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{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,7,8]],"date-time":"2023-07-08T14:40:44Z","timestamp":1688827244849},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"3-4","license":[{"start":{"date-parts":[[2022,8,3]],"date-time":"2022-08-03T00:00:00Z","timestamp":1659484800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,8,3]],"date-time":"2022-08-03T00:00:00Z","timestamp":1659484800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100009117","name":"Technische Universit\u00e4t Chemnitz","doi-asserted-by":"crossref"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Ann."],"published-print":{"date-parts":[[2023,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>With <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\vec {\\Delta }_j\\ge 0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mrow>\n <mml:msub>\n <mml:mover>\n <mml:mi>\u0394<\/mml:mi>\n <mml:mo>\u2192<\/mml:mo>\n <\/mml:mover>\n <mml:mi>j<\/mml:mi>\n <\/mml:msub>\n <mml:mo>\u2265<\/mml:mo>\n <mml:mn>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 <jats:italic>j<\/jats:italic>-forms on a geodesically complete Riemannian manifold <jats:italic>M<\/jats:italic> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\nabla $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mi>\u2207<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> the Levi-Civita covariant derivative, we prove among other things<jats:list list-type=\"bullet\">\n <jats:list-item>\n <jats:p>a Gaussian heat kernel bound for <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\nabla \\mathrm {e}^{ -t\\vec {\\Delta }_j }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mrow>\n <mml:mi>\u2207<\/mml:mi>\n <mml:msup>\n <mml:mrow>\n <mml:mi>e<\/mml:mi>\n <\/mml:mrow>\n <mml:mrow>\n <mml:mo>-<\/mml:mo>\n <mml:mi>t<\/mml:mi>\n <mml:msub>\n <mml:mover>\n <mml:mi>\u0394<\/mml:mi>\n <mml:mo>\u2192<\/mml:mo>\n <\/mml:mover>\n <mml:mi>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 <jats:italic>M<\/jats:italic> and its covariant derivative are bounded,<\/jats:p>\n <\/jats:list-item>\n <jats:list-item>\n <jats:p>an exponentially weighted <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^p$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:msup>\n <mml:mi>L<\/mml:mi>\n <mml:mi>p<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-bound for the heat kernel of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\nabla \\mathrm {e}^{ -t\\vec {\\Delta }_j }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mrow>\n <mml:mi>\u2207<\/mml:mi>\n <mml:msup>\n <mml:mrow>\n <mml:mi>e<\/mml:mi>\n <\/mml:mrow>\n <mml:mrow>\n <mml:mo>-<\/mml:mo>\n <mml:mi>t<\/mml:mi>\n <mml:msub>\n <mml:mover>\n <mml:mi>\u0394<\/mml:mi>\n <mml:mo>\u2192<\/mml:mo>\n <\/mml:mover>\n <mml:mi>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 <jats:italic>M<\/jats:italic> and its covariant derivative are bounded,<\/jats:p>\n <\/jats:list-item>\n <jats:list-item>\n <jats:p>that <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\nabla \\mathrm {e}^{ -t\\vec {\\Delta }_j }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mrow>\n <mml:mi>\u2207<\/mml:mi>\n <mml:msup>\n <mml:mrow>\n <mml:mi>e<\/mml:mi>\n <\/mml:mrow>\n <mml:mrow>\n <mml:mo>-<\/mml:mo>\n <mml:mi>t<\/mml:mi>\n <mml:msub>\n <mml:mover>\n <mml:mi>\u0394<\/mml:mi>\n <mml:mo>\u2192<\/mml:mo>\n <\/mml:mover>\n <mml:mi>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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^p$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:msup>\n <mml:mi>L<\/mml:mi>\n <mml:mi>p<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for all <jats:inline-formula><jats:alternatives><jats:tex-math>$$1\\le p<\\infty $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mrow>\n <mml:mn>1<\/mml:mn>\n <mml:mo>\u2264<\/mml:mo>\n <mml:mi>p<\/mml:mi>\n <mml:mo><<\/mml:mo>\n <mml:mi>\u221e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, if the curvature tensor of <jats:italic>M<\/jats:italic> and its covariant derivative are bounded,<\/jats:p>\n <\/jats:list-item>\n <jats:list-item>\n <jats:p>a second order Davies-Gaffney estimate (in terms of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\nabla $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mi>\u2207<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\vec {\\Delta }_j$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:msub>\n <mml:mover>\n <mml:mi>\u0394<\/mml:mi>\n <mml:mo>\u2192<\/mml:mo>\n <\/mml:mover>\n <mml:mi>j<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) for <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathrm {e}^{ -t\\vec {\\Delta }_j }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:msup>\n <mml:mrow>\n <mml:mi>e<\/mml:mi>\n <\/mml:mrow>\n <mml:mrow>\n <mml:mo>-<\/mml:mo>\n <mml:mi>t<\/mml:mi>\n <mml:msub>\n <mml:mover>\n <mml:mi>\u0394<\/mml:mi>\n <mml:mo>\u2192<\/mml:mo>\n <\/mml:mover>\n <mml:mi>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 <jats:italic>j<\/jats:italic>-th degree Bochner-Lichnerowicz potential <jats:inline-formula><jats:alternatives><jats:tex-math>$$V_j=\\vec {\\Delta }_j-\\nabla ^{\\dagger }\\nabla $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mrow>\n <mml:msub>\n <mml:mi>V<\/mml:mi>\n <mml:mi>j<\/mml:mi>\n <\/mml:msub>\n <mml:mo>=<\/mml:mo>\n <mml:msub>\n <mml:mover>\n <mml:mi>\u0394<\/mml:mi>\n <mml:mo>\u2192<\/mml:mo>\n <\/mml:mover>\n <mml:mi>j<\/mml:mi>\n <\/mml:msub>\n <mml:mo>-<\/mml:mo>\n <mml:msup>\n <mml:mi>\u2207<\/mml:mi>\n <mml:mo>\u2020<\/mml:mo>\n <\/mml:msup>\n <mml:mi>\u2207<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of <jats:italic>M<\/jats:italic> is bounded from below (where <jats:inline-formula><jats:alternatives><jats:tex-math>$$V_1=\\mathrm {Ric}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mrow>\n <mml:msub>\n <mml:mi>V<\/mml:mi>\n <mml:mn>1<\/mml:mn>\n <\/mml:msub>\n <mml:mo>=<\/mml:mo>\n <mml:mi>Ric<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>), which is shown to fail for large time, if <jats:inline-formula><jats:alternatives><jats:tex-math>$$V_j$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:msub>\n <mml:mi>V<\/mml:mi>\n <mml:mi>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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\nabla (\\vec {\\Delta }_j+\\kappa )^{-1\/2}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mrow>\n <mml:mi>\u2207<\/mml:mi>\n <mml:msup>\n <mml:mrow>\n <mml:mo>(<\/mml:mo>\n <mml:msub>\n <mml:mover>\n <mml:mi>\u0394<\/mml:mi>\n <mml:mo>\u2192<\/mml:mo>\n <\/mml:mover>\n <mml:mi>j<\/mml:mi>\n <\/mml:msub>\n <mml:mo>+<\/mml:mo>\n <mml:mi>\u03ba<\/mml:mi>\n <mml:mo>)<\/mml:mo>\n <\/mml:mrow>\n <mml:mrow>\n <mml:mo>-<\/mml:mo>\n <mml:mn>1<\/mml:mn>\n <mml:mo>\/<\/mml:mo>\n <mml:mn>2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^p$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:msup>\n <mml:mi>L<\/mml:mi>\n <mml:mi>p<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for all <jats:inline-formula><jats:alternatives><jats:tex-math>$$1\\le p<\\infty $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mrow>\n <mml:mn>1<\/mml:mn>\n <mml:mo>\u2264<\/mml:mo>\n <mml:mi>p<\/mml:mi>\n <mml:mo><<\/mml:mo>\n <mml:mi>\u221e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (which we prove for <jats:inline-formula><jats:alternatives><jats:tex-math>$$1\\le p\\le 2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mrow>\n <mml:mn>1<\/mml:mn>\n <mml:mo>\u2264<\/mml:mo>\n <mml:mi>p<\/mml:mi>\n <mml:mo>\u2264<\/mml:mo>\n <mml:mn>2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>), and explain its implications to geometric analysis, such as the <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^p$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:msup>\n <mml:mi>L<\/mml:mi>\n <mml:mi>p<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-Calder\u00f3n-Zygmund inequality. Our main technical tool is a Bismut derivative formula for <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\nabla \\mathrm {e}^{ -t\\vec {\\Delta }_j }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n <mml:mrow>\n <mml:mi>\u2207<\/mml:mi>\n <mml:msup>\n <mml:mrow>\n <mml:mi>e<\/mml:mi>\n <\/mml:mrow>\n <mml:mrow>\n <mml:mo>-<\/mml:mo>\n <mml:mi>t<\/mml:mi>\n <mml:msub>\n <mml:mover>\n <mml:mi>\u0394<\/mml:mi>\n <mml:mo>\u2192<\/mml:mo>\n <\/mml:mover>\n <mml:mi>j<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00208-022-02409-5","type":"journal-article","created":{"date-parts":[[2022,8,3]],"date-time":"2022-08-03T19:07:46Z","timestamp":1659553666000},"page":"1753-1798","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Estimates for the covariant derivative of the heat semigroup on differential forms, and covariant Riesz transforms"],"prefix":"10.1007","volume":"386","author":[{"given":"Robert","family":"Baumgarth","sequence":"first","affiliation":[]},{"given":"Baptiste","family":"Devyver","sequence":"additional","affiliation":[]},{"given":"Batu","family":"G\u00fcneysu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,8,3]]},"reference":[{"key":"2409_CR1","doi-asserted-by":"crossref","unstructured":"Auscher, P., Coulhon, T., Duong, X. 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