Harmonic Measures for Leafwise Elliptic Operators Along Foliations

Abstract

Let N be a smooth compact manifold of dimension n ≥ 1 and let L be a second order elliptic differential operator on N with continuous coefficients and without terms of order zero. It is well known that there is a unique harmonic measure for L, i.e., there is a unique Borel-probability measure η on N such that \(\int {\left( {Lf} \right)d\eta = 0} \) for every smooth function f on N (see e.g., [I-W]). Moreover η is contained in the Lebesgue measure class. Namely, if λ is any smooth measure on N, then the adjoint of L with respect to the L 2-inner product defined by λ on the space of smooth functions on N is again a second order elliptic differential operator C on N whose kernel is spanned by a positive square integrable function α on N which we may assume to be normalized such that (math). Then αλ is the unique harmonic measure for L (up to a constant).