Asymptotic behaviour of the Hessian Take $f \in C^{2}$ with bounded integral. If the Hessian $D^{2}f$ is $O(|x|^{-4-\alpha})$ can we say something about the behaviour at infinity of the gradient? For example neglecting boundary terms in integration by parts, i.e. saying that
$\int_{S_{r}}|x|^{2}\nabla f \cdot \nu  d\sigma \to 0 $, where $S_{r}$ is the sphere in $\mathbb{R}^{2}$?
 A: In general, you cannot infer anything about the decay of $\nabla f$ from the decay of $D^2 f$. For every $A, B, \beta\in\mathbb R$, the function
$$
f'(t)=A+B\sin(t^{1-\beta}), \qquad t> 0,$$
is such that $f''(t)=O(t^{-\beta})$, but $f'$ does not even decay at $\infty$.
However, if you add the assumption that $\nabla f(x)\to 0$ as $\lvert x \rvert\to \infty$, then you have that $\lvert D^2 f(x)\rvert =O(\lvert x \rvert^{-\beta})$ for a $\beta>1$ implies that $\lvert \nabla f(x)\rvert=O(\lvert x \rvert^{1-\beta})$.
Proof.
If $f$ is a function of one real variable only,
$$
\lvert f'(t)\rvert =\left\lvert \int_{t}^\infty f''(s)\, ds\right\rvert\le C \left\lvert \int_t^\infty s^{-\beta}\, ds\right\rvert=O(t^{1-\beta}), $$
where we used that $f'(t)\to 0$ as $t\to \infty$, and the assumption that
$$
\lvert f''(s)\rvert \le C s^{-\beta}.$$ Note that we need $\beta>1$ to ensure that the integral is convergent. This concludes the one-dimensional case.
In higher dimension, we fix $x\in\mathbb R^n$ with $\lvert x\rvert=1$, and we consider
$$
g(t)=f(tx), \qquad t\ge 0.$$
We have that $g''(t)=x^TD^2f(tx)x=O(t^{-\beta})$. By the previous argument,
$$
\lvert g'(t)\rvert =\lvert x\cdot \nabla f(tx)\rvert \le C(x)t^{1-\beta}, $$
for some $C(x)>0$. Now consider the standard orthonormal basis $e_1, e_2, \ldots, e_n$ (or any other orthonormal basis, actually). Letting
$$
\bar{C}:= \max\{ C(e_1), \ldots, C(e_n)\}, $$
the previous inequality implies that
$$
\left\lvert \frac{\partial f}{\partial x_k}(y)\right\rvert \le  \bar{C} \lvert y\rvert^{1-\beta}, \qquad \forall k=1, 2, \ldots, n.$$
This means exactly that $\nabla f(x)=O(\lvert x \rvert^{1-\beta})$, as we claimed.
