H Boundedness of Determinants of Vector Fields

We consider multilinear operators T (f1, . . . , fl) given by determinants of matrices of the form (Xkfj)1≤j,k≤l, where the Xk’s are C∞ vector fields on Rn. We give conditions on the Xk’s so that the corresponding operator T map products of Lebesgue spaces L p1×· · ·×Lpl into some anisotropic space H1, when 1 p1 + · · · + 1 pl = 1. 0. Introduction and statement of results. A well known Theorem of Coifman, Lions, Meyer, and Semmes [CLMS] states that the Jacobian J(F ) of a map F = (f1, . . . , fn) from R into itself maps the product of Sobolev spaces L1 1 × · · · × Ln 1 into the Hardy space H, when 1 p1 + · · · + 1 pn = 1. J(F ) is given by the determinant of the matrix ( ∂ ∂xk fj)1≤j,k≤n, where { ∂ ∂xk }1≤k≤n is the usual basis of the tangent space of R at every point. Replacing the standard basis { ∂ ∂xk } by general vector fields {Xk}, we form the multilinear operator T ({fj}) = det(Xkfj). We consider the following question: Under what conditions on the Xk’s do we have that T maps products of Lebesgue spaces into some Hardy space H as before? The purpose of this paper is to give a satisfactory answer to the question posed above. If the Xk are taken from the usual basis of the Heisenberg group in R, Rochberg and the author [GR] prove that the corresponding T maps into the group space H. In this work we show that if the vector fields Xk satisfy Hörmander’s condition, then the corresponding T maps suitable products of Lebesgue spaces into the local anisotropic Hardy space H with respect to the metric associated with the vector fields defined by Nagel, Stein, and Wainger [NSW]. Precise statements of results are given in Theorems A and B. Research partially supported by the NSF and the University of Missouri Research Board 1991 Mathematics Classification. Primary 42B30 Typeset by AMS-TEX 1 Suppose that S = {Y1, Y2, . . . , Yl} is a set of smooth vector fields defined on a bounded open connected subset Ω of R for some n ≥ 2. Assume that S is a Hörmander system. This means that there exists an integer s such that the vector fields Y1, . . . , Yl together with their commutators of order at most s span the tangent space of Ω at every point x. [NSW] define a (quasi)metric ρ on Ω by setting ρ(x, y) = inf{t : ∃ piecewise smooth curve γ : [0, t] → R, γ(0) = x, γ(t) = y and γ′(s) = ∑l j=1 βj(s)Yj(γ(s)) with ∑l j=1 ∣∣βj(s)∣∣2 ≤ 1 for all s ∈ [0, t]}, for all x, y in Ω. Intuitively, ρ(x, y) is the least time taken to move from x to y along a path pointing in the directions of the Yj ’s. For x ∈ Ω and δ > 0, let B(x, δ) = {y ∈ Ω : ρ(x, y) < δ} be the ball centered at x with radius δ with respect to the metric ρ. [NSW] prove that these balls satisfy a doubling property for Lebesgue measure. More precisely, they prove that for any compact subset K of Ω, there exist positive constants CK and δ1(K) such that for all 0 < δ < δ1(K) and all x in K (0.1) |B(x, 2δ)| ≤ CK |B(x, δ)|, where | · | denotes Lebesgue measure. As a corollary of (0.1), the local Hardy-Littlewood maximal function (Mf)(x) = (Mδ1(K)f)(x) = sup 0<δ<δ1(K) |B(x, δ)|−1 ∫ B(x,δ) |f(y)| dy maps L(Ω) to L(K) for any K compact subset of Ω and 1 < p < ∞. We now define the space H loc(Ω). Fix a smooth bump φ in the unit ball of R n and let φδ(y) = δ−nφδ(δ−1y). For any x0 in Ω and δ > 0 small enough, the push-forward of φδ by any of the coordinate maps constructed in [NSW] gives a smooth bump ψ0 δ supported in the ball B(x0, δ). One can check that for any compact subset K of Ω and for all j = 1, . . . , l and x ∈ K (0.2) |ψ0 δ (x)| ≤ CK |B(x0, δ)|−1 and |Yj(ψ δ )(x)| ≤ CKδ|B(x0, δ)|−1 when 0 < δ < δ2(K), where δ2(K) is a small constant depending on K. For a function f on Ω and δ > 0, let (0.3) (Mδf)(x0) = sup 0<σ<δ ∣∣∣ ∫ f(z)ψ0 σ (z) dz ∣∣∣. We call Mδ the “smooth” maximal function of f . We say that f lies in H loc(Ω) if for all compact subsets K of Ω, there exists a δ0(K) > 0 such that Mδ0(K)f is in L(K). We define the Hardy-1 space norm of f on K by setting (0.4) ‖f‖H1(K) = ‖Mδ0(K)f‖L1(K). 2 For a C map F = (f1, . . . , fl) : Ω → R, define (0.5) Jac(F ) = Jac{Y1,...,Yl}(F ) = det   Y1f1 Y1f2 . . . Y1fl Y2f1 Y2f2 . . . Y2fl .. .. .. Ylf1 Ylf2 . . . Ylfl   . Our results are: Theorem A. For 1 ≤ j ≤ l, let 1 < pj < ∞ be given with 1 p1 + · · · + 1 pl = 1. Suppose that Jac{Y1,...,Yl}(F ) has integral zero for all C ∞ compactly supported functions F on Ω. Then for any compact subset K of Ω, there exists a constant CK > 0 such that for all C functions F = (f1, . . . , fl) : Ω → R, we have: (0.6) ‖Jac{Y1,...,Yl}(F )‖H1(K) ≤ CK l ∏