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).
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
