Compensated Compactness and the Heisenberg Group

Jacobians of maps on the Heisenberg group are shown to map suitable group Sobolev spaces into the group Hardy space H1. From this result and a weak∗ convergence theorem for the Hardy space H1 of the Heisenberg group, a compensated compactness property for these Jacobians is obtained. 0. Introduction We investigate compensated compactness properties of Jacobians of maps on the Heisenberg group and we prove results analogous to those for the Jacobians of maps on R. Let H = C × R be the Lie group with multiplicative structure (z, t) · (z′, t′) = (z + z′, t+ t′ + 2 Im z · z̄′) where z = (z1, . . . , zn), z′ = (z′ 1, . . . , z′ n) and z · z̄′ = ∑n j=1 zj z̄ ′ j . This is the n order Heisenberg group. It can be shown that the group operation is C∞ on the manifold C × R and hence H is a locally compact Lie group. Let (z1, . . . , zn, t) be coordinates on H. Write zj = xj + iyj and define the vector fields: Xj = ∂ ∂xj + 2yj ∂ ∂t Yj = ∂ ∂yj − 2xj ∂ ∂t T = ∂ ∂t . It turns out that the Xj , Yj and T form a basis for the left invariant vector fields on H . They satisfy the commutation relations [Xj , Yj ] = −4T, j = 1, . . . , n and all other ∗Work partially supported by the National Science Foundation. ∗∗Work partially supported by NSF grant DMS-9007491. Mathematics Subject Classification (1991): 42 Typeset by AMS-TEX 1 commutators vanish. The group H is equipped with a natural dilation structure r(z, t) = (rz, rt), r > 0, which is consistent with the group multiplication. The associated norm |(z, t)| = (|z|4 + t) is homogeneous of degree 1 with respect to this group of dilations. We denote by Br(u0) the Heisenberg group ball {u ∈ H : |u−1u0| < r}. Haar measure on H is the usual Lebesgue measure on R. Convolution on the Heisenberg group H is defined by (f ∗ g)(u) = ∫ f(v)g(v−1u) dv = ∫ f(uv−1)g(v) dv. Fix φ be a smooth bump on H with ∫ φ du = 0. For a distribution f on H define f(u) = sup δ>0 |(f ∗ φδ)(u)| where φδ(u) = δ−2n−2φ(δ−1u). If γ > 2n+2 2n+3 , the Hardy space H (H) = H is the set of all f such that f ∈ L(H). An alternative definition of H(H) can be given via the atomic decomposition. See [FOS] or [CW] for details. Given a C map F = (f1, . . . , fn) from H into R, define (0.3) Jac(F ) = det   L1f1 L1f2 . . . L1fn L2f1 L2f2 . . . L2fn .. .. .. Lnf1 Lnf2 . . . Lnfn   , where Lj is either Xj or Yj . We would like to show that Jac(F ) maps a product of suitable group Sobolev spaces into the spaces H(H) for 1 ≥ γ > 2n+2 2n+3 . The analogous result for the vector fields ∂ ∂xj , 1 ≤ j ≤ n on R n has been proved by P. L. Lions and Y. Meyer when γ = 1 and extended by [CLMS] for 1 ≥ γ > n n+1 . Note that the lower bound for γ in both cases is d d+1 , where d is the homogeneous dimension of the group. Our first result is Theorem 1. Let n ≥ 2 and for 1 ≤ j ≤ n, let 1 < pj < 2n+2. Let γ = ( 1 p1 + · · ·+ 1 pn )−1 be the harmonic mean of the pj’s. We suppose that 2n+2 2n+3 < γ ≤ 1. Then, there exists 2 a constant C > 0 that depends only on n and on the pj’s such that for every map F = (f1, . . . fn) on H we have: (0.4) ‖ Jac(F )‖Hγ ≤ C n ∏