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This question is a bit tricky for me to post because I don't really understand all the symbols and techniques involved - so it is likely that I'll do mistakes. In case it is unclear what I write please point out and I'll try my best to improve the question.

Suppose we have an orthonormal basis $\{\phi_n\}$ for the space $L^2(V)$ of square integrable sections of some vector bundle $V$ over a compact manifold $M$, obtained via a generalised Fourier Series argument.

Suppose we define the function \begin{equation} K(t,x,y) = \sum_n e^{-t\lambda_n} \phi_n(x) \otimes \overline{\phi_n}(y) : V_x \to V_y \end{equation} regarding it as an endomorphism from the fiber of $V$ over $y$ to the fiber of $V$ over x. Now I'd like to estimate $|K|_{\infty,k}$, how do I actually compute this ?

I think I understand how to write down the expression, so i have

\begin{equation} |K(t,x,y)|_{\infty,k} = \text{sup}_{(t,x,y) \in [0,\infty) \times \mathbb{R}^m \times \mathbb{R}^m} \sum_{|j + \alpha + \beta| \leq k} |D^j_t D^\alpha_x D^\beta_y K(t,x,y)| \end{equation} and each term of this sum I think I can write of the form \begin{equation} D^j_t D^\alpha_x D^\beta K(t,x,y) = \sum_n (D^j_t e^{-t\lambda_n})D^\alpha_x \phi_n(x) \otimes D^\beta_y \overline{\phi_n}(y) \end{equation}

Now I am not entirely sure how to handle the tensor product here - in order to estimate these, is it correct to write \begin{equation} |D^j_t D^\alpha_x D^\beta K(t,x,y)|\leq \sum_n |\lambda_n e^{-t\lambda_n}||D^\alpha_x \phi_n(x) \otimes D^\beta_y \overline{\phi_n}(y)| \quad ? \end{equation} if this is the correct way of writing, how can I further compute \begin{equation} |D^\alpha_x \phi_n(x) \otimes D^\beta_y \overline{\phi_n}(y)| \end{equation}

Many thanks for your help !

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