Existence of smooth functions $f$ satisfying $\sup_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq C B^q k^{1/8} q^{q/2}$

Question

$\mathcal{S}^{1/2}_{1/2}(\mathbb{R})$ is defined to be the collection of $C^\infty$ functions $f$ on $\mathbb{R}$ such that \begin{equation} \sup_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq CA^kB^q k^{k/2} q^{q/2} \end{equation} where $k$ and $q$ are nonnegative integers and $A,B,C$ are positive constants depending on $f$.

In fact, I am aware that it is a special case of more general function spaces $\mathcal{S}^{\alpha}_{\beta}(\mathbb{R})$ for nonnegative $\alpha, \beta$.

I am currently trying to find smooth functions satisfying some stronger bounds. For example, are there smooth functions $f$ satisfying the bounds \begin{equation} \sup_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq CB^q k^{1/8} q^{q/2} \end{equation} for positive constants $C, B$ depending on $f$? I suspect that Hermite functions might satisfy these bounds, but cannot prove rigorously.

Could anyone please provide any example?

Answer 1

Start from the estimate (with $q = K$ and $k = K+1$) (valid for $x > 0$) $$ |f^{(K)}(x)| \leq C B^{K} (K+1)^{1/8} K^{K/2} x^{-K-1}$$ Integrate back from infinity $K$ times, you get $$ |f(x)| \leq C \frac{B^K (K+1)^{1/8} K^{K/2}}{K!} \frac{1}{x} $$ Using Stirling's approximation $$ |f(x)| \lesssim \frac{e^K B^K (K+1)^{1/8} K^{K/2}}{K^{1/2} K^K} \frac{1}{x} $$ with the implicit constant independent of $K$.

Take the limit $K \to \infty$ you see that the RHS converges to 0. This shows that the only function with the specified property is $f \equiv 0$.