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- W2071567274 abstract "This paper is concerned with pointwise estimates for the gradient of the heat kernel<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$K_t$ id=E1><mml:msub><mml:mi>K</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$t>0$ id=E2><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 xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$M$ id=E3><mml:mi>M</mml:mi></mml:math>. Under standard assumptions on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$M$ id=E4><mml:mi>M</mml:mi></mml:math>, we show that<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$nabla K_t$ id=E5><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 xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$L^p$ id=E6><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup></mml:math>for some<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$1leq pleq infty$ id=E7><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 xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$nabla K_t$ id=E8><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." @default.
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- W2071567274 date "2006-01-01" @default.
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- W2071567274 title "Some remarks on gradient estimates for heat kernels" @default.
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- W2071567274 doi "https://doi.org/10.1155/aaa/2006/73020" @default.
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