We consider a complete non-compact Riemannian manifold satisfying the volume doubling property and a Gaussian upper bound for its heat kernel (on functions). Let \(\overrightarrow{\Delta }_k \) be the Hodge–de Rham Laplacian on differential k-forms with \(k \ge 1\). By the Bochner decomposition formula, \(\overrightarrow{\Delta }_k = \nabla ^* \nabla + R_k\), where \(\nabla \) denotes the Levi-Civita connection and \(R_k\) is a symmetric section of \(\mathrm{End}(\Lambda ^kT^*M)\). Under the assumption that the negative part \(R_k^-\) is in an enlarged Kato class, we prove that for all \(p \in [1, \infty ]\), \(\Vert e^{-t\overrightarrow{\Delta }_k}\Vert _{p-p} \le C ( t \log t)^{\frac{D}{4}(1- \frac{2}{p})}\) (for large t), where D is a homogeneous “dimension” appearing in the volume doubling property. This estimate can be improved if \(R_k^-\) is strongly sub-critical. In general, \((e^{-t\overrightarrow{\Delta }_k})_{t>0}\) is not uniformly bounded on \(L^p\) for any \(p \not = 2\). We also prove the gradient estimate \(\Vert \nabla e^{-t\Delta }\Vert _{p-p} \le C t^{-\frac{1}{p}}\), where \(\Delta \) is the Laplace–Beltrami operator (acting on functions). Finally we discuss heat kernel bounds on forms and the Riesz transform on \(L^p\) for \(p > 2\).

中文翻译：

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$$ L ^ p $$ Lp-差分形式和相关问题的热半群的估计

我们考虑一个完全的非紧黎曼流形，该流形满足体积加倍属性和其热核（在函数上）的高斯上限。令\（\ overrightarrow {\ Delta} _k \）是带有\（k \ ge 1 \）的微分k形式上的Hodge-de Rham Laplacian 。根据Bochner分解公式，\（\ overrightarrow {\ Delta _k = \ nabla ^ * \ nabla + R_k \），其中\（\ nabla \）表示Levi-Civita连接，\（R_k \）是对称截面的\（\ mathrm {完}（\ LAMBDA ^ KT ^ * M）\） 。假定负部分\（R_k ^-\）在扩大的Kato类中，我们证明对于所有\（p \ in [1，\ infty] \），\（\ Vert e ^ {-t \ overarrowarrow {\ Delta} _k} \ Vert _ {pp} \ le C（t \ log t）^ {\ frac {D} {4}（1- \ frac {2 } {p}）} \）（对于大t而言），其中D是出现在体积加倍属性中的齐次“维数”。如果\（R_k ^-\）非常次临界，则可以改进此估计。通常，对于任何\（p \ not = 2 \ ），\（（e ^ {-t \ overarrowarrow {\ Delta} _k}）_ {t> 0} \）在\（L ^ p \）上的边界不是统一的。 ）。我们还证明了梯度估计\（\ Vert \ nabla e ^ {-t \ Delta} \ Vert _ {pp} \ le C t ^ {-\ frac {1} {p}} \），其中\（\ Delta \）是Laplace–Beltrami运算符（作用于函数）。最后，我们对形式讨论热内核边界和中Riesz变换上\（L ^ P \）为\（P> 2 \） 。