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Consider the standard mollifier $$ \phi(x) = C\exp\left(-\frac{1}{1-x^2}\right), \quad -1<x<1. $$ such that $\int\phi(x) = 1$.

Let $f(x) = |x|$ and consider the convolution $f\ast \phi$. I am interested in a bound in the derivatives $(f\ast\phi)^{(k)}$, say, how large is $\max_{x\in [-1,1]} |(f\ast\phi)^{(k)}|$?

Question 1. What is an upper bound on $\|\phi^{(k)}\|_\infty$? I think this is $C_1(C_2k)^{2k} \cdot k^{C_3}$ but I didn't seem to find it in the literature. Can anyone point me to some reference?

Question 2. Is there a mollifier (not necessarily compactly supported) such that $\int \phi = 1$ and $\|\phi^{(k)}\|_\infty \leq C_1(C_2k)^k k^{C_3}$? (It seems that $k^{2k}$ is the common bound for several mollifiers, see this post on stackexchange and the appendix in this note by Ben Green. I'd like to know if it's possible to have just a bound of $k^k$.)

Question 3. Is there a bound for $\|\phi^{(k)}\|_1$? The trivial bound $\|\phi^{(k)}\|_1\leq 2\|\phi^{(k)}\|_\infty$ may be too cruel because the spike for the bad $\ell_\infty$ norm is very narrow. For instance, $\|\phi^{(4)}\|_\infty\approx 8000$ while numerical integration gives that $\|\phi^{(4)}\|_1 \approx 1076.13$.

Question 4. Here $f(x) = |x|$, the cruel bound $|f\ast \phi^{(k)}| \leq \max_{x\in [-1,1]} |f(x)| \cdot \|\phi^{(k)}\|_1$ seems too loose. While $\|\phi^{(4)}\|_1 \approx 1076.13$, the true $\max_{x\in [-1,1]} |(f\ast\phi)^{(k)}|\approx 15$. What is a tighter bound in general?

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    $\begingroup$ Regarding Q4: $f'' = 2 \delta_0$, so $f * \phi^{(k)} = 2\phi^{(k-2)}$. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Oct 11, 2019 at 6:57
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    $\begingroup$ Q2: For the standard Gaussian mollifier, $\|\phi^{(n)}\|_1 \sim c 2^{n/2} n^{1/4} (n!)^{1/2}$, see this answer. Furthermore, Lemma 2.1 in this paper states that $\|\phi^{(n)}\|_\infty \le n^{-1/12} (n!)^{1/2}$, giving a reference to Lemma 4.5.2 in [S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Math. Notes, vol. 42, Princeton University Press, Princeton, 1993]. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Oct 11, 2019 at 19:13
  • $\begingroup$ For the construction of mollifiers with optimal bounds on derivatives, a very nice reference is Hörmander's The Analysis of Linear Partial Differential Operators, vol I, ch. 1 $\endgroup$
    – Pietro Majer
    Commented Oct 11, 2019 at 20:21
  • $\begingroup$ @MateuszKwaśnicki Thanks for the reference. Two questions: (1) It seems to me that $\phi^{(n)}=(2π)^{-1/2}\exp(−x^2/2)\cdot 2^{-n/2}H_n(x/\sqrt{2})$. Then $\|\phi^{(n)}\|_1$ will depend on $\int |H_n(x)\exp(-x^2)|$ with a different weight function. I think the order of growth should be similar, but is there a straightforward way to derive the result for the weight function $\exp(-x^2)$ from the weight function $\exp(-x^2/2)$? (2) Why is the omitted factor (n!)^{1/2} in the bound from the paper you cited? $\endgroup$
    – user58955
    Commented Oct 12, 2019 at 0:59
  • $\begingroup$ (1) You are right, sorry! Still, this leads to an upper bound for the $L^1$ norm of $\phi^{(n)}$. (2) Not sure if I understand correctly; we have $\phi^{(n)}(x) = (-1)^n \pi^{1/4} 2^{n/2} (n!)^{1/2}e^{-x^2/2} \phi_n(x)$, where $\phi_n$ is the Hermite function defined right after (1.2) in that paper. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Oct 12, 2019 at 8:30

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