Some remarks on gradient estimates for heat kernels
<p>This paper is concerned with pointwise estimates for the gradient of the heat kernel <mml:math alttext="$K_t$"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math>, <mml:math alttext="$t>0$&q...
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| Format: | Article |
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| Language: | English |
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Wiley
2006-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://www.hindawi.com/GetArticle.aspx?doi=10.1155/AAA/2006/73020 |
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| description | <p>This paper is concerned with pointwise estimates for the gradient of the heat kernel <mml:math alttext="$K_t$"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math>, <mml:math alttext="$t>0$"> <mml:mi>t</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn> </mml:math>, of the Laplace operator on a Riemannian manifold <mml:math alttext="$M$"> <mml:mi>M</mml:mi> </mml:math>. Under standard assumptions on <mml:math alttext="$M$"> <mml:mi>M</mml:mi> </mml:math>, we show that <mml:math alttext="$ abla K_t$"> <mml:mo>∇</mml:mo><mml:msub> <mml:mi>K</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> satisfies Gaussian bounds if and only if it satisfies certain uniform estimates or estimates in <mml:math alttext="$L^p$"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> for some <mml:math alttext="$1leq pleq infty$"> <mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>≤</mml:mo><mml:mi>∞</mml:mi> </mml:math>. The proof is based on finite speed propagation for the wave equation, and extends to a more general setting. We also prove that Gaussian bounds on <mml:math alttext="$ abla K_t$"> <mml:mo>∇</mml:mo><mml:msub> <mml:mi>K</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> are stable under surjective, submersive mappings between manifolds which preserve the Laplacians. As applications, we obtain gradient estimates on covering manifolds and on homogeneous spaces of Lie groups of polynomial growth and boundedness of Riesz transform operators.</p> |
| format | Article |
| id | doaj-art-6d0288d6abf747c69d03a67805f7b3c1 |
| institution | OA Journals |
| issn | 1085-3375 |
| language | English |
| publishDate | 2006-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-6d0288d6abf747c69d03a67805f7b3c12025-08-20T02:24:03ZengWileyAbstract and Applied Analysis1085-33752006-01-012006Some remarks on gradient estimates for heat kernels<p>This paper is concerned with pointwise estimates for the gradient of the heat kernel <mml:math alttext="$K_t$"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math>, <mml:math alttext="$t>0$"> <mml:mi>t</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn> </mml:math>, of the Laplace operator on a Riemannian manifold <mml:math alttext="$M$"> <mml:mi>M</mml:mi> </mml:math>. Under standard assumptions on <mml:math alttext="$M$"> <mml:mi>M</mml:mi> </mml:math>, we show that <mml:math alttext="$ abla K_t$"> <mml:mo>∇</mml:mo><mml:msub> <mml:mi>K</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> satisfies Gaussian bounds if and only if it satisfies certain uniform estimates or estimates in <mml:math alttext="$L^p$"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> for some <mml:math alttext="$1leq pleq infty$"> <mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>≤</mml:mo><mml:mi>∞</mml:mi> </mml:math>. The proof is based on finite speed propagation for the wave equation, and extends to a more general setting. We also prove that Gaussian bounds on <mml:math alttext="$ abla K_t$"> <mml:mo>∇</mml:mo><mml:msub> <mml:mi>K</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> are stable under surjective, submersive mappings between manifolds which preserve the Laplacians. As applications, we obtain gradient estimates on covering manifolds and on homogeneous spaces of Lie groups of polynomial growth and boundedness of Riesz transform operators.</p>http://www.hindawi.com/GetArticle.aspx?doi=10.1155/AAA/2006/73020 |
| spellingShingle | Some remarks on gradient estimates for heat kernels Abstract and Applied Analysis |
| title | Some remarks on gradient estimates for heat kernels |
| title_full | Some remarks on gradient estimates for heat kernels |
| title_fullStr | Some remarks on gradient estimates for heat kernels |
| title_full_unstemmed | Some remarks on gradient estimates for heat kernels |
| title_short | Some remarks on gradient estimates for heat kernels |
| title_sort | some remarks on gradient estimates for heat kernels |
| url | http://www.hindawi.com/GetArticle.aspx?doi=10.1155/AAA/2006/73020 |