# Families of Young Functions and Limits of Orlicz Norms

Sullivan MacDonald and Scott Rodney  Sullivan F. MacDonald, MSc student in the
Dept. of Mathematics & Statistics
McMaster University
Hamilton, ON L8S4L8, Canada<EMAIL_ADDRESS>Scott Rodney
Dept. of Mathematics, Physics and Geology
Cape Breton University
Sydney, NS B1Y3V3, Canada<EMAIL_ADDRESS>
(Date: May 2023)

###### Abstract.

Given a $\sigma$-finite measure space $(X,\mu)$, a Young function $\Phi$, and
a one-parameter family of Young functions $\\{\Psi_{q}\\}$, we find necessary
and sufficient conditions for the associated Orlicz norms of any function
$f\in L^{\Phi}(X,\mu)$ to satisfy

$\lim_{q\rightarrow\infty}\|f\|_{L^{\Psi_{q}}(X,\mu)}=C\|f\|_{L^{\infty}(X,\mu)}.$

The constant $C$ is independent of $f$ and depends only on the family
$\\{\Psi_{q}\\}$. Several examples of one-parameter families of Young
functions satisfying our conditions are given, along with counterexamples when
our conditions fail.

S. Rodney was supported by the NSERC Discovery Grant Program. S.F. MacDonald
was supported by the NSERC USRA Program and the Dept. of Mathematics &
Statistics at McMaster University

##  1\. Introduction

It is a well-known result in classical analysis (see e.g. [5] and [6]) that if
$(X,\mu)$ is a measure space and if $f\in L^{r}(X,\mu)$ for some $r\geq 1$,
then

(1.1)
$\lim_{p\rightarrow\infty}\|f\|_{L^{p}(X,\mu)}=\|f\|_{L^{\infty}(X,\mu)}.$

The authors of [3] investigated a similar limiting property of Orlicz norms
associated with a one-parameter family of Young functions $\\{\Psi_{q}\\}$
defined by setting $\Psi_{q}(t)=t^{p}\log(e-1+t)^{q}$, where $p\geq 1$ is
fixed and $q$ can be any positive real number. They showed that if $f$ belongs
to the Orlicz space $L^{\Psi_{q_{0}}}(X,\mu)$ for some $q_{0}>0$ then

(1.2)
$\lim_{q\rightarrow\infty}\|f\|_{L^{\Psi_{q}}(X,\mu)}=\|f\|_{L^{\infty}(X,\mu)}.$

Here, $\|f\|_{L^{\Psi_{q}}(X,\mu)}$ denotes the Luxembourg norm of $f$ with
respect to $\Psi_{q}$ and it is given by

$\|f\|_{L^{\Psi_{q}}(X,\mu)}=\inf\bigg{\\{}\lambda>0:\int_{X}\Psi_{q}\bigg{(}\frac{|f|}{\lambda}\bigg{)}~{}d\mu\leq
1\bigg{\\}}.$

Modifying the proof of [3, Thm. 1], we show that (1.2) holds for any family of
Young functions $\\{\Psi_{q}\\}$ that satisfies Condition 1.1, which we call
_$\delta$ -admissibility_. Like the authors of [3], our efforts were motivated
by an application in partial differential equations, where we sought to employ
a Moser iterative scheme in Orlicz spaces to study regularity of weak
solutions to Poisson’s equation. Ultimately those results were achieved using
alternative techniques in [2].

Nevertheless, our main result Theorem 1.2 may be useful in the study of
related problems. Furthermore, it illustrates a surprising relationship
between Orlicz spaces defined with respect to $(X,\mu)$, and the space
$L^{\infty}(X,\mu)$ of essentially bounded functions on $X$.

Throughout this work we assume that $(X,\mu)$ is a positive measure space with
$\mu(X)>0$. To state our main result concisely, we begin by defining
$\delta$-admissibility.

######  Condition 1.1.

Given $\delta>0$, a one-parameter family of Young functions $\\{\Psi_{q}\\}$
is said to be $\delta$-admissible if
$\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}(t)=\infty$ for $t>\delta$, and
for $0<t<\delta$ one of the following holds:

  * (i)

If $\mu(X)=\infty$ then $\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}(t)=0$,

  * (ii)

If $\mu(X)<\infty$ then
$\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}(t)<\mu(X)^{-1}$.

Perhaps the simplest $\delta$-admissible family is the $1$-admissible
collection obtained by taking $\Psi_{q}(t)=t^{q}$ for $q\geq 1$, and more
examples of $\delta$-admissible families are discussed in Section 3.

Now we state our main result concerning these families of Young functions.

######  Theorem 1.2.

Let $(X,\mu)$ be a $\sigma$-finite measure space and let $\\{\Psi_{q}\\}$ be a
one-parameter family of Young functions. Let $\Phi$ be another Young function
such that for any $k>0$, the composition

(1.3) $\frac{t}{\Psi_{q}^{-1}(\Phi(t))}$

is non-decreasing on the interval $[0,k]$ whenever $q$ is sufficiently large.
Then the identity

(1.4)
$\lim_{q\rightarrow\infty}\|f\|_{L^{\Psi_{q}}(X,\mu)}=\frac{1}{\delta}\|f\|_{L^{\infty}(X,\mu)}$

holds for every $f\in L^{\Phi}(X,\mu)$ if and only if $\\{\Psi_{q}\\}$ is a
$\delta$-admissible family for some $0<\delta<\infty$.

######  Remark 1.3.

  1. (1)

We note that $\sigma$-finiteness of $(X,\mu)$ is only required for the forward
implication. Further, in many cases (1.3) is non-decreasing on all of
$[0,\infty)$ for large $q$. For example, if $\Psi_{q}(t)=t^{q}$ and
$\Phi(t)=t^{r}$ for $r\geq 1$, then (1.3) is non-decreasing on $[0,\infty)$
for $q\geq r$.

  2. (2)

The distinction between cases (i) and (ii) in Condition 1.1 is needed to prove
necessity of $\delta$-admissibility for identity (1.4), but it is not needed
for sufficiency. Indeed, if $\mu(X)<\infty$ and $0<t<\mu(X)^{-1}$ then one
cannot select sets of large enough $\mu$-measure with which to compute the
limit of $\Psi_{q}^{-1}(t)$ as $q\rightarrow\infty$ using (1.4). We also note
that Condition 1.1 is weaker than the closely related sufficient condition
where $\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}(t)=0$ in (i) and (ii).

Usually it is difficult to find the inverse of a given Young function in
closed form. To make (1.3) easier to verify for given examples of $\Phi$ and
$\Psi_{q}$, notice that (1.3) is non-decreasing on $[0,k]$ exactly when the
following function is non-decreasing on $[0,\Phi(k)]$:

$\frac{\Phi^{-1}(t)}{\Psi_{q}^{-1}(t)}.$

So, to check that the conditions of Theorem 1.2 hold for a Young function
$\Phi$ and a given family $\\{\Psi_{q}\\}$, it suffices to compute the inverse
of $\Psi_{q}$ for each $q$, and to know either $\Phi$ or $\Phi^{-1}$.

The theorem stated above implies the main result of [3], and we include a
streamlined proof of this special case in Section 5. Furthermore, we observe
that if $\Phi(t)=t$ then (1.3) is non-decreasing whenever $\Psi_{q}$ is a
Young function, and from Theorem 1.2 we obtain the following result that
involves no growth condition.

######  Corollary 1.4.

If $(X,\mu)$ is a $\sigma$-finite measure space and $\\{\Psi_{q}\\}$ is
$\delta$-admissible for $0<\delta<\infty$, then (1.4) holds for every $f\in
L^{1}(X,\mu)$.

The remainder of this paper is organized as follows. In Section 2 we establish
preliminary results for Young functions and Orlicz spaces, and in Section 3 we
discuss several examples of $\delta$-admissible families to show how Theorem
1.2 can be applied. Section 4 is then devoted to the proof of our main result,
and Section 5 examines the special case of log-bumps. We conclude with Section
6, where we demonstrate that identity (1.4) can fail if the family
$\\{\Psi_{q}\\}$ is not $\delta$-admissible for any finite $\delta>0$.

##  2\. Preliminaries

This section contains a brief introduction to Young functions and Orlicz
spaces. Our discussion is largely expository, and for a complete treatment the
reader is referred to [4]. To begin, a non-negative function
$\psi:[0,\infty)\rightarrow[0,\infty)$ is said to be a density if it is right
continuous, non-decreasing, $\psi(t)=0$ exactly when $t=0$, and
$\psi(t)\rightarrow\infty$ as $t\rightarrow\infty$. Given a density $\psi$,
the associated function $\Psi:[0,\infty)\rightarrow[0,\infty)$ defined by

$\Psi(t)=\int_{0}^{t}\psi(s)ds$

is called a Young function. For our purposes, the important functional
properties of $\Psi$ are that it is continuous, strictly increasing, and
convex on $(0,\infty)$. Moreover, it is clear that $\Psi(0)=0$ and that
$\Psi(t)\rightarrow\infty$ as $t\rightarrow\infty$. Since the function
$\Psi(t)=t$ has constant density it is not a Young function according to the
definition above, however for our purposes it can often be treated as one.

Given a Young function $\Psi$ and a measure space $(X,\mu)$, the Orlicz space
$L^{\Psi}(X,\mu)$ is defined as the collection of $\mu$-measurable functions
$f:X\rightarrow\mathbb{R}$ for which the Luxembourg norm

$\|f\|_{L^{\Psi}(X,\mu)}=\inf\bigg{\\{}\lambda>0:\int_{X}\Psi\left(\frac{|f|}{\lambda}\right)d\mu\leq
1\bigg{\\}}$

is finite. Equipped with this norm $L^{\Psi}(X,\mu)$ is a Banach space; see
[6]. The Orlicz classes generalize the classical Lebesgue spaces, and it is
easy to verify that $\|\cdot\|_{L^{p}(X,\mu)}=\|\cdot\|_{L^{\Psi}(X,\mu)}$
when $\Psi(t)=t^{p}$ for $p\geq 1$. Orlicz spaces can also provide a finer
scale of norms then $L^{p}(X,\mu)$ in the following sense: if $\mu(X)<\infty$
and $\Psi_{q}(t)=t^{p}(1+\log(1+t))^{q}$ for $p\geq 1$ and $q>0$, then for any
$\varepsilon>0$ we have

$L^{p+\varepsilon}(X,\mu)\subsetneq L^{\Psi_{q}}(X,\mu)\subsetneq
L^{p}(X,\mu).$

These inclusions can be verified using Hölder’s inequality, and their
strictness follows from the examples constructed in [1].

In the sections that follow we employ several properties of the Luxembourg
norm which we now establish. The first is a version of Chebyshev’s inequality
on the Orlicz scale. Henceforth we use the notation $\chi_{S}$ to denote the
indicator function of a set $S\subseteq X$.

######  Theorem 2.1 (Chebyshev’s Inequality).

For any $\alpha\geq 0$, a $\mu$-measurable function
$f:X\rightarrow\mathbb{R}$, and a Young function $\Psi$, the following
inequality holds:

(2.1) $\alpha\Psi^{-1}(\mu(\\{x\in
X:|f(x)|\geq\alpha\\})^{-1})^{-1}\leq\|f\|_{L^{\Psi}(X,\mu)}.$

###### Proof.

First we establish a simpler form of (2.1) in the norm of $L^{1}(X,\mu)$ using
a standard argument. Fix $\alpha>0$ and define
$f_{\alpha}=\alpha\chi_{\\{|f|\geq\alpha\\}}$ so that $f_{\alpha}\leq|f|$
holds pointwise and

$\mu(\\{|f|\geq\alpha\\})=\mu(\\{x\in
X:|f(x)|\geq\alpha\\})=\frac{1}{\alpha}\int_{X}f_{\alpha}(x)d\mu\leq\frac{1}{\alpha}\int_{X}|f(x)|d\mu=\frac{1}{\alpha}\|f\|_{L^{1}(X,\mu)}.$

Next we replace $\alpha$ with $\beta=\alpha/\|f\|_{\Psi}$. Using that $\Psi$
is strictly increasing, it follows from the inequality above that

$\mu(\\{|f|\geq\beta\|f\|_{\Psi}\\})=\mu\bigg{(}\bigg{\\{}\Psi\bigg{(}\frac{|f|}{\|f\|_{L^{\Psi}(X,\mu)}}\bigg{)}\geq\Psi(\beta)\bigg{\\}}\bigg{)}\leq\frac{1}{\Psi(\beta)}\int_{X}\Psi\bigg{(}\frac{|f|}{\|f\|_{L^{\Psi}(X,\mu)}}\bigg{)}d\mu.$

It is a well-known property of the Luxembourg norm, established in many
standard references (e.g. [4]), that
$\int_{X}\Psi(|f|/\|f\|_{L^{\Psi}(X,\mu)})d\mu\leq 1$. It follows that
$\mu(\\{|f|\geq\beta\|f\|_{L^{\Psi}(X,\mu)}\\})\leq\Psi(\beta)^{-1}$, and
since $\Psi^{-1}$ is increasing this implies that $\Psi^{-1}(\mu(\\{x\in
X:|f|\geq\alpha\\})^{-1})^{-1}\leq\beta^{-1}$. Writing $\beta$ in terms of
$\alpha$ and $f$ gives (2.1). ∎

Equipped with this result, we can compute the Orlicz norms of indicator
functions exactly.

######  Corollary 2.2.

Let $S$ be a $\mu$-measurable subset of $X$. Then
$\|\chi_{S}\|_{L^{\Psi}(X,\mu)}=\Psi^{-1}(\mu(S)^{-1})^{-1}.$

###### Proof.

The estimate $\Psi^{-1}(\mu(S)^{-1})^{-1}\leq\|\chi_{S}\|_{\Psi}$ follows at
once from Chebyshev’s inequality. For the reverse inequality, observe that

$\int_{X}\Psi\bigg{(}\frac{\chi_{S}}{\Psi^{-1}(\mu(S)^{-1})^{-1}}\bigg{)}d\mu=\int_{S}\Psi\bigg{(}\frac{1}{\Psi^{-1}(\mu(S)^{-1})^{-1}}\bigg{)}d\mu=\int_{S}\frac{1}{\mu(S)}d\mu=1.$

By the definition of the Luxembourg norm, this implies that
$\|\chi_{S}\|_{\Psi}\leq\Psi^{-1}(\mu(S)^{-1})^{-1}$. ∎

In this paper we work with limits of Orlicz norms that are defined by a one-
parameter family of Young functions. Subject to appropriate growth conditions,
these families have useful pointwise properties which we will exploit in the
sections to follow. Our main condition on these families is the following
generalization of Condition 1.1.

######  Condition 2.3.

Given $\alpha\geq 0$ and $\beta\geq\alpha$, a family of Young functions
$\\{\Psi_{q}\\}$ is said to be $(\alpha,\beta)$-admissible if
$\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}(t)=\infty$ for $t>\beta$, and
for $0<t<\alpha$ one of the following holds:

  * (i)

If $\mu(X)=\infty$ then $\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}(t)=0$,

  * (ii)

If $\mu(X)<\infty$ then
$\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}(t)<\mu(X)^{-1}$.

######  Proposition 2.4.

Let $\\{\Psi_{q}\\}$ be an $(\alpha,\beta)$-admissible family. If
$\mu(X)=\infty$ and $t>0$, or if $\mu(X)<\infty$ and $t\geq\mu(X)^{-1}$, then

(2.2)
$\alpha\leq\liminf_{q\rightarrow\infty}\Psi_{q}^{-1}(t)\leq\limsup_{q\rightarrow\infty}\Psi_{q}^{-1}(t)\leq\beta.$

###### Proof.

If $\alpha=0$ then the first inequality in (2.2) holds trivially, so we assume
that $\alpha>0$.

Fix $t>0$ if $\mu(X)=\infty$ and $t\geq\mu(X)^{-1}$ if $\mu(X)<\infty$, and
assume toward a contradiction that there exists $\eta>0$ such that

$\liminf_{q\rightarrow\infty}\Psi_{q}^{-1}(t)\leq\alpha-\eta.$

Given any $0<\varepsilon<\eta$ then, there exists an increasing sequence
$\\{q_{j}\\}$ such that $q_{j}\rightarrow\infty$ and $j\rightarrow\infty$ and
$\Psi_{q_{j}}^{-1}(t)<\alpha-\varepsilon$ for each $j$. Since each $\Psi_{q}$
is strictly increasing for all $q>0$, we find that
$t<\Psi_{q_{j}}(\alpha-\varepsilon)$ for each $j$. Taking the limit as
$j\rightarrow\infty$, we see from Condition 2.3 that $0<t\leq 0$ if
$\mu(X)=\infty$, and $\mu(X)^{-1}\leq t<\mu(X)^{-1}$ if $\mu(X)<\infty$. In
any case this is a contradiction, meaning that

$\alpha-\eta<\liminf_{q\rightarrow\infty}\Psi_{q}^{-1}(t).$

Since $\eta>0$ was arbitrary, we get the first inequality in (2.2). The
estimates for the limit supremum in (2.2) follow in an identical fashion. ∎

If $\alpha=\beta=\delta$ then Condition 2.3 is the same as
$\delta$-admissibility, and Proposition 2.4 gives

$\displaystyle\lim_{q\rightarrow\infty}\Psi^{-1}_{q}(t)=\delta$

for each $t>0$ when $\mu(X)=\infty$, and for each $t\geq\mu(X)^{-1}$ when
$0<\mu(X)<\infty$. In fact, the limit identity above is equivalent to
$\delta$-admissibility.

######  Proposition 2.5.

A family of Young functions $\\{\Psi_{q}\\}$ is $\delta$-admissible if and
only if

(2.3) $\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}^{-1}(t)=\delta$

holds for all $t>0$ if $\mu(X)=\infty$, and for all $t\geq\mu(X)^{-1}$ if
$0<\mu(X)<\infty$.

###### Proof.

Proposition 2.4 gives the forward implication, leaving us to prove that if
(2.3) holds for $t$ in the appropriate range then $\\{\Psi_{q}\\}$ is
$\delta$-admissible. Regardless of whether $\mu(X)$ is finite or infinite, if
$\Psi_{q}(t)\leq M$ for some $M>0$ and large $q$, then $t\leq\Psi_{q}^{-1}(M)$
and (2.3) gives

$t\leq\lim_{q\rightarrow\infty}\Psi_{q}^{-1}(M)=\delta.$

Since $M$ was arbitrary, it follows in the contrapositive that if $t>\delta$,
then $\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}(t)=\infty$.

Assume now that $0<\mu(X)<\infty$ and suppose that (2.3) holds for all
$t\geq\mu(X)^{-1}>0$. If

$\lim_{q\rightarrow\infty}\Psi_{q}(t)\geq\mu(X)^{-1},$

then given $0<\varepsilon<1$ we have $\Psi_{q}(t)>\mu(X)^{-1}(1-\varepsilon)$
whenever $q$ is sufficiently large. By convexity of $\Psi_{q}$ it follows that

$\mu(X)^{-1}\leq\frac{\Psi_{q}(t)}{1-\varepsilon}\leq\Psi_{q}\bigg{(}\frac{t}{1-\varepsilon}\bigg{)},$

and so $t\geq(1-\varepsilon)\Psi_{q}^{-1}(\mu(X)^{-1})$ whenever $q$ is
sufficiently large. Using (2.3) we take the limit to find

$t\geq(1-\varepsilon)\lim_{q\rightarrow\infty}\Psi_{q}^{-1}(\mu(X)^{-1})=(1-\varepsilon)\delta.$

Since $\varepsilon>0$ was arbitrary, we conclude that $t\geq\delta$. Thus,
$\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}(t)<\mu(X)^{-1}$ when
$t<\delta$.

On the other hand, if $\mu(X)=\infty$ and (2.3) is satisfied for $t>0$, and if
$\Psi_{q}(t)\geq\varepsilon$ for some $\varepsilon>0$ and for all large $q$,
then

$t\geq\lim_{q\rightarrow\infty}\Psi_{q}^{-1}(\varepsilon)=\delta.$

Thus, if $t<\delta$ then
$\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}(t)\leq\varepsilon$, and since
$\varepsilon>0$ was arbitrary we have
$\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}(t)=0$. ∎

##  3\. Examples

There are many examples of $\delta$-admissible families of Young functions,
and moreover they are often easy to construct. In this section we showcase
some families to illustrate the utility of our main result, Theorem 1.2.

######  Example 3.1.

If $\Psi_{q}(t)=t^{q}$ and $\Phi(t)=t^{r}$ for some $r\geq 1$, then (1.3) is
non-decreasing for $t>0$ whenever $q\geq r$. Moreover the family
$\\{\Psi_{q}\\}$ is 1-admissible, and an application of Theorem 1.2 gives the
well-known identity (1.1).

######  Example 3.2.

If $\Phi(t)=t^{r}$ for $r\geq 1$ and $\Psi_{q}(t)=t^{p}\log(e-1+t)^{q}$ for
fixed $p\geq 1$, then

$\frac{\Phi^{-1}(\Psi_{q}(t))}{t}=\frac{t^{\frac{p}{r}}\log(e-1+t)^{\frac{q}{r}}}{t}$

fails the growth condition of Theorem 1.2 when $r>p$, and satisfies it when
$r\leq p$, regardless of the value of $q>0$. In the latter case, Theorem 1.2
recovers identity (1.2) for $f\in L^{r}(X,\mu)$.

######  Example 3.3.

Given $N\in\mathbb{N}$ and $p\geq 1$, consider the family of
$N^{\mathrm{th}}$-order iterated log-bumps

$\Psi_{q}(t)=t^{p}\underbrace{\log\cdots\log}_{N\;\mathrm{times}}(c+t)^{q},$

where $c$ is chosen independent of $q$ so that $\Psi_{q}(1)=1$. This family is
$1$-admissible, and a straightforward adaptation of the argument in Section 5
shows that (1.3) is non-decreasing on any interval of the form $[0,k]$ for
$f\in L^{\Psi_{q_{0}}}(X,\mu)$ whenever $q>q_{0}>0$ is sufficiently large.
Thus, Theorem 1.2 applies to the Orlicz norms characterized by the
$N^{\mathrm{th}}$ order iterated log-bumps above, giving

$\lim_{q\rightarrow\infty}\|f\|_{L^{\Psi_{q}}(X,\mu)}=\|f\|_{L^{\infty}(X,\mu)}$

whenever $f\in L^{\Psi_{q_{0}}}(X,\mu)$ for some $q_{0}>0$. We emphasize that
in this example, the convergence of the $\|f\|_{L^{\Psi_{q}}(X,\mu)}$ norms to
$\|f\|_{L^{\infty}(X,\mu)}$ is independent of $p$. Thus for identity (1.4) to
hold when $\\{\Psi_{q}\\}$ is a family of iterated log-bumps, it is not
necessary to assume that $f\in L^{p+\varepsilon}(X,\mu)$ for any
$\varepsilon>0$.

######  Example 3.4.

For any fixed Young function $\Phi$, one can obtain a $1$-admissible family
using the structure of $N^{\mathrm{th}}$-order iterated log-bumps by defining

$\Psi_{q}(t)=\Phi(t)\underbrace{\log\cdots\log}_{N\;\mathrm{times}}(c+t)^{q},$

where $c$ is chosen so that $\Psi_{q}(1)=\Phi(1)$ for all $q$. Indeed, the
iterated log-bumps of the form

(3.1)
$\displaystyle\Psi_{q}(t)=\bigg{(}t\prod_{j=1}^{N}\underbrace{\log\cdots\log}_{j\;\mathrm{times}}(c_{j}+t)\bigg{)}^{p}\underbrace{\log\cdots\log}_{N\;\mathrm{times}}(c_{N}+t)^{q}$

are of this type for $p\geq 1$, provided that the constants
$c_{1},\dots,c_{N}$ are chosen so that the value of $\Psi_{q}(\delta)$ is
independent of $q$ for some $\delta>0$. Once again Theorem 1.2 applies to this
family, allowing us to reproduce the limit in [1, Theorem 6.1]. This result is
proved in [1] by means of a modification of the techniques of [3], which rely
on the properties of iterated logarithms.

As in the last example, a similar calculation to that employed in Section 5
shows that if $\Phi=\Psi_{q_{0}}$ for $\Psi_{q}$ as in (3.1), then (1.3) is
non-decreasing on any bounded interval of the form $[0,k]$ for $k>0$ whenever
$q>q_{0}$ is sufficiently large. Once again, we conclude that the
$L^{\Psi_{q}}(X,\mu)$ norms of $f$ converge to $\|f\|_{L^{\infty}(X,\mu)}$,
provided $f\in L^{\Psi_{q_{0}}}(X,\mu)$ for some $q_{0}>0$.

There are many more families for which $\delta$-admissibility can be
established, and the interested reader is encouraged to construct their own
examples.

##  4\. Proof of Theorem 1.2

Every Orlicz norm used in this section is defined with respect to a fixed
measure space $(X,\mu)$, so we will always write
$\|\cdot\|_{L^{\Psi}(X,\mu)}=\|\cdot\|_{\Psi}$ and
$\|\cdot\|_{L^{\infty}(X,\mu)}=\|\cdot\|_{\infty}$. Fix $\delta>0$ and suppose
that $\\{\Psi_{q}\\}$ is a $\delta$-admissible family of Young functions.
Identity (1.4) is trivial if $f\equiv 0$, and we will treat the case of
unbounded $f$ separately at the end. Thus, we begin by assuming that
$0<\|f\|_{\infty}<\infty$, and we note that it is enough to prove (1.4) when
$\|f\|_{\Phi}=1$. Since (1.3) is non-decreasing on $[0,\|f\|_{\infty}]$ by
hypothesis for $q$ sufficiently large, we have

(4.1)
$\|f\|_{\Psi_{q}}=\left\|\Psi_{q}^{-1}(\Phi(|f|))\frac{|f|}{\Psi_{q}^{-1}(\Phi(|f|))}\right\|_{\Psi_{q}}\leq\left\|\Psi_{q}^{-1}(\Phi(|f|))\right\|_{\Psi_{q}}\frac{\|f\|_{\infty}}{\Psi_{q}^{-1}(\Phi(\|f\|_{\infty}))}.$

Additionally, we see that $\|\Psi_{q}^{-1}(\Phi(|f|))\|_{\Psi_{q}}\leq 1$
since by definition of the Luxembourg norm,

$\int_{X}\Psi_{q}\left(\Psi_{q}^{-1}\left(\Phi(|f(x)|)\right)\right)d\mu=\int_{X}\Phi(|f|)d\mu\leq
1.$

Moreover, in the case $0<\mu(X)<\infty$ we have that

$\Phi(\|f\|_{\infty})\geq\mu(X)^{-1}\int_{X}\Phi(|f|)d\mu=\mu(X)^{-1}.$

Equality holds above since $\|f\|_{\Phi}=1$ and since $f$ is bounded by
assumption (see e.g. [7, Eq. (3.13)]). Using these estimates, we find from
Proposition 2.5 that

$\limsup_{q\rightarrow\infty}\|f\|_{\Psi_{q}}\leq\|f\|_{\infty}\lim_{q\rightarrow\infty}\Psi_{q}^{-1}(\Phi(\|f\|_{\infty}))^{-1}=\frac{\|f\|_{\infty}}{\delta}.$

Next suppose that $0<\varepsilon<\|f\|_{\infty}$ and let
$S=\\{x\in\Omega:|f(x)|\geq\|f\|_{\infty}-\varepsilon\\}$. From the definition
of the essential supremum and Chebyshev’s inequality it follows at once that
$0<\mu(S)\leq\Phi((\|f\|_{\infty}-\varepsilon)^{-1})$, meaning that $\mu(S)$
is finite and nonzero. Moreover, Chebyshev’s inequality with
$\alpha=\|f\|_{\infty}-\varepsilon$ also shows that

$(\|f\|_{\infty}-\varepsilon)\Psi_{q}^{-1}(\mu(S)^{-1})^{-1}\leq\|f\|_{\Psi_{q}}.$

From Proposition 2.5 it follows that
$\Psi_{q}^{-1}(\mu(S)^{-1})^{-1}\rightarrow\delta^{-1}$ as
$q\rightarrow\infty$, since $\mu(S)^{-1}\geq\mu(X)^{-1}$ when
$0<\mu(X)<\infty$. As a result we find that

$\frac{\|f\|_{\infty}-\varepsilon}{\delta}\leq\displaystyle\liminf_{q\rightarrow\infty}\|f\|_{\Psi_{q}}.$

Since $\varepsilon>0$ was arbitrary, this gives
$\displaystyle\delta^{-1}\|f\|_{\infty}\leq\displaystyle\liminf_{q\rightarrow\infty}\|f\|_{\Psi_{q}}$,
proving that (1.4) holds.

In the case where $\|f\|_{\infty}=\infty$, choose $N>1$ and set
$f_{N}=\min\\{|f|,N\\}$ so that $\|f_{N}\|_{\infty}=N$. Applying our work
above, we see that

$\displaystyle\liminf_{q\rightarrow\infty}\|f\|_{\Psi_{q}}\geq\liminf_{q\rightarrow\infty}\|f_{N}\|_{\Psi_{q}}\geq\delta^{-1}\|f_{N}\|_{\infty}=\delta^{-1}N.$

Since $N$ may be chosen arbitrarily large, we find
$\displaystyle\liminf_{q\rightarrow\infty}\|f\|_{\Psi_{q}}=\infty$ as
required.

Now we show that if (1.4) holds for all $f\in L^{\Phi}(X,\mu)$, then the
family $\\{\Psi_{q}\\}$ is $\delta$-admissible. Specifically, we utilize the
characterization of $\delta$-admissible families given by Proposition 2.5 to
recognize that it is enough to show that

$\lim_{q\rightarrow\infty}\Psi^{-1}_{q}(t)=\delta$

for every $t>0$ when $\mu(X)=\infty$, and for $t\geq\mu(X)^{-1}$ when $\mu(X)$
is finite and positive.

Suppose first that $\mu(X)=\infty$. Given $t>0$, we use that $(X,\mu)$ is
$\sigma$-finite to select sets $S_{1},S_{2}\subset X$ of sufficiently large
measure so that with $t_{j}=\mu(S_{j})^{-1}$ we have $0<t_{2}<t_{1}<t$. Using
(1.4) with $f_{j}=\chi_{S_{j}}\in L^{\Phi}(X,\mu)$ we find from Corollary 2.2
that

$\lim_{q\rightarrow\infty}\Psi_{q}^{-1}(t_{j})=\lim_{q\rightarrow\infty}\|f_{j}\|_{L^{\Psi_{q}}(X,\mu)}^{-1}=\delta\|f_{j}\|_{L^{\infty}(X,\mu)}^{-1}=\delta.$

Since we may choose $\lambda\in(0,1)$ so that $\lambda
t_{2}+(1-\lambda)t=t_{1}$, the concavity of $\Psi_{q}^{-1}$ gives

$\lambda\Psi_{q}^{-1}(t_{2})+(1-\lambda)\Psi_{q}^{-1}(t)\leq\Psi_{q}^{-1}(t_{1}).$

Letting $q\rightarrow\infty$ we find after taking a limit supremum and
rearranging that

$\limsup_{q\rightarrow\infty}\Psi_{q}^{-1}(t)\leq\delta.$

More, since $\Psi_{q}^{-1}$ is increasing,
$\delta\leq\displaystyle\liminf_{q\rightarrow\infty}\Psi_{q}^{-1}(t)$. Thus,
$\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}^{-1}(t)=\delta$ for every
$t>0$.

In the case that $0<\mu(X)<\infty$, if $t>\mu(X)^{-1}$ we may proceed exactly
as above, see Remark 1.3. If $t=\mu(X)^{-1}$, the required estimate follows at
once by applying (1.4) to $f=\chi_{X}\in L^{\Phi}(X,\mu)$. In any case we have
established that

$\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}^{-1}(t)=\delta$

for $t\geq\mu(X)^{-1}$ when $0<\mu(X)<\infty$, and for $t>0$ when
$\mu(X)=\infty$. It follows from Proposition 2.5 that $\\{\Psi_{q}\\}$ is
$\delta$-admissible. $\Box$

######  Remark 4.1.

If we use the more general Condition 2.3 in place of Condition 1.1 in Theorem
1.2, simple modifications of the proof above show that one has the estimates

(4.2)
$\frac{1}{\beta}\|f\|_{\infty}\leq\liminf_{q\rightarrow\infty}\|f\|_{\Psi_{q}}\leq\limsup_{q\rightarrow\infty}\|f\|_{\Psi_{q}}\leq\frac{1}{\alpha}\|f\|_{\infty}.$

It may be the case that the limit of the norms $\|f\|_{\Psi_{q}}$ does not
exist for a family $\\{\Psi_{q}\\}$ which is $(\alpha,\beta)$-admissible, as
we show with the following example. Let

$\Psi_{q}(t)=\begin{cases}\hfil\frac{1}{2}t^{q}&0\leq t\leq\frac{1}{2},\\\
\frac{1}{2}(t^{q}+(2t-1)^{2+\sin q})&\frac{1}{2}<t<1,\\\
\hfil\frac{1}{2}(t^{q}+(2t-1)^{3})&\hfill t\geq 1,\end{cases}$

so that $\\{\Psi_{q}\\}$ is $(\frac{1}{2},1)$-admissible. If $f=\chi_{S}$ for
$S$ with $2<\mu(S)<\infty$ then $\|f\|_{\infty}=1$ and

$1\leq\liminf_{q\rightarrow\infty}\|f\|_{\Psi_{q}}\leq\limsup_{q\rightarrow\infty}\|f\|_{\Psi_{q}}\leq
2.$

On the other hand, we can show that
$\displaystyle\lim_{q\rightarrow\infty}\Psi_{q}^{-1}(t)$ does not exist for
$t\in(0,\frac{1}{2})$, meaning that

(4.3)
$\lim_{q\rightarrow\infty}\|f\|_{\Psi_{q}}=\lim_{q\rightarrow\infty}\Psi_{q}^{-1}(\mu(S)^{-1})^{-1}$

does not exist. To see this, assume toward a contradiction that there is a
$t\in(0,\frac{1}{2})$ such that

$\lim_{q\rightarrow\infty}\Psi_{q}^{-1}(t)=d.$

First we show that $d\in(\frac{1}{2},1)$. Given $\varepsilon>0$ we have for
all large $q$ that $d-\varepsilon<\Psi_{q}^{-1}(t)<d+\varepsilon$ and
$\Psi_{q}(d-\varepsilon)<t<\Psi_{q}(d+\varepsilon)$. If $d<\frac{1}{2}$ then
we can choose $\varepsilon$ small enough that $d+\varepsilon\leq\frac{1}{2}$,
meaning that $\Psi_{q}(d+\varepsilon)\rightarrow 0$ as $q\rightarrow\infty$.
Since $t<\Psi_{q}(d+\varepsilon)$ for all large $q$ this gives a contradiction
for large $q$. Likewise if $d>1$ then we can choose $\varepsilon$ such that
$d-\varepsilon\geq 1$, meaning that $\Psi_{q}(d-\varepsilon)\rightarrow\infty$
as $q\rightarrow\infty$. Since $\Psi_{q}(d-\varepsilon)<t<\frac{1}{2}$ this
gives another contradiction, and it follows that $\frac{1}{2}\leq d\leq 1$.

If $d=\frac{1}{2}$ then we have $\frac{1}{2}<d+\varepsilon<1$ when
$\varepsilon$ is small and
$\Psi_{q}(d+\varepsilon)=\frac{1}{2}((\frac{1}{2}+\varepsilon)^{q}+(2\varepsilon)^{2+\sin
q})$. Choosing $\varepsilon<\frac{1}{2}$ so small that $(2\varepsilon)^{2+\sin
q}\leq 2\varepsilon\leq t$ and then taking $q$ so large that
$(\frac{1}{2}+\varepsilon)^{q}\leq t$, we get $\Psi_{q}(d+\varepsilon)\leq t$,
a contradiction. Similarly, if $d=1$ then $\frac{1}{2}<d-\varepsilon<1$ for
small $\varepsilon$ and

$\Psi_{q}(d-\varepsilon)=\frac{(1-\varepsilon)^{q}+(1-2\varepsilon)^{2+\sin
q}}{2}\geq\frac{(1-2\varepsilon)^{2+\sin
q}}{2}\geq\frac{(1-2\varepsilon)^{3}}{2}.$

Since $t<\frac{1}{2}$, we can choose $\varepsilon$ sufficiently small that
$\frac{(1-2\varepsilon)^{3}}{2}\geq t$, another contradiction. It follows that
$\frac{1}{2}<d<1$ and $\frac{1}{2}<d-\varepsilon<d+\varepsilon<1$ for small
$\varepsilon$. Consequently

$\Psi_{q}(d-\varepsilon)=\frac{(d-\varepsilon)^{q}+(2(d-\varepsilon)-1)^{2+\sin
q}}{2}\;\;\textrm{ and
}\;\;\Psi_{q}(d+\varepsilon)=\frac{(d+\varepsilon)^{q}+(2(d+\varepsilon)-1)^{2+\sin
q}}{2}.$

Taking $q=\frac{\pi}{2}+k\pi$ for odd $k\in\mathbb{Z}$ gives
$t>\Psi_{q}(d-\varepsilon)=\frac{1}{2}((d-\varepsilon)^{q}+2(d-\varepsilon)-1)$,
while using even $k$ gives
$t<\Psi_{q}(d+\varepsilon)=\frac{1}{2}((d+\varepsilon)^{q}+(2(d+\varepsilon)-1)^{3})$.
For $\varepsilon$ small and $q$ large we show that this is impossible. By
convexity of the map $t\mapsto t^{q}$ for $q\geq 1$ we have

$(d+\varepsilon)^{q}\leq\frac{1}{2}(d-\varepsilon)^{q}+\frac{1}{2}(d+3\varepsilon)^{q}<(d-\varepsilon)^{q}+(d+3\varepsilon)^{q},$

and moreover, a straightforward calculation shows that

$(2(d+\varepsilon)-1)^{3}=4d(2d-1)(d-1)+4\varepsilon(2+6d^{2}-6d+6d\varepsilon+2\varepsilon^{2}-3\varepsilon)+(2d-2\varepsilon-1).$

Note that $4d(2d-1)(d-1)<0$ when $\frac{1}{2}<d<1$. From the preceding
estimates we get that

$(d+\varepsilon)^{q}+(2(d+\varepsilon)-1)^{3}<(d-\varepsilon)^{q}+(d+3\varepsilon)^{q}+4d(2d-1)(d-1)+4\varepsilon(2+2\varepsilon^{2}+3\varepsilon)+(2d-2\varepsilon-1).$

Taking $\varepsilon$ small and $q$ large, we can ensure that
$(d+3\varepsilon)^{q}+4d(2d-1)(d-1)+4\varepsilon(2+2\varepsilon^{2}+3\varepsilon)\leq
0$, and doing this gives the following contradiction:

$2t<(d+\varepsilon)^{q}+(2(d+\varepsilon)-1)^{3}\leq(d-\varepsilon)^{q}+2(d-\varepsilon)-1)<2t.$

Thus, the limit in (4.3) may not exist when the $\delta$-admissibility
condition fails.

##  5\. Log-Bump Orlicz Norms

Here we show that Theorem 1.2 implies [3, Theorem 1], which states that (1.4)
holds with $\delta=1$ for a specific family of log-bump Young functions. Given
$p\geq 1$, the log-bumps are of the form $\Psi_{q}(t)=t^{p}\log(e-1+t)^{q}$
for $q>0$, and the collection of all these bumps is a $1$-admissible family.
Thus, [3, Theorem 1] follows from Theorem 1.2 once we demonstrate that for
$k>0$ the function (1.3) is non-decreasing on $[0,k]$ when
$\Phi(t)=\Psi_{q_{0}}(t)$ for some $q_{0}>0$ and when $q$ is large. To do
this, we first assume that $q>q_{0}$ and for $t>0$ we define

$F(t)=\frac{t}{\Psi_{q}^{-1}(\Psi_{q_{0}}(t))},$

so that $F$ satisfies the equation $\Psi_{q_{0}}(t)=\Psi_{q}(tF(t)^{-1})$.
Recalling the form of $\Psi_{q}$ we see that

(5.1) $\displaystyle
F(t)^{p}\log(e-1+t)^{q_{0}}=\log\bigg{(}e-1+\frac{t}{F(t)}\bigg{)}^{q}$

for $t>0$. It follows from the definition above that $F$ is continuous on
$(0,\infty)$, and to extend $F$ continuously to zero we observe that

$\log(e-1)^{q_{0}}\lim_{t\rightarrow
0^{+}}F(t)^{p}=\log\bigg{(}e-1+\lim_{t\rightarrow
0^{+}}\Psi_{q}^{-1}(\Psi_{q_{0}}(t))\bigg{)}^{q}=\log(e-1)^{q}.$

Defining $F(0)=\displaystyle\lim_{t\rightarrow
0^{+}}F(t)=\log(e-1)^{\frac{q-q_{0}}{p}}$ thus ensures that $F$ is continuous
on $[0,\infty)$.

######  Lemma 5.1.

If $q>q_{0}$ then $F(t)>F(0)$ for every $t>0$.

###### Proof.

Observe that if $q>q_{0}$ then $\Psi_{q}(t)\geq\Psi_{q_{0}}(t)$ when $t\geq
1$, while $\Psi_{q}(t)<\Psi_{q_{0}}(t)$ when $0<t<1$. In the case $t\geq 1$,
we use that $\Psi_{q}^{-1}$ is strictly increasing to see that
$t\geq\Psi_{q}^{-1}(\Psi_{q_{0}}(t))$, which implies that $F(t)\geq 1>F(0)$.
Likewise, $0<t<1$ gives $F(t)<1$ and by (5.1) we find

$\log(e-1+t)^{q}<\log\bigg{(}e-1+\frac{t}{F(t)}\bigg{)}^{q}=F(t)^{p}\log(e-1+t)^{q_{0}}.$

Rearranging, we see that $F(t)>\log(e-1+t)^{\frac{q-q_{0}}{p}}>F(0)$. ∎

Now fix $k>0$, set $I=[0,k]$ and let $M=\sup_{I}F$. We show that $F$ is
injective on $I$ when $q$ is large enough. To this end, fix $t_{1}\in I$, set
$c=F(t_{1})$, and note that $F(t_{1})=F(0)$ if and only if $t_{1}=0$. On the
other hand, if $t_{1}>0$ then $c\in(F(0),M]$ by Lemma 5.1. More, from (5.1) we
see that $t_{1}$ is a fixed point of the map
$T_{c}:[0,\infty)\rightarrow\mathbb{R}$ defined by

$T_{c}(t)=c\exp\big{(}c^{\frac{p}{q}}\log(e-1+t)^{\frac{q_{0}}{q}}\big{)}-c(e-1).$

Since $F(0)<c$, it is easy to see that $T_{c}(0)>0$. Furthermore, a
straightforward computation shows that $T_{c}^{\prime\prime}(t)<0$ if and only
if

(5.2)
$q_{0}c^{\frac{p}{q}}\log\left(e-1+t\right)^{\frac{q_{0}}{q}}<q\log\left(e-1+t\right)+q-q_{0}.$

For large $q$, this is achieved uniformly in $c\in[0,M]$. To see why, choose
$q\geq q_{0}$ so that

$M^{p}\leq\bigg{(}\frac{q}{q_{0}}\bigg{)}^{q}\log(e-1)^{q-q_{0}}.$

Then
$c^{\frac{p}{q}}\leq\frac{q}{q_{0}}\log(e-1)^{1-\frac{q_{0}}{q}}\leq\frac{q}{q_{0}}\log(e-1+1)^{1-\frac{q_{0}}{q}}$
for each $t\geq 0$, and this shows that (5.2) holds for each $t\geq 0$, and we
see $T_{c}(t)$ is strictly concave on $(0,\infty)$.

For $q$ large as above, we find that $T_{c}(0)>0$ and $T_{c}$ is a continuous
and strictly concave function on $(0,\infty)$. Thus, $T_{c}(t)$ has a unique
fixed point in $[0,\infty)$ and so it is $t_{1}$. This gives $F$ injective on
$I$ since $F(t_{2})=c=F(t_{1})$ shows that $t_{1}$ and $t_{2}$ are fixed
points of $T_{c}(t)$. Since $F$ is a continuous, injective function on $I$,
the Intermediate Value Theorem shows that $F$ is strictly monotone on $I$.
Lastly, since Lemma 5.1 shows that $F(0)<F(t)$ for $t\in I$, we conclude that
$F$ is strictly increasing on $I$ when $q$ is sufficiently large. Thus, with
the hypotheses of Theorem 1.2 verified, we have reproved [3, Theorem 1].

##  6\. Necessity of Admissibility Conditions

Finally, we show that Conditions 1.1 and 2.3 are necessary for the norm limit
to be related to the essential supremum of a function. This means that our
admissibility conditions cannot be weakened in Theorem 1.2 or Remark 4.1.

######  Theorem 6.1.

Let $\\{\Psi_{q}\\}$ be a family of Young functions, let $\Phi$ be a Young
function for which (1.3) is non-decreasing, and assume that there exists $f\in
L^{\Phi}(X,\mu)\cap L^{\infty}(X,\mu)$ such that

(6.1)
$0<\liminf_{q\rightarrow\infty}\|f\|_{\Psi_{q}}\leq\limsup_{q\rightarrow\infty}\|f\|_{\Psi_{q}}<\infty.$

Then $\\{\Psi_{q}\\}$ is $(\alpha,\beta)$-admissible for some $\beta>0$ and
$\alpha\geq 0$.

###### Proof.

First assume to the contrary that $\Psi_{q}(t)\rightarrow\infty$ as
$q\rightarrow\infty$ for each fixed $t>0$. Arguing as in the proof of
Proposition 2.4, we conclude that $\Psi_{q}^{-1}(t)\rightarrow 0$ as
$q\rightarrow\infty$ for each $t>0$. Since the limit infimum in (6.1) is non-
zero we have $0<\|f\|_{\infty}$, and as in the proof of Theorem 1.2 we can
choose $\varepsilon<\|f\|_{\infty}$ to see that

$(\|f\|_{\infty}-\varepsilon)\liminf_{q\rightarrow\infty}\Psi_{q}^{-1}(\mu(\\{x\in\Omega:|f(x)|\geq\|f\|_{\infty}-\varepsilon\\})^{-1})^{-1}\leq\liminf_{q\rightarrow\infty}\|f\|_{\Psi_{q}}.$

By definition of the essential supremum,
$\mu(\\{x\in\Omega:|f(x)|\geq\|f\|_{\infty}-\varepsilon\\})>0$, meaning that
the limit on the left-hand side above diverges and
$\|f\|_{\Psi_{q}}\rightarrow\infty$ as $q\rightarrow\infty$, contradicting the
right-hand limit of (6.1). Thus, there is a $t_{0}>0$ for which
$\displaystyle\limsup_{q\rightarrow\infty}\Psi_{q}(t_{0})<\infty$, and
therefore

$\limsup_{q\rightarrow\infty}\Psi_{q}(t)<\infty$

holds for every $0\leq t\leq t_{0}$ since each Young function is strictly
increasing. This means that $(\alpha,\beta)$-admissibility holds for some
$\alpha\geq 0$ and $\beta>0$.

Similarly, suppose that $\Psi_{q}(t)\rightarrow 0$ as $t\rightarrow\infty$ for
each fixed $t>0$, so that $\Psi_{q}^{-1}(t)\rightarrow\infty$ by the argument
of Proposition 2.4. Arguing as in Section 4 we have

$\limsup_{q\rightarrow\infty}\|f\|_{\Psi_{q}}\leq\|f\|_{\infty}\limsup_{q\rightarrow\infty}\Psi_{q}^{-1}(\Phi(\|f\|_{\infty}))^{-1}=0,$

and once again this contradicts (6.1). Since each $\Psi_{q}$ is strictly
increasing we conclude that
$\displaystyle\liminf_{q\rightarrow\infty}\Psi_{q}(t)>0$ for large $t>0$. Thus
there exists $\alpha\leq\beta<\infty$ satisfying Condition 2.3. ∎

In the case of $\delta$-admissibility, where $\alpha=\beta=\delta$, the
argument above shows that Theorem 1.2 fails when $\delta$ is not both positive
and finite.

######  Corollary 6.2.

Let $\\{\Psi_{q}\\}$ be a family of Young functions such that for every $t>0$,

$\lim_{q\rightarrow\infty}\Psi_{q}(t)=\infty\qquad(\textrm{resp.
}\lim_{q\rightarrow\infty}\Psi_{q}(t)=0).$

If $f$ satisfies the remaining hypotheses of Theorem 1.2 then regardless of
the value of $\|f\|_{\infty}$,

$\lim_{q\rightarrow\infty}\|f\|_{L^{\Psi_{q}}(X,\mu)}=\infty\qquad(\textrm{resp.
}\lim_{q\rightarrow\infty}\|f\|_{L^{\Psi_{q}}(X,\mu)}=0).$

To illustrate, if $f=\chi_{[0,1]}$ and $\Psi_{q}(t)=t^{p}\log(e+t)^{q}$ then
$\|f\|_{L^{\Psi_{q}}(\mathbb{R},dx)}=\Psi_{q}^{-1}(1)^{-1}$, but
$\\{\Psi_{q}\\}$ is not $\delta$-admissible for any $\delta>0$. A
straightforward calculation shows that

$\lim_{q\rightarrow\infty}\|f\|_{L^{\Psi_{q}}(\mathbb{R},dx)}\geq\liminf_{q\rightarrow\infty}\|f\|_{L^{\Psi_{q}}(\mathbb{R},dx)}=\liminf_{q\rightarrow\infty}\Psi_{q}^{-1}(1)^{-1}=\infty>1=\|f\|_{L^{\infty}(\mathbb{R},dx)}.$

Thus, if Condition 1.1 fails then the induced Orlicz norms may not converge to
$\|f\|_{L^{\infty}(X,\mu)}$.

## References

  * [1] A. Mailhot, On The Limit of Orlicz Norms, Cape Breton University Undergraduate Thesis, 2022. See http://faculty.cbu.ca/srodney/pdf/theses/thesisam.pdf
  * [2] D. Cruz-Uribe and S. Rodney, Bounded Weak Solutions to Elliptic PDE with Data in Orlicz Spaces, J. Diff. Eq. (297), (2021), p. 409-432
  * [3] D. Cruz-Uribe and S. Rodney, A Note on the Limit of Orlicz Norms, Real Anal. Exchange (48), (2023), p. 77-82
  * [4] M.N. Rao and Z.D. Ren, Theory of Orlicz Spaces, Marcel-Dekker NYC, (1991)
  * [5] H.L. Royden, Real Analysis, third edition, MacMillan NYC, (1988)
  * [6] W. Rudin, Real and Complex Analysis, third edition, McGraw Hill, (1987)
  * [7] N.S. Trudinger and T.K. Donaldson, Orlicz-Sobolev spaces and Imbedding Theorems, J. Func. Anal. (8), (1971), p. 52-75


We introduce a notion of generic local algorithm which strictly generalizes existing frameworks of local algorithms such as factors of i.i.d. by capturing local quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA).

Motivated by a question of Farhi et al. [arXiv:1910.08187, 2019] we then show limitations of generic local algorithms including QAOA on random instances of constraint satisfaction problems (CSPs).
Specifically, we show that any generic local algorithm whose assignment to a vertex depends only on a local neighborhood with $o(n)$ other vertices (such as the QAOA at depth less than $\epsilon\log(n)$) cannot arbitrarily-well approximate boolean CSPs if the problem satisfies a geometric property from statistical physics called the coupled overlap-gap property (OGP) [Chen et al., Annals of Probability, 47(3), 2019].
We show that the random $\mathsf{MAX}$-$k$-$\mathsf{XOR}$ problem has this property when $k\geq4$ is even by extending the corresponding result for diluted $k$-spin glasses.

Our concentration lemmas confirm a conjecture of Brandao et al. [arXiv:1812.04170, 2018] asserting that the landscape independence of QAOA extends to logarithmic depth—in other words, for every fixed choice of QAOA angle parameters, the algorithm at logarithmic depth performs almost equally well on almost all instances.

One of these concentration lemmas is a strengthening of McDiarmid's inequality, applicable when the random variables have a highly biased distribution, and may be of independent interest.

§ INTRODUCTION

§ PRELIMINARIES

§ LOCALITY AND SHARED RANDOMNESS

§ MAIN THEOREMS

§ CONCENTRATION ANALYSIS FOR GENERIC LOCAL ALGORITHMS

§ PROOFS OF MAIN THEOREMS

§ STRENGTHENED MCDIARMID'S INEQUALITY FOR BIASED DISTRIBUTIONS

§ OVERLAP-GAP PROPERTY FOR GENERAL CASE RANDOM

§ DISCUSSION


11institutetext:  Manipal Institute of Technology,
Manipal Academy of Higher Education, Manipal, India
1 Department of Data Science and Computer Applications
2 Department of Computer Science
3 Department of Mechanical Engineering

# ReFuSeg: Regularized Multi-Modal Fusion for Precise Brain Tumour
Segmentation

Aditya Kasliwal1 Sankarshanaa Sagaram1,** Laven Srivastava2,** Pratinav Seth1
and Adil Khan3

###### Abstract

Semantic segmentation of brain tumours is a fundamental task in medical image
analysis that can help clinicians in diagnosing the patient and tracking the
progression of any malignant entities. Accurate segmentation of brain lesions
is essential for medical diagnosis and treatment planning. However, failure to
acquire specific MRI imaging modalities can prevent applications from
operating in critical situations, raising concerns about their reliability and
overall trustworthiness. This paper presents a novel multi-modal approach for
brain lesion segmentation that leverages information from four distinct
imaging modalities while being robust to real-world scenarios of missing
modalities, such as T1, T1c, T2, and FLAIR MRI of brains. Our proposed method
can help address the challenges posed by artifacts in medical imagery due to
data acquisition errors (such as patient motion) or a reconstruction
algorithm’s inability to represent the anatomy while ensuring a trade-off in
accuracy. Our proposed regularization module makes it robust to these
scenarios and ensures the reliability of lesion segmentation.

###### Keywords:

Brain Lesion Multi-modality Segmentation Missing Modality Learning

<EMAIL_ADDRESS>
††footnotetext: **Authors have contributed equally.

##  1 Introduction

The rise of Artificial Intelligence in healthcare has made AI-based
interventions for brain tumour diagnosis and pre-assessment increasingly
vital. Analyzing brain tumours through AI-driven techniques contributes
significantly and helps in understanding the progression of brain tumour cells
and assisting in surgical groundwork. Characterization of these segmented
tumours can directly aid in predicting the interim duration for diagnosis and
the patient’s overall life expectancy, making brain tumour segmentation
crucial for various applications in this field. Magnetic Resonance Imaging
(MRI) is a reliable diagnostic tool that is crucial in monitoring and planning
brain tumour surgeries. The recent advancements in automated brain tumour
segmentation using MRI have achieved remarkable success and practical utility.
[18] [4] These algorithms typically rely on multiple modalities, with the four
most relevant being T1-weighted images with and without contrast enhancement,
T2-weighted images, and FLAIR images. Combining these complementary 3D MRI
modalities, such as T1, T1 with contrast agent (T1c), T2, and Fluid-attenuated
Inversion Recovery (FLAIR), helps highlight different tissue properties and
regions where the tumour has spread. The integration of multiple modalities is
essential for capturing a comprehensive view of the brain and improving
segmentation accuracy. Each modality provides unique insights into the
underlying tissue properties and pathology, allowing the model to exploit
complementary information for robust and precise lesion segmentation.

While deep learning-based brain tumour segmentation techniques have shown
impressive performance in various benchmarks, they face challenges due to the
limited kernel size in typical image segmentation models. [18] [6] This
limitation hinders their ability to learn long-range dependencies necessary
for accurately segmenting tumours of various shapes and sizes. In clinical
routines, missing MRI sequences due to time constraints or image artifacts can
be a common challenge. Therefore, developing methods that can compensate for
missing modalities and recover segmentation performance is highly desirable,
promoting the broader adoption of these algorithms in clinical practice.

We propose ReFuSeg: Regularized Multi-Modal Fusion for Precise Brain Lesion
Segmentation, our proposed architecture utilizes a novel approach toward
contrastive regularisation to learn features between multiple modalities. Our
approach prevents the model from overfitting to any particular modality and
promotes the learning of complementary information from each modality.
This leads to more robust and generalizable features capable of capturing the
intrinsic characteristics of each modality. All four encoders work
independently, learning individual features amongst each modality which helps
it maintain robustness within its predictions as it is not dependent on any
one singular modality to make accurate segmentation predictions.

The effectiveness of our proposed approach has been validated through
experimental evaluations on the BraTS 23 dataset [1, 11, 16]. Our method
displays robustness in accurately segmenting brain lesions, even in the case
of missing modalities where they exhibit outstanding Dice and Hausdorff-95
scores, even when provided with only limited portions of the original data.
Furthermore, the proposed model is unaffected by the inclusion of noise
artifacts, which are commonly found in everyday clinical usage. Owing to these
promising results, we demonstrate the suitability of our approach for real-
world scenarios.

##  2 Related Works

In Multimodal Image Segmentation previous methodologies have explored various
approaches to address the segmentation challenge. Some studies have
incorporated 3D network convolutions which fused the correlation
representations via attention guided mechanisms [25]. A distinct approach was
taken by Havaei et al. [5], where they constructed a unified model through a
self-supervised training pipeline for each channel. Instead of using
volumetric data, they directly passed 2D slices into the model encoders.
Predictions from multiple channels were combined by merging feature maps, and
mean, and variance were computed to achieve the final segmentation. In another
study [21], a cascaded network [14] was employed. In the first stage, the
tumour was segmented, and subsequent stages focused on learning substructures
for more detailed segmentation.

Missing Modalities A popular technique [3] involves utilizing an adversarial
loss on intermediate feature maps from two domains, facilitating knowledge
transfer between these domains. Similarly, in a different work [15], a class-
specific adversarial loss was employed on feature maps to transfer a learned
network from a source domain to a target domain. Generative models [23] have
also been used in the past to synthesize missing modalities. Furthermore,
self-supervised learning techniques have been utilized [20] by randomly
dropping modalities during training and leveraging the learned combined
feature maps. These feature maps are then adjusted to match any encoder
distributions to compensate for the missing data.

Contrastive learning is a powerful mechanism for representation learning. The
core principle involves training a neural network to map similar inputs closer
together in a learned feature space while simultaneously pushing dissimilar
inputs further apart.[2] Contrastive loss minimizes dissimilarity between
positive examples based on the learned features. Contrastive learning enables
the effective learning of shared and discriminative representations across
these modalities by capitalizing on the inherent similarities and distinctions
across different modalities. A combination of inter and intra-modal feature
learning [22]has also been used to capture similarities and minimize
disagreement between each modality. In biomedical segmentation, [13] a network
was proposed where one encoder learns features from T1 modality while the
secondary segmentation backbone consisting of Convolution blocks takes in the
multi-modal images and minimizes contrastive loss.

##  3 Methodology

Figure 1: Proposed ReFuSeg architecture for brain tumour segmentation.

In this paper, we introduce a novel methodology for brain lesion segmentation
which is robust in the case of missing modalities due to the independent
working of all the encoders, as a solution for the BraTS 2023 Adult Glioma
Segmentation challenge [1, 16, 11] which involves generating segmented tumour
masks by utilizing the four modalities provided in the dataset for each
instance, i.e., T1, T1 with contrast agent (T1c), T2 and Fluid-attenuated
Inversion Recovery (FLAIR). Our proposed approach, illustrated in the figures,
comprises two primary components:

  1. 1.

A multi-modal U-Net [19] based architecture incorporating four ResNet-34 [10]
encoders alongside a feature fusion technique.

  2. 2.

A regularization module that receives final features from the four encoders,
after which contrastive loss is calculated in the backward pass, enabling
robustness in case of missing modality.

In line with the conventional U-Net design, these encoders down-sample the
input image and capture features at multiple levels of abstraction. It’s worth
mentioning that the four encoders work independently to extract distinctive
features.

###  3.1 Contrastive Regularization

In our proposed approach, the features extracted from the encoders are
directed to four distinct contrastive modules. Each contrastive module
comprises a series of layers, including average pooling, fully connected, and
batch normalization, followed by another fully connected layer and batch
normalization. The resulting outputs from these contrastive modules are
instrumental in calculating the contrastive loss similar to the approach used
in [7], which is then added to our final loss function. Notably, the T1 and
T1c modalities have a higher likelihood of containing similar features due to
the nature of their MRI image acquisition methods, i.e., the feature changes
noticeable in both modalities are primarily made in the post-processing of the
MRI scans; the same is true for T2 and FLAIR modalities. However, the learning
process is unaffected by redundant features because contrastive loss is a
regularizing mechanism for these modalities.It promotes efficient selection of
features and enhances individual contributions during the learning process.

###  3.2 Feature Transfer from Encoders to Decoder

Once the features are extracted from all four encoders, a new feature map is
generated by taking the element-wise maximum of the corresponding level
feature maps from each encoder. This ensures that the most relevant features
from each encoder are selected, resulting in a new feature map that
effectively captures pertinent information from all modalities. The decoder,
which comprises several decoding blocks, then receives the new feature map
through skip connections. During the upsampling process, the decoder combines
these upsampled feature maps with their corresponding counterparts from the
encoders, utilizing the skip connections. However, in this modified U-Net
model, the skip connections transfer the newly calculated feature map from the
four encoders rather than individual feature maps from each encoder. By
amalgamating features extracted from the T1, T1c, T2, and FLAIR MRI images
across multiple levels of abstraction, the adapted U-Net model demonstrates
its capability to generate high-quality outputs.

###  3.3 Handling Missing Modalities

When faced with missing modalities, such as the absence of a T1-weighted scan
in the input, our model exhibits robust performance due to the independent
functioning of the four encoders. The model effectively utilizes the available
encoders corresponding to the present modalities in such situations. The
multilevel feature maps extracted from these encoders are directly passed to
the decoder via the skip connections. This approach ensures the model can
still capture relevant features from the available input data and produce
accurate outputs, even in missing modalities.

###  3.4 Loss Function

Dice Loss The Dice loss [17] serves as a metric to assess the overlap between
binary predicted and ground truth masks in image segmentation tasks. Its
objective is to maximize the similarity between these masks. Therefore,
minimizing the Dice loss during model training leads to improved accuracy of
the segmentation model. Additionally, our experiments demonstrate that
combining Dice loss with contrastive loss further enhances the segmentation
performance.

$\text{Dice
Loss}=1-\frac{2\sum_{i=1}^{n}y_{i}\hat{y}i}{\sum_{i=1}^{n}y_{i}^{2}+\sum_{i=1}^{n}\hat{y}_{i}^{2}}$
(1)

where $y_{i}$ represents the ground truth value for the i-th sample, and
$\hat{y}_{i}$ represents the corresponding predicted value from the model. The
summation runs over all n samples in the dataset.

Focal Loss The Focal loss [12] introduces a modulating term to the cross-
entropy [24] loss, aiming to prioritize learning on challenging misclassified
examples. This dynamic scaling of the cross entropy loss involves the scaling
factor diminishing to zero as confidence in the correct class rises. In the
context of image segmentation tasks, the Focal loss assumes a pivotal role in
handling class imbalance and accentuating challenging samples, resulting in
notable improvements in segmentation performance. The Focal loss formula is
given as follows:

$\text{Focal
Loss}=-\frac{1}{n}\sum_{i=1}^{n}\left(\alpha(1-\hat{y}_{i})^{\gamma}y_{i}\log(\hat{y}_{i})+(1-\alpha)\hat{y}_{i}^{\gamma}(1-y_{i})\log(1-\hat{y}_{i})\right)$
(2)

In the formula, $y_{i}$ represents the ground truth value (0 or 1) for the
i-th sample, and $\hat{y}_{i}$ represents the corresponding predicted value
from the model. The summation runs over all n samples in the dataset.

The adjustable hyperparameters, $\alpha$ and $\gamma$, enable us to control
the focusing effect and the rate at which the loss decreases for well-
classified samples. Integrating the Focal loss with other appropriate loss
functions, such as the contrastive loss mentioned in our experiments, can
improve segmentation results.

Contrastive Loss Contrastive loss [8] serves as a fundamental loss function
utilized in machine learning to train models for similarity learning. Its
purpose is to facilitate learning data representations, wherein similar data
points are brought closer together in the representation space while
dissimilar data points are pushed further apart. This enhances image fidelity
and realism by reducing artifacts and noise in the output images. In their
work, [2] applied this contrastive loss to train their model by comparing
pairs of images and calculating a similarity score between them. The
contrastive loss formula is defined as follows:

$l(v_{i},v_{j})=-\log\left(\frac{\exp(\text{sim}(v_{i},v_{j}))}{\sum_{k\neq
i}\exp(\text{sim}(v_{i},v_{k}))}\right)$ (3)

Here, $v_{i}$ and $v_{j}$ represent the representations of two modalities of
the same instance and $\text{sim}(\textperiodcentered,\textperiodcentered)$
denotes the function computing the cosine similarity. The denominator in the
formula represents the sum over all views $k$ in the batch, excluding $i$.
Subsequently, the contrastive loss is computed over the corresponding
instances of two modalities in the batch of size $N$ and is then averaged as:

$L_{C}=\frac{1}{2N}\sum_{i=1}^{N}\left[l(v_{i\text{x}},v_{i\text{y}})+l(v_{i\text{y}},v_{i\text{x}})\right]$
(4)

$\qquad\qquad\qquad\qquad\qquad\qquad\forall\left(x,y\right)=\left[{(T1,T1c),(T2,FLAIR)}\right]$

The resulting $L_{C}$ contributes to the overall loss function, which is
critical in guiding the model’s training process for similarity learning and
image representation enhancement.

Final Loss The final loss employed in our paper, achieved after rigorous
evaluation of different hyperparameter combinations, is represented as
follows:

$L_{Final}=0.5\cdot L_{Dice}+0.5\cdot L_{Focal}+\beta\cdot L_{C}$ (5)

Here, $\beta$ acts as a switch for contrastive loss in the overall loss
function.

##  4 Experimental Analysis

Table 1: Validation Results for BraTS 2023 Dataset. Contrastive | Dice | Hausdorff
---|---|---
Regularisation | ET | TC | WT | ET | TC | WT
$\times$ | 0.792 | 0.828 | 0.909 | 22.9 | 14.07 | 7.34
$\checkmark$ | 0.786 | 0.832 | 0.910 | 21.8 | 9.17 | 7.08

The BraTS [1, 16, 11] 2023 dataset was used in our study, which consisted of
5,880 MRI scans from 1,470 patients with brain diffuse glioma. The BraTS mpMRI
scans were provided in NIfTI format (.nii.gz) and included native (T1) and
post-contrast T1-weighted (T1c), T2-weighted (T2), and T2 Fluid Attenuated
Inversion Recovery (FLAIR) volumes. The training dataset had 1,251 instances,
while the validation dataset had 219 instances. We submitted our predicted
results on the validation dataset to the challenge website to assess our
model’s performance. The original 3D files, each of size 240x240x155, were
preprocessed into 155 2D slices of dimensionality 240x240. This preprocessing
step addressed spatial invariance, reduced computational complexity, and
enhanced anatomical interpretability. Additionally, we applied several
augmentations to the data, including horizontal and vertical flips with
probabilities of 0.5, rotation with a limit of 20 degrees, shift limit of 0.1,
and probability of 0.5, random crop to a size of 224x224, and final resizing
to 240x240.

We used softmax as the activation function in the output layer.The performance
of the model was evaluated using the Dice score and Hausdorff-95 distance. The
Dice score is a metric that measures the overlap between two sets, and a high
Dice score indicates that the model has accurately captured the boundaries and
shapes of the target structure. The Hausdorff-95 distance, on the other hand,
measures the distance between the nearest points of two sets, and it is
specifically used to assess the model’s performance at boundary regions. To
comprehensively evaluate the model, both these metrics were employed in our
analysis. Our model was trained using the Adam optimizer [9] with a learning
rate of $10^{-4}$. During the initial experimentation, the proposed model was
trained for 50 epochs without contrastive regularization. Subsequently,
another training run was conducted with contrastive regularization. The
comparison of results between non-contrastive and contrastive regularization
can be found in Table 1.

To evaluate the model’s robustness when faced with missing modalities, we
performed inference four times, each time excluding one of the four available
modalities. We conducted this inference process for both non-contrastive and
contrastive approaches, and the detailed results are presented in Table 2.

##  5 Results

The validation results, reveal noteworthy insights into the impact of
contrastive regularization on the model’s performance. These findings, as
illustrated in Table 1, demonstrate a remarkable and favorable improvement in
the model’s performance when contrastive regularization is employed, in
comparison to its performance without this technique.

Table 2: Comparing impact of missing modalities on BraTS 2023 validation set, with and without contrastive regularization.  Modality Dropped |  Contrastive Regularisation | Dice | Hausdorff-95
---|---|---|---
ET | TC | WT | ET | TC | WT
T1 | $\times$ | 0.753 | 0.824 | 0.901 | 40.01 | 17.76 | 14.18
T1 | $\checkmark$ | 0.769 | 0.833 | 0.908 | 31.04 | 13.26 | 9.59
T1c | $\times$ | 0.074 | 0.322 | 0.878 | 253.92 | 45.08 | 11.47
T1c | $\checkmark$ | 0.762 | 0.803 | 0.874 | 38.91 | 22.06 | 30.83
T2 | $\times$ | 0.057 | 0.271 | 0.884 | 200.89 | 65.63 | 7.17
T2 | $\checkmark$ | 0.783 | 0.819 | 0.89 | 27.41 | 17.5 | 16.22
FLAIR | $\times$ | 0.745 | 0.703 | 0.401 | 38.00 | 26.85 | 27.96
FLAIR | $\checkmark$ | 0.768 | 0.786 | 0.524 | 27.01 | 23.56 | 21.97

As indicated in Table 1, the model exhibited signs of overfitting on specific
modalities when contrastive regularization was not utilized. In such cases,
the model excessively relied on a limited subset of the available modalities.
As a result, the model’s overall performance suffered significantly during
validation, especially when confronted with missing modalities. Conversely,
incorporating contrastive regularization yielded substantial enhancement in
the model’s performance in the presence of missing modalities, demonstrating
reduced reliance on any single modality (T1, T1c, T2, or T2 flair) and,
instead, effectively harnessing features from all available modalities. This
regularization effect signifies the model’s adept utilization of features
across all modalities, effectively mitigating the overfitting challenges.

##  6 Conclusion

This research paper introduces an innovative framework that integrates data
fusion and regularization techniques for semantic segmentation, utilizing four
encoders within a U-Net-based architecture. The proposed framework serves as
our response to the BraTS 2023 Adult Glioma Segmentation challenge [1, 16,
11], held at the 9th MICCAI Workshop on Brain Lesions (BrainLes). It
represents a straightforward and resource-efficient architecture compared to
other models in this field.

Our architecture remarkably yields outstanding results with Dice scores of
0.786, 0.832, and 0.910, as well as Hausdorff distances of 21.8, 9.17, and
7.08 for the enhancing tumour (ET), tumour core (TC), and whole tumour (WT)
classes respectively, on the validation dataset with contrastive
regularization. The model possesses the ability to be robust in handling
missing data and maintaining its efficacy even when dealing with absent
modalities, as demonstrated in Table 2. Notably, we believe to be the first to
present such a fusion model that effectively addresses missing modalities,
rendering our architecture highly suitable for real-world scenarios with
frequently occurring missing data. Looking ahead, we are eager to explore
further experiments involving the scaling of contrastive regularization.
Additionally, we plan to investigate the applicability of this approach in
diverse domains and industries, broadening its potential impact.

##  7 Acknowledgments

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#  The Modeling of Time-Structured Multiturn Injection
into Fermilab Main Injector
(Microbunch Injection with Parasitic Longitudinal Painting)

Phil S. Yoon, David E. Johnson, and Weiren Chou

Fermi National Accelerator Laboratory, Batavia, IL 60510  E-mail:
<EMAIL_ADDRESS>Work supported by Fermilab Research Alliance (FRA), LLC under
contract No. DE-AC02-07-CH11359 with the United States Department of Energy

(February 2024)

###### Abstract

This paper presents the modeling of time-structured multiturn injection for an
upgraded Main Injector with the 8-GeV Superconducting RF proton driver, or an
ILC-style linac, or a Project-X linac. The Radio-Frequency mismatch between a
linac and the upgraded Main Injector will induce parasitic longitudinal
painting in RF-phase direction. Several different scenarios with a choice of
different RF parameters for single RF system and double RF system in the
presence of longitudinal space charge have been investigated. From the studies
of microbunch injection with the aid of ESME (2003) numerical simulations, it
is found that the dual RF system with a choice of appropriate RF parameters
allows us to overcome the space-charge limitation set by beam intensity during
the multiturn-injection process. A double RF system with a harmonic ratio
($R_{H}=H_{2}/H_{1}$) of 2.0 and a voltage ratio ($R_{V}=V_{2}/V_{1}$) of 0.5
are most favored to reduce both longitudinal and transverse effects of space
charge in the Main Injector.

## Microbunch Injection into the Main Injector

from a Superconducting RF linac into the Main Injector

After studying the method of time-structured multiturn injection from the
present 400-MeV linac to Fermilab’s Booster [1], we have further explored the
method of microbunch injection for applications to the Main Injector: from a
future Superconducting RF (SRF) linac to an upgraded Main Injector (FMI-2)111
We will use Main Injector (FMI) and upgraded Main Injector (FMI-2)
interchangeably herein. including parasitic longitudinal painting. The future
SC linac referred herein can be the 8-GeV SC xsRF linac proton driver [2, 3,
4], or an ILC-style linac, or the Project-X linac [5].

###  Overview of the Main Injector

The Main Injector (MI) is a ring with a circumference of about 3.3 (km). The
central roles of the MI is to connect to the Tevatron, the Booster, the
Antiproton source, switchyard, and the Recycler ring via a number of beam
transport lines in the Fermilab accelerator complex. The MI can accelerate and
decelerate particles between 8 GeV and 150 GeV, depending on the mode of
operation. The harmonic number of the MI is 588 and harmonic RF at injection
is 52.8114 (MHz)222 For the sake of brevity and convenience, MI RF will be
referred to as 53 MHz hereafter.

###  Overview of the 8-GeV Superconducting RF Linac

An 8-GeV SC linac has been proposed as a single-stage $H^{-}$ injector into
the Main Injector as a replacement for the aging 400-MeV Linac and 8-GeV
Booster. This would be the highest energy $H^{-}$ multiturn injection system
in the world. Fermilab has been carrying out design studies [4, 6] of the SC
linac and injection systems[7] for the last several years. The linac design[8]
utilizes a warm-temperature 325-MHz RFQ and rebuncher cavities to bunch the
beam at 325 MHz. At $\beta=0.89$ (about $E_{kin}=1.1~{}GeV$), the RF of the SC
cavities is 1.3 GHz. The ultimate bunch structure required for injection into
the MI will be formed by a 325-MHz fast chopper system[9]. The fast chopper
system will be required to remove individual 325-MHz bunches or groups of
bunches to be matched to the MI RF structure and provide a beam abort notch:
two out of every six microbunches333 The microbunch is referred to as a
325-MHz bunch hereafter. are to be removed.

###  Time Structure of the Main Injector

In the injection modeling for the MI with the aid of ESME[10], a train of four
microbunches, produced by the fast chopper system located at the front end of
the 8-GeV SC RF linac, are injected into the upgraded Main Injector. Figure 1
is a schematic illustrating one MI RF bucket that is populated with an initial
train of four microbunches. The two chopped microbunches are represented by
two consecutive empty 325-MHz RF buckets.

Figure 1:  Three consecutive MI RF buckets of 53 MHz. A train of four
microbunches of 325 MHz are synchronously injected into each standing 53-MHz
bucket. Two chopped microbunches are equivalent to 6 ns.

More details of the time structure of the MI RF bucket after the first
synchronous injection from a SC linac are illustrated in Figure 2.

Figure 2:  The time structure of a single RF system of an upgraded Main
Injector: principal RF harmonic of 52.8 (MHz), sub-harmonic of 325 (MHz), and
a beam abort notch of 6 ns Figure 3:  Single RF system: the injection of 4
microbunches at the $1^{st}$ turn with RF waveform drawn in the background

A total beam notch per MI RF bucket is about 6 ns, which corresponds to two
325-MHz RF buckets. Figure 3 is a phase-space ($\Delta E,\theta$) plot of the
very first train of four microbunches with a RF-voltage waveform drawn in the
background. The following is a list of longitudinal parameters that can be
found in the header of each ESME phase-space plot in this memo.

iter (number of turns), $H_{B}$ (bucket height), $S_{B}$ (bucket area),
$S_{b}$ (bunch area), $V$ (RF voltage), $E_{s}$ (synchronous energy),
$\nu_{s}$ (synchrotron tune), pdot ($dp/dt$), $\eta$ (slip factor), $\tau_{s}$
(revolution period), h (harmonic number), $\Psi$ (synchronous phase), and N
(number of macroparticles)

The horizontal axis is $\Delta\theta$ ($=\theta-\theta_{s}$) in units of
degrees with one full revolution representing 360 (deg), the vertical axis on
the left is $\Delta E$ ($=E-E_{s}$) in units of MeV, and the vertical axis on
the right is $V_{rf}$ in units of kV for the RF waveform.

The azimuthal-density444 The azimuthal density is referred to as the charge
density, or the azimuthal profile of macroparticles. and energy-density
distribution555 The energy-density distribution is referred to as the profile
of macroparticles in energy ($\Delta E$) direction. of one of four
microbunches are plotted in Figures 4 and 5, respectively. The root-mean-
square (RMS) width of one microbunch is 1.96 $\times 10^{-4}$ (deg), which
corresponds to 6.05 (ps). The red area indicates the tail portion of a
microbunch. The fractional tail portion is about 1.34 $\%$. To make the tail
portion of a microbunch stand out, the density distribution is also plotted on
a logarithmic scale. The RMS value of initial energy spread is about 0.26
(MeV) per each microbunch. Regarding the MI to be a 11.1338-$\mu$s, or
360-degree ring, we used the following conversion factor:

$0.032335~{}(deg/ns)\vspace{-0.2cm}$ (1)

Figure 4:  Zero-centered charge density with tail portion; (left) on a linear
scale (right) on a logarithmic scale Figure 5:  Zero-centered energy density;
(left) on a linear scale (right) on a logarithmic scale

The principal MI RF is 53 (MHz) and the SC-linac bunching frequency is 325
(MHz). From the MI RF, we can obtain an integer harmonic number of 588. The
width of a MI RF bucket is 18.935 (ns), into which trains of microbunches are
repeatedly injected over 270 turns, filling in a MI RF bucket in the region of
$\pm$ 6 (ns) around the center of a MI RF bucket. The beam notch that is kept
free of beams is 3.3 (ns) long, following an earlier injection study with a
long bunch[11]. In the following sections, we will explore and discuss several
microbunch-injection scenarios, based upon the RF parameters used in the
present MI operation.

###  Microbunch-Injection Scenarios for the Main Injector

The goals of the modeling of the time-structured multiturn injection into
FMI-2 with a SC linac are threefold:

  1. (1)

To find the optimized Fermilab Main Injector RF system with the following
aspects:

    1. (i)

Efficient RF capture methods with minimum particle losses

    2. (ii)

Main Injector RF system with adequate RF parameters

    3. (iii)

Minimization of space-charge effects in all degrees of freedom

    4. (iv)

Optimization of total capture time and total injection time

  2. (2)

To design a fast beam-chopper system to make nearly loss-free injection
attainable

  3. (3)

To investigate any other limitations on the intensity upgrades from the
present Main Injector

With the above goals in mind, we modeled four different scenarios of
synchronous injection over 270 injection turns, but with different RF harmonic
systems and RF parameters. The following is the parameters of the Main
Injector that are consistently employed in ESME simulations for all scenarios
presented herein.

Table 1: Main Injector Parameters for ESME Simulations Mean Radius | 528.297 (m)
---|---
Beam Momentum (at injection) | 8.889 ($GeV/c$)
$\gamma_{tr}^{2}$ (transition gamma) | 466.53572
Principal Harmonic No. | 588
$\Phi_{s}$ (synchronous phase) | 0.0 (deg)
effective beam radius (at injection) | 0.0050 (m)
effective beam-pipe radius (at injection) | 0.051 (m)
No. of Space-Charge Bins | 64
No. of FFT Bins | 64
Microbunch Intensity | 2.65 $\times 10^{8}$
Total Intensity (after injection is complete) | 1.54 $\times 10^{14}$

SCENARIO (I)

[t]synchronous injection of a train of 4 microbunches into a standing RF
bucket single RF Harmonic (53 MHz) $V_{rf}$ = 800 (kV) (fixed RF voltage with
no ramping) 270 injection turns ($\sim$ 3 ms) $\rightarrow$ 2,700 turns
($\sim$ 30 ms) longitudinal space charge included longitudinal painting in RF-
phase ($\theta$) direction

We begin with a single RF system of 53 (MHz) and the fixed RF voltage
($V_{rf}$) of 800 (kV). Referring to Figure 3, a series of four blobs captured
in a RF bucket represent the first train of microbunches that are
synchronously injected into a standing RF bucket. The dotted sinusoidal curve
for RF voltage waveform in the background of the phase-space figure indicates
the single RF harmonic and the amplitude of RF voltage.

The single turn in the MI in terms of the average machine circumference and
the revolution period can be expressed as follows:

$360~{}(deg)=11.1338~{}(\mu s)=3319.388~{}(m)$ (2)

For one MI RF bucket, the following Eqn. (3) is obtained from Eqn. (2) as
illustrated in Figure 2.

$52.8114~{}(MHz)=18.9353~{}(ns)=0.6122~{}(deg)=5.645~{}(m)$ (3)

Due to the longitudinal mismatch, or RF mismatch between the SC linac and the
FMI-2, an inherent phase slip between 325-MHz microbunch structure and the MI
stationary buckets could be parasitically induced in the form of longitudinal
painting in RF-phase direction. With $f$ and $\lambda$ being radio frequency
and RF wavelength, respectively, we computed the RF ratio
($\mathcal{R}_{rf}$).

$\displaystyle\mathcal{R}_{rf}$
$\displaystyle=\frac{f_{PD}}{f_{MI}}=\frac{325~{}\text{MHz}}{52.8114~{}\text{MHz}}$
(4)
$\displaystyle=\frac{\lambda_{MI}}{\lambda_{PD}}=\frac{5.6453~{}(m)}{0.9173~{}(m)}$
$\displaystyle=6.154$

Referring to Figure 2, Eqn. (4) implies that a total of 6 linac RF buckets of
325 MHz can fit in one MI RF bucket. Since the RF ratio is a non-integer
number, a modulus of the RF ratio is computed as for the case of the Booster
injection modeling[1].

$\mathcal{R}_{rf}\Bigr{|}_{mod}=0.154$ (5)

One can then arrive at the range of phase slip within one MI RF bucket.

$0.155\times 0.0995~{}(deg)=0.0154~{}(deg)$ (6)

Figure 6 shows trains of microbunches and discrete charge distribution after 3
injection turns with phase slips included. For each successive MI RF bucket,
the 325-MHz bunches advance by 0.0154 (deg). Hence, it is estimated that every
6 turns the phase of injected microbunches slips through one 325-MHz bucket.

$\vspace{0.3cm}\delta\theta_{rf}=\frac{0.0995~{}(deg)}{0.0154~{}(deg)}=6.46$
(7)

Having this amount of phase slip included, we simulated the SC linac-to-MI
injection with ESME over 270 injection turns. After the 270 turns, the
longitudinal phase space is painted in RF-phase space as illustrated by Figure
7. As in Fiugre 8, one-peaked but continuous distribution of charge density666
Charge density, or azimuthal density is referred to as charge line density,
for convenience. at the 270-th turn is shown. The evolution of longitudinal
emittance growth for the case of a single RF harmonic is shown in Figure 9. As
the number of macroparticles increases, so does the longitudinal emittance up
to about 0.08 (eV-s). Then, the emittance flattens out until the end of the
simulation run. The induced voltage ($\mathcal{V}_{sc}$) due to space-charge
fields ($\mathcal{E}(z)$) is proportional to the g-factor, which is usually
defined as $1+2\ln(R_{w}/R_{b})$. Hence, from the charge distribution, space-
charge-induced voltage per turn is computed at a specified turn. In
particular, we looked at the induced voltage at the last injection turn of
270-th turn as shown in Figure 11.

$\mathcal{V}_{sc}\propto\frac{1}{\beta\gamma^{2}}\Bigl{[}1+2\ln(R_{w}/R_{b})\Bigr{]}\frac{d\lambda}{dz}$
(8)

where $R_{w}$, and $R_{b}$ denote effective beam pipe radius, and effective
beam radius, respectively. In the case of the Main Injector at injection
energy of 8.0 GeV, the g-factor is about 5.64. In Figure 12, the peak
collective voltage in frequency domain ($\hat{V}_{FD}$) is drawn at each turn.
The collective voltage goes up to about 50.0 (keV) at the end of the
injection.

(a) phase space

(b) charge density

Figure 6: [Scenario (I)] Phase space and discrete charge density after 3
injection turns

(a) 270 turns

(b) 2,700 turns

Figure 7: [Scenario (I)] Synchronous injection of microbunches with a single
RF system (a) after 270 turns (b) after 2,700 turns; Note that LSC in the
figure title stands for Longitudinal Space Charge.

(a) 270 turns

(b) 2,700 turns

Figure 8:  [Scenario (I)] Distribution of charge density with single RF
harmonic at the 270-th turn and at the 2,700-th turn.

Figure 9:  [Scenario (I)] Evolution of longitudinal emittance with a single RF
harmonic over 270 turns Figure 10:  [Scenario (I)] Evolution of the number of
injected macroparticles with a single RF harmonic over 270 turns

Figure 11:  [Scenario (I)] Induced $\Delta E$ per turn due to space charge
($\Delta E/turn$) with a single RF harmonic at $270^{th}$ turn Figure 12:
[Scenario (I)] Time evolution of collective voltage in frequency domain over
270 turns

Figure 13:  [Scenario (I)] Time evolution of bunching factor

As the bunching factor is usually defined as average current
($\langle~{}I~{}\rangle$) over peak current ($\hat{I}$) for convenience, the
value is upper-bounded at unity. Figure 13 shows that the bunching factor
converges to about 0.3 for the case of Scenario (I). SCENARIO (II)

In Scenario (II), the injection process described in Scenario (I) is followed
by ramping up the RF voltage linearly up to 150 $\%$ of the initial RF
voltage. Thus, the RF voltage ramping lasts for additional 27 (ms), and the
total injection time elapses about 30 (ms).

[t]$V_{rf,~{}i}$ = 800 (kV), $V_{rf,~{}f}$ = 1,200 (kV) ramping RF voltage
after the injection is complete

At the end of RF voltage ramping, the macroparticles are well captured in a RF
bucket as shown in Figure 14. Figure 15 shows the variation of RF voltage from
800 (kV) to 1,200 (kV) starting from the 270-th turn through the 2,700-th
turn. In comparison to Figure 8 from Scenario (I), the gradient of charge
distribution ($d\lambda/dz$) are reduced and spread more out with ramping RF
voltage over extended 2,700 turns (cf. Figure 16). However, it still shows a
peaked distribution centered around the origin. The longitudinal emittance
($\varepsilon_{l}$) grows up to 0.085 (eV-s) with a fixed RF voltage at 800.
(kV). Once the RF voltage starts ramping linearly, the emittance continues to
grow gradually up to 0.09 (eV-s) as shown in Figure 17.

In an attempt to reduce further the space-charge-induced voltage and to
produce a more uniform charge distribution, a dual RF harmonic system is
explored in Scenarios (III) and (IV) in the following subsections.

Figure 14:  [Scenario (II)] Synchronous injection of microbunches with a
single RF harmonic and ramping RF voltage Figure 15:  [Scenario (II)] RF
voltage curve over 2,700 turns

Figure 16:  [Scenario (II)] Distribution of charge density with a single RF
harmonic and ramping RF voltage after 2,700 turns Figure 17:  [Scenario (II)]
Time evolution of longitudinal emittance ($\epsilon_{l}$) with a single RF
system

SCENARIO (III)

[t]Dual RF Harmonics:
$f_{rf,~{}1}$ = 53 MHz and $f_{rf,~{}2}$ = 106 MHz $H_{1}~{}=~{}588$ and
$H_{2}~{}=~{}1176$
$R_{H}$ = $H_{2}/H_{1}$ = 2.0 $V_{rf,~{}1}~{}=~{}400.~{}(kV)$ and
$V_{rf,~{}2}~{}=~{}300.~{}(kV)$ (fixed RF voltages)
$R_{H}$ = $V_{rf,~{}2}/V_{rf,~{}1}$ = 0.75

In scenario (III), a dual RF harmonic system of 400 (kV) on 53 (MHz) and 300
(kV) on 106 (MHz) is explored. The higher-order RF harmonic ($H_{2}=1176$) is
twice of the principal RF harmonic ($H_{1}=588$). The secondary RF voltage
($V_{rf,~{}2}$) is 75 $\%$ of the principal RF voltage ($V_{rf,~{}1}$). The RF
voltage waveform of the higher harmonic for a dual RF system can be

$\mathcal{V}=\mathcal{V}_{rf,~{}1}\sin(H_{1}\phi_{1}+\psi_{1})+\mathcal{V}_{rf,~{}2}\sin(H_{2}\phi_{2}+\psi_{2}),$
(9)

where $\mathcal{V}_{rf}$, $H$, $\phi$, and $\psi$ are RF voltage, harmonic
number, phase angle of each macroparticle, and phase of RF cavity,
respectively. As can be seen in Figure 18, the waveform has a negative slope
around the stable phase of 0 (deg). By adding a higher secondary harmonic RF
voltage to a principal harmonic RF voltage, we can create a flat-bottom
potential energy. As a result, the RF-bucket contour with a dual RF system is
also flattened at the top and bottom on a phase-space plot. Hence, the charge
distribution follows the flat contour shape. Also, the principal RF voltage
can be half as high as the RF voltage used for the case of single RF harmonic.
Since the flat-bottom potential energy irons out the peaked charge
distribution, the utilization of a dual RF system gives a great advantage to
the single RF system. Eventually, the dual RF system will help us lower the
beam-current limitations caused by space-charge effects. After 2,700 turns,
the formation of localized macroparticle distributions are observed around
three local bumps of the dual-harmonic voltage waveform. Figure 20 shows the
time evolution of injected macroparticles inside a dual RF bucket. From turn 1
through turn 270, a train of four microbunches are injected with parasitic
phase offsets. After the injection process is complete, in order to reach an
equilibrium state, injected macroparticles are circulated up to 2,700 turns
with no further injection. Note that as time elapses after the completion of
injection process, sporadic white gaps in between lumps of macroparticles are
gradually disappeared. In Figures from 22 through 24 three plots of
longitudinal phase space, charge density, and energy density comprise each row
corresponding to turn number under a phase-space plot. With the choice of RF
voltages $R_{V}=0.75$, a bi-modal charge distribution is created due to a pair
of potential wells around the stable phase of 0 (deg). However, with the help
of longitudinal painting, the contour of charge distribution becomes smoother
as the turn number increases, but the dual peaks remain in the charge
distribution.

Figure 18:  [Scenario (III)] RF bucket and dual RF waveform in the background

(a) after 270 turns

(b) after 2,700 turns

Figure 19:  [Scenario (III)] Distribution of macroparticles with longitudinal
painting

(a) after 270 turns

(b) after 2,700 turns

Figure 20:  [Scenario (II)] Distribution of charge density with longitudinal
painting included

(a) $1^{st}$ turn

(b) $100^{th}$ turn

(c) $200^{th}$ turn

(d) $270^{th}turn$

(e) $600^{th}$ turn

(f) $900^{th}$ turn

(g) $1200^{th}turn$

(h) $1500^{th}$ turn

(i) $1800^{th}$ turn

(j) $2100^{th}turn$

(k) $2400^{th}$ turn

(l) $2700^{th}$ turn

Figure 21: [Scenario (III)] Time evolution of phase space with longitudinal
painting included, starting from the $1^{st}$ injection turn through 2700
turns

(a) $100^{th}$ turn

(b) charge density

(c) energy density

(d) $200^{th}$ turn

(e) charge density

(f) energy density

(g) $270^{th}$ turn

(h) charge density

(i) energy density

(j) $300^{th}$ turn

(k) charge density

(l) energy density

Figure 22: [Scenario (III)] Time evolution of phase space with longitudinal
painting starting from $100^{th}$ turn through $300^{th}$ turns

(a) $600^{th}$ turn

(b) charge density

(c) energy density

(d) $900^{th}$ turn

(e) charge density

(f) energy density

(g) $1200^{th}$ turn

(h) charge density

(i) energy density

(j) $1500^{th}$ turn

(k) charge density

(l) energy density

Figure 23: [Scenario (III)] Time evolution of phase space with longitudinal
painting starting from $600^{th}$ turn through $1500^{th}$ turns

(a) $1800^{th}$ turn

(b) charge density

(c) energy density

(d) $2100^{th}$ turn

(e) charge density

(f) energy density

(g) $2400^{th}$ turn

(h) charge density

(i) energy density

(j) $2700^{th}$ turn

(k) charge density

(l) energy density

Figure 24: [Scenario (III)] Time evolution of phase space with longitudinal
painting starting from $1800^{th}$ turn through $2700^{th}$ turns

SCENARIO (IV)

[t]Dual RF Harmonics:
$f_{rf,~{}1}~{}=~{}53~{}MHz$ and $f_{rf,~{}2}=106~{}MHz$
$H_{2}=1176$ and $H_{1}=588$
$R_{H}=f_{rf,~{}2}/f_{rf,~{}1}=2.0$ $V_{rf,~{}1}$ = 400 (kV) and $V_{rf,~{}2}$
= 200 (kV) (fixed RF voltages)

Scenario (IV) is under the same conditions as in scenario (III), except that
the secondary RF voltage is 50 $\%$ of the principal RF voltage. It is more
advantageous in that the RF waveform has a nice plateau around the stable
phase of 0 (deg) as shown in Figure 25. Through the longitudinal painting at
each turn, a total of 1080 (270 $\times$ 4) microbunches that are injected
over 270 turns and further circulated up to 30 (ms) gradually transform into a
continuous macrobunch, or long bunch spanning in between -0.2 (deg) and +0.2
(deg). Unlike in Figure 20 (a), localization of macroparticles is not observed
in Figure 27 (a) after the injection of 270 turns is complete. After further
circulation of beams up to 2,700 turns with no further injection, the beam
charge distribution becomes smoother in the region of -0.2 (deg) and +0.2
(deg). (see Figures 1.19 and 1.20). As a consequence, the bi-modal
distribution of charge density observed from the outcome of Scenario (III) is
not observed as shown in Figures 22 through 24. As the number of injected
macroparticles increases, longitudinal painting progresses, and time elapses,
the fine structure of charge distribution gradually disappears. Eventually,
after the 2,700 turns, the contour of charge distribution becomes smoother.
Including phase offsets without energy jitter, this effect stands out in
charge distribution, rather than energy distribution as observed in Scenario
(III).

Figure 25:  [Scenario (IV)] RF waveform for a dual RF system with the RF
voltage ratio of 0.5

(a) at the 270-th turn

(b) at the 2,700-th turn

Figure 26:  [Scenario (IV)] Synchronous injection of microbunches with a dual
RF harmonic (a) at $270^{th}$ turns and (b) at $2700^{th}$ turns

(a) at the 270-th turn

(b) at the 2,700-th turn

Figure 27:  [Scenario (IV)] Distribution of charge density with a dual RF
harmonic (a) at $270^{th}$ turns and (b) at $2700^{th}$ turns

(a) $1^{st}$ turn

(b) $100^{th}$ turn

(c) $200^{th}$ turn

(d) $270^{th}turn$

(e) $600^{th}$ turn

(f) $900^{th}$ turn

(g) $1200^{th}turn$

(h) $1500^{th}$ turn

(i) $1800^{th}$ turn

(j) $2100^{th}turn$

(k) $2400^{th}$ turn

(l) $2700^{th}$ turn

Figure 28: [Scenario (IV)] Time evolution of phase space with longitudinal
painting included, starting from the $1^{st}$ injection turn through 2700
turns

(a) $100^{th}$ turn

(b) charge density

(c) energy density

(d) $200^{th}$ turn

(e) charge density

(f) energy density

(g) $270^{th}$ turn

(h) charge density

(i) energy density

(j) $300^{th}$ turn

(k) charge density

(l) energy density

Figure 29: [Scenario (IV)] Time evolution of phase space with longitudinal
painting starting from $100^{th}$ turn through $300^{th}$ turns

(a) $600^{th}$ turn

(b) charge density

(c) energy density

(d) $900^{th}$ turn

(e) charge density

(f) energy density

(g) $1200^{th}$ turn

(h) charge density

(i) energy density

(j) $1500^{th}$ turn

(k) charge density

(l) energy density

Figure 30: [Scenario (IV)] Time evolution of phase space with longitudinal
painting starting from $600^{th}$ turn through $1500^{th}$ turns

(a) $1800^{th}$ turn

(b) charge density

(c) energy density

(d) $2100^{th}$ turn

(e) charge density

(f) energy density

(g) $2400^{th}$ turn

(h) charge density

(i) energy density

(j) $2700^{th}$ turn

(k) charge density

(l) energy density

Figure 31: [Scenario (IV)] Time evolution of phase space with longitudinal
painting starting from $1,800^{th}$ turn through $2,700^{th}$ turns

As Figure 32 shows, the encouraging result is obtained with the dual RF
harmonic system. The longitudinal emittance grows about 50 $\%$ less than in
the cases of a single RF harmonic system. The peak induced voltage
($\hat{\mathstrut\mathcal{V}}_{sc}$) due to space charge is around 40 (kV) per
turn as in Figure 33. Figure 34 shows the evolution of peak voltage induced by
space charge. By using the phase modulation in a controlled fashion, which
will allow us to maneuver charge distribution, we will be able to lower space-
charge voltage further. The bunching factor calculated at each turn for
Scenario (IV) turns out to be close to that of Scenario (I) with a single RF
harmonic as in Figure 35. It should be noted that the bunching factor
calculations are important in that it serves as an indicator of how large tune
spreads will be prior to 3-D space-charge calculations.

Figure 32:  [Scenario (IV)] The growth of longitudinal emittance with a dual
RF system over 2,700 turns

Figure 33:  [Scenario (IV)] Additional $\Delta E$ induced by longitudinal
space charge with a dual RF harmonics after 2,700 turns Figure 34:  [Scenario
(IV)] Evolution of peak voltage in frequency domain with a dual RF harmonics
over 2,700 turns Figure 35:  [Scenario (IV)] Evolution of bunching factor with
a dual RF system

## Concluding Remarks

We have investigated different scenarios of microbunch injection-methods
between a super-conducting linac and the MI ring under the influence of
longitudinal space charge: from the 8-GeV linac proton driver to the Main
Injector

The RF mismatch between a linac and a ring can induce phase shifts with trains
of microbunches, which serve as longitudinal painting in a parasitic fashion.
Hence, it would be rather advantageous to use harmonics of non-integer ratio
between a linac and a ring in order to induce longitudinal painting. Besides,
subsequent charge redistribution in longitudinal direction can be achieved
through additional beam circulation with no further beam injection. Since the
roaming of charge in longitudinal phase space can reduce the gradient of
charge distribution, induced voltage due to space charge can be reduced
accordingly. In addition to painting in phase, future simulations are planned
to include both phase and energy jitters due to errors in the SC RF linac.
Because of the short bunch length of the linac beam, it is anticipated that
the impact of the broad-band impedance may play an important role in the
longitudinal dynamics[13]. An optimized dual RF system and longitudinal
painting can overcome beam-intensity limitations induced by space charge in
high-intensity machines. Four injection scenarios manifest that a double RF
system with the harmonic ratio ($\mathcal{R}_{H}=1176/588$) of 2.0 and the
voltage ratio ($\mathcal{R}_{V}=200kV/400kV$) of 0.5 are most favored. All of
the scenarios for the time-structured multiturn injection including animated
simulation results are available on a Fermilab web site [14].

## Acknowledgement

Authors wish to thank J. Maclachlan of the Proton Source Department and J-P.
Carneiro of Accelerator Physics Center (APC) at Fermilab. Authors had useful
discussions with J. Maclachlan for MI injection modeling with macrobunches at
an early stage. J-P. Carneiro provided us with an input file of the 8-GeV SRF
linac distribution that were generated by the TRACK code[8].

## Bibliography

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Fermilab-Pub-07-672-DO, Dec. 2007

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