# An inequality for the convolutions on unimodular locally compact groups and
the optimal constant of Young’s inequality

Takashi Satomi

###### Abstract

Let $\mu$ be the Haar measure of a unimodular locally compact group $G$ and
$m(G)$ as the infimum of the volumes of all open subgroups of $G$. The main
result of this paper is that

$\displaystyle\int_{G}f\circ\left(\phi_{1}*\phi_{2}\right)\left(g\right)dg\leq\int_{\mathbb{R}}f\circ\left(\phi_{1}^{*}*\phi_{2}^{*}\right)\left(x\right)dx$

holds for any measurable functions $\phi_{1},\phi_{2}\colon
G\to\mathbb{R}_{\geq 0}$ with
$\mu(\mathrm{supp}\;\phi_{1})+\mu(\mathrm{supp}\;\phi_{2})\leq m(G)$ and any
convex function $f\colon\mathbb{R}_{\geq 0}\to\mathbb{R}$ with $f(0)=0$. Here
$\phi^{*}$ is the rearrangement of $\phi$.

Let $Y_{O}(P,G)$ and $Y_{R}(P,G)$ denote the optimal constants of Young’s and
the reverse Young’s inequality, respectively, under the assumption
$\mu(\mathrm{supp}\;\phi_{1})+\mu(\mathrm{supp}\;\phi_{2})\leq m(G)$. Then we
have $Y_{O}(P,G)\leq Y_{O}(P,\mathbb{R})$ and $Y_{R}(P,G)\geq
Y_{R}(P,\mathbb{R})$ as a corollary. Thus, we obtain that $m(G)=\infty$ if and
only if $H(p,G)\leq H(p,\mathbb{R})$ in the case of $p^{\prime}:=p/(p-1)\in
2\mathbb{Z}$, where $H(p,G)$ is the optimal constant of the Hausdorff–Young
inequality.

Keywords: convolution, convexity, locally compact group, rearrangement,
$L^{p}$-space, Young’s inequality, reverse Young’s inequality, Hausdorff–Young
inequality.

MSC2020: Primary 46E30; Secondary 22D15, 28C10, 39B62, 42A85, 43A05.

##  1 Introduction

The main result of this paper is a generalization of an inequality for the
convolution and the rearrangement by Wang–Madiman [WM14, Theorem 7.4] to any
unimodular locally compact group $G$ (Theorem 1.2). As a corollary, the
optimal constants of Young’s inequality (Corollary 1.3 (1)) and the
Hausdorff–Young inequality (Corollary 3.5) are bounded from above, and the
optimal constant of the reverse Young’s inequality is bounded from below
(Corollary 1.3 (2)). These inequalities are best possible for $G=\mathbb{R}$.

The rearrangement of a measurable function is defined as follows.

######  Definition 1.1.

The rearrangement $\phi^{*}\colon\mathbb{R}\to\mathbb{R}_{\geq 0}$ of a
measurable function $\phi\colon G\to\mathbb{R}_{\geq 0}$ on a measure space
$(G,\mu)$ is defined as

$\displaystyle\phi^{*}(x):=\inf\\{t\geq
0\mid\mu(\phi^{-1}(\mathbb{R}_{>t}))\leq 2|x|\\}.$

For $G=\mathbb{R}$, the function $\phi^{*}$ is called the symmetric decreasing
rearrangement of $\phi$. Definition 1.1 was essentially defined by O’Neil
[O’N63, Definition 1.2] to generalize the Lorentz space [Lor49, Section 1] to
any measurable space $G$.

Now, we write $\mathrm{supp}\;\phi:=\\{g\in G\mid\phi(g)\neq 0\\}$ (we assume
that the definition of measurable function includes the assumption that
$\mathrm{supp}\;\phi$ is $\sigma$-finite). We denote by $\mu$ the Haar measure
of a unimodular locally compact group $G$ and by $m(G)$ the infimum of the
volumes of all open subgroups of $G$. The main theorem of this paper is as
follows.

######  Theorem 1.2.

Suppose measurable functions $\phi_{1},\phi_{2}\colon G\to\mathbb{R}_{\geq 0}$
on a unimodular locally compact group $G$ satisfy

$\displaystyle\mu(\mathrm{supp}\;\phi_{1})+\mu(\mathrm{supp}\;\phi_{2})$
$\displaystyle\leq m(G),$ (1.1)
$\displaystyle\phi_{1}*\phi_{2}(g),\phi_{1}^{*}*\phi_{2}^{*}(x)$
$\displaystyle<\infty\;\text{a.e.},$ (1.2)

and a convex function $f\colon\mathbb{R}_{\geq 0}\to\mathbb{R}$ satisfies
$f(0)=0$. If $\int_{G}f\circ\left(\phi_{1}*\phi_{2}\right)\left(g\right)dg$
and
$\int_{\mathbb{R}}f\circ\left(\phi_{1}^{*}*\phi_{2}^{*}\right)\left(x\right)dx$
can be defined, then we have

$\displaystyle\int_{G}f\circ\left(\phi_{1}*\phi_{2}\right)\left(g\right)dg\leq\int_{\mathbb{R}}f\circ\left(\phi_{1}^{*}*\phi_{2}^{*}\right)\left(x\right)dx.$
(1.3)

We note that $m(G)=\infty$ if and only if $G$ has no open compact subgroups
(Remark 2.2). In this case, Theorem 1.2 holds even when
$\mu(\mathrm{supp}\;\phi_{1})=\infty$ or
$\mu(\mathrm{supp}\;\phi_{2})=\infty$.

Theorem 1.2 was proved in the case of $G=\mathbb{R}$ or some cases of $f$ as
Table 1.1. For the Brunn–Minkowski inequality and Kemperman’s result, see the
previous paper [Sat21, Corollary 1.3] of the author. Now, we consider the case
of $G=\mathbb{R}$. For $f(y):=-y^{p}$ with $0<p\leq 1$, Theorem 1.2 was proved
by Brascamp–Lieb [BL76, Proposition 9] to improve the reverse Young’s
inequality (Fact 2.6 (2)) and the Prékopa–Leindler inequality. When $\phi_{1}$
and $\phi_{2}$ are integrable and $f$ is any convex function, Theorem 1.2 was
essentially proved by Wang–Madiman to bound Shannon’s differential entropy of
$\phi_{1}*\phi_{2}$ from below [WM14, Theorem 7.4].

Table 1.1: Comparison of Theorem 1.2 with some known results $f(y)\quad\backslash\quad G$ | $G=\mathbb{R}$ |  $G$: unimodular locally compact group
---|---|---
$\left\\{\begin{aligned} &0&&\text{if}\;y=0\\\ &-1&&\text{if}\;y>0\end{aligned}\right.$ | the Brunn–Minkowski inequality | Kemperman [Kem64, Theorem 1.2]
$-y^{p}\;(0<p\leq 1)$ | Brascamp–Lieb [BL76, Proposition 9] | Theorem 1.2
positive convex | Burchard [Bur94, Corollary 1]
convex function | Wang–Madiman [WM14, Theorem 7.4]
not convex | not hold (Section 4)

In the case of $G=\mathbb{R}^{n}$ for $n\geq 2$, the symmetric decreasing
rearrangement $\phi^{\star}\colon\mathbb{R}^{n}\to\mathbb{R}_{\geq 0}$ defined
in Wang–Madiman’s paper differs from $\phi^{*}$ in Definition 1.1.
Nevertheless, there is a one-to-one correspondence between $\phi^{*}$ and
$\phi^{\star}$. In this case, one can see that Wang–Madiman’s result is
sharper than Theorem 1.2 by replacing $\phi_{1}$ and $\phi_{2}$ in Theorem 1.2
with $\phi_{1}^{\star}$ and $\phi_{2}^{\star}$, respectively.

When $\phi_{1}$ and $\phi_{2}$ are characteristic functions on a unimodular
locally compact group $G$, Theorem 1.2 was previously obtained by the author
[Sat21, Theorem 1.1]. By using this result and the layer cake representation
(Section 5), this result is generalized to any measurable functions $\phi_{1}$
and $\phi_{2}$ as in Theorem 1.2.

By Theorem 1.2, the integral for the convolution on $G$ is bounded by the
integral on $\mathbb{R}$. Thus, some inequalities on $G$ can be obtained from
inequalities on $\mathbb{R}$. For example, we bound the optimal constant of
Young’s inequality from above and the reverse Young’s inequality from below
(Corollary 1.3). Let

$\displaystyle
B(p):=\frac{p^{1/2p}}{|p^{\prime}|^{1/2p^{\prime}}},\quad\frac{1}{p}+\frac{1}{p^{\prime}}=1$

and $\|\cdot\|_{p}$ denotes the $L^{p}$-norm on $G$ for $p\in\mathbb{R}_{>0}$.
For $p_{1},p_{2}>0$ with

$\displaystyle\frac{1}{p_{1}}+\frac{1}{p_{2}}>1,$ (1.4)

we let the real number $q(p_{1},p_{2})>0$ such that

$\displaystyle\frac{1}{q(p_{1},p_{2})}=\frac{1}{p_{1}}+\frac{1}{p_{2}}-1.$
(1.5)

We note that $q(p_{1},p_{2})>1$ holds for any $p_{1},p_{2}>1$ with (1.4). We
have (1.4) and $q(p_{1},p_{2})<1$ for any $0<p_{1},p_{2}<1$. Theorem 1.2 shows
the following corollary.

######  Corollary 1.3.

Suppose measurable functions $\phi_{1},\phi_{2}\colon G\to\mathbb{R}_{\geq 0}$
on a unimodular locally compact group $G$ satisfy (1.1).

  1. (1)

(a stronger version of Young’s inequality) We have

$\displaystyle\|\phi_{1}*\phi_{2}\|_{q(p_{1},p_{2})}\leq\frac{B(p_{1})B(p_{2})}{B\circ
q(p_{1},p_{2})}\|\phi_{1}\|_{p_{1}}\|\phi_{2}\|_{p_{2}}$

for any $p_{1},p_{2}>1$ with (1.4).

  2. (2)

(a stronger version of the reverse Young’s inequality) We have

$\displaystyle\|\phi_{1}*\phi_{2}\|_{q(p_{1},p_{2})}\geq\frac{B(p_{1})B(p_{2})}{B\circ
q(p_{1},p_{2})}\|\phi_{1}\|_{p_{1}}\|\phi_{2}\|_{p_{2}}$

for any $0<p_{1},p_{2}<1$.

When $m(G)=\infty$, Corollary 1.3 can be regarded as an extension of
Fournier’s result [Fou77, Theorem 1] (Subsection 2.2). In this case, we have
an upper bound of the optimal constant of the Hausdorff–Young inequality
(Corollary 3.5) by using Corollary 1.3 (1) and a result of Klein–Russo (Fact
3.4). It was proved that Corollary 1.3 is best possible for $G=\mathbb{R}$
(Fact 2.6). Although Corollary 1.3 was already proved in particular cases of
$G$, Corollary 1.3 includes new cases such as compact groups or semisimple Lie
groups as far as the author knows (Section 2).

Here is the organization of this paper. In Section 2, we will compare
Corollary 1.3 with some known results and show Corollary 1.3 by using Theorem
1.2. In Section 3, we will bound the optimal constant of the Hausdorff–Young
inequality from above by using Corollary 1.3 (1). In Section 4, we will see
that the function $f$ in Theorem 1.2 must be a convex function for
$G=\mathbb{R}$. In Section 5, we will summarize some properties of the
rearrangement to show Theorem 1.2. Sections 6-8 will be devoted to the proof
of Theorem 1.2. In Section 6, we will show Theorem 1.2 for certain convex
functions $f=f_{t}$. In Section 7, we will show Theorem 1.2 in the case where
$\phi_{1}*\phi_{2}$ is integrable. In Section 8, we will complete the proof of
Theorem 1.2.

##  2 The optimal constant of (the reverse) Young’s inequality

In this section, we compare Corollary 1.3 with some known results, and show
Corollary 1.3 by using Theorem 1.2. Here is a summary of this section. We
define the optimal constants of Young’s inequality (Fact 2.3) and the reverse
Young’s inequality (Theorem 2.4) as $\tilde{Y}_{O}(P,G)$ and
$\tilde{Y}_{R}(P,G)$, respectively, for
$P=(p_{1},p_{2})\in\mathbb{R}_{>0}^{2}$. Similarly, we define these optimal
constants as $Y_{O}(P,G)$ and $Y_{R}(P,G)$ under the assumption (1.1). We will
define $\tilde{Y}_{O}(P,G)$, $\tilde{Y}_{R}(P,G)$, $Y_{O}(P,G)$ and
$Y_{R}(P,G)$ precisely in Definition 2.1 later. Similarly, we denote by
$H(p,G)$ the optimal constant of the Hausdorff–Young inequality for $1<p\leq
2$ (Definition 3.1). Some known results about the optimal constants are
summarized in Table 2.1.

Table 2.1: Some known results about the optimal constants $m(G)<\infty$ | $G=\mathbb{R}^{n}$ | $m(G)=\infty$
---|---|---
$Y_{O}(P,G)\leq C(P)$ |  |
Corollary 1.3 (1) | $Y_{O}(P,G)=\tilde{Y}_{O}(P,G)=C(P)^{n}$ | $Y_{O}(P,G)=\tilde{Y}_{O}(P,G)\leq C(P)$
$\tilde{Y}_{O}(P,G)=1$ | Beckner (Fact 2.6 (1)) | Corollary 1.3 (1)
Fournier [Fou77, Theorem 3] |  |
$Y_{R}(P,G)\geq C(P)$ |  |
Corollary 1.3 (2) | $Y_{R}(P,G)=\tilde{Y}_{R}(P,G)=C(P)^{n}$ | $Y_{R}(P,G)=\tilde{Y}_{R}(P,G)\geq C(P)$
$\tilde{Y}_{R}(P,G)=1$ | Brascamp–Lieb [BL76] ($n=1$), | Corollary 1.3 (2)
Theorem 2.4, Remark 2.5 | Barthe (Fact 2.6 (2)) |
|  |  $H(p,G)\leq B(p)$ if $p^{\prime}\in 2\mathbb{Z}$
$H(p,G)=1$ | $H(p,G)=B(p)^{n}$ | Russo [Rus78, Theorem 3 (b)],
Russo [Rus74, Theorem 1] | Babenko [Bab61] ($p^{\prime}\in 2\mathbb{Z}$), | Corollary 3.5
| Beckner [Bec75, Theorem 1] | $H(p,G)<1$
|  | Fournier [Fou77, Theorem 2]

Corollary 1.3 implies

$\displaystyle Y_{O}(P,G)\leq C(P),\quad Y_{R}(P,G)\geq C(P),\quad
C(P):=\frac{B(p_{1})B(p_{2})}{B\circ q(p_{1},p_{2})}.$

When $m(G)=\infty$, we have $Y_{O}(P,G)=\tilde{Y}_{O}(P,G)$ and
$Y_{R}(P,G)=\tilde{Y}_{R}(P,G)$ by definition. For $G=\mathbb{R}^{n}$, the
constants $Y_{O}(P,G)$ and $Y_{R}(P,G)$ were given explicitly (Fact 2.6).
Corollary 1.3 can be proved in some cases of $G$ by using a result of
Cowling–Martini–Müller–Parcet (Fact 2.8). On the other hand, when $G$ is a
semisimple Lie group, Corollary 1.3 has not been known as far as the author
knows.

When $m(G)<\infty$, we have $\tilde{Y}_{O}(P,G)=1$ and $\tilde{Y}_{R}(P,G)=1$
(Remark 2.5). So, we consider $Y_{O}(P,G)$ and $Y_{R}(P,G)$ instead of
$\tilde{Y}_{O}(P,G)$ and $\tilde{Y}_{R}(P,G)$. When $G$ is a Lie group and
both of $\mathrm{supp}\;\phi_{1}$ and $\mathrm{supp}\;\phi_{2}$ are
sufficiently small (even smaller than ones in (1.1)), the optimal constant is
determined only by $\dim G$ (Fact 2.9). This fact gives a stronger upper bound
than Corollary 1.3. On the other hand, there are more functions which can be
applied to Corollary 1.3.

In Subsection 2.1, we will define the optimal constants $\tilde{Y}_{O}(P,G)$,
$\tilde{Y}_{R}(P,G)$, $Y_{O}(P,G)$, and $Y_{R}(P,G)$. In Subsection 2.2, we
will compare Corollary 1.3 with some known results for $N=2$. In Subsection
2.3, we will see that similar results are valid for any $N$. In Subsection
2.4, we will show Corollary 1.3 by using Theorem 1.2.

###  2.1 The definition of the optimal constants

In this subsection, we define the optimal constant of (the reverse) Young’s
inequality.

######  Definition 2.1.

Let $N\in\mathbb{Z}_{\geq 1}$. For
$P:=(p_{1},p_{2},\cdots,p_{N})\in\mathbb{R}_{>0}^{N}$ with

$\displaystyle\sum_{k=1}^{N}\frac{1}{p_{k}}>N-1,$

we let the real number $q(P)>0$ such that

$\displaystyle\frac{1}{q(P)}=1-N+\sum_{k=1}^{N}\frac{1}{p_{k}}.$

  1. (1)

When $p_{1},p_{2},\cdots,p_{N},q(P)\neq 1$, we define

$\displaystyle C(P):=\frac{1}{B\circ q(P)}\prod_{k=1}^{N}B(p_{k}).$

  2. (2)

For a unimodular locally compact group $G$, we define

$\displaystyle\tilde{\mathcal{B}}:=\\{(\phi_{1},\phi_{2},\cdots,\phi_{N})\mid\phi_{1},\phi_{2},\cdots,\phi_{N}\colon
G\to\mathbb{R}_{\geq
0},\;\|\phi_{1}\|_{p_{1}}=\|\phi_{2}\|_{p_{2}}=\cdots=\|\phi_{N}\|_{p_{N}}=1\\}$

and

$\displaystyle\tilde{Y}_{O}(P,G)$
$\displaystyle:=\sup\\{\|\phi_{1}*\phi_{2}*\cdots*\phi_{N}\|_{q(P)}\mid(\phi_{1},\phi_{2},\cdots,\phi_{N})\in\tilde{\mathcal{B}}\\},$
$\displaystyle\tilde{Y}_{R}(P,G)$
$\displaystyle:=\inf\\{\|\phi_{1}*\phi_{2}*\cdots*\phi_{N}\|_{q(P)}\mid(\phi_{1},\phi_{2},\cdots,\phi_{N})\in\tilde{\mathcal{B}}\\}.$

  3. (3)

Let $G$ and $\tilde{\mathcal{B}}$ be as in (2). For $N=2$, we define
$\mathcal{B}$ as the set of all elements of $\tilde{\mathcal{B}}$ with (1.1)
and

$\displaystyle Y_{O}(P,G)$
$\displaystyle:=\sup\\{\|\phi_{1}*\phi_{2}\|_{q(P)}\mid(\phi_{1},\phi_{2})\in\mathcal{B}\\},$
$\displaystyle Y_{R}(P,G)$
$\displaystyle:=\inf\\{\|\phi_{1}*\phi_{2}\|_{q(P)}\mid(\phi_{1},\phi_{2})\in\mathcal{B}\\}.$

We note that $C(P)<1<q(P)$ holds for any $p_{1},p_{2},\cdots,p_{N}>1$ and
$C(P)>1>q(P)$ holds for any $0<p_{1},p_{2},\cdots,p_{N}<1$. When
$m(G)=\infty$, the assumption (1.1) always holds. Thus, for any unimodular
locally compact group $G$, we have

$\displaystyle Y_{O}(P,G)\leq\tilde{Y}_{O}(P,G),\quad
Y_{R}(P,G)\geq\tilde{Y}_{R}(P,G)$ (2.1)

and these equalities hold if $m(G)=\infty$. Actually, these equalities hold if
and only if $m(G)=\infty$ (Remark 2.5).

######  Remark 2.2 ([Sat21], Remark 2.4 (3)).

The following three conditions (1), (2), and (3) are equivalent for any
locally compact group $G$.

  1. (1)

One has $m(G)<\infty$.

  2. (2)

The locally compact group $G$ has an open compact subgroup.

  3. (3)

The identity component $G_{0}$ of $G$ is compact.

If the equivalent conditions (1), (2), and (3) are satisfied, then we have
$m(G)=\mu(G_{0})$. Thus, (1.1) holds if and only if at least one of the
following two conditions (a) and (b) are satisfied.

  1. (a)

One has
$\mu(\mathrm{supp}\;\phi_{1})+\mu(\mathrm{supp}\;\phi_{2})\leq\mu(G_{0})$.

  2. (b)

The identity component $G_{0}$ is not compact.

In particular, if $G_{0}$ is open (e.g., $G$ is a Lie group), then (1.1) and
(a) are equivalent.

###  2.2 The case of two functions

In the case of $N=1$, we have $q(p_{1})=p_{1}$ and hence

$\displaystyle
Y_{O}(p_{1},G)=\tilde{Y}_{O}(p_{1},G)=Y_{R}(p_{1},G)=\tilde{Y}_{R}(p_{1},G)=C(p_{1})=1.$

Thus, the case of $N=2$ is the simplest example such that these constants are
not trivial. In this subsection, we consider the case of $N=2$ in Definition
2.1. We have the following statements.

######  Fact 2.3 (Young’s inequality [Wei40]).

Let $G$, $P=(p_{1},p_{2})\in\mathbb{R}_{\geq 1}^{2}$, and $\tilde{Y}_{O}(P,G)$
be as in Definition 2.1 for $N=2$. Then we have $\tilde{Y}_{O}(P,G)\leq 1$.

######  Theorem 2.4 (The reverse Young’s inequality).

Let $G$, $P=(p_{1},p_{2})\in(0,1]^{2}$, and $\tilde{Y}_{R}(P,G)$ be as in
Definition 2.1 for $N=2$. Then we have $\tilde{Y}_{R}(P,G)\geq 1$.

In the case of $G=\mathbb{R}$ or $G=S^{1}$, Theorem 2.4 was proved by Leindler
[Lei72, Theorems 1 and 2].

###### Proof of Theorem 2.4.

It suffices to show $\|\phi_{1}*\phi_{2}\|_{p}^{p}\geq 1$ for any measurable
functions $\phi_{1},\phi_{2}\colon G\to\mathbb{R}_{\geq 0}$ with
$\|\phi_{1}\|_{p_{1}}=\|\phi_{2}\|_{p_{2}}=1$. Since

$\displaystyle\frac{1}{p_{1}^{\prime}}+\frac{1}{p_{2}^{\prime}}+\frac{1}{p}=1,\quad\phi_{1}(g)\phi_{2}(g^{-1}g^{\prime})=\phi_{1}(g)^{p_{1}/p_{2}^{\prime}}\phi_{2}(g^{-1}g^{\prime})^{p_{2}/p_{1}^{\prime}}(\phi_{1}(g)^{p_{1}}\phi_{2}(g^{-1}g^{\prime})^{p_{2}})^{1/p}$

for any $g,g^{\prime}\in G$ by (1.5), we have

$\displaystyle\phi_{1}*\phi_{2}(g^{\prime})\geq\|\phi_{1}\|_{p_{1}}^{p_{1}/p_{2}^{\prime}}\|\phi_{2}\|_{p_{2}}^{p_{2}/p_{1}^{\prime}}\left(\int_{G}\phi_{1}(g)^{p_{1}}\phi_{2}(g^{-1}g^{\prime})^{p_{2}}dg\right)^{1/p}=\left(\int_{G}\phi_{1}(g)^{p_{1}}\phi_{2}(g^{-1}g^{\prime})^{p_{2}}dg\right)^{1/p}$

by Hölder’s inequality. Thus,

$\displaystyle\|\phi_{1}*\phi_{2}\|_{p}^{p}\geq\int_{G}\int_{G}\phi_{1}(g)^{p_{1}}\phi_{2}(g^{-1}g^{\prime})^{p_{2}}dgdg^{\prime}=\|\phi_{1}\|_{p_{1}}^{p_{1}}\|\phi_{2}\|_{p_{2}}^{p_{2}}=1$

is obtained. ∎

######  Remark 2.5.

When $m(G)<\infty$, the equalities in Fact 2.3 and Theorem 2.4 hold. In fact,
there is an open compact subgroup $G^{\prime}\subset G$ by Remark 2.2 in this
case. Thus, the equalities in Fact 2.3 and Theorem 2.4 hold for
$\phi_{1}=\phi_{2}=1_{G^{\prime}}$, where $1_{G^{\prime}}$ is a characteristic
function of $G^{\prime}$. On the other hand, we have

$\displaystyle Y_{O}(P,G)\leq C(P)<1=\tilde{Y}_{O}(P,G)$

for $p_{1},p_{2}>1$ by Corollary 1.3 (1) and similarly

$\displaystyle Y_{R}(P,G)\geq C(P)>1=\tilde{Y}_{R}(P,G)$

for $p_{1},p_{2}<1$ by Corollary 1.3 (2). That is, the equality of (2.1) does
not hold when $m(G)<\infty$. Thus, the equality of (2.1) holds if and only if
$m(G)=\infty$.

We fix $P=(p_{1},p_{2})$ with $p_{1},p_{2}>1$. Fournier proved that there is a
real number $c(P)<1$, which is independent in $G$, such that $Y_{O}(P,G)\leq
c(P)$ for any $G$ with $m(G)=\infty$ [Fou77, Theorem 1]. Corollary 1.3 (1)
implies that The minimum of $c(P)$ satisfying this claim is $C(P)$.

For $G=\mathbb{R}^{n}$, Beckner and Brascamp–Lieb explicitly determined the
values of $Y_{O}(P,G)$ and $Y_{R}(P,G)$, respectively, as follows.

######  Fact 2.6.

Let $G$, $P=(p_{1},p_{2})$, $Y_{O}(P,G)$, $Y_{R}(P,G)$, and $C(P)$ be as in
Definition 2.1 for $N=2$, and $n\in\mathbb{Z}_{\geq 1}$.

  1. (1)

(Beckner, [Bec75, Theorem 3]) If $p_{1},p_{2}>1$, then we have
$Y_{O}(P,\mathbb{R}^{n})=C(P)^{n}$.

  2. (2)

(Barthe, [Bar98a, Theorem 1]) If $p_{1},p_{2}<1$, then we have
$Y_{R}(P,\mathbb{R}^{n})=C(P)^{n}$.

Fact 2.6 (2) was proved by Brascamp–Lieb when $n=1$ [BL76, Theorem 8].

######  Remark 2.7.

There are some proofs of Fact 2.6 (1).

  1. (1)

Beckner proved it by using the fact that pairs of Gaussian functions are
solutions of functional equations which are necessary for
$\|\phi_{1}^{*}*\phi_{2}^{*}\|_{q(P)}$ to be the maximum [Bec75, Theorem 3].

  2. (2)

Brascamp–Lieb proved it by showing
$Y_{O}(P,\mathbb{R}^{n})=Y_{O}(P,\mathbb{R})^{n}$ and by observing the
behavior of the limit of an upper bound for $Y_{O}(P,\mathbb{R}^{n})$ as
$n\to\infty$ [BL76, Section 2.5].

  3. (3)

Barthe gave a direct proof, which utilizes the change of variable by
Henstock–Macbeath [HM53, Section 5] and the weighted AM-GM inequality [Bar98a,
Theorem 1] (see also [Bar98]).

  4. (4)

Carlen–Lieb–Loss implicitly proved it by considering heat equations which have
initial values $\phi_{1}^{p_{1}}$ and $\phi_{2}^{p_{2}}$, and by showing that
the $L^{q(P)}$-norm of the convolution of these functions is increasing with
respect to time [CLL04, Theorem 3.1]. A similar argument appears in the papers
of Bennett–Carbery–Christ–Tao [Ben+08, Example 1.5] and Bennett–Bez [BB09,
Section 1.1].

  5. (5)

Cordero-Erausquin–Ledoux proved it by estimating $\exp\circ
Y_{O}(P,\mathbb{R})$ from above by a linear combination of Shannon’s
differential entropy of functions, and by calculating the Fisher information
of these functions [CEL10, Theorem 6].

Brascamp–Lieb [BL76, Theorem 8], Barthe [Bar98a, Theorem 1], and Bennett–Bez
[BB09, Section 1.1] also proved Fact 2.6 (2) by similar arguments.

By Fact 2.6, Corollary 1.3 is best possible for $G=\mathbb{R}$. Corollary 1.3
(1) can be shown by using Fact 2.6 and the following fact in some cases of
$G$.

######  Fact 2.8 (Cowling–Martini–Müller–Parcet, [Cow+19, Proposition 2.2
(iii)]).

Let $G$, $P=(p_{1},p_{2})\in\mathbb{R}_{\geq 1}^{2}$, and $\tilde{Y}_{O}(P,G)$
be as in Definition 2.1 for $N=2$. Then we have
$\tilde{Y}_{O}(P,G)\leq\tilde{Y}_{O}(P,G^{\prime})\tilde{Y}_{O}(P,G/G^{\prime})$
for any closed normal unimodular subgroup $G^{\prime}\subset G$.

When $G$ is expressed as $G=G^{\prime}\rtimes G/G^{\prime}$, Fact 2.8 was
proved by Klein–Russo [KR78, Lemma 2.4]. We can show Corollary 1.3 (1) for
some cases of $G$ such as non-compact nilpotent Lie groups by Fact 2.6 (1) and
Fact 2.8 (see also [Nie94]). On the other hand, when $G$ is a semisimple Lie
group, Corollary 1.3 cannot be proved only by this argument. As far as the
author knows, Corollary 1.3 has not been known in some cases of $G$ such as
semisimple Lie groups or compact groups.

When $G$ is a Lie group, the optimal constant of Young’s inequality depends
only on the dimension $\dim G$ under a stronger assumption than (1.1).

######  Fact 2.9 (Bennett–Bez–Buschenhenke–Cowling–Flock, [Ben+20, Corollary
2.4]).

Let $P=(p_{1},p_{2})\in\mathbb{R}_{>1}^{2}$ and $C(P)$ be as in Definition 2.1
for $N=2$. Then for any unimodular Lie group $G$ and $Y>C(P)^{\dim G}$, there
exists a non-empty open subset $U\subset G$ such that

$\displaystyle\|\phi_{1}*\phi_{2}\|_{p}\leq
Y\|\phi_{1}\|_{p_{1}}\|\phi_{2}\|_{p_{2}}$ (2.2)

holds for any measurable functions $\phi_{1},\phi_{2}\colon
G\to\mathbb{R}_{\geq 0}$ with
$\mathrm{supp}\;\phi_{1},\mathrm{supp}\;\phi_{2}\subset U$.

In the original paper [Ben+20, Corollary 2.4], Fact 2.9 was proved only when
$G$ is connected. However, this connectedness is not necessary because we can
take $U$ such as $U\subset G_{0}$. By the result of
Cowling–Martini–Müller–Parcet [Cow+19, Proposition 2.4 (i)], the constant
$C(P)^{\dim G}$ in Fact 2.9 is the best possible for any $G$.

Since the optimal constant of Fact 2.9 is smaller than that of Corollary 1.3
(when $\dim G\geq 2$), Fact 2.9 gives a stronger bound than Corollary 1.3. On
the other hand, the assumption (1.1) in Corollary 1.3 is weaker than the
assumption in Fact 2.9 and hence there are more functions which can be applied
to Corollary 1.3. In this sense, Corollary 1.3 is a new result for some
locally compact groups such as $G=S^{1}$ as far as the author knows.

To point out the difference between Corollary 1.3 and Fact 2.9 more precisely,
we consider the case of $G=\mathbb{R}\times S^{1}$. Fact 2.9 implies that
(2.2) holds for any $Y>C(P)^{2}$, $\phi_{1}$, and $\phi_{2}$ when
$\mathrm{supp}\;\phi_{1}$ and $\mathrm{supp}\;\phi_{2}$ are sufficiently
small. On the other hand, Corollary 1.3 implies that (2.2) holds for $Y=C(P)$
and any measurable functions $\phi_{1}$ and $\phi_{2}$. In fact, we can find
examples such that the equality of (2.2) holds by considering
$S^{1}$-invariant functions [Bec75, Section IV.5].

###  2.3 The case of more than two functions

For $N\geq 3$, similar statements hold by repeating the arguments in
Subsection 2.2. For example, we consider the case of $N=3$. Since

$\displaystyle\frac{1}{q(q(p_{1},p_{2}),p_{3})}=\frac{1}{q(p_{1},p_{2})}+\frac{1}{p_{3}}-1=\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}-2=\frac{1}{q(p_{1},p_{2},p_{3})},$

we have $q(q(p_{1},p_{2}),p_{3})=q(p_{1},p_{2},p_{3})$. Thus,

$\displaystyle C(p_{1},p_{2},p_{3})=\frac{B(p_{1})B(p_{2})B(p_{3})}{B\circ
q(p_{1},p_{2},p_{3})}=\frac{B(p_{1})B(p_{2})}{B\circ
q(p_{1},p_{2})}\cdot\frac{B\circ q(p_{1},p_{2})B(p_{3})}{B\circ
q(q(p_{1},p_{2}),p_{3})}=C(p_{1},p_{2})C(q(p_{1},p_{2}),p_{3})$

holds and hence we have similar statements by repeating the arguments in
Subsection 2.2. Similarly, we can generalize these statements for any
$N\in\mathbb{Z}_{\geq 1}$.

Corollary 1.3 can be generalized for any $N$ by weakening the assumption (1.1)
to $m(G)=\infty$. That is, we have the following statement by Corollary 1.3.

######  Corollary 2.10 (Corollary 1.3 for any $N$).

Let $N$, $G$, $P$, $\tilde{Y}_{O}(P,G)$, $\tilde{Y}_{R}(P,G)$, and $C(P)$ be
as in Definition 2.1 with $m(G)=\infty$.

  1. (1)

If $P\in\mathbb{R}_{>1}^{N}$, then we have $\tilde{Y}_{O}(P,G)\leq C(P)$.

  2. (2)

If $P\in(0,1)^{N}$, then we have $\tilde{Y}_{R}(P,G)\geq C(P)$.

Similarly, the other statements in Subsection 2.2 can be generalized to $N\geq
3$. That is, the following statements hold.

######  Fact 2.11 (Fact 2.3 for any $N$, Klein–Russo, [KR78, Corollary 2.3]).

Let $N$, $G$, $P\in\mathbb{R}_{\geq 1}^{N}$, and $\tilde{Y}_{O}(P,G)$ be as in
Definition 2.1. Then we have $\tilde{Y}_{O}(P,G)\leq 1$.

######  Corollary 2.12 (Theorem 2.4 for any $N$).

Let $N$, $G$, $P\in(0,1]^{N}$, and $\tilde{Y}_{R}(P,G)$ be as in Definition
2.1. Then we have $\tilde{Y}_{R}(P,G)\geq 1$.

######  Fact 2.13 (Fact 2.6 for any $N$).

We let $N$, $G$, $P$, $\tilde{Y}_{O}(P,G)$, $\tilde{Y}_{R}(P,G)$, and $C(P)$
be as in Definition 2.1, and $n\in\mathbb{Z}_{\geq 1}$.

  1. (1)

(Beckner, [Bec75, Theorem 4]) If $P\in\mathbb{R}_{>1}^{N}$, then we have
$\tilde{Y}_{O}(P,\mathbb{R}^{n})=C(P)^{n}$.

  2. (2)

If $P\in(0,1)^{N}$, then we have $\tilde{Y}_{R}(P,\mathbb{R}^{n})=C(P)^{n}$.

######  Corollary 2.14 (Fact 2.9 for any $N$).

Let $N$, $P\in\mathbb{R}_{>1}^{N}$, $q(P)$, and $C(P)$ be as in Definition
2.1. Then for any unimodular Lie group $G$ and $Y>C(P)^{\dim G}$, there exists
a non-empty open subset $U\subset G$ such that

$\displaystyle\|\phi_{1}*\phi_{2}*\cdots*\phi_{N}\|_{q(P)}\leq
Y\prod_{k=1}^{N}\|\phi_{k}\|_{p_{k}}$

holds for any measurable functions $\phi_{1},\phi_{2},\cdots,\phi_{N}\colon
G\to\mathbb{R}_{\geq 0}$ with
$\mathrm{supp}\;\phi_{1},\mathrm{supp}\;\phi_{2},\cdots,\mathrm{supp}\;\phi_{N}\subset
U$.

Fact 2.8 cannot be generalized for $N\geq 3$ only by repeating Fact 2.8.
Nevertheless, Cowling–Martini–Müller–Parcet proved Fact 2.8 for any $N$ as
follows.

######  Fact 2.15 (Fact 2.8 for any $N$, Cowling–Martini–Müller–Parcet,
[Cow+19, Proposition 2.2 (iii)]).

Let $N$, $G$, $P\in\mathbb{R}_{>1}^{N}$, and $\tilde{Y}_{O}(P,G)$ be as in
Definition 2.1. Then we have
$\tilde{Y}_{O}(P,G)\leq\tilde{Y}_{O}(P,G^{\prime})\tilde{Y}_{O}(P,G/G^{\prime})$
for any closed normal unimodular subgroup $G^{\prime}\subset G$.

###  2.4 A proof of Corollary 1.3

In this subsection, we show Corollary 1.3 by Theorem 1.2.

###### Proof of Corollary 1.3.

  1. (1)

We have
$\|\phi_{1}*\phi_{2}\|_{q(p_{1},p_{2})}\leq\|\phi_{1}^{*}*\phi_{2}^{*}\|_{q(p_{1},p_{2})}$
by applying $f(y):=y^{q(p_{1},p_{2})}$ to Theorem 1.2. Since
$\|\phi_{1}^{*}\|_{p_{1}}=\|\phi_{1}\|_{p_{1}}$ and
$\|\phi_{2}^{*}\|_{p_{2}}=\|\phi_{2}\|_{p_{2}}$, we obtain

$\displaystyle\|\phi_{1}*\phi_{2}\|_{q(p_{1},p_{2})}\leq\|\phi_{1}^{*}*\phi_{2}^{*}\|_{q(p_{1},p_{2})}\leq
C(p_{1},p_{2})\|\phi_{1}^{*}\|_{p_{1}}\|\phi_{2}^{*}\|_{p_{2}}=C(p_{1},p_{2})\|\phi_{1}\|_{p_{1}}\|\phi_{2}\|_{p_{2}}$

by Fact 2.6 (1).

  2. (2)

We have
$\|\phi_{1}*\phi_{2}\|_{q(p_{1},p_{2})}\geq\|\phi_{1}^{*}*\phi_{2}^{*}\|_{q(p_{1},p_{2})}$
by applying $f(y):=-y^{q(p_{1},p_{2})}$ to Theorem 1.2. Since
$\|\phi_{1}^{*}\|_{p_{1}}=\|\phi_{1}\|_{p_{1}}$ and
$\|\phi_{2}^{*}\|_{p_{2}}=\|\phi_{2}\|_{p_{2}}$, we obtain

$\displaystyle\|\phi_{1}*\phi_{2}\|_{q(p_{1},p_{2})}\geq\|\phi_{1}^{*}*\phi_{2}^{*}\|_{q(p_{1},p_{2})}\geq
C(p_{1},p_{2})\|\phi_{1}^{*}\|_{p_{1}}\|\phi_{2}^{*}\|_{p_{2}}=C(p_{1},p_{2})\|\phi_{1}\|_{p_{1}}\|\phi_{2}\|_{p_{2}}$

by Fact 2.6 (2). ∎

##  3 The Hausdorff–Young inequality

In this section, we give an upper bound of the optimal constant $H(p,G)$ of
the Hausdorff–Young inequality (Corollary 3.5). Klein–Russo proved that
$H(p,G)$ is determined only by the optimal constant of Young’s inequality when
$p^{\prime}\in 2\mathbb{Z}$ (Fact 3.4). Thus, Corollary 1.3 bounds $H(p,G)$
from above in this case.

There are several equivalent definitions of $H(p,G)$ (Remark 3.2). In this
paper, we adopt the definition by Russo (Definition 3.1) based on Kunze’s
argument [Kun58, Section 5]. Let $l_{\phi}\colon L^{2}(G)\to L^{2}(G)$ be the
convolution operator defined as $l_{\phi}(\xi):=\phi*\xi$ for any function
$\phi\colon G\to\mathbb{C}$ on a unimodular locally compact group $G$. For
$\phi\in L^{1}(G)$, $l_{\phi}$ is the bounded operator defined on all elements
of $L^{2}(G)$ by Fact 2.3. We define the optimal constant $H(p,G)$ of the
Hausdorff–Young inequality as follows.

######  Definition 3.1 (Russo, [Rus74, Section 1]).

We define the constant $H(p,G)$ as

$\displaystyle H(p,G):=\sup\\{\|\gamma\|_{2}^{2/p^{\prime}}\mid\gamma\in
L^{2}(G),\;\exists\phi\in
L^{1}(G),\;\|\phi\|_{p}=1,\;l_{\gamma}=|l_{\phi}|^{p^{\prime}/2}\\}$

for any unimodular locally compact group $G$ and $1<p\leq 2$.

######  Remark 3.2.

There are some equivalent definitions of $H(p,G)$.

  1. (1)

Kunze defined a space $\mathcal{F}L^{p^{\prime}}(G)$ of operators on
$L^{2}(G)$ to formulate the Hausdorff–Young inequality (Fact 3.3) by an
argument of the gage space on $L^{2}(G)$ by Segal [Seg53, Section 5]. Russo
defined $H(p,G)$ as the optimal constant of this inequality (Definition 3.1).
Haagerup generalized the definition of $\mathcal{F}L^{p^{\prime}}(G)$ for any
von Neumann algebra [Haa79] (see also [Nel74] and [Ter17]).

  2. (2)

When $G$ is of type-I, Lipsman formulated the Hausdorff–Young inequality by
integrating the Schatten norm of the Fourier transform on the Plancherel
measure [Lip74, Section 2]. Since the $L^{2}$-Schatten norm of the Fourier
transform corresponds to the $L^{2}$-norm of the Haar measure (Plancherel’s
theorem), the optimal constant of this inequality is equal to that of (1)
[Rus76, Section 1].

  3. (3)

Kosaki defined $L^{p^{\prime}}(M)$ as the interpolation space between von
Neumann algebra $M$ and its predual $M_{*}$, and proved that
$L^{p^{\prime}}(M)$ corresponds to $\mathcal{F}L^{p^{\prime}}(G)$ defined by
Haagerup [Kos84]. When $M$ is the group von Neumann algebra $VN(G)$ of $G$,
there is an isometric isomorphism between $M_{*}$ and the Fourier algebra
$A(G)\subset L^{\infty}(G)$ [Eym64]. Thus, $\mathcal{F}L^{p^{\prime}}(G)$ is
isometrically isomorphic to the interpolation space between $VN(G)$ and
$A(G)\cong VN(G)_{*}$ [FLS11, Section 6] and hence $H(p,G)$ can be also
defined by using this normed space (see also [Ter82], [Izu97], [Daw11], and
[Cas13]).

  4. (4)

Cowling–Martini–Müller–Parcet gave $H(p,G)$ explicitly [Cow+19, Proposition
2.1]:

$\displaystyle H(p,G)=\sup\\{\||l_{\phi}|^{p^{\prime}}\|_{L^{1}(G)\to
L^{\infty}(G)}^{1/p^{\prime}}\mid\|\phi\|_{p}=1\\}.$

The following fact is called the Hausdorff–Young inequality.

######  Fact 3.3 (the Hausdorff–Young inequality, [Kun58, Theorem 6 (5)]).

We have $H(p,G)\leq 1$ for any unimodular locally compact group $G$ and
$1<p\leq 2$.

For $G=\mathbb{R}/\mathbb{Z}$, Fact 3.3 was proved by Young for particular
cases of $p$ [You13, Section 6] and by Hausdorff for any $1<p\leq 2$ [Hau23].
When $G$ is abelian, Fact 3.3 was proved by Weil [Wei40]. There are many works
on the estimate of $H(p,G)$ [Bab61] [Rus74] [Bec75] [Rus76] [Fou77] [Rus77]
[KR78] [Rus78] [Ino92] [BSL03] [Bak+07] [BI14] [Cow+19] [BI20].

The constant $H(p,G)$ is determined only by $\tilde{Y}_{O}(P,G)$ for
$p^{\prime}\in 2\mathbb{Z}$ as follows.

######  Fact 3.4 (Klein–Russo, [KR78, Lemma 2.6]).

We have

$\displaystyle
H(p,G)=\tilde{Y}_{O}((\underbrace{p,p,\cdots,p}_{p^{\prime}/2}),G)^{2/p^{\prime}}$

for any unimodular locally compact group $G$ and any $1<p\leq 2$ with
$p^{\prime}\in 2\mathbb{Z}$.

The following statement holds by Corollary 2.10 (1) and Fact 3.4.

######  Corollary 3.5.

We have $H(p,G)\leq B(p)$ for any unimodular locally compact group $G$ with
$m(G)=\infty$ and any $1<p\leq 2$ with $p^{\prime}\in 2\mathbb{Z}$.

When $G$ is separable and any irreducible unitary representation of $G$ is
finite dimensional, Corollary 3.5 was proved by Russo [Rus78, Theorem 3 (b)].

###### Proof of Corollary 3.5.

We have $H(p,G)=\tilde{Y}_{O}(P,G)^{2/p^{\prime}}$ for $P:=(p,p,\cdots
p)\in\mathbb{R}^{p^{\prime}/2}$ by Fact 3.4. Since $m(G)=\infty$, we have
$\tilde{Y}_{O}(P,G)\leq C(P)$ by Corollary 2.10 (1). Here

$\displaystyle
q(P)=\left(1-\frac{p^{\prime}}{2}+\frac{p^{\prime}}{2}\cdot\frac{1}{p}\right)^{-1}=2$

holds and hence we have

$\displaystyle C(P)=\frac{B(p)^{p^{\prime}/2}}{B(2)}=B(p)^{p^{\prime}/2}$

by $B(2)=1$. Thus,

$\displaystyle H(p,G)=\tilde{Y}_{O}(P,G)^{2/p^{\prime}}\leq
C(P)^{2/p^{\prime}}=B(p)$

is obtained. ∎

##  4 The necessity of the convexity of $f$

In this section, we see the necessity of the convexity of $f$ for
$G=\mathbb{R}$. That is, the following example implies that a function $f$
satisfying Theorem 1.2 needs to be a convex function. In the case of

$\displaystyle\phi_{1}:=\frac{1_{(-1,1)}}{2},\quad\phi_{2}:=y_{1}1_{(-5,-3)\cup(-1-2\lambda,-1)\cup(1,1+2\lambda)\cup(3,5)}+y_{2}1_{(-3,-1-2\lambda)\cup(-1,1)\cup(1+2\lambda,3)}$

for $0\leq\lambda\leq 1$ and $0\leq y_{1}\leq y_{2}$, we have

$\displaystyle\phi_{1}^{*}=\frac{1_{(-1,1)}}{2},\quad\phi_{2}^{*}=y_{1}1_{(-5,2\lambda-3)\cup(3-2\lambda,5)}+y_{2}1_{(2\lambda-3,3-2\lambda)}$

(Figure 4.1, Figure 4.2, and Figure 4.3). Thus,

$\displaystyle\phi_{1}*\phi_{2}(x)=\left\\{\begin{aligned}
&y_{2}-\frac{y_{2}-y_{1}}{2}|x|&&\text{if}\;|x|\leq 2\lambda\\\ &\lambda
y_{1}+(1-\lambda)y_{2}&&\text{if}\;2\lambda\leq|x|\leq 2\lambda+2\\\
&2y_{2}-y_{1}-\frac{y_{2}-y_{1}}{2}|x|&&\text{if}\;2\lambda+2\leq|x|\leq 4\\\
&y_{1}\left(3-\frac{|x|}{2}\right)&&\text{if}\;4\leq|x|\leq 6\\\
&0&&\text{if}\;6\leq|x|\end{aligned}\right.$

(Figure 4.4) and

$\displaystyle\phi_{1}^{*}*\phi_{2}^{*}(x)=\left\\{\begin{aligned}
&y_{2}&&\text{if}\;|x|\leq 2-2\lambda\\\
&(\lambda-1)y_{1}+(2-\lambda)y_{2}-\frac{y_{2}-y_{1}}{2}|x|&&\text{if}\;2-2\lambda\leq|x|\leq
4-2\lambda\\\ &y_{1}&&\text{if}\;4-2\lambda\leq|x|\leq 4\\\
&y_{1}\left(3-\frac{|x|}{2}\right)&&\text{if}\;4\leq|x|\leq 6\\\
&0&&\text{if}\;6\leq|x|\end{aligned}\right.$

(Figure 4.5) hold and hence

$\displaystyle\int_{\mathbb{R}}f\circ(\phi_{1}^{*}*\phi_{2}^{*})(x)dx-\int_{\mathbb{R}}f\circ(\phi_{1}*\phi_{2})(x)dx=4(\lambda
f(y_{1})+(1-\lambda)f(y_{2})-f(\lambda y_{1}+(1-\lambda)y_{2})).$

Since

$\displaystyle\lambda f(y_{1})+(1-\lambda)f(y_{2})-f(\lambda
y_{1}+(1-\lambda)y_{2})\geq 0$

by the convexity of $f$, we obtain (1.3).

This example shows that a measurable function $f\colon\mathbb{R}_{\geq
0}\to\mathbb{R}$ with $f(0)=0$ satisfies Theorem 1.2 only if $f$ is a convex
function.

Figure 4.1: $\phi_{1}(x)=\phi_{1}^{*}(x)$$-1$$1$$\displaystyle\frac{1}{2}$$x$
Figure 4.2:
$\phi_{2}(x)$$-5$$-3$$-1-2\lambda$$-1$$1$$1+2\lambda$$3$$5$$y_{1}$$y_{2}$$x$
Figure 4.3: $\phi_{2}^{*}(x)$$-5$$2\lambda-3$$3-2\lambda$$5$$y_{1}$$y_{2}$$x$
Figure 4.4:
$\phi_{1}*\phi_{2}(x)$$-6$$-4$$-2-2\lambda$$-2\lambda$$2\lambda$$2+2\lambda$$4$$6$$y_{1}$$\lambda
y_{1}+(1-\lambda)y_{2}$$y_{2}$$x$ Figure 4.5:
$\phi_{1}^{*}*\phi_{2}^{*}(x)$$-6$$-4$$2\lambda-4$$2\lambda-2$$2-2\lambda$$4-2\lambda$$4$$6$$y_{1}$$y_{2}$$x$

##  5 Some properties of the rearrangement

In this section, we show some essential properties of the rearrangement and
the layer cake representation to prove Theorem 1.2.

######  Lemma 5.1.

Let $\phi\colon G\to\mathbb{R}_{\geq 0}$ be a measurable function on a measure
space $(G,\mu)$.

  1. (1)

We have

$\displaystyle(\phi^{*})^{-1}(\mathbb{R}_{>t})=\left(-\frac{\mu(\phi^{-1}(\mathbb{R}_{>t}))}{2},\frac{\mu(\phi^{-1}(\mathbb{R}_{>t}))}{2}\right)$

for any $t\geq 0$.

  2. (2)

We have

$\displaystyle 1_{A}^{-1}(\mathbb{R}_{>t})=\left\\{\begin{aligned}
&A&&\text{if}\;t<1\\\ &\emptyset&&\text{if}\;t\geq 1\end{aligned}\right.$

for any $t\geq 0$ and subset $A\subset G$.

  3. (3)

We have

$\displaystyle
1_{\phi^{-1}(\mathbb{R}_{>t})}^{*}=1_{(\phi^{*})^{-1}(\mathbb{R}_{>t})}=1_{(-\mu(\phi^{-1}(\mathbb{R}_{>t}))/2,\mu(\phi^{-1}(\mathbb{R}_{>t}))/2)}$

for any $t\geq 0$.

  4. (4)

We have

$\displaystyle\phi(g)=\int_{0}^{\infty}1_{\phi^{-1}(\mathbb{R}_{>t})}(g)dt$

for any $g\in G$.

  5. (5)

If $\mu$ is a Haar measure on a unimodular locally compact group $G$, then we
have

$\displaystyle\phi_{1}*\phi_{2}(g)=\int_{0}^{\infty}\int_{0}^{\infty}1_{L_{1}(t_{1})}*1_{L_{2}(t_{2})}(g)dt_{1}dt_{2}$

for any $g\in G$ and measurable functions $\phi_{1},\phi_{2}\colon
G\to\mathbb{R}_{\geq 0}$, where we write
$L_{1}(t):=\phi_{1}^{-1}(\mathbb{R}_{>t})$ and
$L_{2}(t):=\phi_{2}^{-1}(\mathbb{R}_{>t})$.

  6. (6)

If a pointwise increasing sequence of functions $\phi_{n}\colon
G\to\mathbb{R}_{\geq 0}$ converges pointwise to $\phi$, then $\phi_{n}^{*}$ is
a pointwise increasing sequence converging pointwise to $\phi^{*}$.

###### Proof.

  1. (1)

Since

$\displaystyle x\in(\phi^{*})^{-1}(\mathbb{R}_{>t})$
$\displaystyle\Longleftrightarrow\phi^{*}(x)>t$
$\displaystyle\Longleftrightarrow\inf\\{t^{\prime}>0\mid\mu(\phi^{-1}(\mathbb{R}_{>t^{\prime}}))\leq
2|x|\\}>t$
$\displaystyle\Longleftrightarrow\mu(\phi^{-1}(\mathbb{R}_{>t}))>2|x|$
$\displaystyle\Longleftrightarrow
x\in\left(-\frac{\mu(\phi^{-1}(\mathbb{R}_{>t}))}{2},\frac{\mu(\phi^{-1}(\mathbb{R}_{>t}))}{2}\right)$

for any $x\in\mathbb{R}$, we obtain

$\displaystyle(\phi^{*})^{-1}(\mathbb{R}_{>t})=\left(-\frac{\mu(\phi^{-1}(\mathbb{R}_{>t}))}{2},\frac{\mu(\phi^{-1}(\mathbb{R}_{>t}))}{2}\right).$

  2. (2)

We have

$\displaystyle 1_{A}^{-1}(\mathbb{R}_{>t})=\\{g\in G\mid
1_{A}(g)>t\\}=\left\\{\begin{aligned} &A&&\text{if}\;t<1\\\
&\emptyset&&\text{if}\;t\geq 1\end{aligned}\right..$

  3. (3)

Since

$\displaystyle
1_{(\phi^{*})^{-1}(\mathbb{R}_{>t})}=1_{(-\mu(\phi^{-1}(\mathbb{R}_{>t}))/2,\mu(\phi^{-1}(\mathbb{R}_{>t}))/2)}$

by (1), it suffices to show

$\displaystyle
1_{\phi^{-1}(\mathbb{R}_{>t})}^{*}=1_{(-\mu(\phi^{-1}(\mathbb{R}_{>t}))/2,\mu(\phi^{-1}(\mathbb{R}_{>t}))/2)}.$
(5.1)

We have

$\displaystyle
1_{\phi^{-1}(\mathbb{R}_{>t})}^{*}(x)=\inf\\{t^{\prime}>0\mid\mu(1_{\phi^{-1}(\mathbb{R}_{>t})}^{-1}(\mathbb{R}_{>t^{\prime}}))\leq
2|x|\\}$

for any $x\in\mathbb{R}$. Since

$\displaystyle\mu(1_{\phi^{-1}(\mathbb{R}_{>t})}^{-1}(\mathbb{R}_{>t^{\prime}}))=\left\\{\begin{aligned}
&\mu(\phi^{-1}(\mathbb{R}_{>t}))&&\text{if}\;t^{\prime}<1\\\
&0&&\text{if}\;t^{\prime}\geq 1\end{aligned}\right.$

by (2), we have

$\displaystyle\inf\\{t^{\prime}\geq
0\mid\mu(1_{\phi^{-1}(\mathbb{R}_{>t})}^{-1}(\mathbb{R}_{>t^{\prime}}))\leq
2|x|\\}$ $\displaystyle=\left\\{\begin{aligned}
&1&&\text{if}\;2|x|<\mu(\phi^{-1}(\mathbb{R}_{>t}))\\\
&0&&\text{if}\;2|x|\geq\mu(\phi^{-1}(\mathbb{R}_{>t}))\end{aligned}\right.$
$\displaystyle=1_{(-\mu(\phi^{-1}(\mathbb{R}_{>t}))/2,\mu(\phi^{-1}(\mathbb{R}_{>t}))/2)}(x).$

Thus, we obtain (5.1).

  4. (4)

We obtain

$\displaystyle\phi(g)=\int_{0}^{\phi(g)}dt=\int_{0}^{\infty}1_{\phi^{-1}(\mathbb{R}_{>t})}(g)dt.$

  5. (5)

We have

$\displaystyle\phi_{1}*\phi_{2}(g)=\int_{G}\phi_{1}(g^{\prime})\phi_{2}(g^{\prime-1}g)dg^{\prime}.$

Since

$\displaystyle\phi_{1}(g^{\prime})=\int_{0}^{\infty}1_{L_{1}(t_{1})}(g^{\prime})dt_{1},\quad\phi_{2}(g^{\prime-1}g)=\int_{0}^{\infty}1_{L_{2}(t_{2})}(g^{\prime-1}g)dt_{2}$

hold by (4), we have

$\displaystyle\int_{G}\phi_{1}(g^{\prime})\phi_{2}(g^{\prime-1}g)dg^{\prime}=\int_{G}\int_{0}^{\infty}1_{L_{1}(t_{1})}(g^{\prime})dt_{1}\int_{0}^{\infty}1_{L_{2}(t_{2})}(g^{\prime-1}g)dt_{2}dg^{\prime}.$

We get

$\displaystyle\int_{G}\int_{0}^{\infty}1_{L_{1}(t_{1})}(g^{\prime})dt_{1}\int_{0}^{\infty}1_{L_{2}(t_{2})}(g^{\prime-1}g)dt_{2}dg^{\prime}=\int_{0}^{\infty}\int_{0}^{\infty}\int_{G}1_{L_{1}(t_{1})}(g^{\prime})1_{L_{2}(t_{2})}(g^{\prime-1}g)dg^{\prime}dt_{1}dt_{2}$

by Fubini’s theorem. Since

$\displaystyle\int_{G}1_{L_{1}(t_{1})}(g^{\prime})1_{L_{2}(t_{2})}(g^{\prime-1}g)dg^{\prime}=1_{L_{1}(t_{1})}*1_{L_{2}(t_{2})}(g),$

we obtain

$\displaystyle\phi_{1}*\phi_{2}(g)=\int_{0}^{\infty}\int_{0}^{\infty}\int_{G}1_{L_{1}(t_{1})}(g^{\prime})1_{L_{2}(t_{2})}(g^{\prime-1}g)dg^{\prime}dt_{1}dt_{2}=\int_{0}^{\infty}\int_{0}^{\infty}1_{L_{1}(t_{1})}*1_{L_{2}(t_{2})}(g)dt_{1}dt_{2}.$

  6. (6)

Since
$\phi_{n}^{-1}(\mathbb{R}_{>t})\subset\phi_{n+1}^{-1}(\mathbb{R}_{>t})\subset\phi^{-1}(\mathbb{R}_{>t})$
for any $n$, we have $\phi_{n}^{*}\leq\phi_{n+1}^{*}\leq\phi^{*}$.

Here $\mu(\phi^{-1}(\mathbb{R}_{>\phi^{*}(x)-\epsilon}))>2|x|$ holds for any
$x\in\mathbb{R}$ and $\epsilon>0$. We fix $x\in\mathbb{R}$ and $\epsilon>0$.
Since the pointwise increasing sequence $\phi_{n}$ converges pointwise to
$\phi$, there exists (sufficiently large) $n$ with
$\mu(\phi_{n}^{-1}(\mathbb{R}_{>\phi^{*}(x)-\epsilon}))>2|x|$. Thus,

$\displaystyle\phi^{*}(x)-\epsilon\leq\phi_{n}^{*}(x)\leq\phi^{*}(x)$

holds. Therefore, we have

$\displaystyle\lim_{n\to\infty}\phi_{n}^{*}(x)=\phi^{*}(x)$

and hence $\phi_{n}^{*}$ is a pointwise increasing sequence converging
pointwise to $\phi^{*}$. ∎

######  Example 5.2.

  1. (1)

Let $\mu$, $G$, $\phi_{1}$, $\phi_{2}$, $L_{1}$, and $L_{2}$ be as in Lemma
5.1 (5). We write

$\displaystyle J_{1}(t):=\frac{\mu(L_{1}(t))}{2},\quad
J_{2}(t):=\frac{\mu(L_{2}(t))}{2}$

for $t\geq 0$. Since

$\displaystyle
1_{L_{1}(t)}^{*}=1_{(\phi_{1}^{*})^{-1}(\mathbb{R}_{>t})}=1_{(-J_{1}(t),J_{1}(t))},\quad
1_{L_{2}(t)}^{*}=1_{(\phi_{2}^{*})^{-1}(\mathbb{R}_{>t})}=1_{(-J_{2}(t),J_{2}(t))}$
(5.2)

by Lemma 5.1 (3), we have

$\displaystyle\phi_{1}^{*}*\phi_{2}^{*}(x)=\int_{0}^{\infty}\int_{0}^{\infty}1_{(-J_{1}(t_{1}),J_{1}(t_{1}))}*1_{(-J_{2}(t_{2}),J_{2}(t_{2}))}(x)dt_{1}dt_{2}$
(5.3)

for any $x\in\mathbb{R}$ by Lemma 5.1 (5). For any $J_{1},J_{2}\geq 0$,

$\displaystyle
1_{(-J_{1},J_{1})}*1_{(-J_{2},J_{2})}(x)=\left\\{\begin{aligned}
&2\min(J_{1},J_{2})&&\text{if}\;|x|\leq|J_{1}-J_{2}|\\\
&J_{1}+J_{2}-|x|&&\text{if}\;|J_{1}-J_{2}|\leq|x|\leq J_{1}+J_{2}\\\
&0&&\text{if}\;J_{1}+J_{2}\leq|x|\end{aligned}\right.$

is a continuous even function and it is decreasing on $x\geq 0$. Thus,
$\phi_{1}^{*}*\phi_{2}^{*}(x)$ is also an even function and it is decreasing
on $x\geq 0$.

  2. (2)

In (1), there exist pointwise increasing sequences $\phi_{1,n}$ and
$\phi_{2,n}$ of measurable functions such that $\phi_{1,n}$ and $\phi_{2,n}$
converge pointwise to $\phi_{1}$ and $\phi_{2}$, respectively. Thus,
$\phi_{1,n}^{*}$, $\phi_{2,n}^{*}$, $\phi_{1,n}*\phi_{2,n}$, and
$\phi_{1,n}^{*}*\phi_{2,n}^{*}$ are pointwise increasing sequences of
measurable functions which converge to $\phi_{1}^{*}$, $\phi_{2}^{*}$,
$\phi_{1}*\phi_{2}$ and $\phi_{1}^{*}*\phi_{2}^{*}$, respectively, by Lemma
5.1 (6).

In the case of Example 5.2 (1), the following lemma holds when
$\phi_{1}^{*}*\phi_{2}^{*}(x)$ is finite almost everywhere.

######  Lemma 5.3.

Suppose measurable functions $\phi_{1},\phi_{2}\colon G\to\mathbb{R}_{\geq 0}$
on a unimodular locally compact group $G$ satisfy that
$\phi_{1}^{*}*\phi_{2}^{*}(x)$ is finite almost everywhere.

  1. (1)

$\phi_{1}^{*}*\phi_{2}^{*}$ is finite (everywhere) on
$\mathbb{R}\setminus\\{0\\}$.

  2. (2)

$\phi_{1}^{*}*\phi_{2}^{*}$ is continuous (everywhere) on
$\mathbb{R}\setminus\\{0\\}$.

  3. (3)

If $\phi_{1}^{*}*\phi_{2}^{*}(0)$ is also finite, then
$\phi_{1}^{*}*\phi_{2}^{*}(x)$ is continuous at $x=0$.

###### Proof.

  1. (1)

it suffices to show $\psi(x):=\phi_{1}^{*}*\phi_{2}^{*}(x)<\infty$ for any
$x\in\mathbb{R}\setminus\\{0\\}$. The function $\psi$ is finite almost
everywhere and hence there exists $0<x^{\prime}<|x|$ with
$\psi(x^{\prime})<\infty$. Since $\psi$ is an even function and decreasing on
$\mathbb{R}_{\geq 0}$ by Example 5.2 (1), we obtain

$\displaystyle\psi(x)\leq\psi(x^{\prime})<\infty.$

  2. (2)

Let $x,x^{\prime}\in\mathbb{R}\setminus\\{0\\}$ with
$|x-x^{\prime}|\leq|x|/2$. We denote
$1_{(-J_{1}(t_{1}),J_{1}(t_{1}))}*1_{(-J_{2}(t_{2}),J_{2}(t_{2}))}$ by
$\alpha_{t_{1},t_{2}}$ in Example 5.2 (1). Thus,

$\displaystyle\psi(x^{\prime})=\int_{0}^{\infty}\int_{0}^{\infty}\alpha_{t_{1},t_{2}}(x^{\prime})dt_{1}dt_{2}$

holds by (5.3). Since $\alpha_{t_{1},t_{2}}$ is an even function and
decreasing on $\mathbb{R}_{\geq 0}$ by Example 5.2 (1), we have

$\displaystyle\alpha_{t_{1},t_{2}}(x^{\prime})\leq\alpha_{t_{1},t_{2}}\left(\frac{x}{2}\right).$

Since $\alpha_{t_{1},t_{2}}$ is continuous and $\psi(x/2)<\infty$ holds by
(1), we obtain

$\displaystyle\lim_{x^{\prime}\to x}\psi(x^{\prime})=\lim_{x^{\prime}\to
x}\int_{0}^{\infty}\int_{0}^{\infty}\alpha_{t_{1},t_{2}}(x^{\prime})dt_{1}dt_{2}=\int_{0}^{\infty}\int_{0}^{\infty}\alpha_{t_{1},t_{2}}(x)dt_{1}dt_{2}=\psi(x)$

by the dominated convergence theorem. Thus, $\psi$ is continuous at $x$.

  3. (3)

We have

$\displaystyle\psi(0)=\int_{0}^{\infty}\int_{0}^{\infty}\alpha_{t_{1},t_{2}}(0)dt_{1}dt_{2}$

by (5.3). Since $\alpha_{t_{1},t_{2}}(x)$ is a continuous even function which
is decreasing on $x\geq 0$, we have

$\displaystyle\lim_{x\to 0}\psi(x)=\lim_{x\to
0}\int_{0}^{\infty}\int_{0}^{\infty}\alpha_{t_{1},t_{2}}(x)dt_{1}dt_{2}=\int_{0}^{\infty}\int_{0}^{\infty}\alpha_{t_{1},t_{2}}(0)dt_{1}dt_{2}=\psi(0)$

by the monotone convergence theorem and (5.3). Thus, $\psi(x)$ is also
continuous at $x=0$. ∎

##  6 The proof of Theorem 1.2 in the case of $f=f_{t}$

We define the convex function $f_{t}\colon\mathbb{R}_{\geq
0}\to\mathbb{R}_{\geq 0}$ as

$\displaystyle f_{t}(y):=\left\\{\begin{aligned} &y-t&&\text{if}\;y\geq t\\\
&0&&\text{if}\;y\leq t\end{aligned}\right.$

for $t\geq 0$. In this section, we show Theorem 1.2 for $f=f_{t}$. That is, we
show

$\displaystyle\|f_{t}\circ(\phi_{1}*\phi_{2})\|\leq\|f_{t}\circ(\phi_{1}^{*}*\phi_{2}^{*})\|$
(6.1)

for any $t\geq 0$, where we denote the $L^{1}$-norm $\|\cdot\|_{1}$ by simply
$\|\cdot\|$. In Subsection 6.1, we will prepare a lemma (Lemma 6.1) to prove
Theorem 1.2. In Subsection 6.2, we will show Theorem 1.2 by using this lemma.

###  6.1 The lemma of Theorem 1.2

In this subsection, we show the following lemma to prove Theorem 1.2.

######  Lemma 6.1.

We have

$\displaystyle
f_{t}\left(\int_{\Omega}\alpha(\omega)d\omega\right)\leq\int_{\Omega}f_{T(\omega)}\circ\alpha(\omega)d\omega,\quad
t:=\int_{\Omega}T(\omega)d\omega$

for any measurable functions $T,\alpha\colon\Omega\to\mathbb{R}$ on a measure
space $\Omega$. Furthermore, the equality holds if
$T(\omega)\leq\alpha(\omega)$ almost everywhere.

###### Proof.

When $\int_{\Omega}\alpha(\omega)d\omega\leq t$, we have

$\displaystyle
f_{t}\left(\int_{\Omega}\alpha(\omega)d\omega\right)=0\leq\int_{\Omega}f_{T(\omega)}\circ\alpha(\omega)d\omega.$

When $\int_{\Omega}\alpha(\omega)d\omega\geq t$, we have

$\displaystyle
f_{t}\left(\int_{\Omega}\alpha(\omega)d\omega\right)=\int_{\Omega}\alpha(\omega)d\omega-t=\int_{\Omega}(\alpha(\omega)-T(\omega))d\omega\leq\int_{\Omega}f_{T(\omega)}\circ\alpha(\omega)d\omega.$

Furthermore, the equality holds if $T(\omega)\leq\alpha(\omega)$ almost
everywhere. ∎

######  Example 6.2.

Let $\mu$, $G$, $\phi_{1}$, $\phi_{2}$, $L_{1}$, $L_{2}$, $J_{1}(t)$, and
$J_{2}(t)$ be as in Example 5.2 (1). When we fix $x_{0}\geq 0$, we have

$\displaystyle
t:=\phi_{1}^{*}*\phi_{2}^{*}(x_{0})=\int_{0}^{\infty}\int_{0}^{\infty}T(t_{1},t_{2})dt_{1}dt_{2}$

by (5.3), where
$T(t_{1},t_{2}):=1_{(-J_{1}(t_{1}),J_{1}(t_{1}))}*1_{(-J_{2}(t_{2}),J_{2}(t_{2}))}(x_{0})$.
We apply Lemma 6.1 to $\Omega=\mathbb{R}_{\geq 0}\times\mathbb{R}_{\geq 0}$.

  1. (1)

We fix $g\in G$ and let
$\alpha(t_{1},t_{2}):=1_{L_{1}(t_{1})}*1_{L_{2}(t_{2})}(g)$.

$\displaystyle\int_{0}^{\infty}\int_{0}^{\infty}\alpha(t_{1},t_{2})dt_{1}dt_{2}=\phi_{1}*\phi_{2}(g)$

holds by Lemma 5.1 (5). Thus, we have

$\displaystyle
f_{t}\circ(\phi_{1}*\phi_{2})(g)\leq\int_{0}^{\infty}\int_{0}^{\infty}f_{T(t_{1},t_{2})}\circ(1_{L_{1}(t_{1})}*1_{L_{2}(t_{2})})(g)dt_{1}dt_{2}$
(6.2)

by Lemma 6.1.

  2. (2)

We fix $x\in\mathbb{R}$ with $|x|\leq x_{0}$ and let
$\alpha(t_{1},t_{2}):=1_{(-J_{1}(t_{1}),J_{1}(t_{1}))}*1_{(-J_{2}(t_{2}),J_{2}(t_{2}))}(x)$.
By (5.3), $\phi_{1}^{*}*\phi_{2}^{*}(x)$ is given as

$\displaystyle\phi_{1}^{*}*\phi_{2}^{*}(x)=\int_{0}^{\infty}\int_{0}^{\infty}\alpha(t_{1},t_{2})dt_{1}dt_{2}.$
(6.3)

Since $1_{(-J_{1}(t_{1}),J_{1}(t_{1}))}*1_{(-J_{2}(t_{2}),J_{2}(t_{2}))}$ is
an even function and decreasing on $\mathbb{R}_{\geq 0}$, we have
$T(t_{1},t_{2})\leq\alpha(t_{1},t_{2})$ by $|x|\leq x_{0}$. Thus,

$\displaystyle
f_{t}\circ\left(\int_{0}^{\infty}\int_{0}^{\infty}\alpha(t_{1},t_{2})dt_{1}dt_{2}\right)=\int_{0}^{\infty}\int_{0}^{\infty}f_{T(t_{1},t_{2})}\circ(\alpha(t_{1},t_{2}))(x)dt_{1}dt_{2}$

holds by applying Lemma 6.1 and hence we have

$\displaystyle
f_{t}\circ(\phi_{1}^{*}*\phi_{2}^{*})(x)=\int_{0}^{\infty}\int_{0}^{\infty}f_{T(t_{1},t_{2})}\circ(1_{(-J_{1}(t_{1}),J_{1}(t_{1}))}*1_{(-J_{2}(t_{2}),J_{2}(t_{2}))})(x)dt_{1}dt_{2}$
(6.4)

by (6.3).

###  6.2 The proof of Theorem 1.2

In this subsection, we show Theorem 1.2 for $f=f_{t}$ in three steps.

  1. (1)

In the case where $\phi_{1}$ and $\phi_{2}$ are the characteristic functions
of any measurable sets of finite volume:

It suffices to show (6.1) when $\|\phi_{1}\|\leq\|\phi_{2}\|$. If
$\|\phi_{1}\|\leq t$, then we have $\phi_{1}*\phi_{2}\leq t$ by $\phi_{2}\leq
1$ and hence we obtain

$\displaystyle\|f_{t}\circ(\phi_{1}*\phi_{2})\|=0\leq\|f_{t}\circ(\phi_{1}^{*}*\phi_{2}^{*})\|.$

If $t\leq\|\phi_{1}\|$, then

$\displaystyle\|f_{t}\circ(\phi_{1}*\phi_{2})\|\leq(\|\phi_{1}\|-t)(\|\phi_{2}\|-t)=\|f_{t}\circ(\phi_{1}^{*}*\phi_{2}^{*})\|$

holds by (1.1) [Sat21, Theorem 1.1].

  2. (2)

In the case where $\phi_{1}$ and $\phi_{2}$ are the characteristic functions
of any measurable sets (which are not necessary of finite volume):

There exist increasing sequences $A_{1}^{(n)},A_{2}^{(n)}\subset G$ of
measurable sets of finite volume such that $1_{A_{1}^{(n)}}$ and
$1_{A_{2}^{(n)}}$ converge pointwise to $\phi_{1}$ and $\phi_{2}$,
respectively. Thus, we have

$\displaystyle\lim_{n\to\infty}\left\|f_{t}\circ\left(1_{A_{1}^{(n)}}*1_{A_{2}^{(n)}}\right)\right\|=\|f_{t}\circ(\phi_{1}*\phi_{2})\|,\quad\lim_{n\to\infty}\left\|f_{t}\circ\left(1_{A_{1}^{(n)}}^{*}*1_{A_{2}^{(n)}}^{*}\right)\right\|=\|f_{t}\circ(\phi_{1}^{*}*\phi_{2}^{*})\|$

by Example 5.2 (2) and the monotone convergence theorem. In addition,

$\displaystyle\left\|f_{t}\circ\left(1_{A_{1}^{(n)}}*1_{A_{2}^{(n)}}\right)\right\|\leq\left\|f_{t}\circ\left(1_{A_{1}^{(n)}}^{*}*1_{A_{2}^{(n)}}^{*}\right)\right\|$

holds for any $n$ by (1) and hence we obtain

$\displaystyle\|f_{t}\circ(\phi_{1}*\phi_{2})\|=\lim_{n\to\infty}\left\|f_{t}\circ\left(1_{A_{1}^{(n)}}*1_{A_{2}^{(n)}}\right)\right\|\leq\lim_{n\to\infty}\left\|f_{t}\circ\left(1_{A_{1}^{(n)}}^{*}*1_{A_{2}^{(n)}}^{*}\right)\right\|=\|f_{t}\circ(\phi_{1}^{*}*\phi_{2}^{*})\|.$

  3. (3)

In the general case:

When

$\displaystyle\lim_{x\to\infty}\phi_{1}^{*}*\phi_{2}^{*}(x)<t\leq\phi_{1}^{*}*\phi_{2}^{*}(0)$
(6.5)

does not hold, it can be reduced to the case of (6.5) (by using the monotone
convergence theorem if necessary) or we have
$\|f_{t}\circ(\phi_{1}^{*}*\phi_{2}^{*})\|=\infty$. In the case of (6.5),
there exists $x_{0}\geq 0$ with $t=\phi_{1}^{*}*\phi_{2}^{*}(x_{0})$ by Lemma
5.3 and the intermediate value theorem. Let $T(t_{1},t_{2})$ be as in Example
6.2. Then

$\displaystyle\|f_{t}\circ(\phi_{1}*\phi_{2})\|\leq\int_{0}^{\infty}\int_{0}^{\infty}\|f_{T(t_{1},t_{2})}\circ(1_{L_{1}(t_{1})}*1_{L_{2}(t_{2})})\|dt_{1}dt_{2}$

holds by integrating (6.2) over $g\in G$. We have

$\displaystyle\|f_{T(t_{1},t_{2})}\circ(1_{L_{1}(t_{1})}*1_{L_{2}(t_{2})})\|\leq\|f_{T(t_{1},t_{2})}\circ(1_{L_{1}(t_{1})}^{*}*1_{L_{2}(t_{2})}^{*})\|$

by (2) and hence

$\displaystyle\|f_{T(t_{1},t_{2})}\circ(1_{L_{1}(t_{1})}*1_{L_{2}(t_{2})})\|\leq\|f_{T(t_{1},t_{2})}\circ(1_{(-J_{1}(t_{1}),J_{1}(t_{1}))}*1_{(-J_{2}(t_{2}),J_{2}(t_{2}))})\|$

by (5.2). Thus, we obtain

$\displaystyle\|f_{t}\circ(\phi_{1}*\phi_{2})\|\leq\int_{0}^{\infty}\int_{0}^{\infty}\|f_{T(t_{1},t_{2})}\circ(1_{(-J_{1}(t_{1}),J_{1}(t_{1}))}*1_{(-J_{2}(t_{2}),J_{2}(t_{2}))})\|dt_{1}dt_{2}=\|f_{t}\circ(\phi_{1}^{*}*\phi_{2}^{*})\|$

by (6.4).

Therefore, we have (6.1) and hence Theorem 1.2 is proved for $f=f_{t}$.

##  7 The proof of Theorem 1.2 when $\phi_{1}*\phi_{2}$ is integrable

In this section, we show Theorem 1.2 by an argument of Wang–Madiman when
$\phi_{1}*\phi_{2}$ is integrable.

######  Fact 7.1 ([WM14, Section VII]).

Suppose a convex function $f\colon\mathbb{R}_{\geq 0}\to\mathbb{R}$ with
$f(0)=0$ is continuous at $0$ and satisfies $f_{+}^{\prime}(0)>-\infty$, where
$f_{+}^{\prime}$ denotes the right derivative. Then there exists a Borel
measure $\nu$ on $\mathbb{R}_{>0}$ such that
$\nu((t_{1},t_{2}])=f_{+}^{\prime}(t_{2})-f_{+}^{\prime}(t_{1})$ for any
$0\leq t_{1}\leq t_{2}$. Furthermore, we have

$\displaystyle f(y)=f_{+}^{\prime}(0)y+\int_{0}^{\infty}f_{t}(y)d\nu(t)$

for any $y\geq 0$.

######  Example 7.2.

Suppose $G$, $\phi_{1}$, and $\phi_{2}$ are as in Theorem 1.2 and
$\phi_{1}*\phi_{2}$ is integrable. Since

$\displaystyle
f\circ(\phi_{1}*\phi_{2})(g)=f_{+}^{\prime}(0)\phi_{1}*\phi_{2}(g)+\int_{0}^{\infty}f_{t}\circ(\phi_{1}*\phi_{2})(g)d\nu(t)$

by Fact 7.1, we have

$\displaystyle\int_{G}f\circ(\phi_{1}*\phi_{2})(g)dg=f_{+}^{\prime}(0)\|\phi_{1}*\phi_{2}\|+\int_{0}^{\infty}\|f_{t}\circ(\phi_{1}*\phi_{2})\|d\nu(t)$
(7.1)

by integrating over $g\in G$. Similarly, we have

$\displaystyle\int_{\mathbb{R}}f\circ(\phi_{1}^{*}*\phi_{2}^{*})(x)dx=f_{+}^{\prime}(0)\|\phi_{1}^{*}*\phi_{2}^{*}\|+\int_{0}^{\infty}\|f_{t}\circ(\phi_{1}^{*}*\phi_{2}^{*})\|d\nu(t).$
(7.2)

Since $\phi_{1}*\phi_{2}$ and $\phi_{1}^{*}*\phi_{2}^{*}$ are integrable by
Fubini’s theorem and

$\displaystyle\|\phi_{1}*\phi_{2}\|=\|\phi_{1}^{*}*\phi_{2}^{*}\|,$ (7.3)

(7.1) and (7.2) are well-defined. Thus, we obtain (1.3) by (6.1) and (7.3).

Theorem 1.2 follows from Example 7.2 when $\phi_{1}*\phi_{2}$ is integrable,
$f(y)$ is continuous at $y=0$, and $f_{+}^{\prime}(0)>-\infty$. Here we show
Theorem 1.2 by using Example 7.2 when $f(y)$ is not continuous at $y=0$ or
$f_{+}^{\prime}(0)=-\infty$. We define $f_{(n)}\colon\mathbb{R}_{\geq
0}\to\mathbb{R}$ as

$\displaystyle f_{(n)}(y):=\left\\{\begin{aligned}
&nf\left(\frac{1}{n}\right)y&&\text{if}\;y\leq\frac{1}{n}\\\
&f(y)&&\text{if}\;y\geq\frac{1}{n}\end{aligned}\right..$

The function $f_{(n)}(y)$ is convex and continuous at $y=0$. We have

$\displaystyle f_{(n)+}^{\prime}(0)=nf\left(\frac{1}{n}\right)>-\infty.$ (7.4)

Since $f$ is not continuous at $y=0$ or $f_{+}^{\prime}(0)=-\infty$, there
exists $n^{\prime}\in\mathbb{Z}_{\geq 1}$ such that $f$ is negative on
$(0,1/n^{\prime}]$. We have $0\geq f_{(n)}\geq f$ on $[0,1/n^{\prime}]$ for
any $n\geq n^{\prime}$ by the convexity of $f$.

######  Remark 7.3.

Since Theorem 1.2 is clear when
$\int_{G}f\circ(\phi_{1}*\phi_{2})(g)dg=-\infty$ or
$\int_{\mathbb{R}}f\circ(\phi_{1}^{*}*\phi_{2}^{*})(x)dx=\infty$, it suffices
to show Theorem 1.2 when

$\displaystyle\int_{G}f\circ(\phi_{1}*\phi_{2})(g)dg$ $\displaystyle>-\infty,$
(7.5) $\displaystyle\int_{\mathbb{R}}f\circ(\phi_{1}^{*}*\phi_{2}^{*})(x)dx$
$\displaystyle<\infty.$ (7.6)

For any $n\geq n^{\prime}$, we have $0\geq f_{(n)}\geq f$ on
$[0,1/n^{\prime}]$ and hence

$\displaystyle\int_{G}f_{(n)}\circ(\phi_{1}*\phi_{2})(g)dg\leq\int_{\mathbb{R}}f_{(n)}\circ(\phi_{1}^{*}*\phi_{2}^{*})(x)dx<\infty$
(7.7)

by Example 7.2, (7.4) and (7.6). Since the pointwise decreasing sequence
$f_{(n)}$ converges to $f$, we have

$\displaystyle\lim_{n\to\infty}\int_{G}f_{(n)}\circ(\phi_{1}*\phi_{2})(g)dg$
$\displaystyle=\int_{G}f\circ(\phi_{1}*\phi_{2})(g)dg,$ (7.8)
$\displaystyle\lim_{n\to\infty}\int_{\mathbb{R}}f_{(n)}\circ(\phi_{1}^{*}*\phi_{2}^{*})(x)dx$
$\displaystyle=\int_{\mathbb{R}}f\circ(\phi_{1}^{*}*\phi_{2}^{*})(x)dx$ (7.9)

by the monotone convergence theorem. Thus, we have (1.3) by (7.7), (7.8) and
(7.9) and hence we obtain Theorem 1.2 when $\phi_{1}*\phi_{2}$ is integrable.

##  8 The proof of Theorem 1.2 in the general case

In this section, we show Theorem 1.2 when $\phi_{1}*\phi_{2}$ may not be
integrable. We will show Theorem 1.2 when $f\circ(\phi_{1}^{*}*\phi_{2}^{*})$
is positive in Subsection 8.1, and when $f$ is monotonically decreasing in
Subsection 8.2. In Subsection 8.3, we will complete the proof of Theorem 1.2
by showing that the cases of Section 7, Subsection 8.1, and Subsection 8.2
exhaust all the cases of Theorem 1.2.

###  8.1 The case where $f\circ(\phi_{1}^{*}*\phi_{2}^{*})$ is positive

When $f\circ(\phi_{1}^{*}*\phi_{2}^{*})$ is positive, Theorem 1.2 can be
obtained by a similar argument as in Section 7. In this case, we may assume
that $f$ is positive by replacing $f$ by $\max(f,0)$, and (7.6) holds by
Remark 7.3. Since $f(0)=0$ and $f$ is positive, $f(y)$ is continuous at $y=0$
and $f_{+}^{\prime}(0)\geq 0$ holds. Thus, there exists a Borel measure $\nu$
in Fact 7.1. Since (7.2) holds by a similar argument as in Example 7.2 (both
sides of (7.2) are well-defined by (7.6)), both sides of (7.1) are well-
defined and (7.1) holds by (6.1) and the dominated convergence theorem. Thus,
we have (1.3) by (6.1) and hence we obtain Theorem 1.2 when
$f\circ(\phi_{1}^{*}*\phi_{2}^{*})$ is positive.

###  8.2 The case where $f$ is monotonically decreasing

In this subsection, we show Theorem 1.2 when $f$ is monotonically decreasing.
There exist pointwise increasing sequences $\phi_{1,n},\phi_{2,n}\colon
G\to\mathbb{R}$ of integrable functions converging to $\phi_{1}$ and
$\phi_{2}$, respectively. By Example 5.2 (2), $\phi_{1,n}*\phi_{2,n}$ and
$\phi_{1,n}^{*}*\phi_{2,n}^{*}$ converge pointwise to $\phi_{1}*\phi_{2}$ and
$\phi_{1}^{*}*\phi_{2}^{*}$, respectively. Since $f$ is monotonically
decreasing, $f\circ(\phi_{1,n}*\phi_{2,n})$ and
$f\circ(\phi_{1,n}^{*}*\phi_{2,n}^{*})$ are pointwise decreasing sequences.
The convex function $f$ is negative by $f(0)=0$ and hence

$\displaystyle\int_{G}f\circ(\phi_{1}*\phi_{2})(g)dg$
$\displaystyle=\lim_{n\to\infty}\int_{G}f\circ(\phi_{1,n}*\phi_{2,n})(g)dg,$
$\displaystyle\int_{\mathbb{R}}f\circ(\phi_{1}^{*}*\phi_{2}^{*})(x)dx$
$\displaystyle=\lim_{n\to\infty}\int_{\mathbb{R}}f\circ(\phi_{1,n}^{*}*\phi_{2,n}^{*})(x)dx$

by the monotone convergence theorem. Since $\phi_{1,n}*\phi_{2,n}$ is
integrable for any $n$, we have

$\displaystyle\int_{G}f\circ(\phi_{1,n}*\phi_{2,n})(g)dg\leq\int_{\mathbb{R}}f\circ(\phi_{1,n}^{*}*\phi_{2,n}^{*})(x)dx$

by Section 7. Thus, (1.3) holds and hence we obtain Theorem 1.2 when $f$ is
monotonically decreasing.

###  8.3 The completion of Theorem 1.2

Theorem 1.2 was obtained in some cases in Section 7, Subsection 8.1, and
Subsection 8.2. In this subsection, we complete the proof of Theorem 1.2 by
showing that these cases exhaust all the cases of Theorem 1.2. That is, we
show the following lemma.

######  Lemma 8.1.

Suppose a convex function $f\colon\mathbb{R}_{\geq 0}\to\mathbb{R}$ with
$f(0)=0$ is not monotonically decreasing. We let $t>0$ with $f(t)<0$.

  1. (1)

There exists $t^{\prime}>t$ with $f(t^{\prime})=0$. Furthermore,

$\displaystyle f(y)\leq 0$ (8.1)

holds for any $0\leq y\leq t^{\prime}$, and

$\displaystyle f(y)\geq 0$ (8.2)

holds for any $y\geq t^{\prime}$.

  2. (2)

Let $t^{\prime}$ be as in (1). Then we have

$\displaystyle
y\leq\frac{t\max(-f(y),0)}{-f(t)}+\frac{t^{\prime}f_{t}(y)}{t^{\prime}-t}$

for any $y\geq 0$.

  3. (3)

Suppose measurable functions $\phi_{1},\phi_{2}\colon G\to\mathbb{R}_{\geq 0}$
on a unimodular locally compact group $G$ satisfy (1.2) and (7.6). Then we
have

$\displaystyle\|f_{\phi_{1}^{*}*\phi_{2}^{*}(x_{0})}\circ(\phi_{1}*\phi_{2})\|\leq\|f_{\phi_{1}^{*}*\phi_{2}^{*}(x_{0})}\circ(\phi_{1}^{*}*\phi_{2}^{*})\|<\infty$

for any $x_{0}\in\mathbb{R}$ with
$f\circ(\phi_{1}^{*}*\phi_{2}^{*})(x_{0})<0$.

  4. (4)

Suppose there exists $x_{0}\in\mathbb{R}$ in (3). If (7.5) holds, then
$\phi_{1}*\phi_{2}$ is integrable.

###### Proof.

  1. (1)

Since the convex function $f$ is not monotonically decreasing, there exists
$t^{\prime\prime}>t$ with $f(t^{\prime\prime})>0$. The convex function $f$ is
continuous on $\mathbb{R}_{>0}$ and hence there exists $t^{\prime}>t$ with
$f(t^{\prime})=0$ by the intermediate value theorem. Since
$f(0)=f(t^{\prime})=0$ holds and $f$ is convex, we obtain $f(y)\leq 0$ for any
$0\leq y\leq t^{\prime}$, and $f(y)\geq 0$ for any $y\geq t^{\prime}$.

  2. (2)

When $0\leq y\leq t$, we have (8.1) and hence

$\displaystyle
y\leq\frac{tf(y)}{f(t)}=\frac{t\max(-f(y),0)}{-f(t)}\leq\frac{t\max(-f(y),0)}{-f(t)}+\frac{t^{\prime}f_{t}(y)}{t^{\prime}-t}$

by the convexity of $f$.

When $t\leq y\leq t^{\prime}$, we have

$\displaystyle
y=\frac{t(t^{\prime}-y)+t^{\prime}(y-t)}{t^{\prime}-t}=\frac{t(t^{\prime}-y)+t^{\prime}f_{t}(y)}{t^{\prime}-t}.$
(8.3)

Since $f$ is convex,

$\displaystyle\frac{t^{\prime}-y}{t^{\prime}-t}\leq\frac{f(y)}{f(t)}$

holds by (8.1). Thus, we obtain

$\displaystyle
y=\frac{t(t^{\prime}-y)+t^{\prime}f_{t}(y)}{t^{\prime}-t}\leq\frac{t\max(-f(y),0)}{-f(t)}+\frac{t^{\prime}f_{t}(y)}{t^{\prime}-t}.$

When $t^{\prime}\leq y$, we also have (8.3) and hence we obtain

$\displaystyle
y\leq\frac{t^{\prime}f_{t}(y)}{t^{\prime}-t}\leq\frac{t\max(-f(y),0)}{-f(t)}+\frac{t^{\prime}f_{t}(y)}{t^{\prime}-t}.$

  3. (3)

Let $\beta:=\phi_{1}*\phi_{2}$, $\psi:=\phi_{1}^{*}*\phi_{2}^{*}$ and
$t:=\psi(x_{0})$. Since $\psi$ is an even function by Example 5.2 (1), we may
assume $x_{0}\geq 0$. We have $\|f_{t}\circ\beta\|\leq\|f_{t}\circ\psi\|$ by
(6.1) and hence it suffices to show $\|f_{t}\circ\psi\|<\infty$. Let
$t^{\prime}$ be as in (1). We have

$\displaystyle f_{t^{\prime}}(y)\leq\frac{(t^{\prime}-t)f(y)}{-f(t)}$

for any $y\geq t^{\prime}$ by the convexity of $f$. Thus,

$\displaystyle\|f_{t^{\prime}}\circ\psi\|\leq\frac{t^{\prime}-t}{-f(t)}\int_{G}f\circ\psi(g)dg$

holds and hence we have $\|f_{t^{\prime}}\circ\psi\|<\infty$ by (7.6). Since
$\psi$ is an even function which is monotonically decreasing on
$\mathbb{R}_{\geq 0}$ by Example 5.2 (1), we obtain

$\displaystyle\|f_{t}\circ\psi\|=\int_{-x_{0}}^{x_{0}}(\psi(x)-t)dx\leq\|f_{t^{\prime}}\circ\psi\|+(t^{\prime}-t)\int_{-x_{0}}^{x_{0}}dx<\infty.$

  4. (4)

Let $\beta$, $t$, and $t^{\prime}$ be as in (3). Since

$\displaystyle\|\beta\|\leq\frac{t\|\max(-f\circ\beta,0)\|}{-f(t)}+\frac{t^{\prime}\|f_{t}\circ\beta\|}{t^{\prime}-t}$

by (2) and (8.2), we obtain $\|\beta\|<\infty$ by (3) and (7.5). ∎

By Lemma 8.1, the cases of Section 7, Subsection 8.1, and Subsection 8.2
exhaust all the cases of Theorem 1.2. Thus, we complete the proof of Theorem
1.2.

## Acknowledgement

This work was supported by JSPS KAKENHI Grant Number JP19J22628 and Leading
Graduate Course for Frontiers of Mathematical Sciences and Physics (FMSP). The
author would like to thank his advisor Toshiyuki Kobayashi for his support.
The author is also grateful to Yuichiro Tanaka, Toshihisa Kubo and the
anonymous referees for their careful comments.

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Takashi Satomi: Graduate School of Mathematical Sciences, The University of
Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan.

E-mail<EMAIL_ADDRESS>


# Integrating Regular Expressions with Neural Networks via DFA

Shaobo Li1, Qun Liu2, Xin Jiang2, Yichun Yin2
Chengjie Sun1, Bingquan Liu1, Zhenzhou Ji1 and Lifeng Shang2
1Harbin Institute of Technology
2Huawei Noah’s Ark Lab
<EMAIL_ADDRESS>{sunchengjie, liubq<EMAIL_ADDRESS>
{qun.liu, Jiang.Xin, yinyichun<EMAIL_ADDRESS>

###### Abstract

Human-designed rules are widely used to build industry applications. However,
it is infeasible to maintain thousands of such hand-crafted rules. So it is
very important to integrate the rule knowledge into neural networks to build a
hybrid model that achieves better performance. Specifically, the human-
designed rules are formulated as Regular Expressions (REs), from which the
equivalent Minimal Deterministic Finite Automatons (MDFAs) are constructed. We
propose to use the MDFA as an intermediate model to capture the matched RE
patterns as rule-based features for each input sentence and introduce these
additional features into neural networks. We evaluate the proposed method on
the ATIS intent classification task. The experiment results show that the
proposed method achieves the best performance compared to neural networks and
four other methods that combine REs and neural networks when the training
dataset is relatively small.

##  1 Introduction

Although Neural Network (NN) based approaches have been widely used in various
natural language processing tasks and achieved remarkable results Young et al.
(2018), there are still limitations of NNs faced by the community, such as the
data-hungry nature, lacking interpretability ability and vulnerability to
adversarial attacks. In most cases, the NN-based models cannot be directly
applied to the scenario where there are limited training samples, where the
rule-based methods can still work properly and are widely used. Integrating
human-designed symbolic knowledge into NN-based models is believed to be a
promising and practical way to alleviate these limitations Garcez et al.
(2019), and many works have been explored in this direction Liang et al.
(2017); Xie et al. (2019); Arora et al. (2020); Luo et al. (2018). However, it
is still an open problem to effectively integrate the highly abstract human
knowledge encoded by discrete rules with the data-driven neural models.

In this paper, we propose a novel method that adopts the regular expression
(RE) as human-designed rules and combines it with the NN-based model (i.e.,
NN-based sentence classifier). The key problem is how to effectively represent
the regular expression into numeric features, and then integrate them with
neural models. In conventional rule-based representations Luo et al. (2018),
the final matching results of input sentences to REs (i.e., accept or reject)
or the occurrences of some key patterns in REs are directly used as the
additional features.

In this work, REs are converted into MDFAs, and MDFAs are used as intermediate
models to capture more fine-grained MDFA-based features from the input
sentence. The MDFA-based features easily integrate with neural networks and
can be absorbed effectively. Furthermore, we propose two RE-NN hybrid models
that incorporate the MDFA-based features into NN at different levels.
Comprehensive experiments are conducted on ATIS intent classification dataset
Hemphill et al. (1990) with training sets of various sizes. The experiment
results demonstrate that the proposed hybrid models improve the performance of
NN and also perform better than the existing models which combine NNs with
REs.

##  2 Method

Figure 1: The proposed method integrates REs with NN through MDFA.

Figure 1 shows the proposed method. Each RE is first turned into MDFA with the
algorithms in Hopcroft et al. (2001). Based on the obtained MDFA, a sequence
of transition states can be obtained for each input sentence, which presents
local matching patterns defined by the RE occurring in the sentence. Then we
transform this sentence-dependent state sequence into MDFA-based features and
integrate them into NN as additional input at the instance level or word
level. For the NN-based component, Recurrent Neural Network (RNN) Jain and
Medsker (1999) is used as the backbone.

The instance-level integration aims to encourage the MDFA-based features to
interact with high-level NN-based representations of an entire sentence.
Differently, the word-level integration associates the MDFA-based features
with smaller-granularity semantic, i.e., word embeddings Mikolov et al.
(2013). The word-level integration strategy adopts MDFA-based features to
guide and regularize RNN internal representations word by word, which is
motivated by the recent findings Michalenko et al. (2018) that there exists a
strong structural relationship between internal representations of RNNs and
MDFAs when recognizing formal languages.

###  2.1 Encode MDFA State Sequences into Numeric MDFA-based Features

In this section, we describe how to encode MDFA state sequences into numeric
features that contain the sentence-dependent information from rules. A regular
expression (RE) can be transformed into an equivalent MDFA that embodies the
human-designed rules in RE Hopcroft et al. (2001). Formally, an MDFA can be
represented as $M=\\{\mathcal{S},\mathcal{E},\delta,s_{0},\mathcal{F}\\}$,
which contains a state set $\mathcal{S}$, an input symbol set $\mathcal{E}$, a
transition function
$\delta:\mathcal{S}\times\mathcal{E}\rightarrow\mathcal{S}$, an initial state
$s_{0}\in\mathcal{S}$ and an end state set $\mathcal{F}\subset\mathcal{S}$.

Matching RE with a sentence can be converted to a sequence of MDFA state
transition. A sentence $W=\\{w_{1},w_{2},\ldots,w_{n}\\}$ that contains $n$
words is fed into the MDFA word by word. Start from the initial state $s_{0}$,
the MDFA will transfer from the current state to a target state depending on
the current received word and the transition function $\delta$. In this way, a
sequence of target state $S$ can be observed sequentially until one end state
is reached or all the words in $W$ are exhausted. If $S$ ends with an end
state in $\mathcal{F}$, it means that the RE (or equivalently, the MDFA)
accepts the sentence $W$111The conventional RE is character-based and results
in a character-triggered MDFA that contains massive fine-grained states. To
reduce the number of states and make these states more informative, we employ
the word-based RE instead of character-based by treating a whole word as a
single symbol..

The state sequence $S$ contains a detailed procedure about how a RE examines
the sentence. We encode $S$ as MDFA-based features for the sentence. Suppose
there are $p$ REs for different labels, we can obtain $p$ MDFA state sequences
by feeding $W$ into each RE. We use superscript $k$ to identify the $k$-th RE
and represent the MDFA state sequence produced by the $k$-th as $S^{k}$.

We propose two types of encoding, named instance-level encoding and word-level
encoding, which encode MDFA state sequences as instance-level features or
word-level features respectively. The instance-level encoding is to encode
each MDFA state sequence into a single feature vector. Specifically, for the
$k$-th RE, all the different states in the state set of the constructed MDFA
are indexed with consecutive integers. Then the $S^{k}$ is converted into a
one-hot vector sequence by converting each state in $S^{k}$ by a one-hot
vector based on the integral index of it. Finally, the one-hot vector sequence
is aggregated into a single vector $\mathbf{u}^{k}$ via max-pooling operation.
Figure 2 illustrates an example of calculating $\mathbf{u}^{k}$.

Figure 2: Encode the MDFA state sequence as an instance-level feature vector.

The word-level encoding encodes the MDFA state sequences as word-level
features, by which every word in $W$ is assigned with binary tags from
different REs. When matching the $k$-th RE with $W$, the state transition is
triggered word by word by feeding the words in $W$ into MDFA orderly. We
denote the state arrived at after MDFA consuming the $i$-th word $w_{i}$ in
$W$ as $s_{w_{i}}^{k}$ and align it with $w_{i}$. Inspired by the BIO
(Beginning-Inside-Outside) tagging format Ramshaw and Marcus (1999), the state
$s_{w_{i}}^{k}$ is simplified to a binary tag $v_{w_{i}}$ for $w_{i}$ as:

  * •

$w_{i}$ is tagged with 1 ($v_{w_{i}}=1$) if the $k$-th RE accepts $W$ and
$s_{w_{i}}^{k}\in S^{k}$, which means the $w_{i}$ is inside of the matching
procedure and results in acceptance.

  * •

$w_{i}$ is tagged with 0 ($v_{w_{i}}=0$) otherwise.

Because of the high-precision low-recall nature of human-designed rules, the
“accept” output by RE is more believable than the “reject”. Therefore, the
MDFA-based encoding is only activated when the input sentence is accepted by
the RE, set to zero otherwise. In this way, binary tag sequence
$v^{k}=\\{v_{w_{1}}^{k},v_{w_{2}}^{k},\ldots,v_{w_{n}}^{k}\\}$ can be obtained
for $W$ based on the $k$-th RE.

Finally, we can obtain two types of MDFA-based numeric features,
$\\{\mathbf{u}^{k}\\}$ and $\\{v^{k}\\}$ ($k={1,\ldots,p}$), which will be
further incorporated into the NN-based component at the instance level or word
level respectively.

###  2.2 NN-based Component

We use a pure NN-based sentence classifier, that consists of a BLSTM
(Bidirectional LSTM) with attention mechanism as feature extractor Zhou et al.
(2016) and a MLP (Multi-Layer Perceptron) as classifier Amendolia et al.
(2003), as the NN-based component in the proposed hybrid models. Specifically,
for the input sentence $W=\\{w_{1},w_{2},\ldots,w_{n}\\}$, BLSTM first
generates a hidden state sequence
$\mathbf{H}=\\{\mathbf{h}_{1},\mathbf{h}_{2},\ldots,\mathbf{h}_{n}\\}$ as

$\mathbf{H}={\rm BLSTM}(\\{\mathbf{e}(w_{1}),\ldots,\mathbf{e}(w_{n})\\}),$
(1)

where $\mathbf{e}(\cdot)$ denotes the embedding vector of the input word.
$\mathbf{H}$ is transformed to a fixed-length feature vector $\mathbf{f}$
through attention mechanism. Then $\mathbf{f}$ is fed to the MLP classifier to
calculate the class probabilities $\mathbf{y}$:

$\alpha_{i}=\frac{\exp(\mathbf{h}_{i}\mathbf{W}\mathbf{h}_{n})}{\sum_{j}{\exp(\mathbf{h}_{j}\mathbf{W}\mathbf{h}_{n})}},\enspace\enspace\mathbf{f}=\sum_{i=1}^{n}{\alpha_{i}}\mathbf{h}_{i}$
(2) $\mathbf{y}={\rm MLP}(\mathbf{f})$ (3)

This NN-based component is also used as our NN baseline, referred to as nnsc.

###  2.3 Introduce the MDFA-Based Features into the NN-based Component

We import two types of MDFA-based features into the NN-based component at the
instance or word level to build two hybrid models. In the first hybrid model,
referred to as instance, $\\{\mathbf{u}^{k}\\}$ is introduced into MLP
classifier companying with the NN-based feature $\mathbf{f}$, which aims to
enrich the NN-based representation of the entire input sentence with instance-
level features. This hybrid model calculates the probabilities by

$\mathbf{y}={\rm MLP}([\mathbf{f};\mathbf{u}^{1};\ldots;\mathbf{u}^{p}])$ (4)

instead of Equation (3) in the nnsc. $[;]$ denotes concatenating of vectors.

The other hybrid model named word incorporates the word-level features
$\\{v^{k}\\}$ into word embedding. $p$ REs produce $p$ binary tag sequences
$\\{v^{k}\\}$ and each word in the input sentence corresponds to $p$ binary
tags. All these $p$ binary tags are appended to the corresponding word
embedding vector. For word $w_{i}$ in the input sentence $W$, the word
embedding $\mathbf{e}(w_{i})$ is expanded to MDFA-enhanced word embedding
${\mathbf{e}}_{v}(w_{i})$:

$\mathbf{e}_{v}(w_{i})=[\mathbf{e}(w_{i});v_{w_{i}}^{1};\ldots;v_{w_{i}}^{p}]$
(5)

And Equation (1) in nnsc is modified to

$\mathbf{H}={\rm
BLSTM}(\\{\mathbf{e}_{v}(w_{1}),\ldots,\mathbf{e}_{v}(w_{n})\\})$ (6)

in this hybrid model when calculating $\mathbf{y}$.

##  3 Experiments

Figure 3: Comparisons of the REs, NN-based model (denoted as nnsc) and the
proposed hybrid models (denoted as instance and word).

###  3.1 Settings

We evaluate the models in few-shot settings, where the training samples are
insufficient to training the NN-based model to achieve commendable
performance, to see if the hybrid model boosts the performance by introducing
human-designed rules. To further show the effectiveness of the proposed MDFA-
based integration, we compared our hybrid models with other hybrid models that
combine REs with NN without using MDFA.

We employ the widely used ATIS intent classification
dataset222https://pfllo.github.io/data/ACL18-data_split.zip. This dataset
contains 4,978 training samples and 893 test samples with 18 intent labels.
There are 54 manually written REs for intent classification obtained from Luo
et al. (2018)333https://pfllo.github.io/data/ACL18-REs.zip. We build few-shot
training sets by randomly selecting $q$ samples for each class from the full
training set. The models are trained on each training set respectively and
evaluated on the official test set.

###  3.2 Comparison with NN and Other RE-NN Hybrid Models

Model |  $q$=5 |  $q$=10 |  $q$=20 |  $q$=5 |  $q$=10 |  $q$=20
---|---|---|---|---|---|---
|  |  |  w/300 |  w/300 |  w/300
mlp-i* |  63.72 |  73.46 |  83.20 |  94.23 |  94.90 |  95.71
mlp-o* |  58.68 |  77.83 |  89.25 |  91.82 |  97.09 |  97.15
attn (luo et al.) |  75.36 |  85.44 |  88.80 |  92.05 |  96.98 |  97.76
fol (hu et al.) |  56.22 |  68.42 |  84.10 |  91.94 |  96.75 |  97.42
nnsc (ours) |  65.28 |  73.90 |  86.89 |  94.06 |  96.97 |  98.20
instance (ours) |  81.97 |  84.99 |  91.04 |  94.51 |  97.64 |  97.87
word (ours) |  80.17 |  86.67 |  90.25 |  94.62 |  97.20 |  98.32
Table 1:  Comparisons with the other methods that combine REs with NN without
MDFA and the NN-based Model (denoted as nnsc). * indicates the baseline hybrid
models described in Luo et al. (2018). The results are reported in accuracy.

Our hybrid models are compared with the pure NN-based model (denoted as nnsc)
and four RE-NN hybrid models that introduce REs into NN without using MDFA, as
shown in Table 1444“w/300” represents 300 extra training samples from each top
3 frequency classes are used additionally. These training sets are used in Luo
et al. (2018) and imported for fair comparisons.. mlp-i feeds the final
matching results of REs into the MLP classifier as additional features, while
hybrid model instance introduces the more fine-grained MDFA-based encodings.
mlp-o add the RE matching result to the probabilities output by the MLP
classifier, attn uses the keywords in REs to regularize the attention weights
in nnsc. fol converts the REs into FOL (First-Order-Logic) rules and distills
the knowledge from FOL rules into NN using the method proposed by (Hu et al.,
2016).

It can be seen that instance and word achieve better performances. We
conjecture that this is because the MDFA-based features can capture more fine-
grained information from REs, and these features can be well absorbed by
neural networks with the proposed integration strategies.

###  3.3 Experiments on Training Sets of Different Sizes

To further investigate the performance of the models on training sets of
different sizes, we selected 69 values for $q$ from 1 to 400 and constructed
the training sets accordingly. To reduce the impact of randomness, we repeat
each selection three times and the experiment on each training set 5 times.
The performances of the proposed models and nnsc are graphically shown in
Figure 3, with the shadow representing the 95% confidence interval.

Note that the proposed hybrid models achieve significant improvements when the
amount of training data is small (e.g., $q\leq 30$). Due to the elements in
the MDFA-based features has a small value set (i.e., 0, 1), such features have
fewer combinations than natural language constituted by thousands of different
words. Therefore, the MDFA-based features can be learned even with fewer
training samples. However, in the case when the training dataset is large
enough, the REs (i.e., 65.7% accuracy shown as the dashed line in Figure 3)
are much weaker than nnsc. The performance is therefore hard to be boosted by
additionally utilizing rules, and the improvements over nnsc decrease as the
training set size increases.

The average accuracy improvement over all the constructed training sets that
the proposed instance and word models achieved are 2.55% and 3.99%,
respectively. The word-level integration performs relatively better than the
instance-level. We think it is because there exist rich and flexible
interactions between RNN and rule-based features.

##  4 Conclusions

In this paper, we tried to incorporate human-designed REs into NN-based
models. The REs are transformed into MDFAs to provide MDFA-based features that
can be easily absorbed by NN. Moreover, we proposed two kinds of hybrid models
that incorporate the MDFA-based features into NN at word level or instance
level. The experiment results on the ATIS intent classification task have
shown that the proposed hybrid models improve the performance of NN when the
training data is insufficient and further outperform the other RE-NN hybrid
models without using MDFA.

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