I have a conjecture which, if it's right, will help me be to solve an exercise.
Let $C^{\infty}_b(\mathbb{R}^n)$ be the space of smooth functions whose partial derivatives are all bounded on $\mathbb{R}^n$.
Choose $p\in C^{\infty}(\mathbb{R}^{2n})$ such that for all $y\in\mathbb{R}^{n}$, $x\mapsto p(x,y)\in C^{\infty}_b(\mathbb{R}^n)$.
Define $f:y\in S^{n-1}\mapsto \text{sup}\{p(x,y),\;x\in\mathbb{R}^n\}$, where $S^{n-1}:=\{x\in\mathbb{R}^n,\;||x||=1\}$.
Is it true that $f$ is at least continuous with the hypotheses mentioned above. Thanks in advance for your help.
Note that $f$ might not be defined: it’s possible that the sets do not have a maximum but only a supremum. Note, however, that in the counter-example we provide, every $x \longmapsto p(x,y)$ converges to zero at infinity, so $f$ is defined with maxima.
Define, for $y >0$, $g_y: x \in (0,\infty) \longmapsto y^{-y/2} x^ye^{-x}$. Then, as $y \rightarrow \infty$, if $u,v \geq 0$, $P \in \mathbb{R}[y]$, $P(y)(\partial^u_x \partial^v_y)(g_y)(x)$ converges locally uniformly in $x$ to zero, while $\|g_y\|_{\infty} \geq y^{y/2}e^{-y} \rightarrow \infty$.
So let $\chi$ be a smooth function with support inside $(0,\infty)$ which is equal to $1$ on $[1,\infty)$.
Consider $p((x_1,x_2),(y_1,y_2))=\chi(x_1)g_{1/|y_1|}(x_1)$. (So $p(\cdot,(0,y_2))=0$, and $p(x,y)=0$ if $x_1 < 0$).
Then $p$ satisfies the assumptions above, but $p_{| S^1 \times \mathbb{R}^2}$ is unbounded.