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Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$. Let $\Delta$ be the Laplacian on tensor fields, which locally satisfies $$ \Delta A = g^{ij}(\nabla _{\partial_i}\nabla _{\partial_j}A - \nabla_{\nabla_{\partial_i}\partial_j}A) $$ for any tensor field $A$.

From this question, we know that $$\nabla(\Delta A)-\Delta(\nabla A)=\nabla Rm *A+Rm*\nabla A,$$ where for tensors $T_1$ and $T_2$, the notation $T_1*T_2$ means some linear combination of contractions of $T_1$ and $T_2$.

Question: Do there exist constants $C_1$ and $C_2$ such that $$\|\nabla(\Delta A)-\Delta(\nabla)A\|\leq C_1\|\nabla Rm\|\|A\|+C_2\|Rm\|\|\nabla A\|,$$ where the various norms are induced from the Riemannian metric $g$? Here $C_1$ and $C_2$ should only depend on the type $(p,q)$ of the tensor $A$.

Clarification: Here $\|\cdot\|$ denotes either the pointwise or the $L^2$-norm. We can assume $M$ to be compact if that helps.

geometricK
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  • Is $|\cdot|$ the pointwise norm? – Deane Mar 20 '22 at 15:08
  • @Deane I'm interested both when $|\cdot|$ is the pointwise norm or the $L^2$-norm. I should say I'm really only interested in the case when $(M,g)$ has bounded geometry (or even compact $M$), but I'm not sure if that makes any difference. – geometricK Mar 21 '22 at 06:34
  • Pointwise: 1) Write everything out in components with respect to an orthonormal frame. 2) Observe that, given $a, b \in \mathbb{R}$, $|ab| = |a||b|$. 3) Observe that, with respect to an orthonormal frame, the components of a tensor $T$ satisfy (say, for a 4-order tensor) $|T_{ijkl}|\le |T|$. – Deane Mar 21 '22 at 15:24
  • For $L^2$, this does not work, because $|fg|_2$ cannot be bounded by $|f|_2|g|_2$. – Deane Mar 21 '22 at 15:26

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