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Ann."],"published-print":{"date-parts":[[2023,8]]},"abstract":"Abstract<\/jats:title>With $$\\vec {\\Delta }_j\\ge 0$$<\/jats:tex-math>\n \n \n \n \u0394<\/mml:mi>\n \u2192<\/mml:mo>\n <\/mml:mover>\n j<\/mml:mi>\n <\/mml:msub>\n \u2265<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the uniquely determined self-adjoint realization of the Laplace operator acting on j<\/jats:italic>-forms on a geodesically complete Riemannian manifold M<\/jats:italic> and $$\\nabla $$<\/jats:tex-math>\n \u2207<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> the Levi-Civita covariant derivative, we prove among other things\n \n a Gaussian heat kernel bound for $$\\nabla \\mathrm {e}^{ -t\\vec {\\Delta }_j }$$<\/jats:tex-math>\n \n \u2207<\/mml:mi>\n \n \n e<\/mml:mi>\n <\/mml:mrow>\n \n -<\/mml:mo>\n t<\/mml:mi>\n \n \n \u0394<\/mml:mi>\n \u2192<\/mml:mo>\n <\/mml:mover>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, if the curvature tensor of M<\/jats:italic> and its covariant derivative are bounded,<\/jats:p>\n <\/jats:list-item>\n \n an exponentially weighted $$L^p$$<\/jats:tex-math>\n \n L<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-bound for the heat kernel of $$\\nabla \\mathrm {e}^{ -t\\vec {\\Delta }_j }$$<\/jats:tex-math>\n \n \u2207<\/mml:mi>\n \n \n e<\/mml:mi>\n <\/mml:mrow>\n \n -<\/mml:mo>\n t<\/mml:mi>\n \n \n \u0394<\/mml:mi>\n \u2192<\/mml:mo>\n <\/mml:mover>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, if the curvature tensor of M<\/jats:italic> and its covariant derivative are bounded,<\/jats:p>\n <\/jats:list-item>\n \n that $$\\nabla \\mathrm {e}^{ -t\\vec {\\Delta }_j }$$<\/jats:tex-math>\n \n \u2207<\/mml:mi>\n \n \n e<\/mml:mi>\n <\/mml:mrow>\n \n -<\/mml:mo>\n t<\/mml:mi>\n \n \n \u0394<\/mml:mi>\n \u2192<\/mml:mo>\n <\/mml:mover>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is bounded in $$L^p$$<\/jats:tex-math>\n \n L<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for all $$1\\le p<\\infty $$<\/jats:tex-math>\n \n 1<\/mml:mn>\n \u2264<\/mml:mo>\n p<\/mml:mi>\n <<\/mml:mo>\n \u221e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, if the curvature tensor of M<\/jats:italic> and its covariant derivative are bounded,<\/jats:p>\n <\/jats:list-item>\n \n a second order Davies-Gaffney estimate (in terms of $$\\nabla $$<\/jats:tex-math>\n \u2207<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\vec {\\Delta }_j$$<\/jats:tex-math>\n \n \n \u0394<\/mml:mi>\n \u2192<\/mml:mo>\n <\/mml:mover>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) for $$\\mathrm {e}^{ -t\\vec {\\Delta }_j }$$<\/jats:tex-math>\n \n \n e<\/mml:mi>\n <\/mml:mrow>\n \n -<\/mml:mo>\n t<\/mml:mi>\n \n \n \u0394<\/mml:mi>\n \u2192<\/mml:mo>\n <\/mml:mover>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for small times, if the j<\/jats:italic>-th degree Bochner-Lichnerowicz potential $$V_j=\\vec {\\Delta }_j-\\nabla ^{\\dagger }\\nabla $$<\/jats:tex-math>\n \n \n V<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n \n \n \u0394<\/mml:mi>\n \u2192<\/mml:mo>\n <\/mml:mover>\n j<\/mml:mi>\n <\/mml:msub>\n -<\/mml:mo>\n \n \u2207<\/mml:mi>\n \u2020<\/mml:mo>\n <\/mml:msup>\n \u2207<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of M<\/jats:italic> is bounded from below (where $$V_1=\\mathrm {Ric}$$<\/jats:tex-math>\n \n \n V<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n Ric<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>), which is shown to fail for large time, if $$V_j$$<\/jats:tex-math>\n \n V<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is bounded.<\/jats:p>\n <\/jats:list-item>\n <\/jats:list> Based on these results, we formulate a conjecture on the boundedness of the covariant local Riesz-transform $$\\nabla (\\vec {\\Delta }_j+\\kappa )^{-1\/2}$$<\/jats:tex-math>\n \n \u2207<\/mml:mi>\n \n \n (<\/mml:mo>\n \n \n \u0394<\/mml:mi>\n \u2192<\/mml:mo>\n <\/mml:mover>\n j<\/mml:mi>\n <\/mml:msub>\n +<\/mml:mo>\n \u03ba<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n -<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in $$L^p$$<\/jats:tex-math>\n \n L<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for all $$1\\le p<\\infty $$<\/jats:tex-math>\n \n 1<\/mml:mn>\n \u2264<\/mml:mo>\n p<\/mml:mi>\n <<\/mml:mo>\n \u221e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (which we prove for $$1\\le p\\le 2$$<\/jats:tex-math>\n \n 1<\/mml:mn>\n \u2264<\/mml:mo>\n p<\/mml:mi>\n \u2264<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>), and explain its implications to geometric analysis, such as the $$L^p$$<\/jats:tex-math>\n \n L<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-Calder\u00f3n-Zygmund inequality. 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