Suppose $f\in C^{\infty}(\mathbb{R}^n)$ is real analytic and $\Delta f(o)\not=0$. Are there pointwise estimates for $\frac{\partial^{\alpha}f}{\partial x^{\alpha}}$ in terms of $\Delta=\sum_{k=1}^n\frac{\partial^{2}f}{\partial x_k^2}$ in the sense :$\exists C>0$ s.t $|\frac{\partial^{\alpha}f}{\partial x^{\alpha}}(o)|\leq C|\Delta f(o)|$ for all $\alpha$?
How about we pose the same question and are interested in existence of $C>0$ s.t $|\frac{\partial^{\alpha}f}{\partial x^{\alpha}}(o)|\leq C^{|\alpha|}|\Delta f(o)|$ for all $\alpha$? How can one tie up the norm partial derivative of a nice function to the norm of its Laplacian?
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