# Optimal $C^{\infty}$-approximation of functions with exponentially or sub-
exponentially integrable derivative

Luigi Ambrosio  Scuola Normale Superiore, 56126 Pisa, Italy
<EMAIL_ADDRESS>,  Sebastiano Nicolussi Golo  Department of Mathematics
and Statistics, 40014 University of Jyväskylä, Finland
<EMAIL_ADDRESS>and  Francesco Serra Cassano  Dipartimento
di Matematica, Università di Trento, via Sommarive, 14, 38123 Trento, Italy
<EMAIL_ADDRESS>
###### Abstract.

We discuss Meyers-Serrin’s type results for smooth approximations of functions
$b=b(t,x):\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}^{m}$, with convergence
of an energy of the form

$\int_{\mathbb{R}}\int_{\mathbb{R}^{n}}w(t,x)\varphi\left(|Db(t,x)|\right)\mathrm{d}x\mathrm{d}t\,,$

where $w>0$ is a suitable weight function, and
$\varphi:[0,\infty)\to[0,\infty)$ is a convex function with $\varphi(0)=0$
having exponential or sub-exponential growth.

###### Key words and phrases:

Meyers-Serrin, Sobolev–Orlicz Spaces

###### 2010 Mathematics Subject Classification:

46E30, 35A35

L.A. and F.S.C. have been supported by the PRIN 2017 project “Gradient flows,
Optimal Transport and Metric Measure Structures”. S.N.G. has been supported by
the Academy of Finland (grant 328846 “Singular integrals, harmonic functions,
and boundary regularity in Heisenberg groups”, grant 322898 “Sub-Riemannian
Geometry via Metric-geometry and Lie-group Theory”, grant 314172 “Quantitative
rectifiability in Euclidean and non-Euclidean spaces”), S.N.G. and F.S.C. have
been supported by the INdAM – GNAMPA Project 2019 “Rectifiability in Carnot
groups”. Data sharing not applicable to this article as no datasets were
generated or analysed during the current study.

###### Contents

  1. 1 Introduction and results
  2. 2 Recalls of some density results of smooth functions in Orlicz and Orlicz-Sobolev spaces
    1. 2.1 N-functions
    2. 2.2 The Orlicz class $K_{\varphi}(\Omega)$
    3. 2.3 The Orlicz space $L_{\varphi}(\Omega)$
    4. 2.4 Convergences in $L_{\varphi}(\Omega)$
    5. 2.5 The vector space $E_{\varphi}(\Omega)$
    6. 2.6  Orlicz-Sobolev spaces and density results of smooth functions.
  3. 3 Exponential and sub-exponential N-functions
  4. 4 Proof of the approximation results

##  1\. Introduction and results

In this note we deal with the approximation of functions
$b=b(t,x):I\times\Omega\to\mathbb{R}^{m}$ by smooth ones, where
$I\subset\mathbb{R}$ and $\Omega\subset\mathbb{R}^{n}$ are an open interval
and an open set, respectively. Our main motivation comes from [ANGSC], where
$m=n$ and $b$ is a possibly nonautonomous vector field. In that paper we are
dealing with a priori upper bounds for Sobolev norms of the flow of $b$ by
means of quantities $N_{\varphi,w}(|Db|)$ that depend on the spatial
derivative $Db$ of $b$. The quantities $N_{\varphi,w}(|Db|)$ are _energies_ of
the form

$N_{\varphi,w}(|Db|):=\,\int_{I}\int_{\Omega}w(t,x)\varphi\left(|Db(t,x)|\right)\,\mathrm{d}x\,\mathrm{d}t\,,$

where $w>0$ is a suitable weight function (related to ${\rm
dist}(x,\partial\Omega)$, or to the length of the maximal interval of the ODE
associated to $b$) and $\varphi:[0,\infty)\to[0,\infty)$ is a convex function
with $\varphi(0)=0$ having exponential or sub-exponential growth, the model
case being $\exp_{*}(t):=\exp(t)-1$. When one tries to extend the a priori
estimates from the case of smooth vector fields $b$ to those having only a
Sobolev spatial regularity, one faces the difficulty of passing the quantity
$N_{\varphi,w}$ to the limit.

In this context, if $\varphi$ had polynomial growth, a weighted and
$t$-dependent version of the celebrated Meyers-Serrin Theorem [MS] would be
applicable, providing even a smooth approximation $(b_{h})_{h}$ with
$N_{\varphi,w}(|D(b_{h}-b)|)\to 0$. In general, as the discussion below shows,
this kind of approximation fails when $\varphi$ does not satisfy a doubling
condition. However, we realized that the exponential (or subexponential) case
is a borderline one. Indeed, thanks to the weak subadditivity condition

$\varphi(a+b)\leq\,k\,[(1+\varphi(b))\,\varphi(a)+\varphi(b)]$

we are able to prove convergence of the energy $N_{\varphi,w}$ and density of
smooth functions with respect to modular convergence when $\varphi$ is
strictly convex. This kind of convergence in energy, though weaker than
convergence with respect to Luxenburg norm (or modular convergence, when
$\varphi$ is not strictly convex), should be compared with the theory of $BV$
functions, where smooth functions are not dense in $BV$ norm, but dense in
energy. Moreover, this convergence will be sufficient to pass our a priori
estimates to the limit in the paper [ANGSC].

In the classical setting of Orlicz spaces (see Section 2) the weighted
$\varphi$-energy $N_{\varphi,w}$ is called a modular and we put some of our
results in this context. The main approximation result of the note reads as
follows.

######  Theorem 1.

Let $I\subset\mathbb{R}$ be an open interval and $\Omega\subset\mathbb{R}^{n}$
an open set. Let $w:\,I\times\Omega\to(0,\infty)$ be a Borel function
uniformly bounded from above and from below on compact subsets of
$I\times\Omega$. Let $\varphi:\,[0,\infty)\to[0,\infty)$ be a convex function
satisfying $\varphi(0)=\,0$ and and for which there exists a positive constant
$k_{\varphi}$ such that

(1)
$\varphi(a+b)\leq\,k_{\varphi}[\varphi(a)\,\varphi(b)+\,\varphi(a)+\varphi(b)]\qquad\text{for
all $a,\,b\in[0,\infty)$}\,.$

Let $b\in L^{1}_{\rm loc}(I;W^{1,1}_{\rm loc}(\Omega;\mathbb{R}^{m}))\cap
C^{0}(I\times\Omega;\mathbb{R}^{m})$ satisfy

(2) $N_{\varphi,w}(|Db|)<\infty\,.$

Then there exist $b_{h}\in C^{\infty}(I\times\Omega;\mathbb{R}^{m})$
satisfying

(3) $b_{h}\to b\text{ in }L^{1}_{\rm loc}(I\times\Omega;\mathbb{R}^{m}),\qquad
Db_{h}\to Db\text{ in }L^{1}_{\rm loc}(I\times\Omega;\mathbb{R}^{nm})\,,$

and

(4) $w\varphi\left(|Db_{h}|\right)\to w\varphi\left(|Db|\right)\text{ in
}L^{1}(I\times\Omega)\,.$

In particular,

(5) $\lim_{h\to\infty}N_{\varphi,w}(|Db_{h}|)=N_{\varphi,w}(|Db|)\,.$

Besides the model case of $\varphi=\exp_{*}$, we are able to consider the
functions

(6) $\exp_{\gamma,\tau}^{*}(t):=\,\exp_{\gamma,\tau}(t)-1,$

where $0\leq\,\gamma\leq\,1$, $\tau>0$ and

(7)
$\exp_{\gamma,\tau}(t):=\exp\left(\displaystyle\frac{t}{(\log(t+\tau))^{\gamma}}\right).$

It is easy to see that a convex function $\varphi$ with polynomial growth
satisfies (1), see Remark 6. We will show in Lemma 21 that
$\exp_{\gamma,\tau}^{*}$ satisfies the conditions in Theorem 1 for $\varphi$
with $k_{\varphi}=1$, if $\tau$ is sufficiently large. The functions
$\exp_{\gamma,\tau}$, though convex, do not have null derivative at $0$, and
therefore do not fit exactly in the theory of $N$-functions. Therefore, in
order to provide a bridge with the theory of $N$-functions of Orlicz spaces,
we will also consider the modified functions

(8)
$\widetilde{\exp}_{\gamma,\tau}(t):=\,\exp_{\gamma,\tau}(t)-1-\frac{t}{(\log\tau)^{\gamma}}=\exp_{\gamma,\tau}^{*}(t)-\frac{t}{(\log\tau)^{\gamma}}\,,$

which are indeed $N$-functions and can be treated it by comparison with
$\exp_{\gamma,\tau}^{*}$.

######  Corollary 2.

Let $w:\,I\times\Omega\to(0,\infty)$ be a Borel function uniformly bounded
from above and from below on compact subsets of $I\times\Omega$. Let $b\in
L^{1}_{\rm loc}(I;W^{1,1}_{\rm loc}(\Omega;\mathbb{R}^{m}))\cap
C^{0}(I\times\Omega;\mathbb{R}^{m})$ satisfy (2) with

(9) $\varphi=\,\widetilde{\exp}_{\gamma,\tau}$ with $\tau$ sufficiently large

and

(10) $\text{either $w|Db|\in L^{1}(I\times\Omega)$, or $w\in
L^{1}(I\times\Omega)$}\,.$

Then there exist $b_{h}\in C^{\infty}(I\times\Omega;\mathbb{R}^{m})$
satisfying (3), (4) and (5).

Notice that, when the functions $\varphi$ we are dealing with have a more than
polynomial growth at infinity and the weight function $w$ is uniformly bounded
from 0 on compact subsets, Corollary 2, the Sobolev Embedding Theorem grants
continuity of $b$ with respect to the spatial variable. But, in the proof of
Theorem 1, it seems that the continuity of $b$ with respect to $t$ is also
needed (cf. the estimate of the term $z_{\delta}$). However, if we assume the
weight $w$ to be time-independent, we can adapt the proof of Theorem 1 to drop
the continuity assumption on $b$, and we obtain the following extension of
Corollary 2.

######  Theorem 3.

Let $w:\,\Omega\to(0,\infty)$ be a Borel function uniformly bounded from above
and from below on compact subsets of $\Omega$. Assume that either
$\varphi=\exp^{*}_{\gamma,\tau}$ or $\varphi=\widetilde{\exp_{\gamma,\tau}}$
and (10) holds, with $\tau$ sufficiently large and let $b\in L^{1}_{\rm
loc}(I;W^{1,1}_{\rm loc}(\Omega;\mathbb{R}^{m}))$ satisfy (2). Then there
exist $b_{h}\in C^{\infty}(I\times\Omega;\mathbb{R}^{m})$ satisfying (3), (4)
and (5).

When the function $\varphi$ is strictly convex, from the $\varphi$-energy
convergence of the Jacobian matrices of autonomous vector fields, one can
obtain the modular convergence (see Definition 10).

######  Theorem 4.

Let $\varphi:[0,\infty)\to[0,\infty)$ be a strictly convex function, and
assume that $b_{h}\to b$ in $L^{1}_{\rm loc}(\Omega;\mathbb{R}^{m})$ with

(11)
$\int_{\Omega}\varphi(|Db_{h}|)\,\mathrm{d}x\to\int_{\Omega}\varphi(|Db|)\,\mathrm{d}x<\infty\,.$

Then

(12) $\int_{\Omega}\varphi\left(\frac{|Db_{h}-Db|}{2}\right)\,\mathrm{d}x\to
0\,.$

Finally, if $\varphi=\,\widetilde{\exp}_{\gamma,\tau}$, being
$\widetilde{\exp}_{\gamma,\tau}$ a $N$-function (see Lemma 21.(ii)), we can
set our result in the classical setting of Orlicz-Sobolev spaces (see Section
2). Therefore, an immediate consequence of Theorem 3 is the following
approximation result in the Orlicz-Sobolev class
$W^{1}K_{\widetilde{\exp}_{\gamma,\tau}}(\Omega)$.

######  Theorem 5.

Let $\varphi=\,\widetilde{\exp}_{\gamma,\tau}$ be the $N$-function in (8) with
$\tau$ given by Lemma 21 and $u\in
W^{1}K_{\widetilde{\exp}_{\gamma,\tau}}(\Omega)$. Suppose that

(13) either $|Du|\in L^{1}(\Omega)$, or $\Omega$ has finite measure .

Then there exists $(u_{h})_{h}\subset C^{\infty}(\Omega)\cap
W^{1}K_{\widetilde{\exp}_{\gamma,\tau}}(\Omega)$ such that $(u_{h})_{h}$ is
mean convergent to $u$ (with respect to the modular
$N_{\widetilde{\exp}_{\gamma,\tau}}$) and $(|Du_{h}|)_{h}$ is
${\widetilde{\exp}_{\gamma,\tau}}$-energy convergent to $|Du|$, that is,

(14)
$\lim_{h\to\infty}N_{\widetilde{\exp}_{\gamma,\tau}}\left(u_{h}-u\right)=\,0\text{
and
}\lim_{h\to\infty}N_{\widetilde{\exp}_{\gamma,\tau}}\left(|Du_{h}|\right)=N_{\widetilde{\exp}_{\gamma,\tau}}\left(|Du|\right)\,.$

To our knowledge, the previous result does not seem be a consequence of the
well-known results about approximation by smooth functions in Orlicz-Sobolev
spaces (see Section 2.6), even in the classical case with $\gamma=0$.

##  2\. Recalls of some density results of smooth functions in Orlicz and
Orlicz-Sobolev spaces

We will quickly recall here the notions of Orlicz and Orlicz-Sobolev spaces
and some their main properties. In particular, we will focus on the main
density results of smooth functions in Orlicz and Orlicz-Sobolev spaces. We
will mainly use the notation from [A, Ch. VIII].

###  2.1. N-functions

A function $\varphi:[0,\infty)\to[0,\infty)$ is called a $N$-function, if

$\varphi(t):=\,\int_{0}^{t}a(s)\,\,\mathrm{d}s\text{ if }t\geq\,0,$

with $a:\,[0,\infty)\to[0,\infty)$ satisfying:

  * •

$a(0)=\,0$, $a(t)>\,0$ if $t>\,0$, and $\lim_{t\to\infty}a(t)=\,\infty$;

  * •

$a$ is nondecreasing, that is, if $t\geq\,s\geq\,0$, then $a(t)\geq\,a(s)$;

  * •

$a$ is right continuous, that is, if $t\geq\,0$, then $\lim_{s\to
t^{+}}a(s)=\,a(t)$.

Given a $N$-function $\varphi$ and $\lambda>\,0$, we denote by
$\varphi_{\lambda}:[0,\infty)\to[0,\infty)$ the function

$\varphi_{\lambda}(t):=\,\varphi\left(\frac{t}{\lambda}\right)\text{ if
}t\geq\,0\,,$

which is still a $N$-function.

A function $\varphi$ is said to satisfy a global $\Delta_{2}$-condition  if
there exists $k>\,0$ such that

$\varphi(2t)\leq\,k\,\varphi(t)\text{ for each }t\geq\,0\,.$

A function $\varphi$ is said to satisfy a $\Delta_{2}$-condition near infinity
if there exist $k,t_{0}>\,0$ such that

$\varphi(2t)\leq\,k\,\varphi(t)\text{ for each }t\geq\,t_{0}\,.$

######  Remark 6.

Observe that a convex function $\varphi$ satisfying a global
$\Delta_{2}$-condition trivially fulfills condition (1). Indeed, by the
convexity and $\Delta_{2}$-condition, we can get the following estimate

$\varphi(a+b)\leq\,\frac{1}{2}\varphi(2a)+\frac{1}{2}\varphi(2b)\leq\,\frac{k}{2}\left(\varphi(a)+\varphi(b)\right)\text{
for each }a,\,b\in\mathbb{R}\,.$

Given $\Omega\subset\mathbb{R}^{n}$ and a N-function $\varphi$ , a pair
$(\varphi,\Omega)$ is said to be $\Delta$-regular if

  * •

$\varphi$ satisfies a global $\Delta_{2}$-condition,

or

  * •

$\varphi$ satisfies a $\Delta_{2}$-condition near infinity and $\Omega$ has
finite measure.

###  2.2. The Orlicz class $K_{\varphi}(\Omega)$

Let $\Omega\subset\mathbb{R}^{n}$ be an open set and let $\varphi$ be a
$N$-function. The Orlicz class  $K_{\varphi}(\Omega)$ is the set of all
(equivalence classes modulo equality a.e. on $\Omega$ of) measurable functions
$u:\Omega\to\mathbb{R}$ such that

$N_{\varphi}(u):=\,\int_{\Omega}\varphi(|u(x)|)\,\,\mathrm{d}x<\,\infty\,.$

In the theory of modular spaces, the map $u\mapsto N_{\varphi}(u)$ is called a
modular ([RR, pg. 82]). A comprehensive account of modular function spaces can
be found in [K]. We treat the case of real-valued functions for simplicity,
but all results have an obvious extension to the case of
$\mathbb{R}^{m}$-valued maps.

Let us recall some properties of the Orlicz class $K_{\varphi}(\Omega)$.

######  Proposition 7.

Given an open set $\Omega\subset\mathbb{R}^{n}$ and a $N$-function $\varphi$,
the following statements hold:

  * (i)

$K_{\varphi}(\Omega)$ is a convex set of measurable functions.

  * (ii)

$K_{\varphi_{\lambda}}(\Omega)\supseteq K_{\varphi}(\Omega)$ if
$\lambda\geq\,1$ and $K_{\varphi_{\lambda}}(\Omega)\subseteq
K_{\varphi}(\Omega)$ if $\lambda\leq\,1$, where
$\varphi_{\lambda}(t):=\varphi(t/\lambda)$ is a $N$-function for all
$\lambda>0$.

  * (iii)

If $f,\,g\in K_{\varphi}(\Omega)$, then $f+g\in K_{\varphi_{2}}(\Omega)$ and

$N_{\varphi_{2}}(f+g)\leq\,\frac{1}{2}N_{\varphi}(f)+\frac{1}{2}N_{\varphi}(g)\,.$

  * (iv)

If $f\in K_{\varphi}(\Omega)$ and $\lambda>\,0$, then $\lambda f\in
K_{\varphi_{\lambda}}(\Omega)$.

  * (v)

If $\Omega$ has finite measure, then

$L^{\infty}(\Omega)\subset K_{\varphi}(\Omega)\subsetneq L^{1}(\Omega).$

  * (vi)

If $\Omega$ has finite measure, then for every $u\in L^{1}(\Omega)$ there is a
$N$-function $\varphi$ such that $u\in K_{\varphi}(\Omega)$.

###### Proof.

Properties (i), (ii), (iii) and (iv) are immediate consequences of the
definition of $K_{\varphi}(\Omega)$ and the convexity of $\varphi$. For the
proof of properties (v) and (vi) see, for instance, [KR]. ∎

######  Lemma 8 ([A, Lem. 8.8] or [KR, Ch. III, Th. 8.2]).

$K_{\varphi}(\Omega)$ is a vector space if and only if $(\varphi,\Omega)$ is
$\Delta$-regular .

###  2.3. The Orlicz space $L_{\varphi}(\Omega)$

The Orlicz space $L_{\varphi}(\Omega)$ is defined to be the linear hull of the
Orlicz class $K_{\varphi}(\Omega)$, that is the smallest vector subspace of
$L^{1}_{\rm loc}(\Omega)$ containing $K_{\varphi}(\Omega)$. It is easy to see
that, since $K_{\varphi}(\Omega)$ is convex, one has

$L_{\varphi}(\Omega):=\,\left\\{\lambda\,u:\,\lambda\in\mathbb{R},\,u\in
K_{\varphi}(\Omega)\right\\}\,.$

Moreover, from Lemma 8, $K_{\varphi}(\Omega)=L_{\varphi}(\Omega)$ if and only
if $(\varphi,\Omega)$ is $\Delta$-regular.

We can endow $L_{\varphi}(\Omega)$ with the following norm, called Luxemburg
norm,

$\|u\|_{\varphi}=\,\|u\|_{\varphi,\Omega}:=\,\inf\left\\{\lambda>\,0:\,\int_{\Omega}\varphi\left(\frac{|u(x)|}{\lambda}\right)\,\,\mathrm{d}x\leq\,1\right\\}\,.$

######  Theorem 9 ([A, Thm. 8.10]).

$(L_{\varphi}(\Omega),\|\cdot\|_{\varphi})$ is a Banach space.

###  2.4. Convergences in $L_{\varphi}(\Omega)$

The typical convergences that apply in Orlicz spaces are the following.

######  Definition 10.

A sequence of functions $(u_{h})_{h}\subset L_{\varphi}(\Omega)$ is said to be
norm convergent to $u\in L_{\varphi}(\Omega)$ if

$\|u_{h}-u\|_{\varphi}\to 0\text{ as }h\to\infty\,.$

A sequence of functions $(u_{h})_{h}\subset L_{\varphi}(\Omega)$ is said to be
modular convergent to $u\in L_{\varphi}(\Omega)$ if there exists $\lambda>\,0$
such that

(15) $N_{\varphi}\left(\frac{u_{h}-u}{\lambda}\right)\to 0\text{ as
}h\to\infty.$

If $\lambda=1$ in (15), $(u_{h})_{h}$ is said to be mean convergent to $u\in
L_{\varphi}(\Omega)$. A sequence of functions $(u_{h})_{h}\subset
K_{\varphi}(\Omega)$ is said to be $\varphi$-energy convergent to $u\in
K_{\varphi}(\Omega)$ if

(16) $N_{\varphi}(u_{h})\to N_{\varphi}(u)\text{ as }h\to\infty.$

Norm and modular convergences are classical in the theory of Orlicz spaces
(see, for instance, [A, KR]). We do not know whether the $\varphi$-energy
convergence has been already named in the literature.

The following implications between norm, mean, modular and $\varphi$-energy
convergence hold.

######  Proposition 11.

Let $(u_{h})_{h}$ and $u$ be in $L_{\varphi}(\Omega)$.

  * (i)

Suppose that $(u_{h})_{h}$ is norm convergent to $u$. Then it is also mean
convergent. The converse implication in general does not hold. It holds if
$(\varphi,\Omega)$ is $\Delta$-regular.

  * (ii)

Suppose that $(\varphi,\Omega)$ is $\Delta$-regular, $\varphi$ is strictly
convex, $u_{h}\to u$ a.e. in $\Omega$ and $(u_{h})_{h}$ is $\varphi$-energy
convergent to $u$. Then $(u_{h})_{h}$ is norm convergent to $u$.

  * (iii)

$(u_{h})_{h}$ is norm convergent to $u$ if and only if, for each
$\lambda>\,0$,

$N_{\varphi}\left(\frac{u_{h}-u}{\lambda}\right)\to 0\text{ as }h\to\infty\,.$

  * (iv)

Suppose that $(\varphi,\Omega)$ is $\Delta$-regular and $(u_{h})_{h}\subset
K_{\varphi}(\Omega)$ is mean convergent to $u\in K_{\varphi}(\Omega)$. Then
$(u_{h})_{h}$ is $\varphi$-energy convergent to $u$.

  * (v)

Suppose that $(2u_{h})_{h}\subset K_{\varphi}(\Omega)$ is mean convergent to
$2u\in K_{\varphi}(\Omega)$ (with respect to the modular $N_{\varphi}$). Then
$(u_{h})_{h}\subset K_{\varphi}(\Omega)$, $u\in K_{\varphi}(\Omega)$ and
$(u_{h})_{h}$ is $\varphi$-energy convergent to $u$.

  * (vi)

Suppose that $2u\in K_{\varphi}(\Omega)$ and $(u_{h})_{h}$ is norm convergent
to $u$. Then $u_{h}\in K_{\varphi}(\Omega)$ for $h$ large and $(u_{h})$ is
also $\varphi$-energy convergent to $u$.

###### Proof.

(i) and (ii) are proven in [RR, Chap. III, Sect. 3.4, Thm. 12]. The proof of
(iii) is somehow elementary, see for instance [AFY, Lem. 2.7] and [K, pg. 4].

We prove (iv) when $\varphi$ satisfies a global $\Delta_{2}$-condition: in the
other case, when $\Omega$ has finite measure and the $\varphi$ satisfies a
$\Delta_{2}$-condition near infinity, has a similar proof. From the
assumptions, the sequence $(\varphi(|u_{h}-u|))_{h}\subset L^{1}(\Omega)$
converges in $L^{1}(\Omega)$ to $0$. Thus, up to a subsequence, we can assume
that

$\varphi(|u_{h}-u|)\to 0\text{ a.e. in }\Omega,\text{ as $h\to\infty$}\,.$

Since $\varphi$ is a $N$-function, then $\varphi:\,[0,\infty)\to[0,\infty)$ is
bijective and $\varphi^{-1}:\,[0,\infty)\to[0,\infty)$ is still continuous.
Thus, we also get that

(17)
$|u_{h}-u|=\,\varphi^{-1}(\varphi(|u_{h}-u|))\to\varphi^{-1}(\varphi(0))=\,0\text{
a.e.\leavevmode\nobreak\ in }\Omega,\text{ as $h\to\infty$}\,.$

From the convexity of $\varphi$ and the global $\Delta_{2}$-condition of
$\varphi$, it follows that

(18) $\displaystyle\varphi(|u_{h}|)$
$\displaystyle\leq\,\frac{1}{2}\varphi(2|u_{h}-u|)+\frac{1}{2}\varphi(2|u|)\,$
$\displaystyle\leq\,\frac{k}{2}\left(\varphi(|u_{h}-u|)+\varphi(|u|)\right).$

By (17) and (18), we can apply Vitali’s convergence theorem and then

$\varphi(|u_{h}|)\to\varphi(|u|)\text{ in }L^{1}(\Omega)\text{, as
}h\to\infty\,.$

Thus (16) follows.

For (v), we get at once that $(u_{h})_{h}\subset K_{\varphi}(\Omega)$ and
$u\in K_{\varphi}(\Omega)$, because $\varphi$ is increasing. We can show (17)
as in the proof of claim (iv), and the convexity of $\varphi$ implies

$\varphi(|u_{h}|)\leq\,\frac{1}{2}\varphi(2|u_{h}-u|)+\frac{1}{2}\varphi(2|u|)$

Thus, applying Vitali’s convergence theorem, we still get (16).

Finally, we prove (vi). From the norm convergence and (iii), we can infer
that, up to a subsequence,

$\varphi(|u_{h}-u|)\to 0\text{ a.e. in }\Omega\text{, as }h\to\infty\,,$

and

$\varphi(2|u_{h}-u|)\to 0\text{ in }L^{1}(\Omega)\text{, as }h\to\infty\,.$

We can show again (17) as in claim (iv) and
$\varphi(|u_{h}|)\leq\frac{1}{2}\varphi(2|u_{h}-u|)+\frac{1}{2}\varphi(2|u|)$
from the convexity of $\varphi$. Thus, applying Vitali’s convergence theorem,
we get (16). ∎

######  Example 12.

From items (iv) and (v) of Proposition 11, one could get the wrong impression
that mean convergence implies $\varphi$-energy convergence. We show that this
is not the case if $(\varphi,\Omega)$ is not $\Delta$-regular: for
$\Omega=\,(0,1)$ and $\varphi=\,\widetilde{\exp}_{0}$ (cf. (8)), we give a
sequence of functions $u_{h}\in K_{\varphi}((0,1))$ that is mean convergent to
$u\in K_{\varphi}((0,1))$, but that is not $\varphi$-energy convergent.

Let $f_{h},f:(0,1)\to\mathbb{R}$ be the functions

$\displaystyle{f_{h}(x):=\begin{cases}\frac{2\sqrt{h}}{\log h}&\text{ if
}0<\,x<\,\frac{1}{h}\\\ \frac{1}{\log h}\frac{1}{\sqrt{x}}&\text{ if
}\frac{1}{h}\leq\,x<\,1\end{cases}},\quad f(x):=\frac{1}{\sqrt{x}}.$

Direct computations show that

$\displaystyle\int_{0}^{1}f(x)\,\mathrm{d}x$ $\displaystyle=2,$
$\displaystyle\int_{0}^{1}\log(f(x))\,\mathrm{d}x$
$\displaystyle=\frac{1}{2},$ $\displaystyle\int_{0}^{1}f_{h}(x)\,\mathrm{d}x$
$\displaystyle\overset{h\to\infty}{\longrightarrow}0,$
$\displaystyle\int_{0}^{1}\log(1+f_{h})\,\mathrm{d}x$
$\displaystyle\overset{h\to\infty}{\longrightarrow}0,$
$\displaystyle\int_{0}^{1}f(x)\,f_{h}(x)\,\mathrm{d}x$
$\displaystyle=\frac{4}{\log(h)}+1\overset{h\to\infty}{\longrightarrow}1.$

Define

$u:=\log(f)\quad\text{ and }\quad u_{h}:=\log(f)+\log(1+f_{h}).$

Then, for $\varphi(s)=\exp(s)-1-s$, we have

$\displaystyle N_{\varphi}(u)$
$\displaystyle=\int_{0}^{1}f(x)\,\mathrm{d}x-1-\int_{0}^{1}\log(f(x))\,\mathrm{d}x,$
$\displaystyle N_{\varphi}(u_{h})$
$\displaystyle=\int_{0}^{1}f(x)\,\mathrm{d}x-1-\int_{0}^{1}\log(f(x))\,\mathrm{d}x$
$\displaystyle\qquad+\int_{0}^{1}f(x)\,f_{h}(x)\,\mathrm{d}x-\int_{0}^{1}\log(1+f_{h})\,\mathrm{d}x,$
$\displaystyle N_{\varphi}(u_{h}-u)$
$\displaystyle=\int_{0}^{1}f_{h}(x)\,\mathrm{d}x-\int_{0}^{1}\log(1+f_{h}(x))\,\mathrm{d}x.$

We conclude that $u,u_{h}\in K_{\varphi}((0,1))$, $N_{\varphi}(|u-u_{h}|)\to
0$ but

$N_{\varphi}(|u_{h}|)-N_{\varphi}(|u|)=\int_{0}^{1}f(x)\,f_{h}(x)\,\mathrm{d}x-\int_{0}^{1}\log(1+f_{h})\,\mathrm{d}x\overset{h\to\infty}{\longrightarrow}1,$

that is, $u_{h}$ is not $\varphi$-energy convergent to $u$. Notice that the
key fact is that $f_{h}\to 0$ in $L^{1}((0,1))$ but $f\cdot f_{h}\not\to 0$.
Let us also observe that $(2u_{h})_{h}\subset K_{\varphi}(\Omega)$, but
neither $2u\in K_{\varphi}(\Omega)$ nor $(2u_{h})_{h}$ is mean convergent to
$2u$ with respect to $N_{\varphi}$.

###  2.5. The vector space $E_{\varphi}(\Omega)$

Let $E_{\varphi}(\Omega)$ denote the closure in
$(L_{\varphi}(\Omega),\|\cdot\|_{\varphi})$ of the space of functions $u$
which are bounded in $\Omega$ with bounded support in $\Omega$.

One can see ([A, Sect. 8.14]) that
$E_{\varphi}(\Omega)\subset\,K_{\varphi}(\Omega)$ and that, if
$(\varphi,\Omega)$ is $\Delta$-regular, then

$E_{\varphi}(\Omega)=\,K_{\varphi}(\Omega)=\,L_{\varphi}(\Omega)\,.$

Moreover the following characterization of $E_{\varphi}(\Omega)$ holds ([A,
Lemma 8.15]).

######  Lemma 13.

$E_{\varphi}(\Omega)$ is the maximal linear subspace of $K_{\varphi}(\Omega)$.

######  Corollary 14.

If $(\varphi,\Omega)$ is not $\Delta$-regular, it holds that

$E_{\varphi}(\Omega)\subsetneq\,K_{\varphi}(\Omega)\subsetneq
L_{\varphi}(\Omega)\,.$

###### Proof.

By Lemma 8, $K_{\varphi}(\Omega)$ cannot be a vector space. Thus, by Lemma 13,
we get the desired conclusions. ∎

Let us now recall some density results in
$(E_{\varphi}(\Omega),\|\cdot\|_{A})$.

######  Theorem 15 ([A, Thm. 8.20]).

Let $\Omega\subset\mathbb{R}^{n}$ be an open set and let $\varphi$ be a
$N$-function.

  * (i)

$C^{\infty}_{c}(\Omega)$ are dense in
$(E_{\varphi}(\Omega),\|\cdot\|_{\varphi})$.

  * (ii)

$(E_{\varphi}(\Omega),\|\cdot\|_{\varphi})$ is separable.

  * (iii)

Let us extend $u\in E_{\varphi}(\Omega)$ to the whole $\mathbb{R}^{n}$ so as
to vanish outside $\Omega$ and let $(\rho_{\varepsilon})_{\varepsilon}$ be a
family of mollifiers on $\mathbb{R}^{n}$. Then

$\rho_{\varepsilon}*u\to u\text{ in
}(E_{\varphi}(\Omega),\|\cdot\|_{\varphi})\text{, as }\varepsilon\to 0\,.$

An immediate consequence of Theorem 15 is that, if $(\varphi,\Omega)$ is not
$\Delta$-regular, then $C^{0}_{c}(\Omega)$ is not dense in
$(L_{\varphi}(\Omega),\|\cdot\|_{\varphi})$. In fact, one can prove the
following stronger result:

######  Theorem 16 ([KR, Chap. II, Thm. 10.2]).

If the pair $(\varphi,\Omega)$ is not $\Delta$-regular, then
$(L_{\varphi}(\Omega),\|\cdot\|_{\varphi})$ is not separable.

Let us also point out some density results in $K_{\varphi}(\Omega)$ with
respect to the modular convergence.

######  Theorem 17.

Let $\Omega\subset\mathbb{R}^{n}$ be an open set and let $\varphi$ be a
$N$-function.

  * (i)

The set of bounded functions on $\Omega$ contained in $K_{\varphi}(\Omega)$
with bounded support is dense in $K_{\varphi}(\Omega)$ with respect to the
mean convergence, that is, for each $u\in K_{\varphi}(\Omega)$ there exists a
sequence of bounded functions $(u_{h})_{h}\subset K_{\varphi}(\Omega)$ such
that

$N_{\varphi}(u_{h}-u)\to 0\text{, as }h\to\infty\,.$

  * (ii)

$C^{0}_{c}(\Omega)$ is dense in $K_{\varphi}(\Omega)$ with respect to the
modular convergence with $\lambda=4$. More precisely, for each $u\in
K_{\varphi}(\Omega)$, there is a sequence $(u_{h})_{h}\subset
C^{0}_{c}(\Omega)$ such that

$N_{\varphi}\left(\frac{u_{h}-u}{4}\right)\to 0\text{, as }h\to\infty\,.$

###### Proof.

The proof of part (i) can be found in [KR, Chap. II, pg. 77] or [A, Sect.
8.14].

To prove part (ii), given $u\in K_{\varphi}(\Omega)$ and $\epsilon>0$, we
first notice that, by a standard truncation argument in $\Omega$, there is a
function $\tilde{u}\in K_{\varphi}(\Omega)$ with support compactly contained
in $\Omega$ and with $N_{\varphi}(u-\tilde{u})<\epsilon$.

Next, let $\tilde{u}_{k}:=\max\\{-k,\min\\{\tilde{u},k\\}\\}$ be the standard
truncation of $u$, $F_{k}:=\\{|\tilde{u}|>k\\}$ and $f\in C^{0}_{c}(\Omega)$
with $\sup|f|\leq k$. We estimate

$\displaystyle N_{\varphi_{2}}(\tilde{u}-f)$
$\displaystyle=\int_{\Omega\setminus
F_{k}}\varphi\left(\frac{|\tilde{u}_{k}-f|}{2}\right)\,\mathrm{d}x+\int_{F_{k}}\varphi\left(\frac{|\tilde{u}-f|}{2}\right)\,\mathrm{d}x$
$\displaystyle\leq\int_{\Omega\setminus
F_{k}}\varphi\left(\frac{|\tilde{u}_{k}-f|}{2}\right)\,\mathrm{d}x+\frac{1}{2}\int_{F_{k}}\varphi(|\tilde{u}|)\,\mathrm{d}x+\frac{1}{2}\int_{F_{k}}\varphi(|f|)\,\mathrm{d}x$
$\displaystyle\leq\int_{\Omega\setminus
F_{k}}\varphi\left(\frac{|\tilde{u}_{k}-f|}{2}\right)\,\mathrm{d}x+\int_{F_{k}}\varphi(|\tilde{u}|)\,\mathrm{d}x.$

Now, since $\int_{\Omega}\varphi(|\tilde{u}|)\,\mathrm{d}x<\infty$, we can
choose $k$ so large that the second integral is smaller than $\epsilon/2$.
Since $\tilde{u}_{k}$ has compact support and thanks to Lusin’s theorem, we
can find $f\in C^{0}_{c}(\Omega)$ with $|f|\leq k$ and the Lebesgue measure of
$\\{x\in\Omega:\ \tilde{u}_{k}(x)\neq f(x)\\}$ sufficiently small, in such a
way that also the first integral gives a contribution smaller than
$\epsilon/2$.

In conclusion, for every $\epsilon>0$ we have $f\in C^{0}_{c}(\Omega)$ such
that

$N_{\varphi_{4}}(u-f)\leq\frac{1}{2}(N_{\varphi_{2}}(u-\tilde{u})+N_{\varphi_{2}}(\tilde{u}-f))\leq\frac{1}{2}(N_{\varphi}(u-\tilde{u})/2+\epsilon)\leq\epsilon.$

∎

###  2.6.  Orlicz-Sobolev spaces and density results of smooth functions.

Given a $N$-function $\varphi$, the Orlicz-Sobolev vector space
$W^{1}L_{\varphi}(\Omega)$ consists of those (equivalence classes of)
functions $u\in L_{\varphi}(\Omega)\cap W^{1,1}_{loc}(\Omega)$ whose weak
derivatives $D_{i}u\in L_{\varphi}(\Omega)$ for each $i=1,\ldots,n$. The
vector space $W^{1}E_{\varphi}(\Omega)$ and the convex set
$W^{1}K_{\varphi}(\Omega)$ are defined in analogous fashion. Obviously

$W^{1}E_{\varphi}(\Omega)\subset W^{1}K_{\varphi}(\Omega)\subset
W^{1}L_{\varphi}(\Omega)\,.$

It is easy to see (see, for instance, [A, §8.27]) that
$W^{1}L_{\varphi}(\Omega)$ is a Banach space with respect to the norm

$\|u\|_{1,\varphi}:=\,\max\\{\|u\|_{\varphi},\|D_{1}u\|_{\varphi},\ldots,\|D_{n}u\|_{\varphi}\\}\,.$

Notice also that, since

$\max_{i}|D_{i}u|\leq|Du|\leq\,\sum_{i=1}^{n}|D_{i}u|\text{ a.e. in
}\Omega\,,$

and $L_{\varphi}(\Omega)$ is a linear space, an equivalent norm on
$W^{1}L_{\varphi}(\Omega)$ is given by.

$\|u\|_{\varphi}+\||Du|\|_{\varphi}\,.$

Observe that $W^{1}E_{\varphi}(\Omega)$ turns out to be a closed subspace of
$W^{1}L_{\varphi}(\Omega)$. Moreover $W^{1}E_{\varphi}(\Omega)$ coincides with
$W^{1}L_{\varphi}(\Omega)$ if and only if $(\varphi,\Omega)$ is
$\Delta$-regular. Notice also that, for the applications we have in mind, what
is more relevant is the $\varphi$-integrability of the derivative, rather than
the integrability of the function which, also in view of Sobolev embeddings,
could be qualified in a different way, see also Remark 24.

Celebrated Meyers-Serrin’s result was extended from the classical Sobolev
spaces to the Orlicz-Sobolev space $W^{1}E_{\varphi}(\Omega)$ in [DT] (see
also [AFY]).

######  Theorem 18 ([DT]).

$C^{\infty}(\Omega)\cap W^{1}E_{\varphi}(\Omega)$ is dense in
$(W^{1}E_{\varphi}(\Omega),\|\cdot\|_{1,\varphi})$.

It is easy to see that the previous result also fails for functions in the
Orlicz-Sobolev class $W^{1}K_{\varphi}(\Omega)$, and so also in the Orlicz-
Sobolev space $W^{1}L_{\varphi}(\Omega)$, provided that $(\varphi,\Omega)$ is
not $\Delta$-regular, as the following example shows.

######  Example 19.

Assume that $n=1$, $\Omega=\,(-1,1)$, let $\varphi=\,\widetilde{\exp}_{0}$ be
$N$-function in (8) with $\gamma=0$ and let

$u(x):=\displaystyle{\begin{cases}\frac{x}{2}\,\log\frac{1}{e|x|}&\text{ if
}|x|\leq\,\frac{1}{e}\\\ 0&\text{ if }\frac{1}{e}<\,|x|\,<\,1\end{cases}}\,.$

Then it is easy to see that $u\in W^{1}K_{\varphi}(\Omega)\setminus
W^{1}E_{\varphi}(\Omega)$, since the weak derivative

$u^{\prime}(x):=\begin{cases}\log\displaystyle{\frac{1}{e\,\sqrt{|x|}}}&\text{
if }|x|<\,\frac{1}{e}\\\ 0&\text{ if }\frac{1}{e}<\,|x|\,<\,1\end{cases}\text{
a.e. }x\in\Omega\,$

belongs to $K_{\varphi}(\Omega)\setminus E_{\varphi}(\Omega)$. Indeed

$\int_{-1}^{1}\varphi(|u^{\prime}|)\,\,\mathrm{d}x=\int_{-1}^{1}\left(\exp(|u^{\prime}|)-|u^{\prime}|-1\right)\,\,\mathrm{d}x<\,\infty\,,$

but $2|u^{\prime}|\notin K_{\varphi}(\Omega)$, since

$\int_{-1}^{1}\varphi(2|u^{\prime}|)\,\,\mathrm{d}x=\int_{-1}^{1}\left(\exp(2|u^{\prime}|)-2|u^{\prime}|-1\right)\,\,\mathrm{d}x=\,\infty\,.$

Thus $|u^{\prime}|\notin E_{\varphi}(\Omega)$, since $E_{\varphi}(\Omega)$ is
a linear subspace . By contradiction, assume there exists a sequence
$(u_{h})_{h}\subset C^{\infty}(\Omega)\cap W^{1}L_{\varphi}(\Omega)$ such that
$u_{h}\to u$ in $W^{1}L_{\varphi}(\Omega)$, as $h\to\infty$. In particular, it
also follows that

(19) $u^{\prime}_{h}\to u^{\prime}\text{ in }L_{\varphi}(\Omega)\text{ as
}h\to\infty\,.$

Let $\psi\in C^{0}_{c}(\Omega)$ such that $0\leq\,\psi\leq\,1$ and
$\psi\equiv\,1$ in $(-1/e,1/e)$ and let

$v_{h}:=\,\psi\,u^{\prime}_{h}\,.$

By Proposition 11(iii) and (19), it still holds that

$E_{\varphi}\ni\psi\,u^{\prime}_{h}\to\psi\,u^{\prime}=\,u^{\prime}\text{ in
}L_{\varphi}(\Omega)\text{, as }h\to\infty\,.$

Then a contradiction since $u^{\prime}\notin E_{\varphi}$.

A weaker density result of regular functions in $W^{1}L_{\varphi}(\Omega)$
holds by using the modular convergence, as shown in [G].

######  Theorem 20 ([G]).

Let $u\in W^{1}L_{\varphi}(\Omega)$. Then there exist $\lambda>\,0$ and a
sequence of functions $(u_{h})_{h}\subset C^{\infty}(\Omega)\cap
W^{1}L_{\varphi}(\Omega)$ such that

$N_{\varphi}\left(\frac{u_{h}-u}{\lambda}\right)\to 0\text{ and
}N_{\varphi}\left(\frac{D_{i}u_{h}-D_{i}u}{\lambda}\right)\to 0\text{ as
}h\to\infty\,,$

for each $i=1,\ldots,n$. In particular it suffices to choose $\lambda$ such
that $\frac{16}{\lambda}D_{i}u\in K_{\varphi}(\Omega)$.

##  3\. Exponential and sub-exponential N-functions

It is easy to see that a convex function $\varphi$ with polynomial growth
satisfies (1), see Remark 6. We will show in Lemma 21 that
$\exp_{\gamma,\tau}^{*}$ satisfies the conditions in Theorem 1 for $\varphi$
with $k_{\varphi}=1$, if $\tau$ is sufficiently large.

Recall from (6) and (7) that we have set

$\exp_{\gamma,\tau}(t):=\exp\left(\displaystyle\frac{t}{(\log(t+\tau))^{\gamma}}\right),\quad\text{and}\quad\exp_{\gamma,\tau}^{*}(t):=\,\exp_{\gamma,\tau}(t)-1.$

The functions $\exp_{\gamma,\tau}^{*}$, though convex, do not have null
derivative at $0$, and therefore do not fit exactly in the theory of
$N$-functions. Therefore, in order to provide a bridge with the theory of
$N$-functions of Orlicz spaces, we will also consider the modified functions

$\widetilde{\exp}_{\gamma,\tau}(t):=\,\exp_{\gamma,\tau}(t)-1-\frac{t}{(\log\tau)^{\gamma}}=\exp_{\gamma,\tau}^{*}(t)-\frac{t}{(\log\tau)^{\gamma}}\,,$

which are indeed $N$-functions and can be treated by comparison with
$\exp_{\gamma,\tau}^{*}$.

######  Lemma 21.

There exists $\tau_{0}>0$ such that, for all $\tau\geq\tau_{0}$ and
$0\leq\gamma\leq 1$, one has

  * (i)

$\exp_{\gamma,\tau}$ is a smooth strictly convex increasing function.
Moreover, for all $t,s\in[0,\infty)$,

(20)
$\exp_{\gamma,\tau}(t+s)\leq\,\exp_{\gamma,\tau}(t)\,\exp_{\gamma,\tau}(s),$

and $\exp_{\gamma,\tau}^{*}$ satisfies (1) with $k_{\varphi}=1$, that is,

(21)
$\exp_{\gamma,\tau}^{*}(t+s)\leq\,\exp_{\gamma,\tau}^{*}(t)\,\exp_{\gamma,\tau}^{*}(s)+\exp_{\gamma,\tau}^{*}(t)+\exp_{\gamma,\tau}^{*}(s).$

  * (ii)

$\widetilde{\exp}_{\gamma,\tau}$ is a $N$-function satisfying

(22) $\widetilde{\exp}_{\gamma,\tau}(t)\leq\,\exp_{\gamma,\tau}^{*}(t)\text{
if }t\geq\,0\text{ and
}\lim_{t\to\infty}\frac{\widetilde{\exp}_{\gamma,\tau}(t)}{\exp_{\gamma,\tau}^{*}(t)}=\,1\,.$

###### Proof.

(i) By a simple calculation, it is easy to see that, if $\tau>\,1$,
$\exp_{\gamma,\tau}$ is well-defined, $\exp_{\gamma,\tau}\in
C^{\infty}([0,\infty))$ and

$\displaystyle\exp^{\prime}_{\gamma,\tau}(t)$
$\displaystyle=\,\exp\left(\frac{t}{(\log(t+\tau))^{\gamma}}\right)\frac{\log(t+\tau)-\displaystyle{\frac{\gamma\,t}{t+\tau}}}{(\log(t+\tau))^{\gamma+1}},$
$\displaystyle\exp^{\prime\prime}_{\gamma,\tau}(t)$
$\displaystyle=\,\frac{\exp\left(\displaystyle{\frac{t}{(\log(t+\tau))^{\gamma}}}\right)}{(\log(t+\tau))^{2\gamma+2}}\Big{[}\left(\log(t+\tau)-\frac{\gamma\,t}{t+\tau}\right)^{2}$
$\displaystyle\qquad\qquad-(\log(t+\tau))^{\gamma+1}\frac{\gamma\,\tau+\gamma(t+\tau)}{(t+\tau)^{2}}+\frac{\gamma(\gamma+1)t(\log(t+\tau))^{\gamma}}{(t+\tau)^{2}}\Big{]}$

for all $t\geq 0$. Now, observe that, if $\tau>e$ and $t\geq 0$, then

(23)
$\frac{\gamma\,t}{t+\tau}\leq\,\gamma,\qquad\frac{\gamma\,\tau+\gamma(t+\tau)}{(t+\tau)^{2}}\leq\,\frac{2\,\gamma}{\tau},\text{
and }\log(t+\tau)>1\geq\gamma.$

Combining these inequalities, it follows that, if $\tau>e$,

(24) $\exp^{\prime}_{\gamma,\tau}(t)>0\text{ for each }t\geq\,0\,,$

so that $\exp_{\gamma,\tau}$ is strictly increasing on $[0,\infty)$. Let us
now show that, for sufficiently large $\tau$, one has

(25) $\exp^{\prime\prime}_{\gamma,\tau}(t)>\,0\text{ for each }t\geq\,0\,.$

By (23), for each $t\geq\,0$ and $\tau>\,e$, we obtain that

$\begin{split}\left(\log(t+\tau)-\frac{\gamma\,t}{t+\tau}\right)^{2}&-(\log(t+\tau))^{\gamma+1}\frac{\gamma\,\tau+\gamma(t+\tau)}{(t+\tau)^{2}}+\frac{\gamma(\gamma+1)t\log(t+\tau))^{\gamma}}{(t+\tau)^{2}}\\\
&\geq\,\left(\log(t+\tau)-\frac{\gamma\,t}{t+\tau}\right)^{2}-(\log(t+\tau))^{\gamma+1}\frac{\gamma\,\tau+\gamma(t+\tau)}{(t+\tau)^{2}}\\\
&\geq\,\left(\log(t+\tau)-\gamma\right)^{2}-\frac{2\gamma}{\tau}(\log(t+\tau))^{2}\\\
&=\log(t+\tau)\left(\log(\tau)-2\gamma\left(\frac{\log(\tau)}{\tau}+1\right)\right)+\gamma^{2}\\\
&\geq\log(t+\tau)\left(\log(\tau)-2\left(\frac{\log(\tau)}{\tau}+1\right)\right).\end{split}$

It is clear that, there is $\tau_{0}>0$ (independent on $\gamma$), so that the
latter quantity is positive for all $\tau\geq\tau_{0}$ and $t\geq 0$. Hence,
(25) follows and $\exp_{\gamma,\tau}$ is strictly convex on $[0,\infty)$.

Let us show (20), that is, for every $t,s\in[0,\infty)$,

(26)
$\begin{split}\exp_{\gamma,\tau}(t+s)&=\,\exp\left(\frac{t+s}{(\log(t+s+\tau))^{\gamma}}\right)\\\
&\leq\,\exp_{\gamma,\tau}(t)\,\exp_{\gamma,\tau}(s)\\\
&=\,\exp\left(\frac{t}{(\log(t+\tau))^{\gamma}}+\frac{s}{(\log(s+\tau))^{\gamma}}\right)\,.\end{split}$

Observe that

$\begin{split}\frac{t+s}{(\log(t+s+\tau))^{\gamma}}&=\,\frac{t}{(\log(t+s+\tau))^{\gamma}}+\frac{s}{(\log(t+s+\tau))^{\gamma}}\\\
&\leq\,\frac{t}{(\log(t+\tau))^{\gamma}}+\frac{s}{(\log(s+\tau))^{\gamma}}\,,\end{split}$

whence (26) follows, being the exponential function nondecreasing. Inequality
(21) follows by using (20) and the fact that
$\exp_{\gamma,\tau}(t)=\,\exp_{\gamma,\tau}^{*}(t)+1$.

(ii) Notice that, if

$a(t):=\,\exp^{\prime}_{\gamma}(t)-\exp^{\prime}_{\gamma}(0)\text{ if
}t\geq\,0\,,$

by (24) and (25), $a$ is continuous, (strictly) increasing, $a(0)=\,0$,
$a(t)>\,0$ if $t>\,0$ and $\lim_{t\to\infty}a(t)=\,\infty$. Moreover, being
$a$ increasing, we have for $t\geq 0$,

$\widetilde{\exp}_{\gamma,\tau}(t)=\,\exp_{\gamma,\tau}(t)-1-\frac{t}{(\log\tau)^{\gamma}}=\,\int_{0}^{t}\left(\exp^{\prime}_{\gamma}(s)-\exp^{\prime}_{\gamma}(0)\right)\,ds=\,\int_{0}^{t}a(s)\,ds.$

Thus, $\widetilde{\exp}_{\gamma,\tau}$ is (strictly) convex.

Finally, since the functions have a more than linear growth, it is clear that
$\widetilde{\exp}_{\gamma,\tau}(t)/\exp_{\gamma,\tau}(t)$ tends to 1 as
$t\to\infty$. ∎

######  Remark 22.

Notice that, by (22), the pair $(\widetilde{\exp}_{\gamma,\tau},\Omega)$ is
never $\Delta$-regular for any $\Omega\subset\mathbb{R}^{n}$. In particular,
by Lemma 8, $K_{\varphi}(\Omega)$ is never a vector space if
$\varphi=\widetilde{\exp}_{\gamma,\tau}$.

######  Remark 23.

Exponential growth functions fall also in the class treated by Theorem 1. More
precisely, the functions

$e_{\alpha}^{*}(t):=\exp(\alpha t)-1,$

for $\alpha>0$ satisfy the conditions on $\varphi$ given in Theorem 1.

##  4\. Proof of the approximation results

In this section we are going to show our results.

###### Proof of Theorem 1.

Uniform positivity of $w$ on compact subsets gives

(27)
$\,\int_{R}\varphi\left(|Db(s,x)|\right)\,\mathrm{d}s\,\mathrm{d}x<\infty\qquad\text{whenever
$R\Subset I\times\Omega$}.$

We are going to exploit an adaptation of the technique of the proof of Meyers-
Serrin’s theorem (see, for instance, [AFP, Thm. 3.9]). Let
$Q:=\,I\times\Omega$ and let $U_{j}$, $j=0,1,\ldots$, be the nondecreasing
sequence of open subsets

$U_{0}:=\,\emptyset,\quad U_{j}:=\,\left\\{(s,x)\in Q:\,{\rm
dist}((s,x),\partial Q)>\frac{1}{j},\,|s|+|x|<j\right\\}\,\,\,(j=1,2,\ldots),$

and let

$Q_{j}:=\,U_{j+1}\setminus\overline{U}_{j-1}\quad j=1,2,\ldots\,.$

Then $\cup_{j}Q_{j}=Q$, each $Q_{j}$ has compact closure in $Q$ and any point
of $Q$ belongs to at most four sets $Q_{j}$. More specifically, if $j\geq\,3$
and $x\in Q_{j}$, then $x$ may belong at most to $Q_{j-1}$ and $Q_{j+1}$.

Let $(\zeta_{j})_{j}$ be a partition of unity relative to the covering
$(Q_{j})$, that is, nonnegative functions $\zeta_{j}\in C^{\infty}_{c}(Q_{j})$
such that $\sum_{j=1}^{\infty}\zeta_{j}\equiv\,1$ in $Q$. Moreover, let
$\psi_{j}\in C^{\infty}_{c}(Q)$ be cut-off functions such that
$0\leq\,\psi_{j}\leq\,1$ in $Q$ and $\psi_{j}\equiv\,1$ in $Q_{j}$.

For each $j=1,2,\ldots$, let
$b_{j}:\,\mathbb{R}^{n+1}=\,\mathbb{R}_{s}\times\mathbb{R}^{n}_{x}\to\mathbb{R}^{m}$
denote

$b_{j}(s,x):=\begin{cases}\,\psi_{j}(s,x)\,b(s,x)&\text{ if }(s,x)\in
I\times\Omega\\\ 0&\text{ if }(s,x)\in\mathbb{R}^{n+1}\setminus
I\times\Omega\,,\end{cases}$

so that it is clear that ${\rm spt}(b_{j})\Subset Q$, $b_{j}\in
L^{1}(\mathbb{R}_{s};W^{1,1}(\mathbb{R}^{n}_{x};\mathbb{R}^{m}))\cap
C^{0}_{\rm c}(\mathbb{R}^{n+1},\mathbb{R}^{m})$, and

(28) $Db_{j}=\,D(\psi_{j}\,b)=\,\psi_{j}\,Db+\nabla\psi_{j}\otimes b\text{
a.e.\leavevmode\nobreak\ in }Q\,.$

In particular,

(29) $b_{j}=\,b\text{ and }Db_{j}=\,Db\text{ a.e.\leavevmode\nobreak\ in
}Q_{j}\,,$ (30) $Db_{j}\in L^{1}(\mathbb{R}^{n+1};\mathbb{R}^{nm})\,.$

The monotonicity of $\varphi$ and the weak subadditivity condition (1) give

(31)
$\begin{split}\varphi\left(|Db_{j}|\right)&\leq\,\varphi\left(\psi_{j}\,|Db|+|\nabla\psi_{j}\otimes
b|\right)\\\
&\leq\,k_{\varphi}\left[\varphi\left(\psi_{j}\,|Db|\right)\,\varphi\left(|\nabla\psi_{j}\otimes
b|\right)+\varphi\left(\psi_{j}\,|Db|\right)+\varphi\left(|\nabla\psi_{j}\otimes
b|\right)\right].\end{split}$

Since, by (27),

$\varphi\left(\psi_{j}\,|Db|\right)\in L^{1}(\mathbb{R}^{n+1})\quad\text{ and
}\quad\varphi\left(|\nabla\psi_{j}\otimes b|\right)\in
L^{1}(\mathbb{R}^{n+1})\cap L^{\infty}(\mathbb{R}^{n+1})\,,$

we obtain from (31) that

(32) $\varphi\left(|Db_{j}|\right)\in L^{1}(\mathbb{R}^{n+1}).$

Let $(\rho_{\varepsilon}(s,x))_{\varepsilon}$ be space-time mollifiers on
$\mathbb{R}^{n+1}=\,\mathbb{R}_{s}\times\mathbb{R}^{n}_{x}$. For each
$\delta\in(0,1)$ we will make a suitable choice of $0<\varepsilon_{j}<\delta$
and define $b_{\delta}:\,Q\to\mathbb{R}^{m}$ as

$b_{\delta}(s,x):=\sum_{j=1}^{\infty}\zeta_{j}(s,x)\,(\rho_{\epsilon_{j}}*b_{j})(s,x)\,.$

Since the sum is locally finite, $b_{\delta}$ is well defined. Moreover, by
construction, $b_{\delta}\in C^{\infty}(I\times\Omega;\mathbb{R}^{m})$ and one
has

(33) $b_{\delta}\to b\text{ in }L^{1}_{\rm loc}(Q;\mathbb{R}^{m})\text{, as
}\delta\to 0\,.$

Notice now that

(34)
$\begin{split}Db_{\delta}&=\,\sum_{j=1}^{\infty}D(\zeta_{j}(\rho_{\varepsilon_{j}}*b_{j}))=\,\sum_{j=1}^{\infty}\zeta_{j}(\rho_{\varepsilon_{j}}*Db_{j})+\sum_{j=1}^{\infty}\nabla\zeta_{j}\otimes(\rho_{\varepsilon_{j}}*b)\\\
&=\,v_{\delta}+z_{\delta}\text{ in }Q\,,\end{split}$

where

$v_{\delta}:=\sum_{j=1}^{\infty}\zeta_{j}(\rho_{\varepsilon_{j}}*Db_{j})\,,$

and

$z_{\delta}:=\sum_{j=1}^{\infty}\left(\nabla\zeta_{j}\otimes(\rho_{\varepsilon_{j}}*b_{j})-\nabla\zeta_{j}\otimes
b_{j}\right)\,,$

where we used the fact that, since $\sum_{j=1}^{\infty}\nabla\zeta_{j}\equiv
0$, we have $\sum_{j=1}^{\infty}\nabla\zeta_{j}\otimes b\equiv 0$.

For each $\delta>\,0$ and $j=1,2,\ldots$, we can find
$0<\varepsilon_{j}<\delta$ such that

(35)
$\int_{Q}|\zeta_{j}\,(\rho_{\varepsilon_{j}}*b_{j})-\zeta_{j}\,b|\,\mathrm{d}x\,\mathrm{d}s<\frac{\delta}{2^{j}}\,$

and

(36)
$\|\zeta_{j}(\rho_{\varepsilon_{j}}*Db_{j})-\zeta_{j}Db_{j}\|_{L^{1}(\mathbb{R}^{n+1};\mathbb{R}^{m})}<\frac{\delta}{2^{j+1}}\,.$

In addition, setting

$M_{j}:=\,\max\\{1,\sup_{Q_{j}}w\\}\,,$

we can also ensure that

(37)
$\begin{split}&\|\nabla\zeta_{j}\otimes(\rho_{\varepsilon_{j}}*b_{j})-\nabla\zeta_{j}\otimes
b\|_{L^{p}(Q_{j};\mathbb{R}^{nm})}\\\
&=\,\|\nabla\zeta_{j}\otimes(\rho_{\varepsilon_{j}}*b_{j})-\nabla\zeta_{j}\otimes
b\|_{L^{p}(\mathbb{R}^{n+1};\mathbb{R}^{nm})}<\frac{\delta}{2^{j+1}M_{j}}\text{
if }p=1,\infty\,\end{split}$

and

(38)
$\|\rho_{\varepsilon_{j}}*\varphi(|Db_{j}|)-\varphi(|Db_{j}|)\|_{L^{1}(\mathbb{R}^{n+1})}<\frac{\delta}{2^{j}\,M_{j}}\,.$

Notice that, by assumption on $w$, $M_{j}<\infty$. Notice also that the
choices in (35) and (37) are possible thanks to the continuity of $b$ (in
particular, for (37) with $p=\infty$, we are using also continuity of $b$ with
respect to the time variable), while the choices in (36) and (38) are possible
thanks to the classical properties of convolution together with (30) and (32),
respectively. From (37), it follows that

(39) $\|z_{\delta}\|_{L^{p}(Q;\mathbb{R}^{nm})}<\frac{\delta}{2}\text{ if
}p=1,\infty,\text{
and}\int_{Q}|z_{\delta}|\,w\,\mathrm{d}x\,\mathrm{d}s<\delta\,.$

Moreover, by (39) with $p=\,\infty$ and the convexity of $\varphi$, if we set
$L:=\varphi(1)$, then $\varphi(0)=0$ and the monotonicity of difference
quotients give

(40) $\sigma_{\delta}:=\varphi(|z_{\delta}|)\leq\,L|z_{\delta}|\text{ in
}Q\,.$

Notice now that, since
$Db=\,\sum_{j=1}^{\infty}\zeta_{j}Db=\,\sum_{j=1}^{\infty}\zeta_{j}Db_{j}$, by
(36) and (39) with $p=1$, we have

(41)
$\begin{split}\|Db_{\delta}-Db\|_{L^{1}(Q;\mathbb{R}^{m})}&=\,\|v_{\delta}+z_{\delta}-Db\|_{L^{1}(Q;\mathbb{R}^{m})}\\\
&\leq\,\|v_{\delta}-Db\|_{L^{1}(Q;\mathbb{R}^{m})}+\,\|z_{\delta}\|_{L^{1}(Q;\mathbb{R}^{m})}\\\
&\leq\,\sum_{j=1}^{\infty}\|\zeta_{j}(\rho_{\varepsilon_{j}}*Db_{j})-\zeta_{j}Db_{j}\|_{L^{1}(\mathbb{R}^{n+1};\mathbb{R}^{m})}+\frac{\delta}{2}\\\
&\leq\,\sum_{j=1}^{\infty}\frac{\delta}{2^{j+1}}+\frac{\delta}{2}=\,\delta\,.\end{split}$

By (41), it follows that

(42) $\lim_{\delta\to 0}\|Db_{\delta}-Db\|_{L^{1}(Q;\mathbb{R}^{m})}=0.$

In particular, by the continuity of $\varphi$ and (42), there exists an
infinitesimal sequence $(\delta_{h})_{h}$ such that, if
$b_{h}:=\,b_{\delta_{h}}$,

(43) $\varphi\left(|Db_{h}|\right)\to\varphi\left(|Db|\right)\text{a.e. in
$Q$, as }h\to\infty\,.$

Let us now show (4). One has, a.e. in $Q$,

$\begin{split}\varphi\left(|Db_{\delta}|\right)&\leq\varphi\left(|z_{\delta}|+|v_{\delta}|\right)\leq\,k_{\varphi}\left(\varphi\left(|z_{\delta}|\right)\,\varphi\left(|v_{\delta}|\right)+\varphi\left(|z_{\delta}|\right)+\varphi\left(|v_{\delta}|\right)\right)\\\
&\leq\,k_{\varphi}((1+\sigma_{\delta})\,\varphi\left(\left|v_{\delta}\right|\right)+\sigma_{\delta})\,,\end{split}$

where $\sigma_{\delta}=\varphi\left(|z_{\delta}|\right)$. Set
$\sigma_{\delta}^{\infty}:=\|\sigma_{\delta}\|_{L^{\infty}(Q)}$, so that, a.e.
in $Q$,

(44)
$\varphi\left(|Db_{\delta}|\right)\leq\,k_{\varphi}((1+\sigma_{\delta}^{\infty})\,\varphi\left(\left|v_{\delta}\right|\right)+\sigma_{\delta})\,.$

By the monotonicity and convexity of $\varphi$, by Jensen’s inequality and
taking into account that $(\zeta_{j})_{j}$ is a partition of unity, we get,
a.e. in $Q$,

(45)
$\begin{split}\varphi\left(\left|v_{\delta}\right|\right)&=\,\varphi\left(\left|\sum_{j=1}^{\infty}\zeta_{j}(\rho_{\varepsilon_{j}}*Db_{j})\right|\right)\leq\,\varphi\left(\sum_{j=1}^{\infty}\zeta_{j}(\rho_{\varepsilon_{j}}*|Db_{j}|)\right)\\\
&\leq\,\sum_{j=1}^{\infty}\zeta_{j}(\rho_{\varepsilon_{j}}*\varphi\left(|Db_{j}|\right))=:\,G_{\delta}\,.\end{split}$

Hence, by (44) and (45), if follows that

(46)
$w\varphi\left(|Db_{\delta}|\right)\leq\,k_{\varphi}((1+\sigma_{\delta}^{\infty})\,wG_{\delta}+\,w\sigma_{\delta})\quad\text{a.e.
in }Q\,.$

It is clear that, by (39) and (40),

(47) $w\,\sigma_{\delta}$ converges to $0$ in $L^{1}(Q)$ and
$\sigma_{\delta}^{\infty}\to 0$, as $\delta\to 0$.

Let us now prove that

(48) $\text{$w\,G_{\delta}$ converges to $w\varphi(|Db|)$ in $L^{1}(Q)$, as
$\delta\to 0$}\,.$

Observe that, by (29),

$\varphi(|Db|)=\,\sum_{j=1}^{\infty}\zeta_{j}\varphi(|Db|)=\,\sum_{j=1}^{\infty}\zeta_{j}\varphi(|Db_{j}|)\text{
a.e. in }Q\,,$

so that (38) gives (48).

The combination of (48) and (46) gives the equi-integrability of
$w\varphi\left(|Db_{\delta}|\right)$. By using (43), Vitali’s form of the
dominated convergence theorem (see, for instance, [AFP, Exercise 1.18]), gives
(4) and the proof is complete. ∎

######  Remark 24.

Let $\Psi:[0,\infty)\to[0,\infty)$ be any continuous function with $\Psi(0)=0$
and linear growth at the origin. Notice that all the terms
$\zeta_{j}(\rho_{\epsilon_{j}}*b_{j})-\zeta_{j}b$ can be made arbitrarily
small not only in $L^{\infty}_{t}(L^{\infty}_{x})$, but also in
$L^{1}_{t}(L^{1}_{x})$, choosing $\epsilon_{j}\ll 1$. Then, using the
representation

$(b_{\delta_{h}}-b)=\sum_{j=1}^{\infty}\zeta_{j}(\rho_{\epsilon_{j}}*b_{j})-\zeta_{j}b$

we can improve the construction to get also

$\lim_{h\to\infty}\int_{I}\int_{\Omega}\Psi(|b_{\delta_{h}}-b|)\,\mathrm{d}x\,\mathrm{d}s=0.$

More precisely, choosing $\epsilon_{j}\ll 1$ properly, we can make arbitrarily
small all terms

$\int_{Q_{j}}\Psi(|\sum_{j=1}^{\infty}\zeta_{j}(\rho_{\epsilon_{j}}*b_{j})-\zeta_{j}b|)\,\mathrm{d}x\,\mathrm{d}s$

since the sum is locally finite and $Q_{j}\Subset I\times\Omega$.

######  Remark 25.

Since the proof of Theorem 1 is based on a convolution argument, we have more
control on the convergence depending on the properties of $b$. We give two
cases that can be of interest.

First, if $b\in L^{1}(I;C(\Omega;\mathbb{R}^{m}))$ is continuous in the
spatial variable, then the approximating sequence $b_{h}\in
C^{\infty}(I\times\Omega;\mathbb{R}^{m})$ in Theorem 1 can be taken so that
$b_{h}\to b$ in $b\in L^{1}(I;C(\Omega;\mathbb{R}^{m}))$.

Second, if there exists a bounded open set $\Omega^{\prime}\Subset\Omega$ such
that

(49) ${\rm spt}(b(t,\cdot))\subset\Omega^{\prime}\text{ for every }t\in I\,,$

then the approximating sequence $b_{h}\in
C^{\infty}(I\times\Omega;\mathbb{R}^{m})$ can be taken with

(50) ${\rm spt}(b_{h}(t,\cdot))\subset\Omega^{\prime}\text{ for each }t\in
I\text{ and }h\in\mathbb{N}\,.$

###### Proof of Corollary 2.

Let us prove that

(51)
$\int_{I}\int_{\Omega}w(s,x)\exp_{\gamma,\tau}^{*}\left(|Db(s,x)|\right)\,\mathrm{d}x\,\mathrm{d}s<\infty\,.$

Notice that, since $b$ satisfies (2) with (9), then
$w\,\widetilde{\exp}_{\gamma,\tau}\left(|Db|\right)\in L^{1}(I\times\Omega)$.
On the one hand, if $w|Db|\in L^{1}(I\times\Omega)$, since

$w\exp_{\gamma,\tau}^{*}\left(|Db|\right)=\,w\widetilde{\exp}_{\gamma,\tau}\left(|Db|\right)+\frac{w|Db|}{(\log\tau)^{\gamma}}$

and $w\,\widetilde{\exp}_{\gamma,\tau}\left(|Db|\right)\in
L^{1}(I\times\Omega)$ we immediately obtain (51). On the other hand, if $w\in
L^{1}(I\times\Omega)$, by (22), there exists $\bar{t}>\,0$ such that

(52)
$\frac{1}{2}\,\exp_{\gamma,\tau}^{*}(t)\leq\,\widetilde{\exp}_{\gamma,\tau}(t)\text{
for each }t\geq\,\bar{t}\,.$

Thus

$\begin{split}\int_{I\times\Omega}w\,\exp_{\gamma,\tau}^{*}\left(|Db|\right)\,\,\mathrm{d}s\,\mathrm{d}x&=\,\int_{\\{|Db|<\,\bar{t}\\}}w\,\exp_{\gamma,\tau}^{*}\left(|Db|\right)\,\,\mathrm{d}s\,\mathrm{d}x\\\
&\qquad\qquad+\int_{\\{|Db|\geq\,\bar{t}\\}}w\,\exp_{\gamma,\tau}^{*}\left(|Db|\right)\,\,\mathrm{d}s\,\mathrm{d}x\\\
&\leq\,\exp_{\gamma,\tau}^{*}\left(\bar{t}\right)\int_{I\times\Omega}w\,\,\mathrm{d}s\,\mathrm{d}x\\\
&\qquad\qquad+2\int_{I\times\Omega}w\,\widetilde{\exp}_{\gamma,\tau}\left(|Db|\right)\,\,\mathrm{d}s\,\mathrm{d}x<\,\infty\end{split}$

and (51) follows once more.

By (51), we can apply Theorem 1 to get the existence of $b_{h}\in
C^{\infty}(I\times\Omega;\mathbb{R}^{m})$ satisfying (3) and (4) with
$\varphi=\,\exp_{\gamma,\tau}^{*}$. Since
$w\,\widetilde{\exp}_{\gamma,\tau}\left(|Db_{h}|\right)\leq\,w\,\exp_{\gamma,\tau}^{*}\left(|Db_{h}|\right)$,
by applying Vitali’s convergence theorem, we obtain again the desired
conclusion. ∎

In the proof of Theorem 3 we will need the following lemma.

######  Lemma 26.

Let $f\in L^{1}_{\rm loc}(\mathbb{R}_{s}\times\Omega)$ and
$(\rho_{\varepsilon}(s))_{\varepsilon}$ be a family of time mollifiers in
$\mathbb{R}_{s}$. Then, for a.e. $x\in\Omega$, the time convolution product
$f^{\varepsilon}(\cdot,x):\mathbb{R}_{s}\to\mathbb{R}$

$\begin{split}f^{\varepsilon}(s,x)&=\,(\rho_{\varepsilon}*f(\cdot,x))(s)\\\
&:=\int_{\mathbb{R}}\rho_{\varepsilon}(s-v)\,f(v,x)\,dv\text{ for each
}s\in\mathbb{R}\end{split}$

and

(53) $f^{\varepsilon}(\cdot,x)\in C^{0}(\mathbb{R}_{s})\text{ for each
$\varepsilon>\,0$, for a.e. }x\in\Omega\,.$

In addition, for any open set $\omega\subset\Omega$ one has

(54)
$\|f^{\varepsilon}\|_{L^{1}(\mathbb{R}_{s}\times\omega)}\leq\,\|f\|_{L^{1}(\mathbb{R}_{s}\times\omega)}\text{
for each $\varepsilon>\,0$}$

and

(55) $f^{\varepsilon}\to f\text{ in }L^{1}(\mathbb{R}_{s}\times\omega)\text{
as }\varepsilon\to 0\,,$

provided that $f\in L^{1}(\mathbb{R}_{s}\times\omega)$.

Finally, if we assume that, for each ball $B(x_{0},r)\Subset\Omega$ one has

(56) $f\in L^{1}\bigl{(}\mathbb{R}_{s};C^{0}(B(x_{0},r))\bigr{)}\,,$

then

(57) $f^{\varepsilon}\in C^{0}(\mathbb{R}_{s}\times\Omega)\text{ for each
$\varepsilon>\,0$}\,.$

###### Proof.

Properties (53), (54) and (55) can be proved as in the case of the global
$(s,x)$-convolution by mollifiers (see, for instance, [AFP, Section 2.1]). Let
us prove (57). Let $(s_{0},x_{0})\in\mathbb{R}_{s}\times\Omega$ and let
$((s_{h},x_{h}))_{h}\subset\mathbb{R}_{s}\times\Omega$ a sequence converging
to $(s_{0},x_{0})$. From (56),

$\int_{\mathbb{R}}F(s)\,ds<\infty\text{ if
}F(s):=\,\sup_{B(x_{0},r)}|f(s,\cdot)|\,,$

and, without loss of generality, we can assume that $(x_{h})_{h}\subset
B(x_{0},r)$ for a fixed $r>0$. Then, since

$|\rho_{\varepsilon}(s_{h}-v)\,f(v,x_{h})|\leq\,\sup_{\mathbb{R}}\rho_{\varepsilon}\,F(v)\quad\text{for
a.e. $v\in\mathbb{R}$},$

by Lebesgue’s dominated convergence theorem, it follows that

$\begin{split}\lim_{h\to\infty}f^{\varepsilon}(s_{h},x_{h})&=\,\lim_{h\to\infty}\int_{\mathbb{R}}\rho_{\varepsilon}(s_{h}-v)\,f(v,x_{h})\,dv\\\
&=\,\int_{\mathbb{R}}\lim_{h\to\infty}(\rho_{\varepsilon}(s_{h}-v)\,f(v,x_{h}))\,dv\\\
&=\,\int_{\mathbb{R}}\rho_{\varepsilon}(s_{0}-v)\,f(v,x_{0})\,dv=\,f^{\varepsilon}(s_{0},x_{0})\,.\end{split}$

∎

###### Proof of Theorem 3.

We extend $b$ to $\mathbb{R}\times\Omega$ setting $b(t,x)=0$ whenever $t\notin
I$. Denoting by $b^{\epsilon}$ the mollified functions with respect to the
time variable, one has

$Db^{\epsilon}(t,x)=\int_{\mathbb{R}}\rho_{\epsilon}(t-s)Db(s)\,\mathrm{d}s$

and therefore Jensen’s inequality gives

$\displaystyle\int_{\Omega}w(x)\varphi(Db^{\epsilon}(t,x))\,\mathrm{d}x$
$\displaystyle\leq$
$\displaystyle\int_{\Omega}w(x)\int_{\mathbb{R}}\rho_{\epsilon}(t-s)\varphi(|Db(s,x)|)\,\mathrm{d}s\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\rho_{\epsilon}(t-s)\int_{\Omega}w(x)\varphi(Db(s,x)|)\,\mathrm{d}x\,\mathrm{d}s\,.$

By integration on $I$, it follows that $N_{\varphi,w}(|Db^{\epsilon}|)\leq
N_{\varphi,w}(|Db|)$. Notice now that $b^{\epsilon}$ satisfy the assumptions
of Corollary 2, thanks to (57). Thus for all $\epsilon>0$ we get the existence
of a sequence $(b^{\epsilon}_{h})_{h}\subset
C^{\infty}(I\times\Omega;\mathbb{R}^{m})$ satisfying (3) and (4). Finally, by
taking a diagonal sequence, we get the desired conclusion. ∎

###### Proof of Theorem 4.

Notice that for any open set $A\subset\Omega$ the weak
$L^{1}(A;\mathbb{R}^{mn})$ convergence of derivatives grants

$\liminf_{h\to\infty}\int_{A}\varphi\left(|Db_{h}|\right)\,\mathrm{d}x\geq\int_{A}\varphi\left(|Db|\right)\,\mathrm{d}x\,.$

Therefore, by applying an elementary lemma (see, for instance, the proof of
Proposition 1.80 in [AFP]) the convergence of the integrals on $\Omega$ can be
localized, getting

$\lim_{h\to\infty}\int_{A}\varphi\left(|Db_{h}|\right)\,\mathrm{d}x=\int_{A}\varphi\left(|Db|\right)\,\mathrm{d}x$

whenever $A\subset\Omega$ is open with Lebesgue negligible boundary. In
particular, choosing $A\Subset\Omega$ with this property, since $A$ has finite
measure we can use the strict convexity of $\varphi$ and [V] to get that
$Db_{h}\to Db$ in $L^{1}(A;\mathbb{R}^{mn})$. It follows that $Db_{h}\to Db$
in $L^{1}_{\rm loc}(I\times\Omega;\mathbb{R}^{nm})$ and therefore, modulo the
extraction of a subsequence, we can assume that $Db_{h}\to Db$ a.e. in
$\Omega$.

Combining the pointwise convergence

$\lim_{h\to\infty}\varphi(|Db_{h}|)=\varphi(|Db|)\quad\text{a.e. in $\Omega$}$

with the convergence of the integrals, Scheffé’s lemma gives that
$\varphi(|Db_{h}|)$ converges in $L^{1}(\Omega)$ to $\varphi(|Db|)$. Now, the
inequality

$\varphi\left(\frac{|Db_{h}-Db|}{2}\right)\leq\frac{1}{2}\varphi\left(|Db_{h}|\right)+\frac{1}{2}\varphi\left(|Db|\right)$

grants the equi-integrability of $\varphi(|Db_{h}-Db|/2)$. Vitali’s
convergence theorem can finally be applied to get the result. ∎

###### Proof of Theorem 5.

Notice that, being $\widetilde{\exp}$ a $N$-function, it has a linear growth
at the origin. Thus by applying Remark 24 with $\Psi=\widetilde{\exp}$ and
Corollary 2 with $b(t,x)=\,u(x)$ and $w\equiv 1$, we get the desired
conclusion. ∎

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