---
abstract: 'We obtain an improvement of the Beckner’s inequality $\| f\|^{2}_{2} -\|f\|^{2}_{p} \leq (2-p) \| \nabla f\|_{2}^{2}$ valid for $p \in [1,2]$ and the Gaussian measure. Our improvement is essential for the intermediate case $p \in (1,2)$, and moreover, we find the natural extension of the inequality for any real $p$.'
address:
- 'Department of Mathematics, Kent State University, Kent, OH 44240'
- 'Department of Mathematics, Michigan State University'
author:
- Paata Ivanisvili and Alexander Volberg
title: 'Improving Beckner’s bound via Hermite functions'
---

[^1]

Introduction
============

The history of the problem
--------------------------

The Poincaré inequality [@JN] for the standard Gaussian measure $d\gamma_{n} = \frac{e^{-|x|^{2}/2}}{\sqrt{(2\pi)^{n}}}dx$ states that $$\begin{aligned}
\label{poincare}
\int_{\mathbb{R}^{n}} f^{2} d\gamma_{n} - \left(\int_{\mathbb{R}^{n}} f d\gamma_{n}\right)^{2} \leq \int_{\mathbb{R}^{n}} | \nabla f |^{2} d\gamma_{n}\end{aligned}$$ for any smooth bounded function $f :\mathbb{R}^{n} \to \mathbb{R}$. Later William Beckner [@WB] generalized (\[poincare\]) for any real power $p$, $1 \leq p \leq 2$ as follows $$\begin{aligned}
\label{beckner}
\int_{\mathbb{R}^{n}} f^{p} d\gamma_{n} - \left(\int_{\mathbb{R}^{n}} f d\gamma_{n}\right)^{p} \leq \frac{p(p-1)}{2}\int_{\mathbb{R}^{n}} f^{p-2}| \nabla f |^{2} d\gamma_{n}\end{aligned}$$ for any smooth bounded $f : \mathbb{R}^{n} \to (0,\infty)$. We caution the reader that in [@WB] inequality (\[beckner\]) was formulated in a slightly different but equivalent form (see Theorem 1, inequality (3) in [@WB]). It should be also mentioned that in case $p=2$ inequality (\[beckner\]) does coincide with (\[poincare\]) for all $f \geq 0$ but it does not imply the Poincaré inequality for the functions taking the negative values, especially when $\int_{\mathbb{R}^{n}} f d \gamma_{n}=0$. If $p \to 1+$ then (\[beckner\]) provides us with log-Sobolev inequality (see [@WB]). In general, the constant $\frac{p(p-1)}{2}$ is sharp in the right hand side of (\[beckner\]) as it can be seen for $n=1$ on the test functions $f(x)=e^{\varepsilon x}$ by sending $\varepsilon \to 0$.

Later Beckner’s inequality (\[beckner\]) was studied by many mathematicians for different measures, in different settings and for different spaces as well. For possible references we refer the reader to [@ABD; @ALS; @BCR1; @BCR2; @BR1; @Bob1; @Bob2; @BBL; @Chaf; @WFY; @RK; @KO].

An analysis done in [@IV] indicates that the right hand side (RHS) of (\[beckner\]) can be improved. In the present paper we address this issue: what is the precise estimate of the difference given in the left hand side (LHS) of (\[beckner\]), and whether the requirement $p \in [1,2]$ can be avoided by slightly changing the RHS of (\[beckner\]).

We give complete answers to these questions. For example, if $p=\frac{3}{2}$ we will obtain an improvement in Beckner’s inequality (\[beckner\]) $$\begin{aligned}
&\int_{\mathbb{R}^{n}} f^{3/2} d\gamma_{n} - \left(\int_{\mathbb{R}^{n}} f d\gamma_{n}\right)^{3/2} \leq \label{b3/2}\\
 &\int_{\mathbb{R}^{n}}\left(   f^{3/2} - \frac{1}{\sqrt{2}}(2f-\sqrt{f^{2}+ |\nabla f|^{2}})\sqrt{f+\sqrt{f^{2}+|\nabla f|^{2}}} \right)d\gamma_{n}. \nonumber\end{aligned}$$ The LHS of (\[b3/2\]) coincides with the LHS of (\[beckner\]) for $p=3/2$, but the RHS of (\[b3/2\]) is strictly smaller than the RHS in (\[beckner\]). Indeed, notice that we have the following *pointwise* inequality $$\begin{aligned}
\label{impr1}
x^{3/2} - \frac{1}{\sqrt{2}}(2x-\sqrt{x^{2}+ y^{2}})\sqrt{x+\sqrt{x^{2}+y^{2}}} \leq \frac{3}{8} x^{-1/2}y^{2} \quad \text{for all} \quad  x, y \geq 0,\end{aligned}$$ which follows from the homogeneity, i.e., take $x=1$. As one can see the improvement of Beckner’s inequality (\[beckner\]) is essential. Indeed, if $y \to \infty$ then the RHS of (\[impr1\]) increases as $y^{2}$ whereas the LHS of (\[impr1\]) increases as $y^{3/2}$. Also notice that if $x \to 0$ then the difference in (\[impr1\]) tends to infinity. The only place where the quantities in (\[impr1\]) are comparable is when $y/x \to 0$.

Main results
------------

Let $k$ be a real parameter. Let $H_{k}(x)$ be the Hermite function such that it satisfies the Hermite differential equation $$\begin{aligned}
\label{hermite}
H_{k}''-xH_{k}'+k H_{k}=0,  \quad x \in \mathbb{R},\end{aligned}$$ and which grows relatively slowly $H_{k}(x) = x^{k}+o(x^{k})$ as $x \to +\infty$. If $k$ is a nonnegative integer then $H_{k}$ is the *probabilists’* Hermite polynomial of degree $k$ with the leading coefficient $1$, for example, $H_{0}(x)=1, H_{1}(x)=x, H_{2}(x)=x^{2}-1$ etc. In general, for arbitrary $k \in \mathbb{R}$ one should think that $H_{k}$ is the analytic extension of the Hermite polynomials in $k$ (existence and many other properties will be mentioned in Section \[prf\]).

For $k \in \mathbb{R}$ let $R_{k}$ be the rightmost zero of $H_{k}(x)$ (see Lemma \[EMMlemma\]). If $k \leq 0$ then we set $R_{k}=-\infty$. Define $F_{k}(x)$ as follows $$\begin{aligned}
\label{bdef}
F_{k}\left(\left| \frac{H'_{k}(q)}{H_{k}(q)} \right|\right) = \frac{H_{k+1}(q)}{H^{1+\frac{1}{k}}_{k}(q)} \quad  \text{for} \quad q \in (R_{k}, \infty).\end{aligned}$$

We will see in the next section $F_{k} \in C^{2}([0,\infty))$ is well-defined and $F_{k}(0)=1$. Moreover, if $k > -1$ then $F_{k}$ will be decreasing concave function, and if $k<-1$ then $F_{k}$ will be increasing convex function.

One may observe that $$\begin{aligned}
F_{1}(y)=1-y^{2}; \quad F_{2}(y)=\frac{1}{\sqrt{2}}(2-\sqrt{1+ y^{2}})\sqrt{1+\sqrt{1+y^{2}}}. \end{aligned}$$

If $k=0$ then definition (\[bdef\]) should be understood in the limiting sense as follows $$\begin{aligned}
F_{\exp}(H_{-1}(q)) = q\exp\left( \alpha - \int_{1}^{q} H_{-1}(s)ds\right) \quad \text{for all} \quad q \in \mathbb{R}, \end{aligned}$$ where $$\begin{aligned}
\label{kap}
\alpha = \int_{1}^{\infty}\left(H_{-1}(s)-\frac{1}{s}\right)ds \approx  - 0.266\ldots . \end{aligned}$$

\[better\] For any $p \in \mathbb{R} \setminus [0,1]$ and any smooth bounded $f\geq 0$ with $\int_{\mathbb{R}^{n}} f^{p} d\gamma_{n}<\infty$ we have $$\begin{aligned}
\label{our}
\int_{\mathbb{R}^{n}}f^{p} F_{\frac{1}{p-1}}\left(\frac{|\nabla f|}{f}\right) d\gamma_{n} \leq  \left( \int_{\mathbb{R}^{n}} f d\gamma_{n} \right)^{p}.\end{aligned}$$ The inequality is reversed if $p \in (0,1)$.

The theorem improves Beckner’s inequality (\[beckner\]). This will follow by taking the first two nonzero Taylor terms of $F_{\frac{1}{p-1}}(t)$ as its lower estimate.

\[prob\] We have pointwise improvement in Beckner’s inequality (\[beckner\]), i.e., $$\begin{aligned}
\label{bbeckner}
1-\frac{p(p-1)}{2}t^{2} \leq  F_{\frac{1}{p-1}}\left(t\right)    \quad \text{for all} \quad t \geq 0, \; p \in (1,2].\end{aligned}$$

The improvement will be essential when $t \to \infty$. For example, it will become clear in the next section that as $t \to \infty$ we have $$\begin{aligned}
&F_{\frac{1}{p-1}}(t) \sim -t^{p} \left( H'_{\frac{1}{p-1}}(R_{\frac{1}{p-1}})\right)^{1-p} \quad \text{for} \quad p>1; \label{a1}\\
&F_{\frac{1}{p-1}}(t) \sim \left( \frac{p}{1-p}\right) \left(\frac{e^{t^{2}/2} \, \sqrt{2 \pi}}{t \Gamma(\frac{1}{1-p}) }\right)^{1-p} \quad \text{for} \quad p<1, \quad p \neq 0. \label{a2}\end{aligned}$$

Our theorem interpolates several inequalities. If $p\to 1+$ then (\[our\]) gives log-Sobolev inequality. If $p=2$ then (\[our\]) provides us with Poincaré inequality. If $p\to \pm \infty$ then we obtain *$e$-Sobolev inequality*:

\[e-sob\] For any smooth bounded $f$ we have $$\begin{aligned}
\int_{\mathbb{R}^{n}} \exp(f)\,  F_{\exp}(|\nabla f|) d\gamma_{n} \leq \exp\left(  \int_{\mathbb{R}^{n}} f d\gamma_{n}\right). \end{aligned}$$

Finally if $p \to 0$ we obtain *negative log-Sobolev* inequality:

\[nlog\] For any smooth bounded $f\geq 0$ with $\int_{\mathbb{R}^{n}} \ln f d\gamma_{n}>-\infty$ we have $$\begin{aligned}
\int_{\mathbb{R}^{n}} - \ln f d\gamma_{n}  + \ln \left(\int_{\mathbb{R}^{n}} f d\gamma_{n} \right) \leq  \int_{\mathbb{R}^{n}}- F_{-\ln}\left(\frac{|\nabla f|}{f}\right)d\gamma_{n}\end{aligned}$$ where $F_{-\ln}(t)$ is defined as follows $$\begin{aligned}
F_{-\ln} \left( \frac{H_{-2}(x)}{H_{-1}(x)}\right) = \int_{1}^{x}H_{-1}(s)ds-c + \ln H_{-1}(x), \quad x \in \mathbb{R}.\end{aligned}$$

It is worth mentioning that the current paper provides with estimates of $\Phi$-entropy (see [@Chaf]): $$\begin{aligned}
 \boldsymbol{\mathrm{Ent}}_{\gamma_{n}}^{\Phi}(f) \stackrel{\mathrm{def}}{=} \int_{\mathbb{R}^{n}}\Phi(f)d\gamma_{n} - \Phi\left(\int_{\mathbb{R}^{n}} f d\gamma_{n} \right)
 \end{aligned}$$ for the following fundamental examples: $$\begin{aligned}
  &\Phi(x) = x^{p} \quad \text{for}  \quad p \in \mathbb{R}\setminus [0,1] \quad \text{Theorem~\ref{better}};\\
  &\Phi(x) = -x^{p} \quad \text{for}  \quad p \in (0,1) \quad \text{Theorem~\ref{better}};\\
  &\Phi(x) = e^{x},  \quad \text{Corollary~\ref{e-sob}, or $p \to \pm \infty$ in Theorem~\ref{better}};\\
  &\Phi(x) = -\ln x,  \quad \text{Corollary~\ref{nlog}, or $p \to 0$ in Theorem~\ref{better}};\\
  &\Phi(x) = x\ln x,  \quad \text{$p \to 1$ in Theorem~\ref{better}}.
 \end{aligned}$$

The proof of the theorem {#prf}
========================

The proof of the theorem amounts to check that the real valued function $$\begin{aligned}
\label{f1}
M(x,y)=x^{p}F_{k}\left( \frac{y}{x} \right)\end{aligned}$$ defined on $[\varepsilon,\infty)\times [0, \infty)$ for any $\varepsilon>0$ obeys necessary smoothness condition, it has a boundary condition $M(x,0)=x^{p}$ and it satisfies the following partial differential inequality $$\begin{aligned}
\label{matrica}
\begin{pmatrix}
M_{xx}+\frac{M_{y}}{y}  & M_{xy}\\
M_{xy} & M_{yy}
\end{pmatrix} \leq 0,\end{aligned}$$ with reversed inequality in (\[matrica\]) if $p\in (0,1)$. Then by Theorem 1 in [@IV] we obtain that $$\begin{aligned}
\int_{\mathbb{R}^{n}} f^{p}F_{k}\left( \frac{|\nabla f|}{f}\right) d\gamma_{n} = \int_{\mathbb{R}^{n}} M(f, |\nabla f| )d\gamma_{n} \leq M\left(\int_{\mathbb{R}^{n}}f d\gamma_{n},0 \right) = \left( \int_{\mathbb{R}^{n}} fd\gamma_{n}\right)^{p} \end{aligned}$$ for any smooth bounded $f \geq \varepsilon$ which is the statement of the theorem we want to prove (except we need to justify the passage to the limit $\varepsilon \to 0$ and this will be done later). Notice that the inequality is reversed if $p \in (0,1)$, indeed, in this case we should work with $-M(x,y)$ instead of $M(x,y)$.

Next we will need some tools regarding the Hermite functions $H_{k}$.

Properties of Hermite functions
-------------------------------

$H_{k}$ can be defined (see [@HO]) by $$\begin{aligned}
\label{hyp1}
H_{k}(x) =  -\frac{2^{-k/2} \sin (\pi k)\;   \Gamma(k+1)}{2\pi} \sum_{n=0}^{\infty} \frac{\Gamma((n-k)/2)}{n!} (-x \sqrt{2} )^{n},\end{aligned}$$ or in terms of the confluent hypergeometric functions (see [@LD]) by $$\begin{aligned}
H_{k}(x) =  &\sqrt{\frac{2^{k}}{\pi}}\left[\cos\left( \frac{\pi k}{2}\right) \, \Gamma\left( \frac{k+1}{2}\right)\,  {}_{1}F_{1}\left(-\frac{k}{2}, \frac{1}{2}; \frac{x^{2}}{2}\right)\right. \label{hyp2} \\
&+\left. t\sqrt{2} \sin \left( \frac{\pi k}{2} \right) \, \Gamma\left( \frac{k}{2}+1\right)\,  {}_{1}F_{1}\left(\frac{1-k}{2}, \frac{3}{2}; \frac{x^{2}}{2}\right) \right].\nonumber\end{aligned}$$ If $k$ is a nonnegative integer then one should understand (\[hyp1\]) and (\[hyp2\]) in the limiting sense. Notice the following recurrence properties:$$\begin{aligned}
&H'_{k}(x) = k H_{k-1}(x); \label{pirveli}\\
&H_{k+1}(x) = x H_{k}(x) - H'_{k}(x). \label{meore}\end{aligned}$$ These properties follow from (\[hyp1\]) and the fact that $\Gamma(z+1)=z\Gamma(z)$.

We also notice that $$\begin{aligned}
H_{k}(x) := e^{x^{2}/4}D_{k}(x),\end{aligned}$$ where $D_{k}(x)$ is the *parabolic cylinder function*, i.e., it is the solution of the equation $$\begin{aligned}
D''_{k}+\left( k+\frac{1}{2}-\frac{x^{2}}{4}\right)D_{k}=0. \end{aligned}$$ Since $H_{k}(x)$ is an entire function in $x$ and $k$ (see [@temme] for the parabolic cylinder function) sometimes it will be convenient to write $H(x,k)$ instead of $H_{k}(x)$. The precise asymptotic for $x \to +\infty$, $x>0$ and any $k \in \mathbb{R}$ is given as follows $$\begin{aligned}
\label{loran}
H_{k}(x) \sim x^{k} \cdot  \sum_{n=0}^{\infty} (-1)^{n} \frac{(-k)_{2n}}{n! (2x^{2})^{n}}.\end{aligned}$$ Here $(a)_{n}=1$ if $n=0$ and $(a)_{n}=a(a+1)\ldots(a+n-1)$ if $n>0$. When $x \to -\infty$ we have $$\begin{aligned}
\label{loran-}
 H_{k}(x) \sim |x|^{k}\cos(k \pi) \sum_{n=0}^{\infty} (-1)^{n} \frac{(-k)_{2n}}{n! (2x^{2})^{n}}  + \frac{\sqrt{2 \pi}}{\Gamma(-k)} |x|^{-k-1}e^{x^{2}/2}\sum_{n=0}^{\infty} \frac{(1+k)_{2n}}{n! (2x^{2})^{n}}.
 \end{aligned}$$ We refer the reader to [@temme; @NIST]. For instance, for (\[loran\]) we can use the asymptotic formula (12.9.1) in [@NIST] for the parabolic cylinder function. To verify (\[loran-\]) we can express $H_{k}(-x)$ as a linear combination of two parabolic cylinder functions but having argument $x$ instead of $-x$ (see (12.2.15) in [@NIST]), and then we can use (12.9.1) and (12.9.2) in [@NIST].

Next we will need the result of Elbert–Muldoon [@EMM1] which describes the behavior of the real zeros of $H_{k}(x)$ for any real $k$.

\[EMMlemma\] For $k\leq 0$, $H_{k}(x)$ has no real zeros, and it is positive on the real axis. For $n<k \leq n+1$, $n=0,1,\ldots, $ $H_{k}(x)$ has $n+1$ real zeros. Each zero is increasing function of $k$ on its interval of definition.

The proof of the lemma is Theorem 3.1 in [@EMM1]. It is explained in the paper that as $k$ passes through each nonnegative integer $n$ a new leftmost zero appears at $-\infty$ while the right-most zero passes through the largest zero of $H_{k}(x)$. More precise information about the asymptotic behavior of the zeros as $k \to \infty$ can be found in [@EMM2].

Further we will need Turán’s inequality for $H_{k}$ for any real $k$.

\[turl\] We have the following Turán’s inequality: $$\begin{aligned}
 H_{k}^{2}(x)-H_{k-1}(x)H_{k+1}(x)> 0 \quad \text{for all} \quad k \in \mathbb{R}, \; x \geq  L_{k} \label{tu1}
 \end{aligned}$$ where $L_{k}$ denotes the leftmost zero of $H_{k}$. If $k\leq 0$ then $L_{k}=-\infty$.

The lemma is known as Turán’s inequality when $k$ is a nonnegative integer. Unfortunately we could not find the reference in the case when $k$ is different from a positive integer therefore we decided to include the proof of the lemma.

The following is borrowed from [@MT].

Take $f(x) = e^{-\frac{x^{2}}{2}}(H_{k}^{2}(x)-H_{k-1}(x)H_{k+1}(x))$. Asymptotic formulas (\[loran\]) and (\[loran-\]) imply that $$\begin{aligned}
&\lim_{x \to +\infty} f(x) =0 \quad \text{for all} \quad k \in \mathbb{R};\nonumber\\
&f(x) \sim \sqrt{2 \pi} |x| >0\quad  \text{for} \quad x \to -\infty, \quad k =0;\nonumber\\
&f(x) \sim \frac{2\pi e^{x^{2}/2}}{\Gamma(-k)\Gamma(-k+1)}|x|^{-2k-2} \quad \text{for} \quad x \to -\infty, \quad k \notin \{0\}\cup \mathbb{N}. \label{atinf}\end{aligned}$$ On the other hand notice that $$\begin{aligned}
\label{der}
f'(x) = -e^{-\frac{x^{2}}{2}} H_{k}H_{k-1}.\end{aligned}$$ If $k\leq 0$ then by Lemma \[EMMlemma\] $f'<0$, and because of the conditions $f(-\infty)=+\infty$ and $f(\infty)=0$ we obtain that $f>0$ on $\mathbb{R}$. To verify the statement for $k>0$ we notice that $$\begin{aligned}
\label{der1}
f''(x) = e^{-\frac{x^{2}}{2}}(H_{k}^{2} -k H_{k-1}^{2}).\end{aligned}$$ Now we notice that if $H_{k}(c)=0$ then $H_{k-1}(c)\neq 0$. Indeed, assume contrary $H_{k-1}(c)=0$. Then by (\[pirveli\]) we have $H'_{k}(c)=0$ and by (\[hermite\]) we obtain $H''_{k}(c)=0$, and again taking derivative in (\[pirveli\]) we obtain that $H_{k-2}(c)=0$. Repeating this process we obtain that $H_{k-N}(c)=0$ for any large integer $N>0$. But this contradicts to Lemma \[EMMlemma\]. Thus by (\[der\]) and (\[der1\]) we obtain that $c$ is a point of the local minimum of $f$ if and only if $ H_{k-1}(c)=0$. Then $f(c) = e^{-x^{2}/2}H_{k}^{2}(c)>0$. Finally we obtain that $f : [L_{k}, \infty) \to \mathbb{R}$ is positive on its local minimum points, $f(\infty)=0$ and $f(L_{k}) >0$ (because $H_{k-1}, H_{k+1}$ have opposite signs at zeros of $H_{k}$ by (\[meore\])). Therefore $f > 0$ on $[L_{k}, \infty) \to \mathbb{R}$ and the lemma is proved.

If $k \in \mathbb{N}$ then $H_{k}$ is the probabilists’ Hermite polynomial of degree $k$, so $f(x)$ will be even and inequality (\[tu1\]) will hold for all $x \in \mathbb{R}$ which confirms the classical Turán’s inequality. However, if $k>0$ but $k \notin \mathbb{N}$ then (\[tu1\]) fails when $x \to -\infty$ (see (\[atinf\])).

Finally the next corollary together with Lemma \[EMMlemma\] implies that $\left|\frac{H'_{k}}{H_{k}}\right|=\mathrm{sign}(k) \frac{H'_{k}(q)}{H_{k}(q)} $ is positive and decreasing for $q \in (R_{k}, \infty)$ and $k\in \mathbb{R}\setminus\{0\}$.

\[cor\] For any $x \ge L_{k}$ and any $k \in \mathbb{R}\setminus\{0\}$ we have $$\begin{aligned}
\mathrm{sign}[(H'_{k})^{2}-H_{k}H''_{k}]=\mathrm{sign}(k).\end{aligned}$$

The proof follows from Lemma \[turl\] and the following identity $$\begin{aligned}
\label{identity1}
k(H_{k}^{2}-H_{k-1}H_{k+1}) = (H'_{k})^{2}-H_{k}H''_{k}\end{aligned}$$ from (\[hermite\]), (\[pirveli\]) and (\[meore\]).

Checking the partial differential inequality
--------------------------------------------

Let $p=1+\frac{1}{k}$. Further we assume $k \neq 0, -1$. Define $F=F_{k}$ as in the introduction: $$\begin{aligned}
\label{opr}
F(t) = \frac{H_{k+1}(q)}{H_{k}^{1+1/k}(q)} \quad \text{where} \quad \left| \frac{H'_{k}(q)}{H_{k}(q)}\right|=t, \quad q \in (R_{k}, \infty), \quad t \in (0, \infty).\end{aligned}$$

Notice that by Corollary \[cor\] function $\left| \frac{H'_{k}(q)}{H_{k}(q)}\right|=\mathrm{sign}(k) \frac{H'_{k}(q)}{H_{k}(q)}$ is positive decreasing in $q$ for $q \in (R_{k}, \infty)$, moreover by (\[loran\]) we have $\frac{H'_{k}(q)}{H_{k}(q)} \sim \frac{k}{q}$ when $q \to +\infty$. From the same asymptotic formulas it follows that when $t \to 0+$ we have $$\begin{aligned}
F(t)= 1-\frac{p(p-1)}{2}\, t^{2} +O(t^{4}).\end{aligned}$$ Therefore $F$ is well-defined function and $F \in C^{2}([0,\infty))$.

Take a positive $\varepsilon>0$ and define $M(x,y)$ as in (\[f1\]): $$\begin{aligned}
\label{k1}
M(x,y) :=x^{p} F\left(\frac{y}{x}\right) \quad \text{for} \quad  y\geq 0, \quad x > \varepsilon >0.\end{aligned}$$ Clearly $M(x,\sqrt{y}) \in C^{2}([\varepsilon, \infty)\times[0, \infty))$. By Theorem 1 in [@IV] we have inequality $$\begin{aligned}
\label{ivo1}
\int_{\mathbb{R}^{n}} M(f,|\nabla f|) d\gamma_{n} \leq M\left(\int_{\mathbb{R}^{n}}f d\gamma_{n}, 0 \right)\end{aligned}$$ for all smooth bounded $f \geq \varepsilon$ if (\[matrica\]) holds. In terms of $F$ (see (\[k1\])) condition (\[matrica\]) takes the form $$\begin{aligned}
&tFF''p(p-1)+F'F''-t(p-1)^{2}(F')^{2} \geq 0  \qquad \text{i.e., the determinant of (\ref{matrica}) is nonnegative}\label{determinant}\\
& F'' (t+t^{3})+F'(2t^{2}+1-2pt^{2})+Fp(p-1)t\leq 0 \qquad \text{i.e., the trace of (\ref{matrica}) is nonpositive} \label{trace}\end{aligned}$$ where $t = \frac{y}{x}$ is the argument of $F$. In fact we will show that we have equality in (\[determinant\]) instead of inequality therefore the sign of (\[matrica\]) will depend on the sign of trace (\[trace\]). We will see that inequality (\[trace\]) will be reversed for $p \in (0,1)$.

From (\[opr\]), (\[identity1\]) and (\[tu1\]) we obtain $$\begin{aligned}
 &F'(t) =-\frac{k+1}{|k|} \frac{1}{H_{k}^{1/k}};\label{odin}\\
 &F''(t) = \frac{F'}{|k|}\cdot   \frac{H_{k}H_{k-1}}{H_{k}^{2}-H_{k+1}H_{k-1}};\label{dva}\\
 &F(t) = -\frac{|k|}{k+1} \frac{H_{k+1}}{H_{k}}\, F'. \label{tri}\end{aligned}$$

If we plug (\[dva\]) and (\[tri\]) into (\[determinant\]) we obtain that the left hand side of (\[determinant\]) is zero. If we plug (\[dva\]) and (\[tri\]) into (\[trace\]) we obtain $$\begin{aligned}
\text{LHS of}\; (\ref{trace}) = \left[ \frac{(kH_{k-1}^{2}-H_{k}^{2}+H_{k-1}H_{k+1})^{2}+H_{k-1}^{2}H_{k}^{2}}{H_{k}^{2}(H_{k}^{2}-H_{k+1}H_{k-1})}\right]\, F'. \end{aligned}$$ Thus the sign of LHS of (\[trace\]) coincides with the sign of $F'$ which coincides with $\mathrm{sign}(-(k+1))$. The condition $p \in \mathbb{R}\setminus [0,1]$ implies that $k>-1$ and therefore (\[matrica\]) holds. The condition $p \in (0,1)$ implies that $k<-1$ and therefore inequality in (\[matrica\]) is reversed.

Thus we have obtained (\[ivo1\]) for smooth bounded functions $f \geq \varepsilon$. Next we claim that for an arbitrary smooth bounded $f\geq 0$ with $\int_{\mathbb{R}^{n}}f^{p} d\gamma_{n}<\infty$ we can apply the inequality to $f_{\varepsilon}:= f +\varepsilon$ and send $\varepsilon$ to $0$ in (\[our\]). Indeed, it follows from (\[bdef\]) and (\[loran\]) that as $t \to \infty$ we have $$\begin{aligned}
&F(t) \sim -t^{1+\frac{1}{k}} (H'_{k}(R_{k}))^{-\frac{1}{k}} \quad \text{for} \quad k>0;\\
&F(t) \sim \mathrm{sign}(-1-k) \left(\frac{e^{t^{2}/2} \, \sqrt{2 \pi}}{t \Gamma(-1-k) }\right)^{-\frac{1}{k}} |1+k|^{1+\frac{1}{k}}\quad \text{for} \quad k<0, \quad k \neq -1. \end{aligned}$$ Thus for $p>1$ (i.e., $k>0$) the claim about the limit follows from the estimate $|F(t)| \leq C_{1}+ C_{2} t^{p}$ together with the Lebesgue dominated convergence theorem.

If $p <0$ (i.e., $k \in (-1,0)$) we rewrite (\[our\]) in a standard way as follows $$\begin{aligned}
\label{lim}
\int_{\mathbb{R}^{n}} f_{\varepsilon}^{p} d\gamma_{n} - \left( \int_{\mathbb{R}^{n}} f_{\varepsilon} d\gamma_{n} \right)^{p} \leq \int_{\mathbb{R}^{n}}f_{\varepsilon}^{p}\left(1-
F\left(\frac{|\nabla f |}{f_{\varepsilon}}\right)\right) d\gamma_{n}.\end{aligned}$$ Since $f$ is bounded, $f\geq 0$ and $\int_{\mathbb{R}^{n}} f^{p} d\gamma_{n} <\infty$ there is no issue with the left hand side of (\[lim\]) when $\varepsilon \to 0$. For the right hand side of (\[lim\]) we notice that the function $x^{p}(1-F(y/x))$ is nonnegative and decreasing in $x$ then the claim follows from the monotone convergence theorem. The non negativity follows from the observation that $F(0)=1$ and $F'<0$ (see (\[odin\]) where we have $k>-1$). The monotonicity follows from (\[bdef\]), (\[odin\]), (\[pirveli\]) and the straightforward computations $$\begin{aligned}
\label{hh4}
\frac{\partial }{\partial x} \left(x^{p}(1-F(y/x)) \right)=x^{p-1}\left(p-pF(t)+tF'(t) \right)=x^{p-1} p \left[1-\frac{q}{H_{k}^{\frac{1}{k}}(q)} \right],\end{aligned}$$ where $|k| \frac{H_{k-1}(q)}{H_{k}(q)}=t=\frac{y}{x}$ and $q\in (R_{k}, \infty)$. The last expression in (\[hh4\]) is negative because $$\begin{aligned}
1 \geq F(t) = \frac{H_{k+1}}{H_{k}^{1+\frac{1}{k}}}=\frac{qH_{k}-kH_{k-1}}{H_{k}^{1+\frac{1}{k}}} >\frac{q}{H_{k}^{\frac{1}{k}}}. \end{aligned}$$

Finally if $p \in (0,1)$ (i.e., $k <-1$) we have the opposite inequality in (\[lim\]). In this case the situation is absolutely the same as for $k \in (-1,0)$ except now we should consider the function $x^{p}(F(y/x)-1)$ which is nonnegative and decreasing in $x$ (see (\[hh4\])). This finishes the proof of the theorem.

Now let us show Proposition \[prob\]. Since $F(0)=1$ it is enough to show a stronger inequality, namely $F'+p(p-1)t \geq 0$. From (\[odin\]) and the fact that $k\geq 1$ (since $p \in [1,2]$) we obtain that it is enough to show the following inequality $$\begin{aligned}
 -\frac{p}{H_{k}^{1/k}}+p(p-1)\frac{H'_{k}}{H_{k}}\geq 0 \quad \text{for all} \quad  k \geq 1, \; q \in (R_{k}, \infty).
 \end{aligned}$$ Using (\[pirveli\]) and $p=1+\frac{1}{k}$ we notice that the inequality can be rewritten as follows $1>\frac{H_{k}(q)}{H_{k-1}^{\frac{k}{k-1}}(q)}$ for all $q \in (R_{k}, \infty)$. To verify the last inequality we remind that $F(0)=1$ and $F'(t) <0$. Therefore $F(t)\leq 1$. We recall the definition of $F(t)$ (see (\[opr\])). It follows that $1\geq F = \frac{H_{k+1}}{H_{k}^{1+1/k}}$ for all $k>0$. The last inequality is the same as $$\begin{aligned}
1>\frac{H_{k}(q)}{H_{k-1}^{\frac{k}{k-1}}(q)} \quad \text{for all} \quad q \in (R_{k}, \infty), \quad k \geq 1.\end{aligned}$$ This finishes the proof of the theorem.

Proof of Corollary \[e-sob\] and Corollary \[nlog\]:
----------------------------------------------------

Notice that as $t \to 0$ we have $$\begin{aligned}
F_{\exp}(y) = 1-\frac{y^{2}}{2}+O(y^{4}) \quad \text{and} \quad F_{-\ln}(y) = -\frac{y^{2}}{2}+O(y^{4}). \end{aligned}$$ There are two ways to obtain the corollaries.

### The first way:

One can check that $$\begin{aligned}
&M_{\exp}(x,y)=e^{x}F_{\exp}(y), \quad M_{\exp}(x,0)=e^{x}, \quad M_{\exp}(x,\sqrt{y})\in C^{2}(\mathbb{R}\times \mathbb{R}_{+}); \\
&M_{-\ln}(x,y)=-\ln(x)+F_{-\ln}\left(\frac{y}{x}\right), \quad M_{-\ln }(x,0)=-\ln x, \quad x>0, \end{aligned}$$ and $M_{-\ln}(x,\sqrt{y}) \in C^{2}([\varepsilon, \infty) \times \mathbb{R}^{+})$ for any $\varepsilon>0$. By straightforward computations we notice that if we set $\psi(q) = \alpha - \int_{1}^{q} H_{-1}(s)ds$ then using the identity $1=qH_{-1}(q)+H_{-2}(q)$ we obtain $$\begin{aligned}
F_{\exp}(H_{-1}) = q e^{\psi}, \quad F'_{\exp}(H_{-1})=-e^{\psi} \quad \text{and}  \quad F''_{\exp}(H_{-1}) = - \frac{H_{-1}}{H_{-2}}.\end{aligned}$$ Similarly we compute that $$\begin{aligned}
F'_{-\ln}\left(\frac{H_{-2}}{H_{-1}}\right)=-H_{-1}\quad \text{and} \quad F''_{-\ln}\left(\frac{H_{-2}}{H_{-1}}\right)=- \frac{H_{-2}H_{-1}^{2}}{H_{-1}^{2}-H_{-2}}.\end{aligned}$$ Next one notices that $M_{\exp}$ and $M_{-\ln}$ satisfy (\[matrica\]) (in fact the determinant of (\[matrica\]) is zero). Then by Theorem 1 in [@IV] we obtain the corollaries. The passage to the limit for $M_{-\ln}(x,y)$ when $\varepsilon \to 0$ follows from the monotone convergence theorem. Indeed, we notice that $-F_{-\ln}(y/x) \geq 0$ is decreasing in $x$. We apply Corollary \[nlog\] to $f_{\varepsilon}=f+\varepsilon$ and send $\varepsilon \to 0$.

### The second way:

We will obtain the corollaries as a limiting case of Theorem \[better\]. Indeed, to verify Corollary \[e-sob\] let $f^{p}=e^{g}$ in (\[our\]). Then (\[our\]) takes the form $$\begin{aligned}
\label{ee-sob}
\int_{\mathbb{R}^{n}}e^{g} F_{\frac{1}{p-1}}\left( \frac{|\nabla g|}{p}\right)d\gamma_{n} \leq \left( \int_{\mathbb{R}^{n}} e^{g/p} d\gamma_{n}\right)^{p}.\end{aligned}$$ Now we take $p \to \infty$. The RHS of (\[ee-sob\]) tends to $\exp(\int_{\mathbb{R}^{n}}  g d\gamma_{n})$. For the LHS of (\[ee-sob\]) we should compute the limit $$\begin{aligned}
F_{\exp}(t) :=\lim_{p \to \infty} F_{\frac{1}{p-1}}\left( \frac{t}{p}\right) = \lim_{p \to \infty} F_{\frac{1}{p-1}}\left( \frac{t}{p-1}\right) = \lim_{k \to 0+} F_{k}(tk). \end{aligned}$$ It is clear that $F_{\exp}(0)=1$. Next if we take $k \to 0+$ in (\[bdef\]) we obtain $$\begin{aligned}
\lim_{k \to 0+} F_{k}\left(\left| \frac{H'_{k}}{H_{k}} \right| \right) = \lim_{k \to 0+} F_{k}\left(k \frac{H_{k-1}}{H_{k}}  \right) = \lim_{k \to 0+} F_{k}\left(k \frac{H_{-1}}{H_{0}}  \right) = F_{\exp}(H_{-1})\end{aligned}$$

On the other hand for the RHS of (\[bdef\]) we have $$\begin{aligned}
\lim_{k \to 0+}  \frac{H_{k+1}(q)}{H_{k}^{1+\frac{1}{k}}} = q \lim_{k \to 0+} H_{k}^{-1/k}.\end{aligned}$$ Here we have used $H_{0}(q)=1$ and $H_{1}(q)=q$. Thus it remains to find $\lim_{k \to 0+}H_{k}^{-1/k}$. Notice that $H(x,k):=H_{k}(x)$ is an entire function in $x$ and $k$ (see [@temme] for the Parabolic cylinder function). If we take derivative in $k$ of (\[pirveli\]) we obtain $H_{xk}(x,k)=H(x,k-1)+kH_{k}(x,k)$ (here subindices denote partial derivatives). Now taking $k=0$ we obtain $H_{xk}(x,0)=H(x,-1)$. Thus $H_{k}(x,0)$ is an antiderivative of $H(x,-1)=H_{-1}$. So $$\begin{aligned}
\lim_{k \to 0+} H_{k}^{-1/k}= \lim_{k \to 0+} \exp\left( -\frac{1}{k}\ln (1+kH_{k}(x,0)+o(k))\right) = \exp\left(- \int H_{-1}(s) ds \right).\end{aligned}$$ Finally we obtain $$\begin{aligned}
\label{hh1}
F_{\exp}(H_{-1}(q)) = q \exp \left(C - \int_{1}^{q}H_{-1} \right)\end{aligned}$$

In order to satisfy the condition $F_{\exp}(0)=1$ the constant $c$ must be chosen as follows $C= \int_{1}^{\infty}(H_{-1}-\frac{1}{s})ds$ (indeed send $q \to \infty$ in (\[hh1\])). This finishes the proof of Corollary \[e-sob\]. It is worth mentioning that we have also obtained (see (\[kap\])) $$\begin{aligned}
H_{k}(x,0)=\int_{1}^{x}H_{-1}(s)ds - \alpha.\end{aligned}$$

To verify Corollary \[nlog\] let $F(x,k):=F_{k}(x)$. Let $F_{k}(x,k)$ denotes the partial derivative in $k$ of $F(x,k)$. If we send $p \to 0, p<0$ in (\[our\]) and compare the terms of order $p$ we obtain $$\begin{aligned}
\int_{\mathbb{R}^{n}}\left(  \ln f - F_{k}\left(\frac{|\nabla f|}{f}, -1\right) \right)d\gamma_{n} \geq \ln \left(\int_{\mathbb{R}^{n}} f d\gamma_{n} \right)\end{aligned}$$ It remains to find the function $F_{k}(x,-1)$. Let us equate terms of order $(k+1)$ as $k \to -1, k<-1$ in the following equality $$\begin{aligned}
F\left(\frac{H_{x}(x,k)}{H(x,k)},k \right) = \frac{H(x,k+1)}{H(x,k)^{1+\frac{1}{k}}}. \end{aligned}$$ The straightforward computation shows that $$\begin{aligned}
F_{k}\left( \frac{H_{-2}(x)} {H_{-1}(x)},-1\right) = H_{k}(x,0)+\ln H_{-1}(x)=\int_{1}^{x}H_{-1}(s)ds - \alpha + \ln H_{-1}(x)\end{aligned}$$ where $$\begin{aligned}
\alpha=\int_{1}^{\infty}\left( H_{-1}(s) - \frac{1}{s}\right)ds.\end{aligned}$$

Concluding remarks
==================

The reader may wander how we guessed the choice (\[f1\]). Of course it was not a random guess. Function (\[f1\]) is the best possible in the sense that the determinant of (\[matrica\]) is identically zero $$\begin{aligned}
\label{monge}
&M_{yy}(M_{xx}+\frac{M_{y}}{y})-M_{xy}^{2}=0,\\
&M(x,0)=x^{p} \quad \text{for} \quad   x \geq 0. \nonumber\end{aligned}$$ Initially this was the way we started looking for $M(x,y)$ as the solution of the Monge–Ampère equation with a drift (\[monge\]). By a proper change of variables the equation reduces to the backwards heat equation (see [@IV] for more details where the connection with R. Bryant, Ph. Griffiths theory of exterior differential systems was exploited) $$\begin{aligned}
 &u_{xx}+u_{t}=0, \label{h1}\\
 &u(x,0)=C x^{\frac{p}{p-1}} \label{h2} \quad \text{for} \quad  x\geq 0.
 \end{aligned}$$ One can notice that the Hermite polynomials do satisfy (\[h1\]) and (\[h2\]) when $\frac{p}{p-1}$ is a positive integer. In general, one should invoke Hermite functions and this is the reason of appearance of these functions in our theorem. Another possibility is to assume that $M(x,y)$ should be homogeneous of degree $p$ which enforces $M$ to have the form (\[k1\]) for some $F$. Next setting $h=\frac{F}{F'}$ and further by a subtle change of variables one obtains Hermite differential equation (\[hermite\]).

Nevertheless, for the formal proof of Theorem \[better\] we do not need to go through the details. We have $M(x,y)$ defined by (\[f1\]) and we just need to check that it satisfies the desired properties.

The fact that $M(x,y)$ (see (\[f1\])) satisfies (\[matrica\]) makes it possible to extend Theorem \[better\] in a semigroup setting for uniformly log-concave measures. Indeed, let $d\mu = e^{-U}dx$ where $\mathrm{Hess}\, U \geq R\cdot Id,$ $R>0$. Let $L=\Delta - \nabla U \cdot \nabla$, and let $P_{t} = e^{tL}$ be the semigroup with generator $L$ (see [@IV; @BGL]).

For any $p \in \mathbb{R}\setminus [0,1]$ and any smooth bounded $f\geq0$ with $\int_{\mathbb{R}^{n}} f^{p} d\mu <\infty$ we have $$\begin{aligned}
 P_{t} \left[f^{p} F_{\frac{1}{p-1}}\left(\frac{| \nabla f|}{f\sqrt{R}}\right) \right] \leq (P_{t} f)^{p} F_{\frac{1}{p-1}}\left(\frac{| \nabla P_{t} f|}{P_{t} f \sqrt{R}}\right).
 \end{aligned}$$ The inequality is reversed if $p\in (0,1)$.

Notice that $\tilde{M}(x,y) = M(x,\frac{y}{\sqrt{R}})$ satisfies (\[matrica\]). Now it remains to use inequality (2.3) from [@IV].

Next by taking $t \to \infty$ and using the fact that $| \nabla P_{t} f | \leq e^{-tR} P_{t} | \nabla f |$ we obtain the following corollary

Let $d\mu =e^{-U}dx$ where $\mathrm{Hess}\, U \geq R \cdot Id$ for some $R>0$. For any $p \in \mathbb{R}\setminus [0,1]$ and any smooth bounded $f \geq  0$ with $\int_{\mathbb{R}^{n}} f^{p} d\mu<\infty$ we have $$\begin{aligned}
 \int_{\mathbb{R}^{n}} f^{p} F_{\frac{1}{p-1}}\left(\frac{| \nabla f|}{f\sqrt{R}}\right) d\mu  \leq  \left( \int_{\mathbb{R}^{n}} f d\mu \right)^{p}.
  \end{aligned}$$ The inequality is reversed if $p \in (0,1)$.

See Corollary 1 in [@IV].

The limiting cases of these inequalities when $p \to \pm \infty$ and $p \to 0$ should be understood in the sense of functions $M_{\exp}$ and $M_{-\ln}$ as in Corollary \[e-sob\] and Corollary \[nlog\].

Finally we would like to mention that having characterization (\[matrica\]) of functional inequalities (\[ivo1\]) makes approach to the problem (\[our\]) systematic. Very similar *local estimates* happen to rule some *global inequalities*. We refer the reader to our recent papers on this subject [@IV01; @IV02; @IV03].

Acknowledgements {#acknowledgements .unnumbered}
----------------

We are very grateful to Robert Bryant who suggested a change of variables in (\[determinant\]).

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[^1]: AV is partially supported by the NSF grant DMS-1600065 and by the Hausdorff Institute for Mathematics, Bonn, Germany
