = 0. This defines an r-step Carnot group. As usual, elements of G will be identified with left-invariant vectors fields on G. We adopt when possible the elegant notation from [5]. We fix a left-invariant Riemannian metric g on G with g(Xi, Xj) = δij. We denote the inner product with respect to this metric, as well as all other inner products, by 〈, 〉. We assume that dimV1 = m ≥ 2 and fix an orthonormal basis of V1 : X1, X2, ..., Xm. The horizontal tangent bundle of G, HT , is the subbundle determined by V1 with horizontal tangent space HTx the fiber span[X1(x), ..., Xm(x)]. We use a fixed global coordinate system as exp : G → G is a diffeomorphism (since G is simply-connected and nilpotent). We extend X1, ..., Xm to an orthonormal basis X1, ..., Xm, T1, ..., TN−m of G. All integrals will be with respect to the bi-invariant Harr measure on G which arises as the push-forward of the Lebesque measure in R under the exponential map. We denote by |E| the measure of a measurable set E.