Second-Order Subelliptic Operators on Lie Groups II: Real Measurable Principal Coefficients

Abstract

Let G be a connected Lie group with Lie algebra \(\mathfrak{g}\) and a 1,…,a d’ an algebraic basis of \(\mathfrak{g}\). Further let A i denote the generators of left translations,acting on the L p -spaces L p (G; dg)formed with left Haar measure dg,in the directions a i . We consider second-order operators $$H = - \sum\limits_{{i,j = 1}}^{{d\prime }} {{{A}_{i}}{{c}_{{ij}}}{{A}_{j}} + \sum\limits_{{i = 1}}^{{d\prime }} {({{c}_{i}}{{A}_{i}} + {{A}_{i}}c_{i}^{\prime }) + {{c}_{0}}I} }$$ corresponding to a quadratic form with real measurable coeffcients c ij and complex c i , c i ′ , c 0 є L ∞. The matrix C = (c ij ) of principal coefficients, which is not necessarily symmetric, is assumed to satisfy the subellipticity condition $$\Re C = {{2}^{{ - 1}}}\left( {C + {{C}^{*}}} \right) \geqslant \mu I > 0$$ uniformly over G.We prove that H generates a strongly continuous holomorphic semigroup S on L 2 with a kernel K which satisfies Gaussian bounds $$|{{K}_{z}}(g;h)| \leqslant a|z{{|}^{{ - D\prime /2}}}{{e}^{{\omega |z|}}}{{e}^{{ - b{{{(|g{{h}^{{ - 1}}}|\prime )}}^{2}}|z{{|}^{{ - 1}}}}}}$$ for g, h є G and z in a subsector Λ(θ) of the sector of holomorphy. Moreover, the kernel is Hölder continuous and there is a ν є(0, 1) such that for all k > 0 one has estimates $$\begin{array}{*{20}{c}} {|{{K}_{z}}({{k}^{{ - 1}}}g;{{l}^{{ - 1}}}h) - {{K}_{z}}(g;h)|} \hfill \\ { \leqslant a|z{{|}^{{ - D\prime /2}}}{{e}^{{\omega |z|}}}{{{\left( {\frac{{|k|\prime + |l|\prime }}{{|z{{|}^{{1/2}}} + |g{{h}^{{ - 1}}}|\prime }}} \right)}}^{v}}{{e}^{{ - b{{{(|g{{h}^{{ - 1}}}|\prime )}}^{2}}|z{{|}^{{ - 1}}}}}}} \hfill \\ \end{array}$$ for g, h, k, l є G and z in the subsector with \(|k|\prime + |l|\prime \leqslant \kappa |z{{|}^{{1/2}}} + {{2}^{{ - 1}}}|g{{h}^{{ - 1}}}|\prime\) In addition, if all the coefficients of H are real-valued then $${{K}_{t}}(g;h) \geqslant a\prime {{t}^{{ - D\prime /2}}}{{e}^{{ - \omega \prime t}}}{{e}^{{ - b\prime {{{(|g{{h}^{{ - 1}}}|\prime )}}^{2}}{{t}^{{ - 1}}}}}}$$ for some a ’, a ’ > 0 and ω ’ ≥ 0 uniformly for g,h є G and t > 0 uniformly overG KeywordsHeat KernelElliptic OperatorRelative EntropyPoincare InequalityAlgebraic BasisThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.