# A multifractal formalism for Hewitt-Stromberg measures

Najmeddine Attia, Bilel Selmi
Analysis, Probability and Fractals Laboratory LR18ES17
Department of Mathematics, Faculty of Sciences of Monastir, University of
Monastir, 5000-Monastir, Tunisia<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In the present work, we give a new multifractal formalism for which the
classical multifractal formalism does not hold. We precisely introduce and
study a multifractal formalism based on the Hewitt-Stromberg measures and that
this formalism is completely parallel to Olsen’s multifractal formalism which
based on the Hausdorff and packing measures.

###### Key words and phrases:

Multifractal analysis, multifractal formalism, multifractal Hausdorff measure,
multifractal packing measure, Hewitt-Stromberg measures, Hausdorff dimension,
packing dimension, doubling measures, inhomogeneous multinomial measures,
Moran measures.

###### 2000 Mathematics Subject Classification:

28A78, 28A80.

##  1\. Introduction

In certain circumstances, a measure $\mu$ gives rise to sets of points where
$\mu$ has local density of exponent $\alpha$. The dimensions of these sets
indicate the distribution of the singularities of the measure. To be more
precise, for a finite measure $\mu$ on $\mathbb{R}^{n}$, the pointwise
dimension at $x$ is defined as follows

${\alpha}_{\mu}(x)=\lim_{r\to 0}\frac{\log\mu(B(x,r))}{\log r},$

whenever this limit exists. For $\alpha\geq 0$, define

$E(\alpha)=\Big{\\{}x\in\operatorname{supp}\mu\;\big{|}\;\quad{\alpha}_{\mu}(x)=\alpha\Big{\\}}$

where $B(x,r)$ is the closed ball with center $x$ and radius $r$. The set
$E(\alpha)$ may be thought of as the set where the local dimension of $\mu$
equals $\alpha$ or as a multifractal component of $\operatorname{supp}\mu$.
The main problem in multifractal analysis is to estimate the size of
$E(\alpha)$. This is done by calculating the functions

$f_{\mu}(\alpha)=\dim_{H}(E(\alpha))\quad\text{ and}\quad
F_{\mu}(\alpha)=\dim_{P}(E(\alpha))\;\;\text{ for}\;\;\alpha\geq 0.$

These functions are generally known as the multifractal spectrum of $\mu$ or
the singularity spectrum  of the measure $\mu$. One of the main problems in
multifractal analysis is to understand the multifractal spectrum and the Rényi
dimensions and their relationship with each other. During the past 25 years
there has been an enormous interest in computing the multifractal spectra of
measures in the mathematical literature. Particularly, the multifractal
spectra of various classes of measures in Euclidean space $\mathbb{R}^{n}$
exhibiting some degree of self-similarity have been computed rigorously. The
reader can be referred to the paper [42], the textbooks [25, 54] and the
references therein. Some heuristic arguments using techniques of Statistical
Mechanics (see [32]) show that the singularity spectrum should be finite on a
compact interval, noted by $\text{Dom}(\mu)$, and is expected to be the
Legendre transform conjugate of the $L^{q}$-spectrum, given by

$\tau_{\mu}(q)=\displaystyle\lim_{r\to
0}\frac{\log\left(\sup\left\\{\displaystyle\sum_{i}\mu(B(x_{i},r))^{q}\right\\}\right)}{-\log
r}$

where the supremum is taken over all centered packing
$\big{(}B(x_{i},r)\big{)}_{i}$ of $\operatorname{supp}\mu$. That is, for all
$\alpha\in\text{Dom}(\mu)$,

$\displaystyle f_{\mu}(\alpha)=\inf_{q\in\mathbb{R}}\Big{\\{}\alpha
q+\tau_{\mu}(q)\Big{\\}}=\tau_{\mu}^{*}(\alpha).$ (1.1)

The multifractal formalism (1.1) has been proved rigorously for random and
non-random self-similar measures [1, 16, 42, 43, 51], for self-conformal
measures [26, 27, 28, 29, 37, 52], for self-affine measures [6, 7, 8, 9, 23,
24, 36, 45] and for Moran measures [61, 62, 63, 64]. We note that the proofs
of the multifractal formalism (1.1) in the above-mentioned references [1, 10,
12, 13, 14, 36, 37, 42, 43, 45, 52] are all based on the same key idea. The
upper bound for $f_{\mu}(\alpha)$ is obtained by a standard covering argument
(involving Besicovitch’s Covering Theorem or Vitali’s Covering Theorem).
However, its lower bound is usually much harder to prove and is related to the
existence of an auxiliary measure (Gibbs measures) which is supported by the
set to be analysed. In an attempt to develop a general theoretical framework
for studying the multifractal structure of arbitrary measures, Olsen [42],
Pesin [53] and Peyrière [55] suggested various ways of defining an auxiliary
measure in a very general setting. This formalism was motivated by Olsen’s
wish to provide a general mathematical setting for the ideas presented by the
physicists Halsey et al. in their seminal paper [32]. In fact, they have been
interested in the concept of multifractal spectrum, that is an interesting
geometric characteristic for discrete and continuous models of statistical
physics. An important thing which should be noted is that there are many
measures for which the multifractal formalism does not hold (some examples
could be found in [11, 13, 42, 64]). An imported question, in which several
theorists are interested, is: can we find a necessary and sufficient condition
for the multifractal formalism to hold? Another one, asked by Olsen in [42]
is: which functions give more information about a multifractal measure, the
dimension functions $b_{\mu}$ and $B_{\mu}$ or the spectra functions $f_{\mu}$
and $F_{\mu}$? Olsen gives examples of measures where the dimension functions
can be used to split measures which have the same spectrum. In doing this, he
implicitly suggests that a return to the physicists’ original idea of
calculating the moments of multifractal measures may be the best way to
characterize them. It always needs some extra conditions to obtain a
minoration for the dimensions of the level sets $E(\alpha)$. Olsen proved the
following statement.

######  Theorem 1.

[42] Let $\mu$ be a Borel probability measure on $\mathbb{R}^{n}$. Define
$\underline{\alpha}=\displaystyle\sup_{0<q}-\frac{{{b}_{\mu}(q)}}{q}$ and
$\overline{\alpha}=\displaystyle\inf_{0>q}-\frac{{{b}_{\mu}(q)}}{q}$. Then,

$\dim_{H}(E(\alpha))\leq{b}_{\mu}^{*}(\alpha)\quad\text{and}\quad\dim_{P}(E(\alpha))\leq{B}_{\mu}^{*}(\alpha)\quad\text{for
all}\quad\alpha\in(\underline{\alpha},\;\overline{\alpha}).$

In general, such a minoration is related to the existence of an auxiliary
measure which is supported by the set to be analyzed. Olsen also gives a
result in such a way and supposes the existence of a Gibbs’ measure (see [42])
at a state $q$ for the measure $\mu$, i.e., the existence of a measure
$\nu_{q}$ on $\operatorname{supp}\mu$ and constants $C>0$, $\delta>0$ such
that for every $x\in\operatorname{supp}\mu$ and every $0<r<\delta$,

$\frac{1}{C}~{}\mu(B(x,r))^{q}~{}(2r)^{{B}_{\mu}(q)}\leq\nu_{q}(B(x,r))\leq
C~{}\mu(B(x,r))^{q}~{}(2r)^{{B}_{\mu}(q)}$

to conclude that

$\dim_{H}(E(\alpha))=\dim_{P}(E(\alpha))={b}_{\mu}^{*}(\alpha)={B}_{\mu}^{*}(\alpha),\quad\text{where}\quad\alpha=-{B}_{\mu}^{\prime}(q).$

In general, one needs some degree of similarity to prove the existence of
Gibbs measures. For example, in dynamic contexts, the existence of such
measures are often natural. For this reason, Ben Nasr et al. in [10, 11, 12,
13] improved Olsen’s result and proposed a new sufficient condition that gives
the lower bound. For more details and backgrounds on multifractal analysis as
well as their applications the readers may be referred also to the following
essential references [4, 5, 15, 17, 18, 19, 39, 41, 44, 47, 56, 57, 58, 59,
60, 61, 62, 63, 64].

In [11, 13, 42, 56, 64], the authors provided some examples for which the
classical multifractal formalism does not hold. Indeed, for such examples, the
functions $b_{\mu}$ and $B_{\mu}$ differ and $\dim_{H}(E(\alpha))$ and
$\dim_{P}(E(\alpha))$ are given respectively by the Legendre transform of
$b_{\mu}$ and $B_{\mu}$. Motivated by the above papers, the authors in [3]
introduced new metric outer measures (multifractal analogues of the Hewitt-
Stromberg measure) $\mathsf{H}^{q,t}_{\mu}$ and $\mathsf{P}^{q,t}_{\mu}$ lying
between the multifractal Hausdorff measure ${\mathcal{H}}^{q,t}_{\mu}$ and the
multifractal packing measure ${\mathcal{P}}^{q,t}_{\mu}$, and they used the
multifractal density theorems to prove the decomposition theorem for the
regularities of these measures. In the present paper, we give a new
multifractal formalism for which the functions $b_{\mu}$ and $B_{\mu}$ differ.
Actually, the main aim of this work is to introduce and study a multifractal
formalism based on the Hewitt-Stromberg measures. However, we point out that
this formalism is completely parallel to Olsen’s multifractal formalism
introduced in [42] which based on the Hausdorff and packing measures. Then, we
prove that the lower and upper multifractal Hewitt-Stromberg functions
$\mathsf{b}_{\mu}$ and $\mathsf{B}_{\mu}$ are intimately related to the
spectra functions. More precisely, we have

$\mathsf{f}_{\mu}(\alpha):=\underline{\dim}_{MB}(E(\alpha))\leq{\mathsf{b}}_{\mu}^{*}(\alpha)\quad\text{and}\quad\mathsf{F}_{\mu}(\alpha):=\overline{\dim}_{MB}(E(\alpha))\leq{\mathsf{B}}_{\mu}^{*}(\alpha)\quad\text{for
some}\quad\alpha\geq 0.$

Here $\underline{\dim}_{MB}$ and $\overline{\dim}_{MB}$ denote, respectively,
the lower and the upper Hewitt-Stromberg dimension (the lower and the upper
modified box-counting dimension), see Section 2.2 for precise definitions of
this. One of our purposes of this paper is to show the following result: if
$\mathsf{H}_{\mu}^{q,{\mathsf{b}}_{\mu}(q)}(E(-{\mathsf{b}_{\mu}}^{\prime}(q)))>0$,
then

$\underline{\dim}_{MB}\left(E\big{(}-{\mathsf{b}}_{\mu}^{{}^{\prime}}(q)\big{)}\right)={\mathsf{b}}_{\mu}^{*}\big{(}-{\mathsf{b}}_{\mu}^{{}^{\prime}}(q)\big{)}$

and, if
$\mathsf{P}_{\mu}^{q,{\mathsf{B}}_{\mu}(q)}(E(-{\mathsf{B}_{\mu}}^{\prime}(q)))>0$,
then

$\overline{\dim}_{MB}\left(E\big{(}-{\mathsf{B}}_{\mu}^{{}^{\prime}}(q)\big{)}\right)={\mathsf{B}}_{\mu}^{*}\big{(}-{\mathsf{B}}_{\mu}^{{}^{\prime}}(q)\big{)}.$

Moreover, we describe a sufficient condition leading to the equalities

$\mathsf{f}_{\mu}(\alpha)=\mathsf{F}_{\mu}(\alpha)={F}_{\mu}(\alpha)\quad\text{for
some}\quad\alpha\geq 0.$

Specifically, if we assume that
$\mathsf{H}^{q,\mathsf{B}_{\mu}(q)}_{\mu}(\operatorname{supp}\mu)>0$, then

$\underline{\dim}_{MB}\left(E\big{(}-{\mathsf{B}}_{\mu}^{{}^{\prime}}(q)\big{)}\right)=\overline{\dim}_{MB}\left(E\big{(}-{\mathsf{B}}_{\mu}^{{}^{\prime}}(q)\big{)}\right)=\mathsf{b}^{*}_{\mu}(-{\mathsf{B}}_{\mu}^{{}^{\prime}}(q))=\mathsf{B}^{*}_{\mu}(-{\mathsf{B}}_{\mu}^{{}^{\prime}}(q)).$

We also observe that this sufficient condition is very close to being a
necessary and sufficient one, see Theorem 7. In particular, we deal with the
case where the lower and upper multifractal Hewitt-Stromberg functions
$\mathsf{b}_{\mu}$ and $\mathsf{B}_{\mu}$ do not necessarily coincide, see
Theorem 8.

We will now give a brief description of the organization of the paper. In the
next section we recall the definitions of the various fractal and multifractal
dimensions and measures investigated in the paper. The definitions of the
Hausdorff and packing measures and the Hausdorff and packing dimensions are
recalled in Section 2.1, and the definitions of the Hewitt-Stromberg measures
are recalled in Section 2.2, while the definitions of the Hausdorff and
packing measures are well-known, we have, nevertheless, decided to include
these-there are two main reasons for this: firstly, to make it easier for the
reader to compare and contrast the Hausdorff and packing measures with the
less well-known Hewitt-Stromberg measures, and secondly, to provide a
motivation for the Hewitt-Stromberg measures. Section 2.3 recalls the
multifractal formalism introduced in [42]. In Section 2.4 we recall the
definitions of the multifractal Hewitt-Stromberg measures and separator
functions, and study their properties. Section 2.5 recalls earlier results on
the values of the multifractal Hausdorff measure, the multifractal packing
measure, the multifractal Hewitt-Stromberg measures and separator functions;
this discussion is included in order to motivate our main results presented in
Section 3. Section 4 contains concrete examples related to these concepts. The
paper is concluded with Section 5 that, lists some open problems.

##  2\. Preliminaries and statements of results

###  2.1. Hausdorff measure, packing measure and dimensions

While the definitions of the Hausdorff and packing measures and the Hausdorff
and packing dimensions are well-known, we have, nevertheless, decided to
briefly recall the definitions below. There are several reasons for this:
firstly, since we are working in general metric spaces, the different
definitions that appear in the literature may not all agree and for this
reason it is useful to state precisely the definitions that we are using;
secondly, and perhaps more importantly, the less well-known Hewitt-Stromberg
measures (see Section 2.2) play an important part in this paper and to make it
easier for the reader to compare and contrast the definitions of the Hewitt-
Stromberg measures and the definitions of the Hausdorff and packing measures
it is useful to recall the definitions of the latter measures; and thirdly, in
order to provide a motivation for the Hewitt-Stromberg measures.

Let $X$ be a metric space, $E\subseteq X$ and $t>0$. The Hausdorff measure is
defined, for $\varepsilon>0$, as follows

$\mathcal{H}_{\varepsilon}^{t}(E)=\inf\left\\{\sum_{i}\Big{(}\text{diam}(E_{i})\Big{)}^{t}\Big{|}\;\;E\subseteq\bigcup_{i}E_{i},\;\;\text{diam}(E_{i})<\varepsilon\right\\}.$

This allows to define first the $t$-dimensional Hausdorff measure
$\mathcal{H}^{t}(E)$ of $E$ by

$\mathcal{H}^{t}(E)=\sup_{\varepsilon>0}\mathcal{H}_{\varepsilon}^{t}(E).$

Finally, the Hausdorff dimension $\dim_{H}(E)$ is defined by

$\dim_{H}(E)=\sup\Big{\\{}t\geq
0\;\big{|}\;\mathcal{H}^{t}(E)=+\infty\Big{\\}}.$

The packing measure is defined, for $\varepsilon>0$, as follows

$\displaystyle\overline{\mathcal{P}}_{\varepsilon}^{t}(E)=\sup\left\\{\sum_{i}\Big{(}2r_{i}\Big{)}^{t}\right\\},$

where the supremum is taken over all closed balls
$\Big{(}B(x_{i},r_{i})\Big{)}_{i}\;\text{such
that}\;r_{i}\leq\varepsilon\;\text{and with}\;x_{i}\in
E\;\text{and}\;d(x_{i},x_{j})\geq\frac{r_{i}+r_{j}}{2}\;\text{for}\;i\neq j$.
The $t$-dimensional packing pre-measure $\overline{\mathcal{P}}^{t}(E)$ of $E$
is now defined by

$\overline{\mathcal{P}}^{t}(E)=\sup_{\varepsilon>0}\overline{\mathcal{P}}_{\varepsilon}^{t}(E).$

This makes us able to define the $t$-dimensional packing measure
${\mathcal{P}}^{t}(E)$ of $E$ as

${\mathcal{P}}^{t}(E)=\inf\left\\{\sum_{i}\overline{\mathcal{P}}^{t}(E_{i})\;\Big{|}\;\;E\subseteq\bigcup_{i}E_{i}\right\\},$

and the packing dimension $\dim_{P}(E)$ is defined by

$\dim_{P}(E)=\sup\Big{\\{}t\geq
0\;\big{|}\;\mathcal{P}^{t}(E)=+\infty\Big{\\}}.$

###  2.2. Hewitt-Stromberg measures and dimensions

Hewitt-Stromberg measures were introduced in [33, Exercise (10.51)]. Since
then, they have been investigated by several authors, highlighting their
importance in the study of local properties of fractals and products of
fractals. One can cite, for example [30, 31, 35, 50, 65]. In particular,
Edgar’s textbook [20, pp. 32-36] provides an excellent and systematic
introduction to these measures. Such measures appear also appears explicitly,
for example, in Pesin’s monograph [54, 5.3] and implicitly in Mattila’s text
[38]. One of the purposes of this paper is to define and study a class of
natural multifractal generalizations of the Hewitt-Stromberg measures. While
Hausdorff and packing measures are defined using coverings and packings by
families of sets with diameters less than a given positive number
$\varepsilon$, say, the Hewitt-Stromberg measures are defined using packings
of balls with a fixed diameter $\varepsilon$. For $t>0$, the Hewitt-Stromberg
pre-measures are defined as follows,

$\overline{\mathsf{H}}^{t}(E)=\liminf_{r\to 0}N_{r}(E)\;(2r)^{t}$

and

$\overline{\mathsf{P}}^{t}(E)=\limsup_{r\to 0}M_{r}(E)\;(2r)^{t},$

where the covering number $N_{r}(E)$ of $E$ and the packing number $M_{r}(E)$
of $E$ are given by

$N_{r}(E)=\inf\left\\{\sharp\\{I\\}\;\Big{|}\;\Big{(}B(x_{i},r)\Big{)}_{i\in
I}\;\text{is a family of closed balls with}\;x_{i}\in
E\;\text{and}\;E\subseteq\bigcup_{i}B(x_{i},r)\right\\}$

and

$M_{r}(E)=\sup\left\\{\sharp\\{I\\}\;\Big{|}\;\Big{(}B(x_{i},r_{i})\Big{)}_{i\in
I}\;\text{is a family of closed balls with}\;x_{i}\in
E\;\text{and}\;d(x_{i},x_{j})\geq r\;\text{for}\;i\neq j\right\\}.$

Now, we define the lower and upper $t$-dimensional Hewitt-Stromberg measures,
which we denote respectively by $\mathcal{\mathsf{H}}^{t}(E)$ and
$\mathcal{\mathsf{P}}^{t}(E)$, as follows

${\mathsf{H}}^{t}(E)=\inf\left\\{\sum_{i}\overline{\mathsf{H}}^{t}(E_{i})\;\Big{|}\;\;E\subseteq\bigcup_{i}E_{i}\right\\}$

and

${\mathsf{P}}^{t}(E)=\inf\left\\{\sum_{i}\overline{\mathsf{P}}^{t}(E_{i})\;\Big{|}\;\;E\subseteq\bigcup_{i}E_{i}\right\\}.$

We recall some basic inequalities satisfied by the Hewitt-Stromberg, the
Hausdorff and the packing measure (see [35, 50, Proposition 2.1])

$\overline{\mathsf{H}}^{t}(E)\leq\overline{\mathsf{P}}^{t}(E)\leq\overline{\mathcal{P}}^{t}(E)$

and

${\mathcal{H}}^{t}(E)\leq{\mathsf{H}}^{t}(E)\leq{\mathsf{P}}^{t}(E)\leq{\mathcal{P}}^{t}(E).$

The lower and upper Hewitt-Stromberg dimension $\underline{\dim}_{MB}(E)$ and
$\overline{\dim}_{MB}(E)$ are defined by

$\underline{\dim}_{MB}(E)=\inf\Big{\\{}t\geq
0\;\Big{|}\;{\mathsf{H}}^{t}(E)=0\Big{\\}}=\sup\Big{\\{}t\geq
0\;\Big{|}\;{\mathsf{H}}^{t}(E)=+\infty\Big{\\}}$

and

$\overline{\dim}_{MB}(E)=\inf\Big{\\{}t\geq
0\;\big{|}\;{\mathsf{P}}^{t}(E)=0\Big{\\}}=\sup\Big{\\{}t\geq
0\;\big{|}\;{\mathsf{P}}^{t}(E)=+\infty\Big{\\}}.$

The lower and upper box dimensions, denoted by $\underline{\dim}_{B}(E)$ and
$\overline{\dim}_{B}(E)$, respectively, are now defined by

$\underline{\dim}_{B}(E)=\liminf_{r\to 0}\frac{\log N_{r}(E)}{-\log
r}=\liminf_{r\to 0}\frac{\log M_{r}(E)}{-\log r}$

and

$\overline{\dim}_{B}(E)=\limsup_{r\to 0}\frac{\log N_{r}(E)}{-\log
r}=\limsup_{r\to 0}\frac{\log M_{r}(E)}{-\log r}.$

These dimensions satisfy the following inequalities,

$\dim_{H}(E)\leq\underline{\dim}_{MB}(E)\leq\overline{\dim}_{MB}(E)\leq\dim_{P}(E),$
$\dim_{H}(E)\leq\dim_{P}(E)\leq\overline{\dim}_{B}(E)$

and

$\dim_{H}(E)\leq\underline{\dim}_{B}(E)\leq\overline{\dim}_{B}(E).$

The reader is referred to [22] for an excellent discussion of the Hausdorff
dimension, the packing dimension, lower and upper Hewitt-Stromberg dimension
and the box dimensions. In particular, we have (see [22, 40])

$\underline{\dim}_{MB}(E)=\inf\left\\{\sup_{i}\underline{\dim}_{B}(E_{i})\;\Big{|}\;E\subseteq\bigcup_{i}E_{i},\;\;E_{i}\;\;\text{are
bounded in }\;X\right\\}$

and

$\overline{\dim}_{MB}(E)=\inf\left\\{\sup_{i}\overline{\dim}_{B}(E_{i})\;\Big{|}\;E\subseteq\bigcup_{i}E_{i},\;\;E_{i}\;\;\text{are
bounded in }\;X\right\\}.$

###  2.3. Multifractal Hausdorff measure and packing measure

We start by introducing the generalized centered Hausdorff measure
${\mathcal{H}}^{q,t}_{\mu}$ and the generalized packing measure
${\mathcal{P}}^{q,t}_{\mu}$. We fix an integer $n\geq 1$ and denote by
$\mathcal{P}(\mathbb{R}^{n})$ the family of compactly supported Borel
probability measures on $\mathbb{R}^{n}$. Let
$\mu\in\mathcal{P}(\mathbb{R}^{n})$, $q,t\in\mathbb{R}$,
$E\subseteq{\mathbb{R}}^{n}$ and $\delta>0$. We define the generalized packing
pre-measure by

$\overline{{\mathcal{P}}}^{q,t}_{\mu}(E)=\displaystyle\inf_{\delta>0}\displaystyle\sup\left\\{\sum_{i}\mu\big{(}B(x_{i},{r_{i}})\big{)}^{q}(2r_{i})^{t}\;\Big{|}\;\Big{(}B(x_{i},r_{i})\Big{)}_{i}\;\text{is
a centered}\;\delta\text{-packing of}\;E\right\\}.$

In a similar way, we define the generalized Hausdorff pre-measure by

$\overline{{\mathcal{H}}}^{q,t}_{\mu}(E)=\displaystyle\sup_{\delta>0}\displaystyle\inf\left\\{\sum_{i}\mu\big{(}B(x_{i},r_{i})\big{)}^{q}(2r_{i})^{t}\;\Big{|}\;\Big{(}B(x_{i},r_{i})\Big{)}_{i}\;\text{is
a centered}\;\delta\text{-covering of}\;E\right\\},$

with the conventions $0^{q}=\infty$ for $q\leq 0$ and $0^{q}=0$ for $q>0$.

The function $\overline{{\mathcal{H}}}^{q,t}_{\mu}$ is $\sigma$-subadditive
but not increasing and the function $\overline{{\mathcal{P}}}^{q,t}_{\mu}$ is
increasing but not $\sigma$-subadditive. That is the reason for which Olsen
introduced the following modifications of the generalized Hausdorff and
packing measures ${\mathcal{H}}^{q,t}_{\mu}$ and ${\mathcal{P}}^{q,t}_{\mu}$:

${\mathcal{H}}^{q,t}_{\mu}(E)=\displaystyle\sup_{F\subseteq
E}\overline{{\mathcal{H}}}^{q,t}_{\mu}(F)\quad\text{and}\quad{\mathcal{P}}^{q,t}_{\mu}(E)=\inf_{E\subseteq\bigcup_{i}E_{i}}\sum_{i}\overline{\mathcal{P}}^{q,t}_{\mu}(E_{i}).$

The functions ${\mathcal{H}}^{q,t}_{\mu}$ and ${\mathcal{P}}^{q,t}_{\mu}$ are
metric outer measures and thus measures on the Borel family of subsets of
$\mathbb{R}^{n}$. Moreover, there exists an integer $\xi\in\mathbb{N}$, such
that ${\mathcal{H}}^{q,t}_{\mu}\leq\xi{\mathcal{P}}^{q,t}_{\mu}.$ The measure
${\mathcal{H}}^{q,t}_{\mu}$ is of course a multifractal generalization of the
centered Hausdorff measure, whereas ${\mathcal{P}}^{q,t}_{\mu}$ is a
multifractal generalization of the packing measure. In fact, it is easily seen
that, for $t\geq 0$, one has

$2^{-t}{\mathcal{H}}^{0,t}_{\mu}\leq{\mathcal{H}}^{t}\leq{\mathcal{H}}^{0,t}_{\mu}\quad\text{and}\quad{\mathcal{P}}^{0,t}_{\mu}={\mathcal{P}}^{t},$

where ${\mathcal{H}}^{t}$ and ${\mathcal{P}}^{t}$ denote respectively the
$t$-dimensional Hausdorff and $t$-dimensional packing measures.

We now define the family of doubling measures. For
$\mu\in\mathcal{P}(\mathbb{R}^{n})$ and $a>1$, we write

$P_{a}(\mu)=\limsup_{r\searrow
0}\left(\sup_{x\in\operatorname{supp}\mu}\displaystyle\frac{\mu\big{(}B(x,ar)\big{)}}{\mu\big{(}B(x,r)\big{)}}\right).$

We say that the measure $\mu$ satisfies the doubling condition if there exists
$a>1$ such that $P_{a}(\mu)<\infty$. It is easily seen that the exact value of
the parameter $a$ is unimportant:

$P_{a}(\mu)<\infty,\;\;\text{for some}\;\;a>1\;\;\text{ if and only
if}\;\;P_{a}(\mu)<\infty,\;\;\text{ for all}\;\;a>1.$

Also, we denote by $\mathcal{P}_{D}(\mathbb{R}^{n})$ the family of Borel
probability measures on $\mathbb{R}^{n}$ which satisfy the doubling condition.
We can cite as classical examples of doubling measures, the self-similar
measures and the self-conformal ones [42]. In particular, if
$\mu\in\mathcal{P}_{D}(\mathbb{R}^{n})$ then
${\mathcal{H}}^{q,t}_{\mu}\leq{\mathcal{P}}^{q,t}_{\mu}.$

The measures ${\mathcal{H}}^{q,t}_{\mu}$ and ${\mathcal{P}}^{q,t}_{\mu}$ and
the pre-measure ${\overline{\mathcal{P}}}^{q,t}_{\mu}$ assign in a usual way a
multifractal dimension to each subset $E$ of $\mathbb{R}^{n}$. They are
respectively denoted by $\dim_{\mu}^{q}(E)$, $\operatorname{Dim}_{\mu}^{q}(E)$
and $\Delta_{\mu}^{q}(E)$ (see [42]) and satisfy

$\begin{array}[]{lllcr}\dim_{\mu}^{q}(E)&=&\inf\Big{\\{}t\in\mathbb{R}\;\big{|}\;\quad{\mathcal{H}}^{{q},t}_{\mu}(E)=0\Big{\\}}=\sup\Big{\\{}t\in\mathbb{R}\;\big{|}\;\quad{\mathcal{H}}^{{q},t}_{\mu}(E)=+\infty\Big{\\}},\\\
\\\
\operatorname{Dim}_{\mu}^{q}(E)&=&\inf\Big{\\{}t\in\mathbb{R}\;\big{|}\;\quad{\mathcal{P}}^{{q},t}_{\mu}(E)=0\Big{\\}}=\sup\Big{\\{}t\in\mathbb{R}\;\big{|}\;\quad{\mathcal{P}}^{{q},t}_{\mu}(E)=+\infty\Big{\\}},\\\
\\\
\Delta_{\mu}^{q}(E)&=&\inf\Big{\\{}t\in\mathbb{R}\;\big{|}\;\quad\overline{\mathcal{P}}^{{q},t}_{\mu}(E)=0\Big{\\}}=\sup\Big{\\{}t\in\mathbb{R}\;\big{|}\;\quad\overline{\mathcal{P}}^{{q},t}_{\mu}(E)=+\infty\Big{\\}}.\end{array}$

The number $\dim_{\mu}^{q}(E)$ is an obvious multifractal analogue of the
Hausdorff dimension $\dim_{H}(E)$ of $E$ whereas
$\operatorname{Dim}_{\mu}^{q}(E)$ and $\Delta_{\mu}^{q}(E)$ are obvious
multifractal analogues of the packing dimension $\dim_{P}(E)$ and the pre-
packing dimension $\Delta(E)$ of $E$ respectively. In fact, it follows
immediately from the definitions that

$\dim_{H}(E)=\dim_{\mu}^{0}(E),\;\;\;\dim_{P}(E)=\operatorname{Dim}_{\mu}^{0}(E)\quad\text{and}\quad\Delta(E)=\Delta_{\mu}^{0}(E).$

We define the functions

$b_{\mu}(q)=\dim_{\mu}^{q}(\operatorname{supp}\mu)\quad\text{and}\quad
B_{\mu}(q)=\operatorname{Dim}_{\mu}^{q}(\operatorname{supp}\mu).$

It is well known that the functions $b_{\mu}$ and $B_{\mu}$ are decreasing and
$B_{\mu}$ is convex and satisfying $b_{\mu}\leq B_{\mu}.$

###  2.4. Multifractal Hewitt-Stromberg measures and separator functions

In the following, we will set up, for $q,t\in\mathbb{R}$ and
$\mu\in{\mathcal{P}}(\mathbb{R}^{n})$, the lower and upper multifractal
Hewitt-Stromberg measures ${\mathsf{H}}_{\mu}^{q,t}$ and
${\mathsf{P}}_{\mu}^{q,t}$.
For $E\subseteq\operatorname{supp}\mu$, the pre-measure of $E$ is defined by

${\mathsf{C}}_{\mu}^{q,t}(E)=\limsup_{r\to 0}M_{\mu,r}^{q}(E)(2r)^{t},$

where

$M_{\mu,r}^{q}(E)=\sup\left\\{\displaystyle\sum_{i}\mu(B(x_{i},r))^{q}\;\Big{|}\;\Big{(}B(x_{i},r)\Big{)}_{i}\;\text{is
a centered packing of}\;E\right\\}.$

It’s clear that ${\mathsf{C}}_{\mu}^{q,t}$ is increasing and
${\mathsf{C}}_{\mu}^{q,t}(\emptyset)=0$. However it’s not $\sigma$-additive.
For this, we introduce the ${\mathsf{P}}_{\mu}^{q,t}$-measure defined by

${\mathsf{P}}_{\mu}^{q,t}(E)=\inf\left\\{\displaystyle\sum_{i}{\mathsf{C}}_{\mu}^{q,t}(E_{i})\;\Big{|}\;E\subseteq\bigcup_{i}E_{i}\;\text{and
the}\;E_{i}^{\prime}\text{s are bounded}\right\\}.$

In a similar way we define

${\mathsf{L}}_{\mu}^{q,t}(E)=\liminf_{r\to 0}N_{\mu,r}^{q}(E)(2r)^{t},$

where

$N_{\mu,r}^{q}(E)=\inf\left\\{\displaystyle\sum_{i}\mu(B(x_{i},r))^{q}\;\Big{|}\;\Big{(}B(x_{i},r)\Big{)}_{i}\;\text{is
a centered covering of}\;E\right\\}.$

Since ${\mathsf{L}}_{\mu}^{q,t}$ is not increasing and not countably
subadditive, one needs a standard modification to get an outer measure. Hence,
we modify the definition as follows

$\overline{\mathsf{H}}_{\mu}^{q,t}(E)=\inf\left\\{\displaystyle\sum_{i}{\mathsf{L}}_{\mu}^{q,t}(E_{i})\;\Big{|}\;E\subseteq\bigcup_{i}E_{i}\;\text{and
the}\;E_{i}^{\prime}\text{s are bounded}\right\\}$

and

$\mathsf{H}_{\mu}^{q,t}(E)=\sup_{F\subseteq
E}\overline{\mathsf{H}}_{\mu}^{q,t}(F).$

The measure $\mathsf{H}^{q,t}_{\mu}$ is of course a multifractal
generalization of the lower $t$-dimensional Hewitt-Stromberg measure
${\mathsf{H}}^{t}$, whereas $\mathsf{P}^{q,t}_{\mu}$ is a multifractal
generalization of the upper $t$-dimensional Hewitt-Stromberg measures
${\mathsf{P}}^{t}$. In fact, it is easily seen that, for $t>0$, one has

$\mathsf{H}^{0,t}_{\mu}={\mathsf{H}}^{t}\quad\text{and}\quad\mathsf{P}^{0,t}_{\mu}={\mathsf{P}}^{t}.$

The following result describes some of the basic properties of the
multifractal Hewitt-Stromberg measures including the fact that
${\mathsf{H}}_{\mu}^{q,t}$ and ${\mathsf{P}}_{\mu}^{q,t}$ are Borel metric
outer measures and summarises the basic inequalities satisfied by the
multifractal Hewitt-Stromberg measures, the generalized Hausdorff measure and
the generalized packing measure.

######  Theorem 2.

[3] Let $q,t\in\mathbb{R}$ and $\mu\in{\mathcal{P}}(\mathbb{R}^{n})$. Then for
every set $E\subseteq\mathbb{R}^{n}$ we have

  1. (1)

the set functions $\mathsf{H}_{\mu}^{q,t}$ and $\mathsf{P}_{\mu}^{q,t}$ are
metric outer measures and thus they are measures on the Borel algebra.

  2. (2)

There exists an integer $\xi\in\mathbb{N}$, such that

${\mathcal{H}}^{q,t}_{\mu}(E)\leq\mathsf{H}_{\mu}^{q,t}(E)\leq\xi\mathsf{P}_{\mu}^{q,t}(E)\leq\xi{\mathcal{P}}^{q,t}_{\mu}(E).$

  3. (3)

When $q\leq 0$ or $q>0$ and $\mu\in{\mathcal{P}}_{D}(\mathbb{R}^{n})$, we have

${\mathcal{H}}^{q,t}_{\mu}(E)\leq\mathsf{H}_{\mu}^{q,t}(E)\leq\mathsf{P}_{\mu}^{q,t}(E)\leq{\mathcal{P}}^{q,t}_{\mu}(E).$

The measures $\mathsf{H}^{q,t}_{\mu}$ and $\mathsf{P}^{q,t}_{\mu}$ and the
pre-measure $C^{q,t}_{\mu}$ assign in the usual way a multifractal dimension
to each subset $E$ of $\mathbb{R}^{n}$. They are respectively denoted by
$\mathsf{b}_{\mu}^{q}(E)$, $\mathsf{B}_{\mu}^{q}(E)$ and
$\mathsf{\Delta}_{\mu}^{q}(E)$,

######  Proposition 1.

Let $q\in\mathbb{R}$, $\mu\in{\mathcal{P}}(\mathbb{R}^{n})$ and
$E\subseteq\mathbb{R}^{n}$. Then

  1. (1)

there exists a unique number ${\mathsf{b}}_{\mu}^{q}(E)\in[-\infty,+\infty]$
such that

$\mathsf{H}^{q,t}_{\mu}(E)=\left\\{\begin{matrix}\infty&\text{if}&t<{\mathsf{b}}_{\mu}^{q}(E),\\\
\\\ 0&\text{if}&{\mathsf{b}}_{\mu}^{q}(E)<t,\end{matrix}\right.$

  2. (2)

there exists a unique number ${\mathsf{B}}_{\mu}^{q}(E)\in[-\infty,+\infty]$
such that

$\mathsf{P}^{q,t}_{\mu}(E)=\left\\{\begin{matrix}\infty&\text{if}&t<{\mathsf{B}}_{\mu}^{q}(E),\\\
\\\ 0&\text{if}&{\mathsf{B}}_{\mu}^{q}(E)<t,\end{matrix}\right.$

  3. (3)

there exists a unique number
${\mathsf{\Delta}}_{\mu}^{q}(E)\in[-\infty,+\infty]$ such that

$\mathsf{C}^{q,t}_{\mu}(E)=\left\\{\begin{matrix}\infty&\text{if}&t<{\mathsf{\Delta}}_{\mu}^{q}(E),\\\
\\\ 0&\text{if}&{\mathsf{\Delta}}_{\mu}^{q}(E)<t.\end{matrix}\right.$

In addition, we have

${\mathsf{b}}_{\mu}^{q}(E)\leq{\mathsf{B}}_{\mu}^{q}(E)\leq{\mathsf{\Delta}}_{\mu}^{q}(E).$

The number ${\mathsf{b}}_{\mu}^{q}(E)$ is an obvious multifractal analogue of
the lower Hewitt-Stromberg dimension $\underline{\dim}_{MB}(E)$ of $E$ whereas
${\mathsf{B}}_{\mu}^{q}(E)$ is an obvious multifractal analogues of the upper
Hewitt-Stromberg dimension $\overline{\dim}_{MB}(E)$ of $E$. In fact, it
follows immediately from the definitions that

${\mathsf{b}}_{\mu}^{0}(E)=\underline{\dim}_{MB}(E)\quad\text{and}\quad{\mathsf{B}}_{\mu}^{0}(E)=\overline{\dim}_{MB}(E).$

######  Remark 1.

It follows from Theorem 2 that

$\dim_{\mu}^{q}(E)\leq{\mathsf{b}}_{\mu}^{q}(E)\leq{\mathsf{B}}_{\mu}^{q}(E)\leq\operatorname{Dim}_{\mu}^{q}(E)\leq\Delta_{\mu}^{q}(E).$

The definition of these dimension functions makes it clear that they are
counterparts of the $\tau_{\mu}$-function which appears in the multifractal
formalism. This being the case, it is important that they have the properties
described by the physicists. The next theorem shows that these functions do
indeed have some of these properties.

######  Theorem 3.

Let $q\in\mathbb{R}$ and $E\subseteq\mathbb{R}^{n}$.

  1. (1)

The functions $q\mapsto\mathsf{H}^{q,t}_{\mu}(E)$,
$\mathsf{P}^{q,t}_{\mu}(E)$, $\mathsf{C}^{q,t}_{\mu}(E)$ are decreasing.

  2. (2)

The functions $t\mapsto\mathsf{H}^{q,t}_{\mu}(E)$,
$\mathsf{P}^{q,t}_{\mu}(E)$, $\mathsf{C}^{q,t}_{\mu}(E)$ are decreasing.

  3. (3)

The functions $q\mapsto{{\mathsf{b}}}^{q}_{\mu}(E)$,
${{\mathsf{B}}}^{q}_{\mu}(E)$, ${{\mathsf{\Delta}}}^{q}_{\mu}(E)$ are
decreasing.

  4. (4)

The functions $q\mapsto{{\mathsf{B}}}^{q}_{\mu}(E)$,
${{\mathsf{\Delta}}}^{q}_{\mu}(E)$ are convex.

###### Proof.

Let $q\in\mathbb{R}$ and $E\subseteq\mathbb{R}^{n}$.
The first and second part of Theorem 3 follows since $x\mapsto a^{x}$ is
decreasing for all $a\in]0,1[$.
Observe that part (3) of Theorem 3 follows immediately from (1).
We will now prove the part (4). Let $\alpha\in[0,1]$ and $p,s,t\in\mathbb{R}$.
Suppose that we have shown that

$\mathsf{C}_{\mu}^{\alpha p+(1-\alpha)q,\alpha
t+(1-\alpha)s}(E)\leq\Big{(}\mathsf{C}_{\mu}^{p,t}(E)\Big{)}^{\alpha}\Big{(}\mathsf{C}_{\mu}^{q,s}(E)\Big{)}^{1-\alpha}.$
(2.1)

Then, for all $\epsilon>0$, we have

$\mathsf{C}_{\mu}^{\alpha
p+(1-\alpha)q,\alpha{{\mathsf{\Delta}}}^{p}_{\mu}(E)+(1-\alpha){{\mathsf{\Delta}}}^{q}_{\mu}(E)+\epsilon}(E)\leq\left(\mathsf{C}_{\mu}^{p,{{\mathsf{\Delta}}}^{p}_{\mu}(E)+\epsilon}(E)\right)^{\alpha}\left(\mathsf{C}_{\mu}^{q,{{\mathsf{\Delta}}}^{q}_{\mu}(E)+\epsilon}(E)\right)^{1-\alpha}=0.$

We therefore conclude that

${{\mathsf{\Delta}}}^{\alpha
p+(1-\alpha)q}_{\mu}(E)\leq\alpha{{\mathsf{\Delta}}}^{p}_{\mu}(E)+(1-\alpha){{\mathsf{\Delta}}}^{q}_{\mu}(E)+\epsilon.$

Finally, letting $\epsilon$ tend to $0$, then the convexity of
$q\mapsto{{\mathsf{\Delta}}}^{q}_{\mu}(E)$ follows.

We now turn towards the proof of (2.1). Put $r>0$ and
$\Big{(}B(x_{i},r)\Big{)}_{i}$ be a centered packing of $E$. It follows from
Hölder inequality that

$\displaystyle\sum_{i}\mu(B(x_{i},r))^{\alpha p+(1-\alpha)q}$ $\displaystyle=$
$\displaystyle\sum_{i}\Big{(}\mu(B(x_{i},r)^{p}\Big{)}^{\alpha}\Big{(}\mu(B(x_{i},r)^{q}\Big{)}^{1-\alpha}$
$\displaystyle\leq$
$\displaystyle\left(\sum_{i}\mu(B(x_{i},r)^{p}\right)^{\alpha}\left(\sum_{i}\mu(B(x_{i},r)^{q}\right)^{1-\alpha}$
$\displaystyle\leq$
$\displaystyle\Big{(}M_{\mu,r}^{p}(E)\Big{)}^{\alpha}\Big{(}M_{\mu,r}^{q}(E)\Big{)}^{1-\alpha}.$

This shows that

$M_{\mu,r}^{\alpha p+(1-\alpha)q}~{}(2r)^{\alpha
t+(1-\alpha)s}\leq\Big{(}M_{\mu,r}^{p}~{}(2r)^{t}\Big{)}^{\alpha}\Big{(}M_{\mu,r}^{q}~{}(2r)^{s}\Big{)}^{1-\alpha}.$

Letting $r$ tend to $0$ we get the result.

We must now show the convexity of $q\mapsto{{\mathsf{B}}}^{q}_{\mu}(E)$. Let
$\eta>0$ and put $t={{\mathsf{B}}}^{p}_{\mu}(E)$ and
$s={{\mathsf{B}}}^{q}_{\mu}(E)$. Since
$\mathsf{P}_{\mu}^{q,s+\eta}(E)=\mathsf{P}_{\mu}^{p,t+\eta}(E)=0$, we can
choose bounded coverings $(H_{i})_{i}$ and $(K_{i})_{i}$ of $E$ such that

$\sum_{i}\mathsf{C}_{\mu}^{q,t+\eta}(H_{i})\leq 1\quad\text{and
}\sum_{i}\mathsf{C}_{\mu}^{q,s+\eta}(K_{i})\leq 1.$

Next, for $n\in\mathbb{N}^{*},$ let
$E_{n}=\displaystyle\bigcup_{i,j=1}^{n}(H_{i}\cap K_{j})$, we clearly have

$\displaystyle\mathsf{P}_{\mu}^{\alpha p+(1-\alpha)q,\alpha
t+(1-\alpha)s+\eta}(E_{n})$ $\displaystyle\leq$
$\displaystyle\sum_{i,j=1}^{n}\mathsf{P}_{\mu}^{\alpha p+(1-\alpha)q,\alpha
t+(1-\alpha)s+\eta}(H_{i}\cap K_{j})$ $\displaystyle\leq$
$\displaystyle\sum_{i,j=1}^{n}\mathsf{C}_{\mu}^{\alpha p+(1-\alpha)q,\alpha
t+(1-\alpha)s+\eta}(H_{i}\cap K_{j})$
$\displaystyle\overset{\eqref{convex}}{\leq}$
$\displaystyle\sum_{i,j=1}^{n}\Big{(}\mathsf{C}_{\mu}^{p,t+\eta}(H_{i}\cap
K_{j})\Big{)}^{\alpha}\Big{(}\mathsf{C}_{\mu}^{q,s+\eta}(H_{i}\cap
K_{j})\Big{)}^{1-\alpha}$ $\displaystyle\overset{\text{H\"{o}lder}}{\leq}$
$\displaystyle\left(\sum_{i,j=1}^{n}\mathsf{C}_{\mu}^{p,t+\eta}(H_{i}\cap
K_{j})\right)^{\alpha}\left(\sum_{i,j=1}^{n}\mathsf{C}_{\mu}^{q,s+\eta}(H_{i}\cap
K_{j})\right)^{1-\alpha}$ $\displaystyle\leq$
$\displaystyle\left(\sum_{i,j=1}^{n}\mathsf{C}_{\mu}^{p,t+\eta}(H_{i})\right)^{\alpha}\left(\sum_{i,j=1}^{n}\mathsf{C}_{\mu}^{q,s+\eta}(K_{j})\right)^{1-\alpha}$
$\displaystyle\leq$
$\displaystyle\left(n\sum_{i=1}^{n}\mathsf{C}_{\mu}^{p,t+\eta}(H_{i})\right)^{\alpha}\left(n\sum_{j=1}^{n}\mathsf{C}_{\mu}^{q,s+\eta}(K_{j})\right)^{1-\alpha}$
$\displaystyle\leq$ $\displaystyle n^{\alpha}\;n^{1-\alpha}=n<\infty.$

We now obtain, for all $n\in\mathbb{N}^{*}$,

${{\mathsf{B}}}_{\mu}^{\alpha p+(1-\alpha)q}(E_{n})\leq\alpha
t+(1-\alpha)s+\eta.$

Since clearly $E\subseteq\bigcup_{n}E_{n}$, we therefore conclude that

$\displaystyle{{\mathsf{B}}}_{\mu}^{\alpha p+(1-\alpha)q}(E)$
$\displaystyle\leq$ $\displaystyle{{\mathsf{B}}}_{\mu}^{\alpha
p+(1-\alpha)q}\left(\bigcup_{n}E_{n}\right)\leq\sup_{n}{{\mathsf{B}}}_{\mu}^{\alpha
p+(1-\alpha)q}(E_{n})$ $\displaystyle\leq$
$\displaystyle\alpha{{\mathsf{B}}}_{\mu}^{p}(E)+(1-\alpha){{\mathsf{B}}}_{\mu}^{q}(E)+\eta.$

Letting $\eta$ tend to $0$ now yields the desired result. This completes the
proof of Theorem 3. ∎

Next we define the multifractal separator functions ${\mathsf{b}}_{\mu}$,
${\mathsf{B}}_{\mu}$ and ${{\mathsf{\Lambda}}}_{\mu}$ :
$\mathbb{R}\rightarrow[-\infty,+\infty]$ by

$\begin{array}[]{lllcr}{\mathsf{b}}_{\mu}:\;q\rightarrow{\mathsf{b}}_{\mu}^{q}(\operatorname{supp}\mu),\quad{\mathsf{B}}_{\mu}:\;q\rightarrow{\mathsf{B}}_{\mu}^{q}(\operatorname{supp}\mu)\quad\text{and}\quad{{\mathsf{\Lambda}}}_{\mu}:\;q\rightarrow{{\mathsf{\Delta}}}_{\mu}^{q}(\operatorname{supp}\mu).\end{array}$

We also obtain the following corollary providing information about the lower
and upper multifractal Hewitt-Stromberg functions.

######  Corollary 1.

Let $q\in\mathbb{R}$. We have

  1. (1)

for $q<1$, ${{\mathsf{b}}}_{\mu}(q)\geq 0$.

  2. (2)

For $q=1$, ${{\mathsf{b}}}_{\mu}(q)={{\mathsf{\Lambda}}}_{\mu}(q)=0$.

  3. (3)

For $q>1$, ${{\mathsf{\Lambda}}}_{\mu}(q)\leq 0$.

###### Proof.

This follow immediately from the above theorem and definitions. ∎

###  2.5. Some characterizations of $\mathsf{b}_{\mu}(q)$ and
$\mathsf{B}_{\mu}(q)$

In this section, we investigate the relation between the lower and upper
multifractal Hewitt-Stromberg functions $\mathsf{b}_{\mu}$ and
$\mathsf{B}_{\mu}$ and the multifractal box dimension, the multifractal
packing dimension and the multifractal pre-packing dimension. We first note
that there exists a unique number ${\Theta}_{\mu}^{q}(E)\in[-\infty,+\infty]$
such that

${\mathsf{L}}^{q,t}_{\mu}(E)=\left\\{\begin{matrix}\infty&\text{if}&t<{\Theta}_{\mu}^{q}(E),\\\
\\\ 0&\text{if}&{\Theta}_{\mu}^{q}(E)<t.\end{matrix}\right.$

######  Proposition 2.

Let $q\in\mathbb{R}$ and $\mu$ be a compact supported Borel probability
measure on $\mathbb{R}^{n}$. Then for every $E\subseteq\operatorname{supp}\mu$
we have

${\Theta}_{\mu}^{q}(E)=\liminf_{r\to 0}\frac{\log N_{\mu,r}^{q}(E)}{-\log
r}\qquad\text{and }\qquad{\mathsf{\Delta}}_{\mu}^{q}(E)=\limsup_{r\to
0}\frac{\log M_{\mu,r}^{q}(E)}{-\log r}.$

###### Proof.

We will prove the first equality, the second one is similar. Suppose that

$\displaystyle\liminf_{r\to 0}\frac{\log N_{\mu,r}^{q}(E)}{-\log
r}>{\Theta}_{\mu}^{q}(E)+\epsilon$

for some $\epsilon>0$. Then we can find $\delta>0$ such that for any
$r\leq\delta$,

$N_{\mu,r}^{q}(E)~{}r^{{\Theta}_{\mu}^{q}(E)+\epsilon}>1$

and then

$\mathsf{L}_{\mu}^{q,{\Theta}_{\mu}^{q}(E)+\epsilon}\geq
2^{{\Theta}_{\mu}^{q}(E)+\epsilon}$

which is a contradiction. We therefore infer

$\displaystyle\liminf_{r\to 0}\frac{\log N_{\mu,r}^{q}(E)}{-\log
r}\leq{\Theta}_{\mu}^{q}(E)+\epsilon\;\;\text{ for any}\;\;\epsilon>0.$

The proof of the following statement

$\displaystyle\liminf_{r\to 0}\frac{\log N_{\mu,r}^{q}(E)}{-\log
r}\geq{\Theta}_{\mu}^{q}(E)-\epsilon\;\;\text{ for any}\;\;\epsilon>0$

is identical to the proof of the above statement and is therefore omitted. ∎

######  Remark 2.

Here we follow the approach of Olsen in [42, 45, 48, 49].

  1. (1)

The multifractal dimensions ${\Theta}_{\mu}^{q}(E)$ and
${\mathsf{\Delta}}_{\mu}^{q}(E)$ of $E$ represent the upper and lower
multifractal box-dimension. In particular, we have

${\Theta}_{\mu}^{0}(E)=\underline{\dim}_{B}(E)\quad\text{and}\quad{\mathsf{\Delta}}_{\mu}^{0}(E)=\overline{\dim}_{B}(E).$

  2. (2)

Let us introduce the multifrcatal generalization of the $q$-dimensions called
also relative Rényi $q$-dimensions based on integral representations. Let
$\mu$ be a probability measure on $\mathbb{R}^{n}$. For
$q\in\mathbb{R}\setminus\\{0\\}$, we write

$\underline{D}_{\mu}^{q}=\displaystyle\liminf_{r\rightarrow 0}\frac{1}{q\log
r}\log\int\mu(B(x,r))^{q}d\mu(x),$

and

$\overline{D}_{\mu}^{q}=\displaystyle\limsup_{r\rightarrow 0}\frac{1}{q\log
r}\log\int\mu(B(x,r))^{q}d\mu(x).$

Now we define the generalized entropies due to Rényi by,

$h^{q}_{r}(\mu)=\frac{1}{q-1}\log
M_{\mu,r}^{q}(\operatorname{supp}\mu)\quad\text{for}\quad q\neq 1$

and

$h^{1}_{r}(\mu)=\inf\left\\{-\sum_{i}\mu(E_{i})\log\mu(E_{i})\;\Big{|}\;(E_{i})_{i}\quad\text{is
a partition of}\quad\operatorname{supp}\mu\right\\}.$

We define the upper and lower Rényi $q$-dimensions $\overline{T}_{\mu}^{q}$
and $\underline{T}_{\mu}^{q}$ of $\mu$ by

$\overline{T}_{\mu}^{q}=\displaystyle\limsup_{r\to 0}\frac{\log
h^{q}_{r}(\mu)}{\log
r}\quad\text{and}\quad\underline{T}_{\mu}^{q}=\displaystyle\liminf_{r\to
0}\frac{\log h^{q}_{r}(\mu)}{\log r}.$

If $\underline{D}_{\mu}^{q}=\overline{D}_{\mu}^{q}$ (respectively
$\underline{T}_{\mu}^{q}=\overline{T}_{\mu}^{q}$) we refer to the common value
as the relative Rényi $q$-dimension of $\mu$ and denote it ${D}_{\mu}^{q}$
(respectively ${T}_{\mu}^{q}$). Finally define
$\underline{\mathcal{D}}_{\mu}(q)$, $\overline{\mathcal{D}}_{\mu}(q)$,
$\underline{\mathcal{T}}_{\mu}(q)$ and $\overline{\mathcal{T}}_{\mu}(q)$ :
$\mathbb{R}\rightarrow[-\infty,+\infty]$ by

$\underline{\mathcal{D}}_{\mu}(q)=(1-q)\underline{{D}}_{\mu}^{q-1},\quad\overline{\mathcal{D}}_{\mu}(q)=(1-q)\overline{{D}}_{\mu}^{q-1}$

and

$\underline{\mathcal{T}}_{\mu}(q)=(1-q)\underline{{T}}_{\mu}^{q},\quad\overline{\mathcal{T}}_{\mu}(q)=(1-q)\overline{{T}}_{\mu}^{q}.$

Let $q\in\mathbb{R}$ and $\mu\in{\mathcal{P}}_{D}(\mathbb{R}^{n})$. Then the
following holds

${\mathsf{\Delta}}_{\mu}(q)=\underline{\mathcal{D}}_{\mu}(q)\vee\overline{\mathcal{D}}_{\mu}(q)=\underline{\mathcal{T}}_{\mu}(q)\vee\overline{\mathcal{T}}_{\mu}(q).$

  3. (3)

We define the multifractal Minkowski volume as follows. Let $E$ be a subset of
$\mathbb{R}^{n}$ and $r>0$. We denote by $B(E,r)$ the open $r$ neighbourhood
of $E$, i.e.

$B(E,r)=\Big{\\{}x\in\mathbb{R}^{n}\;\Big{|}\;\text{dist}(x,E)<r\Big{\\}}.$

For a real number $q$ and a Borel measure $\mu$ on $\mathbb{R}^{n}$, we define
the multifractal Minkowski volume $V_{\mu,r}^{q}(E)$ of $E$ with respect to
the measure $\mu$ by

$V_{\mu,r}^{q}(E)=\frac{1}{r^{n}}\int_{B(E,r)}\mu(B(x,r))^{q}d\mathcal{L}^{n}(x).$

Here $\mathcal{L}^{n}$ denotes the $n$-dimensional Lebesgue measure in
$\mathbb{R}^{n}$. The importance of the Rényi dimensions in multifractal
analysis together with the formal resemblance between the multifractal
Minkowski volume $V_{\mu,r}^{q}(E)$ and the moments
$\int_{E}\mu(B(x,r))^{q-1}d\mu(x)$ used in the definition the Rényi dimensions
may be seen as a justification for calling the quantity $V_{\mu,r}^{q}(E)$ for
the multifractal Minkowski volume. Using the multifractal Minkowski volume we
can define multifractal Monkowski dimensions. For a real number $q$ and a
Borel measure $\mu$ on $\mathbb{R}^{n}$, we define the lower and upper
multifractal Minkowski dimension of $E$, by

$\underline{\dim}_{M,\mu}^{q}(E)=\liminf_{r\to 0}\frac{\log
V_{\mu,r}^{q}(E)}{-\log
r}\quad\text{and}\quad\overline{\dim}_{M,\mu}^{q}(E)=\limsup_{r\to
0}\frac{\log V_{\mu,r}^{q}(E)}{-\log r}.$

We note the close similarity between the multifractal Minkowski dimensions and
${\mathsf{\Delta}}_{\mu}^{q}$. Indeed, the equality (2.2) shows that this
similarity is not merely a formal resemblance. In fact, for $q\geq 1$, the
multifractal Minkowski dimensions and ${\mathsf{\Delta}}_{\mu}^{q}$ coincide,
i.e. for $q\geq 1$ and $\mu\in{\mathcal{P}}_{D}(\mathbb{R}^{n})$, we have

$\displaystyle{\mathsf{\Delta}}_{\mu}(q)=\underline{\dim}_{M,\mu}^{q-1}(\operatorname{supp}\mu)\vee\overline{\dim}_{M,\mu}^{q-1}(\operatorname{supp}\mu).$
(2.2)

######  Proposition 3.

Let $q\in\mathbb{R}$ and $\mu$ be a compact supported Borel probability
measure on $\mathbb{R}^{n}$. Then for every $E\subseteq\operatorname{supp}\mu$
we have

${\mathsf{b}}_{\mu}^{q}(E)=\sup_{F\subseteq
E}\left\\{\inf\left\\{\sup_{i}{\Theta}_{\mu}^{q}(F_{i})\;\Big{|}\;\displaystyle
F\subseteq\bigcup_{i}F_{i},\;\;F_{i}\;\;\text{are bounded in
}\;\;\mathbb{R}^{n}\right\\}\right\\}$

and

${\mathsf{B}}_{\mu}^{q}(E)=\inf\left\\{\sup_{i}{\mathsf{\Delta}}_{\mu}^{q}(E_{i})\;\Big{|}\;\displaystyle
E\subseteq\bigcup_{i}E_{i},\;\;E_{i}\;\;\text{are bounded in
}\;\;\mathbb{R}^{n}\right\\}.$

###### Proof.

Denote

$\beta=\sup_{F\subset
E}\left\\{\inf\left\\{\sup_{i}{\Theta}_{\mu}^{q}(F_{i})\;\Big{|}\;F\subseteq\bigcup_{i}F_{i},\;\;F_{i}\;\;\text{are
bounded in }\;\;\mathbb{R}^{n}\right\\}\right\\}.$

Assume that $\beta<{\mathsf{b}}_{\mu}^{q}(E)$ and take
$\alpha\in(\beta,{\mathsf{b}}_{\mu}^{q}(E))$. Then, for all $F\subseteq E$,
there exists $\\{F_{i}\\}$ of bounded subset of $F$ such that
$F\subseteq\cup_{i}F_{i}$, and $\sup_{i}{\Theta}_{\mu}^{q}(F_{i})<\alpha$. Now
observe that $\mathsf{L}_{\mu}^{q,\alpha}(F_{i})=0$ which implies that
${\overline{\mathsf{H}}}_{\mu}^{q,\alpha}(F)=0$. This implies that
${\mathsf{H}}_{\mu}^{q,\alpha}(E)=0$. It is a contradiction. Now suppose that
${\mathsf{b}}_{\mu}^{q}(E)<\beta$, then, for
$\alpha\in({\mathsf{b}}_{\mu}^{q}(E),\beta),$ we have
${\mathsf{H}}_{\mu}^{q,\alpha}(E)=0$. It follows from this that
${\overline{\mathsf{H}}}_{\mu}^{q,\alpha}(F)=0$ for all $F\subseteq E$. Thus,
there exists $\\{F_{i}\\}$ of bounded subset of $F$ such that
$F\subseteq\cup_{i}F_{i}$, and
$\sup_{i}{\mathsf{L}}_{\mu}^{q,\alpha}(F_{i})<\infty$. We conclude that,
$\sup_{i}{\Theta}_{\mu}^{q}(F_{i})\leq\alpha$. It is also a contradiction.

The proof of the second statement is identical to the proof of the statement
in the first part and is therefore omitted. ∎

######  Proposition 4.

If $q\in\mathbb{R}$ and $\mu\in{\mathcal{P}}_{D}(\mathbb{R}^{n})$, then for
any subset $E$ of $\operatorname{supp}\mu$, we have

${\mathsf{B}}_{\mu}^{q}(E)={B}_{\mu}^{q}(E).$

###### Proof.

This follows easily from Propositions 2 and 3, Propositions 2.19 and 2.22 in
[42] and Lemma 4.1 in [46]. ∎

######  Remark 3.

The results developed by Falconer in [22] are obtained as a special case of
the multifractal results by setting $q=0$.

##  3\. A multifractal formalism for Hewitt-Stromberg measures

Multifractal analysis was proved to be a very useful technique in the analysis
of measures, both in theory and applications. The upper and lower local
dimensions of a measure $\mu$ on $\mathbb{R}^{n}$ at a point $x$ are
respectively given by :

$\overline{\alpha}_{\mu}(x)=\limsup_{r\to 0}\frac{\log\mu(B(x,r))}{\log
r}\quad\text{and}\quad\underline{\alpha}_{\mu}(x)=\liminf_{r\to
0}\frac{\log\mu(B(x,r))}{\log r},$

where $B(x,r)$ denote the closed ball of center $x$ and radius $r$. We refer
to the common value as the local dimension of $\mu$ at $x$, and denote it by
${\alpha}_{\mu}(x)$.

The level set of the local dimension of $\mu$ contains a crucial information
on the geometrical properties of $\mu$. The aim of the multifractal analysis
of a measure is to relate the Hausdorff and packing dimensions of these levels
sets to the Legendre transform of some concave (convex) function (see for
example [2, 10, 11, 12, 13, 17, 41, 42]). For $\alpha\geq 0$, we define the
fractal sets,

${\overline{E}}^{\alpha}=\Big{\\{}x\in\operatorname{supp}\mu\;\;\big{|}\;\;\overline{\alpha}_{\mu}(x)\leq\alpha\Big{\\}};\quad\overline{E}_{\alpha}=\Big{\\{}x\in\operatorname{supp}\mu\;\;\big{|}\;\;\overline{\alpha}_{\mu}(x)\geq\alpha\Big{\\}}$

and

${\underline{E}}^{\alpha}=\Big{\\{}x\in\operatorname{supp}\mu\;\;\big{|}\;\;\underline{\alpha}_{\mu}(x)\leq\alpha\Big{\\}};\quad\underline{E}_{\alpha}=\Big{\\{}x\in\operatorname{supp}\mu\;\;\big{|}\;\;\underline{\alpha}_{\mu}(x)\geq\alpha\Big{\\}}.$

Also, let

$E(\alpha)={\overline{E}}^{\alpha}\cap{\underline{E}}_{\alpha}=\Big{\\{}x\in\operatorname{supp}\mu\;\;\big{|}\;\;\alpha_{\mu}(x)=\alpha\Big{\\}}.$

Theorem 4 allows us to consider the relationship between the lower and upper
multifractal Hewitt-Stromberg functions $\mathsf{b}_{\mu}$ and
$\mathsf{B}_{\mu}$ and the multifractal spectra. We start by giving an upper
bound theorem. For $\mu\in{\mathcal{P}}(\mathbb{R}^{n})$, set

$\alpha_{min}=\sup_{0<q}-\frac{{{\mathsf{b}}_{\mu}(q)}}{q},\quad\alpha_{max}=\inf_{0>q}-\frac{{{\mathsf{b}}_{\mu}(q)}}{q},\quad\beta_{min}=\sup_{0<q}-\frac{{{\mathsf{B}}_{\mu}(q)}}{q},\quad\beta_{max}=\inf_{0>q}-\frac{{{\mathsf{B}}_{\mu}(q)}}{q}.$

Before stating this formally, we remind the reader that if
$\varphi:\mathbb{R}\rightarrow\mathbb{R}$ is a real valued function, then the
Legendre transform $\varphi^{*}:\mathbb{R}\rightarrow[-\infty,+\infty]$ of
$\varphi$ is defined by

$\varphi^{*}(x)=\inf_{y}\Big{(}xy+\varphi(y)\Big{)}.$

Now, we can state our multifractal formalism.

######  Theorem 4.

Let $\alpha\geq 0$, then the following hold

  1. (1)

$\alpha_{min}\leq\inf\overline{\alpha}_{\mu}(x)\leq\sup\overline{\alpha}_{\mu}(x)\leq\beta_{max}$

and

$\beta_{min}\leq\inf\underline{\alpha}_{\mu}(x)\leq\sup\underline{\alpha}_{\mu}(x)\leq\alpha_{max}.$

  2. (2)

$\underline{\dim}_{MB}\;E(\alpha)\begin{cases}\leq{\mathsf{b}}_{\mu}^{*}(\alpha)&\;\text{if}\;\;\alpha\in(\alpha_{min},\alpha_{max})\\\
&\\\ =0&\;\text{if}\;\;\alpha\notin(\alpha_{min},\alpha_{max}).\end{cases}$

  3. (3)

$\overline{\dim}_{MB}\;E(\alpha)\begin{cases}\leq{\mathsf{B}}_{\mu}^{*}(\alpha)&\;\text{if}\;\;\alpha\in(\alpha_{min},\alpha_{max})\\\
&\\\ =0&\;\text{if}\;\;\alpha\notin(\alpha_{min},\alpha_{max}).\end{cases}$

###### Proof.

This theorem follows immediately from the following lemmas. ∎

######  Lemma 1.

If $\mu\in{\mathcal{P}}(\mathbb{R}^{n})$ and $\alpha\geq 0$, then

  1. (1)

${\underline{E}}^{\alpha}=\emptyset$ for $\alpha<\beta_{min}$ and
${\overline{E}}_{\alpha}=\emptyset$ for $\alpha>\beta_{max}$.

  2. (2)

${\underline{E}}_{\alpha}=\emptyset$ for $\alpha>\alpha_{max}$ and
${\overline{E}}^{\alpha}=\emptyset$ for $\alpha<\alpha_{min}$.

###### Proof.

  1. (1)

Let $x\in{\underline{E}}^{\alpha}$ and $\alpha<\beta_{min}$. There exists
$\epsilon>0$ and $q_{0}>0$ such that
$-q_{0}(\alpha+\epsilon)>{\mathsf{B}}_{\mu}(q_{0})$. Since
$x\in{\underline{E}}^{\alpha}$, we can thus choose $(r_{n})_{n}$ such that

$r_{n}\to 0\quad\text{and}\quad\frac{\log\mu(B(x,r_{n}))}{\log
r_{n}}<\alpha+\epsilon.$

For brevity write $t=-q_{0}(\alpha+\epsilon)$, then we obtain

$\mu(B(x,r_{n}))^{q_{0}}~{}(2r_{n})^{t}>2^{t}.$

Hence, for all $n\in\mathbb{N}$,

$M_{\mu,r_{n}}^{q_{0},t}\big{(}\\{x\\}\big{)}~{}(2r_{n})^{t}\geq\mu(B(x,r_{n}))^{q_{0}}~{}(2r_{n})^{t}>2^{t}>0.$

It follows that ${\mathsf{P}}_{\mu}^{q_{0},t}\big{(}\\{x\\}\big{)}>0$. We
therefore conclude that

$t\leq{\mathsf{B}}_{\mu}^{q_{0}}\big{(}\\{x\\}\big{)}\leq{\mathsf{B}}_{\mu}(q_{0})$

which contradicts the fact that
$-q_{0}(\alpha+\epsilon)>{\mathsf{B}}_{\mu}(q_{0})$.

The proof of the second statement is identical to the proof of the first
statement in Part (1) and is therefore omitted.

  2. (2)

Let $x\in{\underline{E}}_{\alpha}$ and $\alpha>\alpha_{max}$. Then, we can
find $\epsilon>0$ and $q_{0}<0$ such that
$-q_{0}(\alpha-\epsilon)>{\mathsf{b}}_{\mu}(q_{0})$. Since
$x\in{\underline{E}}_{\alpha}$, we can choose $r_{0}$ such that for
$0<r<r_{0}$ we have

$\frac{\log\mu(B(x,r))}{\log r}>\alpha-\epsilon.$

Then, for $t=-q_{0}(\alpha-\epsilon)$ and $r<r_{0}$, we have

$\mu(B(x,r))^{q_{0}}~{}(2r)^{t}>2^{t}.$

Therefore, it follows that, for all $r<r_{0}$,

$N_{\mu,r}^{q_{0},t}\big{(}\\{x\\}\big{)}~{}(2r)^{t}=\mu(B(x,r))^{q_{0}}(2r)^{t}>2^{t}>0.$

which implies that ${\mathsf{H}}_{\mu}^{q_{0},t}\big{(}\\{x\\}\big{)}>0$. It
now follows from this that

$t\leq{\mathsf{b}}_{\mu}^{q_{0}}\big{(}\\{x\\}\big{)}\leq{\mathsf{b}}_{\mu}(q_{0})$

which contradicts the fact that
$-q_{0}(\alpha-\epsilon)>{\mathsf{b}}_{\mu}(q_{0})$.

The proof of the second statement in part (2) is very similar to the proof of
the first statement and is therefore omitted.

∎

######  Lemma 2.

Let $\mu\in{\mathcal{P}}(\mathbb{R}^{n})$, $\alpha\geq 0,q,t\in\mathbb{R}$ and
$\delta>0$ such that $\delta\leq\alpha q+t.$ Then the following hold

  1. (1)

    1. (a)

${\mathsf{H}}^{\alpha q+t+\delta}({\overline{E}}^{\alpha})\leq\;2^{\alpha
q+\delta}~{}\mathsf{H}_{\mu}^{q,t}({\overline{E}}^{\alpha})\;\;\text{for}\;\;\;0\leq
q.$

    2. (b)

${\mathsf{H}}^{\alpha q+t+\delta}({\underline{E}}_{\alpha})\leq\;2^{\alpha
q+\delta}~{}\mathsf{H}_{\mu}^{q,t}({\underline{E}}_{\alpha})\;\;\text{for}\;\;\;0\geq
q.$

    3. (c)

If $0\leq\alpha q+{{\mathsf{b}}}_{\mu}(q)$ then

$\underline{\dim}_{MB}({\overline{E}}^{\alpha})\leq\inf_{q\geq 0}\alpha
q+{{\mathsf{b}}}_{\mu}(q)\quad\text{and}\quad\underline{\dim}_{MB}({\underline{E}}_{\alpha})\leq\inf_{q\leq
0}\alpha q+{{\mathsf{b}}}_{\mu}(q).$

In particular $\underline{\dim}_{MB}({\overline{E}}^{\alpha})\leq\alpha$.

  2. (2)

    1. (a)

${\mathsf{P}}^{\alpha q+t+\delta}({\overline{E}}^{\alpha})\leq\;2^{\alpha
q+\delta}~{}\mathsf{P}_{\mu}^{q,t}({\overline{E}}^{\alpha})\;\;\text{for}\;\;\;0\leq
q.$

    2. (b)

${\mathsf{P}}^{\alpha q+t+\delta}({\underline{E}}_{\alpha})\leq\;2^{\alpha
q+\delta}~{}\mathsf{P}_{\mu}^{q,t}({\underline{E}}_{\alpha})\;\;\text{for}\;\;\;0\geq
q.$

    3. (c)

If $0\leq\alpha q+{{\mathsf{b}}}_{\mu}(q)$ then

$\overline{\dim}_{MB}({\overline{E}}^{\alpha})\leq\inf_{q\geq 0}\alpha
q+{{\mathsf{B}}}_{\mu}(q)\quad\text{and}\quad\overline{\dim}_{MB}({\underline{E}}_{\alpha})\leq\inf_{q\leq
0}\alpha q+{{\mathsf{B}}}_{\mu}(q).$

In particular $\overline{\dim}_{MB}({\overline{E}}^{\alpha})\leq\alpha$.

###### Proof.

An exhaustive proof of this lemma would require considerable repetition. To
avoid this we prove (1)-(a) and (2)-(a).

  1. (1)

    1. (a)

Clearly, the statement is true for $q=0$. For $m\in\mathbb{N}$, write

$E_{m}=\left\\{x\in{\overline{E}}^{\alpha}\;\;\Big{|}\;\;\frac{\log\mu(B(x,r))}{\log
r}\leq\alpha+\frac{\delta}{q}\;\;\text{for}\;\;0<r<\frac{1}{m}\right\\}.$

Fix $m\in\mathbb{N}$ and $r>0$ such that $0<r<\frac{1}{m}$. Let
$\Big{(}B(x_{i},r)\Big{)}_{i}$ be a centered covering of $E_{m}$. Next, we
observe that

$\displaystyle\frac{\log\mu(B(x_{i},r))}{\log r}\leq\alpha+\frac{\delta}{q}$
$\displaystyle\Longrightarrow$ $\displaystyle\mu(B(x_{i},r))^{q}\geq
r^{q\alpha+\delta}$ $\displaystyle\Longrightarrow$
$\displaystyle\sum_{i}\mu(B(x_{i},r))^{q}\geq
N_{r}(E_{m})~{}r^{q\alpha+\delta}$ $\displaystyle\Longrightarrow$
$\displaystyle N^{q}_{\mu,r}(E_{m})\geq N_{r}(E_{m})~{}r^{q\alpha+\delta}$
$\displaystyle\Longrightarrow$ $\displaystyle\mathsf{L}_{\mu}^{q,t}(E_{m})\geq
2^{-\alpha q-\delta}~{}\overline{\mathsf{H}}^{q\alpha+\delta+t}(E_{m})$
$\displaystyle\Longrightarrow$
$\displaystyle\mathsf{H}_{\mu}^{q,t}(E_{m})\geq\overline{\mathsf{H}}_{\mu}^{q,t}(E_{m})\geq
2^{-\alpha q-\delta}~{}\overline{\mathsf{H}}^{q\alpha+\delta+t}(E_{m}).$

Now from this and since $E_{m}\nearrow{\overline{E}}^{\alpha}$ we can deduce
that

$\mathsf{H}_{\mu}^{q,t}({\overline{E}}^{\alpha})\geq 2^{-\alpha
q-\delta}~{}\mathsf{H}^{q\alpha+\delta+t}({\overline{E}}^{\alpha}).$

  2. (2)

    1. (a)

Once again for $q=0$ the statement is well known. For $m\in\mathbb{N}$, put

$E_{m}=\left\\{x\in{\overline{E}}^{\alpha}\;\;\Big{|}\;\;\frac{\log\mu(B(x,r))}{\log
r}\leq\alpha+\frac{\delta}{q}\;\;\text{for}\;\;0<r<\frac{1}{m}\right\\}.$

We therefore fix $m\in\mathbb{N}$ and $r>0$ such that $0<r<\frac{1}{m}$. Let
$\Big{(}B(x_{i},r)\Big{)}_{i\in\\{1,\ldots,M_{r}(E_{m})\\}}$ be a packing of
$E_{m}$. Then we have

$\displaystyle\frac{\log\mu(B(x_{i},r))}{\log r}\leq\alpha+\frac{\delta}{q}$
$\displaystyle\Longrightarrow$ $\displaystyle\mu(B(x_{i},r))^{q}\geq
r^{q\alpha+\delta}$ $\displaystyle\Longrightarrow$
$\displaystyle\sum_{i}\mu(B(x_{i},r))^{q}\geq
M_{r}(E_{m})~{}r^{q\alpha+\delta}$ $\displaystyle\Longrightarrow$
$\displaystyle M^{q}_{\mu,r}(E_{m})\geq M_{r}(E_{m})~{}r^{q\alpha+\delta}$
$\displaystyle\Longrightarrow$ $\displaystyle\mathsf{C}_{\mu}^{q,t}(E_{m})\geq
2^{-\alpha q-\delta}~{}\overline{\mathsf{P}}^{q\alpha+\delta+t}(E_{m})$
$\displaystyle\Longrightarrow$ $\displaystyle\mathsf{P}_{\mu}^{q,t}(E_{m})\geq
2^{-\alpha q-\delta}~{}\mathsf{P}^{q\alpha+\delta+t}(E_{m}).$

However, since $E_{m}\nearrow{\overline{E}}^{\alpha},$ we conclude that

$\mathsf{P}_{\mu}^{q,t}({\overline{E}}^{\alpha})\geq 2^{-\alpha
q-\delta}~{}\mathsf{P}^{q\alpha+\delta+t}({\overline{E}}^{\alpha})$

which yields the desired result.

∎

Our purpose of the following theorems is to propose a sufficient condition
that gives the lower bound.

######  Theorem 5.

Let $q,t\in\mathbb{R}$, and $\alpha>0$ such that $\alpha q+t\geq 0$. Let
$A\subseteq E(\alpha)$ is a Borel set.

  1. (1)

If $\mathsf{H}_{\mu}^{q,t}(A)>0$, then

$\underline{\dim}_{MB}\;E(\alpha)\geq\alpha q+t.$

In particular, if the multifractal function ${\mathsf{b}}_{\mu}$ is
differentiable at $q,$ then, provided that
${\mathsf{b}}_{\mu}^{*}\Big{(}-{\mathsf{b}}_{\mu}^{{}^{\prime}}(q)\Big{)}\geq
0$ and
$\mathsf{H}_{\mu}^{q,{\mathsf{b}}_{\mu}(q)}\Big{(}E(-{\mathsf{b}_{\mu}}^{\prime}(q))\Big{)}>0$,
we have

$\underline{\dim}_{MB}E\left(-{\mathsf{b}}_{\mu}^{{}^{\prime}}(q)\right)={\mathsf{b}}_{\mu}^{*}\left(-{\mathsf{b}}_{\mu}^{{}^{\prime}}(q)\right).$

  2. (2)

If $\mathsf{P}_{\mu}^{q,t}(A)>0$, then

$\overline{\dim}_{MB}\;E(\alpha)\geq\alpha q+t.$

In particular, if the multifractal function ${\mathsf{B}}_{\mu}$ is
differentiable at $q,$ then, provided that
${\mathsf{B}}_{\mu}^{*}\left(-{\mathsf{B}}_{\mu}^{{}^{\prime}}(q)\right)\geq
0$ and
$\mathsf{P}_{\mu}^{q,{\mathsf{B}}_{\mu}(q)}\Big{(}E(-{\mathsf{B}_{\mu}}^{\prime}(q))\Big{)}>0$,
we have

$\overline{\dim}_{MB}E\left(-{\mathsf{B}}_{\mu}^{{}^{\prime}}(q)\right)={\mathsf{B}}_{\mu}^{*}\left(-{\mathsf{B}}_{\mu}^{{}^{\prime}}(q)\right).$

###### Proof.

This follows easily from Theorem 4 and the following lemma. ∎

######  Lemma 3.

Let $\mu\in{\mathcal{P}}(\mathbb{R}^{n})$, $\alpha\geq 0,q,t\in\mathbb{R}$ and
$\delta>0$ such that $\delta\leq\alpha q+t.$ Then we have the following

  1. (1)

    1. (a)

If $A\subseteq{\overline{E}}^{\alpha},$ is Borel then ${\mathsf{P}}^{\alpha
q+t-\delta}(A)\geq\;2^{\alpha
q-\delta}~{}\mathsf{P}_{\mu}^{q,t}(A)\;\;\text{for}\;\;\;0\geq q.$

    2. (b)

If $A\subseteq{\underline{E}}_{\alpha}$, is Borel then ${\mathsf{P}}^{\alpha
q+t-\delta}(A)\geq\;2^{\alpha
q-\delta}~{}\mathsf{P}_{\mu}^{q,t}(A)\;\;\text{for}\;\;\;0\leq q.$
In particular, if $\mu(A)>0$ then $\overline{\dim}_{MB}(A)\geq\alpha$.

  2. (2)

    1. (a)

If $A\subseteq{\overline{E}}^{\alpha},$ is Borel then ${\mathsf{H}}^{\alpha
q+t-\delta}(A)\geq\;2^{\alpha
q-\delta}~{}\mathsf{H}_{\mu}^{q,t}(A)\;\;\text{for}\;\;\;0\geq q.$

    2. (b)

If $A\subseteq{\underline{E}}_{\alpha}$, is Borel then ${\mathsf{H}}^{\alpha
q+t-\delta}(A)\geq\;2^{\alpha
q-\delta}~{}\mathsf{H}_{\mu}^{q,t}(A)\;\;\text{for}\;\;\;0\leq q.$
In particular, if $\mu(A)>0$ then $\underline{\dim}_{MB}(A)\geq\alpha$.

###### Proof.

An exhaustive proof of this theorem would require considerable repetition. For
this we only prove (1)-(a) and (2)-(a), the other assertions are similar.

  1. (1)

    1. (a)

Clearly the statement is true for $q=0$. For $m\in\mathbb{N}$, write

$E_{m}=\left\\{x\in A\;\;\Big{|}\;\;\frac{\log\mu(B(x,r))}{\log
r}\leq\alpha-\frac{\delta}{q}\;\;\text{for}\;\;0<r<\frac{1}{m}\right\\}.$

Let $m\in\mathbb{N}$ and $r>0$ such that $0<r<\frac{1}{m}$. Let
$\Big{(}B(x_{i},r)\Big{)}_{i}$ be a centred packing of $E_{m}$. We have

$\displaystyle\frac{\log\mu(B(x_{i},r))}{\log r}\leq\alpha-\frac{\delta}{q}$
$\displaystyle\Longrightarrow$ $\displaystyle\mu(B(x_{i},r))^{q}\leq
r^{q\alpha-\delta}$ $\displaystyle\Longrightarrow$
$\displaystyle\sum_{i}\mu(B(x_{i},r))^{q}\leq
M_{r}(E_{m})~{}r^{q\alpha-\delta}$ $\displaystyle\Longrightarrow$
$\displaystyle M_{\mu,r}^{q}(E_{m})\leq M_{r}(E_{m})~{}r^{q\alpha-\delta}$
$\displaystyle\Longrightarrow$ $\displaystyle\mathsf{C}_{\mu}(E_{m})^{q,t}\leq
2^{-\alpha q+\delta}~{}\overline{\mathsf{P}}^{q\alpha-\delta+t}(E_{m})$
$\displaystyle\Longrightarrow$ $\displaystyle\mathsf{P}_{\mu}^{q,t}(E_{m})\leq
2^{-\alpha q+\delta}~{}\mathsf{P}^{q\alpha-\delta+t}(E_{m}).$

Finally, since $E_{m}\nearrow A$ we conclude that

$\mathsf{P}_{\mu}^{q,t}(A)\leq 2^{-\alpha
q+\delta}~{}\mathsf{P}^{q\alpha-\delta+t}(A).$

  2. (2)

    1. (a)

It is well known that the statement is true for $q=0$. For $m\in\mathbb{N}$,
we define the set $E_{m}$ by

$E_{m}=\left\\{x\in A\;\;\Big{|}\;\;\frac{\log\mu(B(x,r))}{\log
r}\leq\alpha-\frac{\delta}{q}\;\;\text{for}\;\;0<r<\frac{1}{m}\right\\}.$

Next, fix $m\in\mathbb{N}$ and $r>0$ such that $0<r<\frac{1}{m}$. Let
$\Big{(}B(x_{i},r)\Big{)}_{i\in\\{1,\ldots,N_{r}(F)\\}}$ be a centred covering
of $F\subset E_{m}$. We get

$\displaystyle\frac{\log\mu(B(x_{i},r))}{\log r}\leq\alpha-\frac{\delta}{q}$
$\displaystyle\Longrightarrow$ $\displaystyle\mu(B(x_{i},r))^{q}\leq
r^{q\alpha-\delta}$ $\displaystyle\Longrightarrow$
$\displaystyle\sum_{i}\mu(B(x_{i},r))^{q}\leq N_{r}(F)~{}r^{q\alpha-\delta}$
$\displaystyle\Longrightarrow$ $\displaystyle N^{q}_{\mu,r}(F)\leq
N_{r}(F)~{}r^{q\alpha-\delta}$ $\displaystyle\Longrightarrow$
$\displaystyle\mathsf{L}_{\mu}^{q,t}(F)\leq 2^{t-\alpha
q+\delta}~{}\overline{\mathsf{H}}^{q\alpha-\delta+t}(F)$
$\displaystyle\Longrightarrow$
$\displaystyle\overline{\mathsf{H}}_{\mu}^{q,t}(F)\leq 2^{-\alpha
q+\delta}~{}\overline{\mathsf{H}}^{q\alpha-\delta+t}(E_{m}).$

Putting these together we have that

$\mathsf{H}_{\mu}^{q,t}(A)\leq 2^{-\alpha
q+\delta}~{}\mathsf{H}^{q\alpha-\delta+t}(A).$

This proves the lemma.

∎

######  Theorem 6.

Let $q\in\mathbb{R}$ and suppose that
$\mathsf{H}^{q,\mathsf{\Lambda}_{\mu}(q)}_{\mu}(\operatorname{supp}\mu)>0.$
Then,

$\displaystyle\begin{array}[]{ll}\underline{\dim}_{MB}\Big{(}\underline{E}_{\;-\mathsf{\Lambda}_{\mu+}^{\prime}(q)}\cap\overline{E}^{\;-\mathsf{\Lambda}_{\mu-}^{\prime}(q)}\Big{)}\end{array}\geq\left\\{\begin{array}[]{ll}-\mathsf{\Lambda}_{\mu-}^{\prime}(q)q+\mathsf{\Lambda}_{\mu}(q),\quad\text{for}\quad
q\leq 0\hbox{,}\\\ \\\
-\mathsf{\Lambda}_{\mu+}^{\prime}(q)q+\mathsf{\Lambda}_{\mu}(q),\quad\text{for}\quad
q\geq 0\hbox{.}\end{array}\right.$

###### Proof.

It is well known from Lemma 3 that for all $\delta>0$ and $t\in\mathbb{R}$,

$\displaystyle\left\\{\begin{array}[]{ll}\mathsf{H}^{-\mathsf{\Lambda}^{\prime}_{\mu+}(q)q+t-\delta}(E(q))\geq
2^{-\mathsf{\Lambda}^{\prime}_{\mu+}(q)q-\delta}~{}\mathsf{H}^{q,t}_{\mu}(E(q)),\quad\text{for}\quad
q\geq 0\hbox{,}\\\ &\\\
\mathsf{H}^{-\mathsf{\Lambda}^{\prime}_{\mu-}(q)q+t-\delta}(E(q))\geq
2^{-\mathsf{\Lambda}^{\prime}_{\mu-}(q)q-\delta}~{}\mathsf{H}^{q,t}_{\mu}(E(q)),\quad\text{for}\quad
q\leq 0\hbox{}\end{array}\right.$

where the set $E(q)$ is defined by

$E(q)=\underline{E}_{\;-\mathsf{\Lambda}_{\mu+}^{\prime}(q)}\cap\overline{E}^{\;-\mathsf{\Lambda}_{\mu-}^{\prime}(q)}.$

Theorem 6 is then an easy consequence of the following lemma. ∎

######  Lemma 4.

One has
$\mathsf{H}^{q,\mathsf{\Lambda}_{\mu}(q)}_{\mu}\Big{(}\operatorname{supp}\mu\setminus
E(q)\Big{)}=0.$

###### Proof.

Let us introduce, for $\alpha$ and $\beta$ in $\mathbb{R}$

$X_{\alpha}=\operatorname{supp}\mu\setminus\underline{E}_{\alpha}\quad\text{and}\quad
Y^{\beta}=\operatorname{supp}\mu\setminus\overline{E}^{\beta}.$

It clearly suffices to prove that

$\displaystyle\mathsf{H}^{q,\mathsf{\Lambda}_{\mu}(q)}_{\mu}\big{(}X_{\alpha}\big{)}=0,\quad\text{for
all}\quad\alpha<-\mathsf{\Lambda}_{\mu+}^{\prime}(q)$ (3.3)

and

$\displaystyle\mathsf{H}^{q,\mathsf{\Lambda}_{\mu}(q)}_{\mu}\big{(}Y^{\beta}\big{)}=0,\quad\text{for
all}\quad\beta>-\mathsf{\Lambda}_{\mu-}^{\prime}(q).$ (3.4)

Indeed, it is clear that

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\mathsf{H}^{q,\mathsf{\Lambda}_{\mu}(q)}_{\mu}\Big{(}\operatorname{supp}\mu\setminus\big{(}\underline{E}_{\;-\mathsf{\Lambda}_{\mu+}^{\prime}(q)}\cap\overline{E}^{\;-\mathsf{\Lambda}_{\mu-}^{\prime}(q)}\big{)}\Big{)}$
$\displaystyle\leq$
$\displaystyle\mathsf{H}^{q,\mathsf{\Lambda}_{\mu}(q)}_{\mu}\Big{(}\operatorname{supp}\mu\setminus\underline{E}_{\;-\mathsf{\Lambda}_{\mu+}^{\prime}(q)}\Big{)}+\mathsf{H}^{q,\mathsf{\Lambda}_{\mu}(q)}_{\mu}\Big{(}\operatorname{supp}\mu\setminus\overline{E}^{\;-\mathsf{\Lambda}_{\mu-}^{\prime}(q)}\Big{)}$
$\displaystyle\leq$
$\displaystyle\mathsf{H}^{q,\mathsf{\Lambda}_{\mu}(q)}_{\mu}\left(\bigcup_{\alpha<-\mathsf{\Lambda}_{\mu+}^{\prime}(q)}\underline{E}_{\alpha}\right)+\mathsf{H}^{q,\mathsf{\Lambda}_{\mu}(q)}_{\mu}\left(\bigcup_{\beta>-\mathsf{\Lambda}_{\mu-}^{\prime}(q)}\overline{E}^{\beta}\right)$
$\displaystyle\leq$
$\displaystyle\sum_{\alpha<-\mathsf{\Lambda}_{\mu+}^{\prime}(q)}\mathsf{H}^{q,\mathsf{\Lambda}_{\mu}(q)}_{\mu}\big{(}X_{\alpha}\big{)}+\sum_{\beta>-\mathsf{\Lambda}_{\mu-}^{\prime}(q)}\mathsf{H}^{q,\mathsf{\Lambda}_{\mu}(q)}_{\mu}\big{(}Y^{\beta}\big{)}=0.$

We only have to prove that (3.3). The proof of (3.4) is identical to the proof
of (3.3) and is therefore omitted.

Let $\alpha<-\mathsf{\Lambda}_{\mu+}^{\prime}(q)$ and $t>0,$ such that
$\mathsf{\Lambda}_{\mu}(q+t)<\mathsf{\Lambda}_{\mu}(q)-\alpha t$, we have

$\mathsf{C}^{q+t,\mathsf{\Lambda}_{\mu}(q)-\alpha
t}_{\mu}\big{(}\operatorname{supp}\mu\big{)}=0.$

For $x\in X_{\alpha}$ and $\delta>0$, we can find $\lambda_{x}\geq 2$ and
$\frac{\delta}{\lambda_{x}}<r_{x}<\delta,$ such that

$\mu(B(x,r_{x}))>r_{x}^{\alpha}.$

The family $\Big{(}B(x,r_{x})\Big{)}_{x\in X_{\alpha}}$ is a centered
$\delta$-covering of $\overline{X}_{\alpha}.$ Then, we can choose a finite
subset $J$ of $\mathbb{N}$ such that the family
$\Big{(}B(x_{i},r_{x_{i}})\Big{)}_{i\in J}$ is a centered $\delta$-covering of
$X_{\alpha}$. Take $\lambda=\sup_{i\in J}\lambda_{x_{i}}$, then for all $i\in
J$, we have

$\mu(B(x_{i},\delta))\geq\mu(B(x_{i},r_{x_{i}}))>r_{x_{i}}^{\alpha}\geq\left(\frac{\delta}{\lambda}\right)^{\alpha}.$

Since $\Big{(}B(x_{i},\delta)\Big{)}_{i\in J}$ is a centered covering of
$\overline{X}_{\alpha}.$ Then, using Besicovitch’s covering theorem, we can
construct $\xi$ finite sub-families $\Big{(}B(x_{1j},\delta)\Big{)}_{j}$,
…,$\Big{(}B(x_{\xi j},\delta)\Big{)}_{j}$, such that each
$X_{\alpha}\subseteq\displaystyle\bigcup_{i=1}^{\xi}\bigcup_{j}B(x_{ij},\delta)$
and $\Big{(}B(x_{ij},\delta)\Big{)}_{j}$ is a packing of $X_{\alpha}$. We
clearly have

$\mu(B(x_{ij},\delta)^{q}~{}\delta^{\mathsf{\Lambda}_{\mu}(q)}\leq\lambda^{\alpha
t}~{}\mu(B(x_{ij},\delta)^{q+t}~{}\delta^{\mathsf{\Lambda}_{\mu}(q)-\alpha
t}.$

It therefore follows that

$N_{\mu}^{q}(X_{\alpha})~{}\delta^{\mathsf{\Lambda}_{\mu}(q)}\leq\lambda^{\alpha
t}~{}\xi~{}M_{\mu}^{q+t}(X_{\alpha})~{}\delta^{\mathsf{\Lambda}_{\mu}(q)-\alpha
t}.$

Letting $\delta\to 0$ now yields

$\overline{\mathsf{H}}_{\mu}^{q,\mathsf{\Lambda}_{\mu}(q)}(X_{\alpha})\leq\mathsf{L}_{\mu}^{q,\mathsf{\Lambda}_{\mu}(q)}(X_{\alpha})\leq
2^{\alpha t}~{}\lambda^{\alpha
t}~{}\xi~{}\mathsf{C}_{\mu}^{q+t,\mathsf{\Lambda}_{\mu}(q)-\alpha
t}(X_{\alpha})\leq 2^{\alpha t}~{}\lambda^{\alpha
t}~{}\xi~{}\mathsf{C}_{\mu}^{q+t,\mathsf{\Lambda}_{\mu}(q)-\alpha
t}(\operatorname{supp}\mu)=0.$

Remark that, in the last inequality, we can replace $X_{\alpha}$ by any
arbitrary subset of $X_{\alpha}.$ Then, we can finally conclude that

${\mathsf{H}}_{\mu}^{q,\mathsf{\Lambda}_{\mu}(q)}(X_{\alpha})\leq 2^{\alpha
t}~{}\lambda^{\alpha
t}~{}\xi~{}\mathsf{C}_{\mu}^{q+t,\mathsf{\Lambda}_{\mu}(q)-\alpha
t}(\operatorname{supp}\mu)=0.$

This completes the proof of (3.3). ∎

The following result proves that the condition
$\mathsf{H}^{q,\mathsf{\Lambda}_{\mu}(q)}_{\mu}(\operatorname{supp}\mu)>0$ is
very close to being a necessary and sufficient condition for the validity of
our multifractal formalism.

######  Theorem 7.

Let $q\in\mathbb{R}$ and $\mu$ be a compact supported Borel probability
measure on $\mathbb{R}^{n}.$ Suppose that one of the following hypotheses is
satisfied,

  1. (1)

$\underline{\dim}_{MB}\Big{(}\underline{E}_{\;-\mathsf{\Lambda}_{\mu+}^{\prime}(q)}\cap\overline{E}^{\;-\mathsf{\Lambda}_{\mu-}^{\prime}(q)}\Big{)}\geq-\mathsf{\Lambda}_{\mu+}^{\prime}(q)q+\mathsf{\Lambda}_{\mu}(q),\quad\text{for}\quad
q\leq 0.$

  2. (2)

$\underline{\dim}_{MB}\Big{(}\underline{E}_{\;-\mathsf{\Lambda}_{\mu+}^{\prime}(q)}\cap\overline{E}^{\;-\mathsf{\Lambda}_{\mu-}^{\prime}(q)}\Big{)}\geq-\mathsf{\Lambda}_{\mu-}^{\prime}(q)q+\mathsf{\Lambda}_{\mu}(q),\quad\text{for}\quad
q\geq 0.$

Then,

$\mathsf{b}_{\mu}(q)=\mathsf{B}_{\mu}(q)=\mathsf{\Lambda}_{\mu}(q).$

In other words,

${\mathsf{H}}^{q,t}_{\mu}(\operatorname{supp}\mu)>0\quad\text{for all}\quad
t<\mathsf{\Lambda}_{\mu}(q).$

###### Proof.

We have, for $q\geq 0$

$\underline{E}_{\;-\mathsf{\Lambda}_{\mu+}^{\prime}(q)}\cap\overline{E}^{\;-\mathsf{\Lambda}_{\mu-}^{\prime}(q)}\subseteq\overline{E}^{\;-\mathsf{\Lambda}_{\mu-}^{\prime}(q)},$

it follows immediately that

$-\mathsf{\Lambda}_{\mu-}^{\prime}(q)q+\mathsf{\Lambda}_{\mu}(q)\leq\underline{\dim}_{MB}\Big{(}\underline{E}_{\;-\mathsf{\Lambda}_{\mu+}^{\prime}(q)}\cap\overline{E}^{\;-\mathsf{\Lambda}_{\mu-}^{\prime}(q)}\Big{)}\leq\underline{\dim}_{MB}\Big{(}\overline{E}^{\;-\mathsf{\Lambda}_{\mu-}^{\prime}(q)}\Big{)}.$

Now, suppose that $\alpha=-\mathsf{\Lambda}_{\mu-}^{\prime}(q)$. We only prove
the case where $q\geq 0$. The other one is very similar and is therefore
omitted. We have

$\underline{\dim}_{MB}\Big{(}\overline{E}^{\alpha}\Big{)}\geq\alpha
q+\mathsf{\Lambda}_{\mu}(q).$

Since
$\mathsf{b}_{\mu}(q)\leq\mathsf{B}_{\mu}(q)\leq\mathsf{\Lambda}_{\mu}(q)$, we
only have to prove that $\mathsf{b}_{\mu}(q)\geq\mathsf{\Lambda}_{\mu}(q)$.
Let $t<\mathsf{\Lambda}_{\mu}(q)$ and choose $\beta,$ such that
$\beta<\alpha$. Then, $\beta q+t<\alpha q+\mathsf{\Lambda}_{\mu}(q)$. For
$p\in\mathbb{N},$ we consider the set

$F_{p}=\left\\{x\in\overline{E}^{\alpha}\;\;\Big{|}\;\;\mu(B(x,r))\geq
r^{\beta},\;0<r<\frac{1}{p}\right\\}.$

It is clear that $F_{p}\nearrow\overline{E}^{\alpha}$ as $p\to\infty$. It
follows that, there exists $p>0$, such that

$\underline{\dim}_{MB}(F_{p})>\beta q+t\Rightarrow\mathsf{H}^{\beta
q+t}(F_{p})>0.$

Let $0<r<\frac{1}{p}$ and $\Big{(}B(x_{i},r)\Big{)}_{i}$ be a centered
covering of $F_{p}$. Then,

$\sum_{i}\mu(B(x_{i},r))^{q}~{}r^{t}\geq\sum_{i}r^{\beta q+t}\geq
N_{r}(F_{p})~{}r^{\beta q+t}.$

We conclude that

$N_{\mu,r}^{q}(F_{p})~{}(2r)^{t}\geq 2^{t}~{}N_{r}(F_{p})~{}r^{\beta q+t}$

and then

$\mathsf{L}_{\mu}^{q,t}(F_{p})\geq 2^{-\beta q}~{}\overline{\mathsf{H}}^{\beta
q+t}(F_{p}).$

This implies that

$\mathsf{H}^{q,t}_{\mu}(\operatorname{supp}\mu)\geq\mathsf{H}^{q,t}_{\mu}(\overline{E}^{\alpha})\geq\mathsf{H}^{q,t}_{\mu}(F_{p})\geq
2^{-\beta q}~{}\mathsf{H}^{\beta q+t}(F_{p})>0.$

It therefore follows that $t\leq\mathsf{b}_{\mu}(q)$. Finally, we get

$\mathsf{b}_{\mu}(q)=\mathsf{B}_{\mu}(q)=\mathsf{\Lambda}_{\mu}(q).$

∎

######  Corollary 2.

Assume that
$\mathsf{H}^{q,\mathsf{\Lambda}_{\mu}(q)}_{\mu}(\operatorname{supp}\mu)>0$
hold for all $q\in\mathbb{R}$ and that $\mathsf{\Lambda}_{\mu}$ is
differentiable at $q$. Let $\alpha=-\mathsf{\Lambda}_{\mu}^{\prime}(q)$, there
holds

$\underline{\dim}_{MB}\big{(}E(\alpha)\big{)}=\overline{\dim}_{MB}\big{(}E(\alpha)\big{)}=\mathsf{b}^{*}_{\mu}(\alpha)=\mathsf{B}^{*}_{\mu}(\alpha)=\mathsf{\Lambda}^{*}_{\mu}(\alpha).$

######  Remark 4.

The results of Theorems 6, 7 and Corollary 2 hold if we replace the
multifractal function $\mathsf{\Lambda}_{\mu}$ by the function
$\mathsf{B}_{\mu}$.

Now, we deal with the case where the lower and upper multifractal Hewitt-
Stromberg functions $\mathsf{b}_{\mu}$ and $\mathsf{B}_{\mu}$ do not
necessarily coincide.

######  Theorem 8.

Let $q\in\mathbb{R}$ and $\mu$ be a compact supported Borel probability
measure on $\mathbb{R}^{n}.$

  1. (1)

If the multifractal function ${\mathsf{b}}_{\mu}$ is differentiable at $q,$
then, provided that
${\mathsf{b}}_{\mu}^{*}\Big{(}-{\mathsf{b}}_{\mu}^{{}^{\prime}}(q)\Big{)}\geq
0$
and
${\mathcal{H}}_{\mu}^{q,{\mathsf{b}}_{\mu}(q)}\Big{(}E\left(-{\mathsf{b}}_{\mu}^{{}^{\prime}}(q)\right)\Big{)}>0$,
we have

$\dim_{H}E\left(-{\mathsf{b}}_{\mu}^{{}^{\prime}}(q)\right)=\underline{\dim}_{MB}E\left(-{\mathsf{b}}_{\mu}^{{}^{\prime}}(q)\right)={b}_{\mu}^{*}\left(-{\mathsf{b}}_{\mu}^{{}^{\prime}}(q)\right)={\mathsf{b}}_{\mu}^{*}\left(-{\mathsf{b}}_{\mu}^{{}^{\prime}}(q)\right).$

  2. (2)

If the multifractal function ${B}_{\mu}$ is differentiable at $q,$ then,
provided that ${B}_{\mu}^{*}\Big{(}-{B}_{\mu}^{{}^{\prime}}(q)\Big{)}\geq 0$
and
${\mathsf{P}}_{\mu}^{q,{B}_{\mu}(q)}\Big{(}E\left(-{B}_{\mu}^{{}^{\prime}}(q)\right)\Big{)}>0$,
we have

$\dim_{P}E\left(-{B}_{\mu}^{{}^{\prime}}(q)\right)=\overline{\dim}_{MB}E\left(-{B}_{\mu}^{{}^{\prime}}(q)\right)={\mathsf{B}}_{\mu}^{*}\left(-{B}_{\mu}^{{}^{\prime}}(q)\right)={B}_{\mu}^{*}\left(-{B}_{\mu}^{{}^{\prime}}(q)\right).$

###### Proof.

The proof is similar to the one of Theorem 5. ∎

##  4\. Examples

In this section, more motivations and examples related to these concepts, will
be discussed.

###  4.1. Example 1

The classical multifractal formalism has been proved rigorously for random and
non-random self-similar measures [42, 43], for self-affine measures [10, 45],
for quasi self-similar measures [41], for quasi-Bernoulli measures [10], for
graph directed self-conformal measures [42] and for some Moran measures [61,
62]. Specifically, we have

${b}_{\mu}(q)={\mathsf{b}}_{\mu}(q)={\mathsf{B}}_{\mu}(q)={B}_{\mu}(q)$

and for some $\alpha\geq 0$, we get

${f}_{\mu}(\alpha)=\mathsf{f}_{\mu}(\alpha)=\mathsf{F}_{\mu}(\alpha)={F}_{\mu}(\alpha)={b}_{\mu}^{*}(\alpha)=\mathsf{b}_{\mu}^{*}(\alpha)=\mathsf{B}_{\mu}^{*}(\alpha)={B}_{\mu}^{*}(\alpha).$

###  4.2. Example 2 : Multifractal formalism of homogeneous Moran measures

We will start by defining the homogeneous Moran sets. Let $\\{n_{k}\\}_{k}$
and $\\{\Phi_{k}\\}_{k\geq 1}$ be respectively two sequences of positive
integers and positive vectors such that

$\Phi_{k}=\Big{(}c_{k_{1}},c_{k_{2}},\ldots,c_{k_{n_{k}}}\Big{)},\qquad\displaystyle\sum_{j=1}^{n_{k}}c_{kj}\leq
1,\;k\in\mathbb{N}.$

For any $m,k\in\mathbb{N}$, such that $m\leq k$, let

$D_{m,k}=\Big{\\{}(i_{m},i_{m+1},\ldots,i_{k})\;\;\big{|}\;\;1\leq i_{j}\leq
n_{j},m\leq j\leq k\Big{\\}}$

and

$D_{k}=D_{1,k}=\Big{\\{}(i_{1},i_{2},\ldots,i_{k})\;\;\big{|}\;\;1\leq
i_{j}\leq n_{j},1\leq j\leq k\Big{\\}}.$

We also set

$D_{0}=\emptyset\qquad\text{and}\qquad D=\displaystyle\cup_{k\geq 0}D_{k},$

Considering $\sigma=(i_{1},i_{2},\ldots,i_{k})\in D_{k}$,
$\tau=(j_{1},j_{2},\ldots,j_{m})\in D_{k+1,m}$, we set

$\sigma*\tau=(i_{1},i_{2},\ldots,i_{k},j_{1},j_{2},\ldots,j_{m}).$

######  Definition 1.

Let $X$ be a complete metric space and $I\subseteq X$ a compact set with no
empty interior (for convenience, we assume that the diameter of $I$ is 1). The
collection ${\mathcal{F}}=\\{I_{\sigma},\sigma\in D\\}$ of subsets of $I$ is
said to have a homogeneous Moran structure, if it satisfies the following
conditions (MSC):

a:

$I_{\emptyset}=I$.

b:

For all $k\geq 1$, $(i_{1},i_{2},\ldots,i_{k-1})\in D_{k-1}$,
$I_{i_{1}i_{2}\ldots i_{k}}\big{(}i_{k}\in\\{1,2,\ldots,n_{k}\\}\big{)}$ are
subsets of $I_{i_{1}i_{2}\ldots i_{k-1}}$ and

$I^{\circ}_{i_{1}i_{2}\ldots i_{k-1},i_{k}}\cap I^{\circ}_{i_{1}i_{2}\ldots
i_{k-1},i_{k}^{\prime}}=\emptyset,\qquad 1\leq i_{k}<i_{k}^{\prime}\leq
n_{k},$

where $I^{\circ}$ denotes the interior of $I$.

c:

For all $k\geq 1$ and $1\leq j\leq n_{k}$, taking
$(i_{1},i_{2},\ldots,i_{k-1},j)\in D_{k}$, we have

$0<c_{kj}=c_{i_{1}i_{2}\ldots
i_{k-1}j}=\displaystyle\frac{|I_{i_{1}i_{2}\ldots
i_{k-1}j}|}{|I_{i_{1}i_{2}\ldots i_{k-1}}|}<1,\quad k\geq 2,$

where $|I|$ denotes the diameter of $I$.

Suppose that ${\mathcal{F}}$ is a collection of subsets of $I$ having a
homogeneous Moran structure. We call $E=\displaystyle{\bigcap_{k\geq
1}\bigcup_{\sigma\in D_{k}}}I_{\sigma},$ a homogeneous Moran set determined by
${\mathcal{F}}$, and call
${\mathcal{F}}_{k}=\Big{\\{}\sigma\;\;\big{|}\;\;\sigma\in D_{k}\Big{\\}}$ the
$k$-order fundamental sets of $E$. $I$ is called the original set of $E$. We
assume $\displaystyle\lim_{k\to\infty}\max_{\sigma\in D_{k}}|I_{\sigma}|=0.$
Then, for all $i\in D$, the set $\left\\{\displaystyle\bigcap_{n\geq
1}I_{i_{1}i_{2}\ldots i_{n}}\right\\}$ is a single point. We use the
abbreviation $w|_{k}$ for the first $k$ elements of the sequence

$w=(i_{1},i_{2},\ldots,i_{k},\ldots)\in D,\qquad
I_{k}(w)=I_{w|_{k}}=I_{i_{1}i_{2}\ldots i_{k}}.$

Here, we consider a class of homogeneous Moran sets $E$ witch satisfy a
special property called the strong separation condition (SSC), i.e., take
$I_{\sigma}\in{\mathcal{F}}$. Let
$I_{\sigma*1},I_{\sigma*2},\ldots,I_{\sigma*n_{k+1}}$ be the $n_{k+1}$ basic
intervals of order $k+1$ contained in $I_{\sigma}$ arranged from the left to
the right, Then we assume that for all $1\leq i\leq n_{k+1}-1$,

$\text{dist}(I_{\sigma*i},I_{\sigma*(i+1)})\geq\delta_{k}|I_{\sigma}|,\quad\text{
for all }i\neq j,$

where $(\delta_{k})_{k}$ is a sequence of positive real numbers, such that

$0<\delta=\displaystyle\inf_{k}\delta_{k}.$

We now define a Moran measure. Let $\Big{\\{}p_{i,j}\Big{\\}}_{j=1}^{n_{i}}$
be the probability vectors, i.e. $p_{i,j}>0$ and $\sum_{j=1}^{n_{i}}p_{i,j}=1$
($i=1,2,3,....$), suppose that $p_{0}=\inf\\{p_{i,j}\\}>0$. Let $\mu$ be a
mass distribution on $E$, such that for any $I_{\sigma}$ ($\sigma\in D_{k}$)

$\mu(I_{\sigma})=p_{1,\sigma_{1}}p_{1,\sigma_{2}}\ldots
p_{1,\sigma_{k}}\quad\text{and}\quad\mu\left(\sum_{\sigma\in
D_{k}}I_{\sigma}\right)=1,$

we call $\mu$ be Moran measure.

Finally we define an auxiliary function $\beta_{k}(q)$ as follows: for all
$k\geq 1$ and $q\in\mathbb{R}$, there is a unique number $\beta_{k}(q)$
satisfying

$\sum_{\sigma\in D_{k}}p_{\sigma}^{q}|I_{\sigma}|^{\beta_{k}(q)}=1.$

Set

$\underline{\beta}(q)=\liminf_{k\to+\infty}\beta_{k}(q)\quad\text{and}\quad\overline{\beta}(q)=\limsup_{k\to+\infty}\beta_{k}(q)$

######  Theorem 9.

Suppose that $E$ is a homogeneous Moran set satisfying (SSC) and $\mu$ is the
Moran measure on $E$,

  1. (1)

then for all $q\in\mathbb{R}$,

${b}_{\mu}(q)={\mathsf{b}}_{\mu}(q)={\Theta}_{\mu}^{q}(\operatorname{supp}\mu)=\underline{\beta}(q)$

and

${B}_{\mu}(q)={\mathsf{B}}_{\mu}(q)={{\mathsf{\Lambda}}}_{\mu}^{q}(\operatorname{supp}\mu)=\overline{\beta}(q).$

  2. (2)

Suppose that $\underline{\beta}^{\prime}(q)$ exists and for this real number
$q$

    1. (a)

there is $k_{0}\in\mathbb{N}$ such that $\underline{\beta}(q)\leq\beta_{k}(q)$
for all $k\geq k_{0}$, or

    2. (b)

there is some $c>0$ and $n_{0}\in\mathbb{N}$ such that
$\beta_{k_{i}}(q)-\underline{\beta}(q)\leq\frac{c}{k_{i}}$ for all $k_{i}\geq
n_{0}$ with $\beta_{k_{i}}(q)<\underline{\beta}(q)$.

    3. (c)

$\underline{\beta}(q)$ is smooth.

Then there exist numbers $0\leq\underline{\alpha}\leq\overline{\alpha}$ such
that

${f}_{\mu}(\alpha)=\mathsf{f}_{\mu}(\alpha)=\left\\{\begin{array}[]{ll}b_{\mu}^{*}(\alpha)={\mathsf{b}}_{\mu}^{*}(\alpha)=\underline{\beta}^{*}(\alpha),&\hbox{if}\;\alpha\in(\underline{\alpha},\overline{\alpha})\\\
\\\
0,&\hbox{if}\;\alpha\notin(\underline{\alpha},\overline{\alpha}).\end{array}\right.$

  3. (3)

Suppose that $\overline{\beta}^{\prime}(q)$ exists and for this real number
$q$

    1. (a)

there is $k_{0}\in\mathbb{N}$ such that $\overline{\beta}(q)\geq\beta_{k}(q)$
for all $k\geq k_{0}$, or

    2. (b)

there is some $c>0$ and $n_{0}\in\mathbb{N}$ such that
$\overline{\beta}(q)-\beta_{k_{i}}(q)\leq\frac{c}{k_{i}}$ for all $k_{i}\geq
n_{0}$ with $\beta_{k_{i}}(q)<\overline{\beta}(q)$.

    3. (c)

$\overline{\beta}(q)$ is smooth.

Then there exist numbers $0\leq\underline{\gamma}\leq\overline{\gamma}$ such
that

${F}_{\mu}(\alpha)=\mathsf{F}_{\mu}(\alpha)=\left\\{\begin{array}[]{ll}B_{\mu}^{*}(\alpha)={\mathsf{B}}_{\mu}^{*}(\alpha)=\overline{\beta}^{*}(\alpha),&\hbox{if}\;\alpha\in(\underline{\gamma},\overline{\gamma})\\\
\\\
0,&\hbox{if}\;\alpha\notin(\underline{\gamma},\overline{\gamma}).\end{array}\right.$

  4. (4)

If the limit $\liminf_{k\to+\infty}\beta_{k}(q)=\beta(q)$ exists, and for all
$k\geq 1$, $k(\beta(q)-\beta_{k}(q))<+\infty$, suppose that
$\alpha=-\beta^{\prime}(q)$ exists, then

$\displaystyle{f}_{\mu}(\alpha)=\mathsf{f}_{\mu}(\alpha)=\mathsf{F}_{\mu}(\alpha)={F}_{\mu}(\alpha)={b}_{\mu}^{*}(\alpha)=\mathsf{b}_{\mu}^{*}(\alpha)=\mathsf{B}_{\mu}^{*}(\alpha)={B}_{\mu}^{*}(\alpha).$

###### Proof.

All of the ideas needed to prove Theorem 9 can be found in [63, 64],
Propositions 2, 3 and 4 and Theorem 8. ∎

####  4.2.1. Moran measures for which the classical multifractal formalism is
valid

Let

$\displaystyle n_{k}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}2,&\hbox{k\;\text{ is odd
number},}\\\ \\\ 3,&\hbox{k\;\text{ is even number}.}\end{array}\right.$
$\displaystyle c_{k}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}r_{1},&\hbox{k\;\text{ is odd
number},}\\\ \\\ r_{2},&\hbox{k\;\text{ is even number,}}\end{array}\right.$

where $0<r_{1}<\frac{1}{2}$ and $0<r_{2}<\frac{1}{3}$. Put

$\displaystyle p_{k,j}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}p_{1,j},&\hbox{k\;\text{ is odd
number},\; }1\leq j\leq 2,\\\ \\\ p_{2,j},&\hbox{k\;\text{ is even number,\;
}}1\leq j\leq 3,\end{array}\right.$

where

$\sum_{j=1}^{2}p_{1,j}=1\quad\text{and}\quad\sum_{j=1}^{3}p_{2,j}=1.$

We therefore conclude that

$\displaystyle\beta_{k}(q)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\frac{-\log\sum_{j=1}^{2}p_{1,j}^{q}-\frac{k-1}{k+1}\log\sum_{j=1}^{3}p_{2,j}^{q}}{\log
r_{1}+\frac{k-1}{k+1}\log r_{2}},&\hbox{k\;\text{ is odd number}, }\\\ \\\ \\\
\displaystyle\frac{-\log\sum_{j=1}^{2}p_{1,j}^{q}-\log\sum_{j=1}^{3}p_{2,j}^{q}}{\log
r_{1}+\log r_{2}},&\hbox{k\;\text{ is even number, }}\end{array}\right.$

and

$\beta(q)=\lim_{k\to+\infty}\beta_{k}(q)=\displaystyle\frac{-\log\sum_{j=1}^{2}p_{1,j}^{q}-\log\sum_{j=1}^{3}p_{2,j}^{q}}{\log
r_{1}+\log r_{2}}.$

This clearly implies that $k(\beta(q)-\beta_{k}(q))<+\infty$ and
$\beta^{\prime}(q)$ exists. Now, it follows immediately from Theorem 9 that

${f}_{\mu}(\alpha)=\mathsf{f}_{\mu}(\alpha)=\mathsf{F}_{\mu}(\alpha)={F}_{\mu}(\alpha)={b}_{\mu}^{*}(\alpha)=\mathsf{b}_{\mu}^{*}(\alpha)=\mathsf{B}_{\mu}^{*}(\alpha)={B}_{\mu}^{*}(\alpha).$

####  4.2.2. Moran measures for which the classical multifractal formalism
does not hold

Let $\\{T_{k}\\}_{k\geq 1}$ be a sequence of integers such that

$T_{1}=1,\quad
T_{k}<T_{k+1}\quad\text{and}\quad\lim_{k\to+\infty}\frac{T_{k+1}}{T_{k}}=+\infty$

We define the family

$\displaystyle n_{i}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}2,&\hbox{\text{if}\;}T_{2k-1}\leq
i<T_{2k},\\\ \\\ 3,&\hbox{\text{if}\;}T_{2k}\leq
i<T_{2k+1}.\end{array}\right.$ $\displaystyle c_{i}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}r_{1},&\hbox{\text{if}\;}T_{2k-1}\leq
i<T_{2k},\\\ \\\ r_{2},&\hbox{\text{if}\;}T_{2k}\leq
i<T_{2k+1},\end{array}\right.$

where $0<r_{1}<\frac{1}{2}$ and $0<r_{2}<\frac{1}{3}$. Put

$\displaystyle p_{i,j}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}p_{1,j},&\hbox{\text{if}\;
}T_{2k-1}\leq i<T_{2k},\;1\leq j\leq 2,\\\ \\\ p_{2,j},&\hbox{\text{if}\;
}T_{2k}\leq i<T_{2k+1},\;1\leq j\leq 3,\end{array}\right.$

where

$\sum_{j=1}^{2}p_{1,j}=1\quad\text{and}\quad\sum_{j=1}^{3}p_{2,j}=1.$

We therefore conclude from this

$\beta_{k}(q)=\frac{\log\sum_{\sigma\in D_{k}}\mu(I_{\sigma})^{q}}{-\log
c_{1}\ldots c_{k}}.$

Finally, if $N_{k}$ is the number of integers $i\leq k$ such that
$p_{i,j}=p_{1,j}$, we have

$\beta_{k}(q)=-\frac{\frac{N_{k}}{k}\log(p_{1,1}^{q}+p_{1,2}^{q})+(1-\frac{N_{k}}{k})\log(p_{2,1}^{q}+p_{2,2}^{q}+p_{2,3}^{q})}{\frac{N_{k}}{k}\log
r_{1}+(1-\frac{N_{k}}{k})\log r_{2}}.$

Observing that

$\liminf_{k\to+\infty}\frac{N_{k}}{k}=0\quad\text{and}\quad\limsup_{k\to+\infty}\frac{N_{k}}{k}=1.$

We can then conclude that

$\liminf_{k\to+\infty}\beta_{k}(q)=\inf\left\\{\frac{\log(p_{1,1}^{q}+p_{1,2}^{q})}{-\log
r_{1}},\;\;\frac{\log(p_{2,1}^{q}+p_{2,2}^{q}+p_{2,3}^{q})}{-\log
r_{2}}\right\\}$

and

$\limsup_{k\to+\infty}\beta_{k}(q)=\sup\left\\{\frac{\log(p_{1,1}^{q}+p_{1,2}^{q})}{-\log
r_{1}},\;\;\frac{\log(p_{2,1}^{q}+p_{2,2}^{q}+p_{2,3}^{q})}{-\log
r_{2}}\right\\}.$

It results that for $0<q<1$, we have

${b}_{\mu}(q)={\mathsf{b}}_{\mu}(q)=\underline{\beta}(q)=\liminf_{k\to+\infty}\beta_{k}(q)=\frac{\log(p_{1,1}^{q}+p_{1,2}^{q})}{-\log
r_{1}}<$
${B}_{\mu}(q)={\mathsf{B}}_{\mu}(q)=\overline{\beta}(q)=\limsup_{k\to+\infty}\beta_{k}(q)=\frac{\log(p_{2,1}^{q}+p_{2,2}^{q}+p_{2,3}^{q})}{-\log
r_{2}}$

and, for $q<0$ or $q>1$,

${b}_{\mu}(q)={\mathsf{b}}_{\mu}(q)=\underline{\beta}(q)=\liminf_{k\to+\infty}\beta_{k}(q)=\frac{\log(p_{2,1}^{q}+p_{2,2}^{q}+p_{2,3}^{q})}{-\log
r_{2}}<$
${B}_{\mu}(q)={\mathsf{B}}_{\mu}(q)=\overline{\beta}(q)=\limsup_{k\to+\infty}\beta_{k}(q)=\frac{\log(p_{1,1}^{q}+p_{1,2}^{q})}{-\log
r_{1}}.$

###  4.3. Example 3

In the following, we give an example of a measure for which the lower and
upper multifractal Hewitt-Stromberg functions are different and the Hausdorff
and packing dimensions of the level sets of the local Hölder exponent
$E(\alpha)$ are given by the Legendre transform respectively of lower and
upper multifractal Hewitt-Stromberg functions. Take $0<p<\hat{p}\leq 1/2$ and
a sequence of integers

$1=t_{0}<t_{1}<\ldots<t_{n}<\ldots,\;\;\text{such
that}\;\;\lim_{n\to+\infty}\frac{t_{n+1}}{t_{n}}=+\infty.$

The measure $\mu$ assigned to the diadic interval of the n-th generation
$I_{\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n}}$ is

$\mu\big{(}I_{\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n}}\big{)}=\prod_{j=1}^{n}\varpi_{j},$

where

$\left\\{\begin{array}[]{ll}\text{if}\;\;t_{2k-1}\leq j<t_{2k}\;\;\text{for
some}\;\;k,\;\varpi_{j}=p\;\text{if}\;\varepsilon_{j}=0,\;\varpi_{j}=1-p\;\;\text{otherwise},&\\\
\\\ \text{if}\;\;t_{2k}\leq j<t_{2k+1}\;\;\text{for
some}\;\;k,\;\varpi_{j}=\hat{p}\;\text{if}\;\varepsilon_{j}=0,\;\varpi_{j}=1-\hat{p}\;\;\text{otherwise.}&\end{array}\right.$

Now, for $q\in\mathbb{R}$ we define,

$\tau(q)=\log_{2}\big{(}p^{q}+(1-p)^{q}\big{)}\quad\text{and}\quad\hat{\tau}(q)=\log_{2}\big{(}\hat{p}^{q}+(1-\hat{p})^{q}\big{)}.$

It results from [11, 13] that

$\left\\{\begin{array}[]{ll}b_{\mu}(q)={\mathsf{b}}_{\mu}(q)=\tau(q)<\hat{\tau}(q)={\mathsf{B}}_{\mu}(q)=B_{\mu}(q),\;\;\text{for}\;\;0<q<1,&\\\
\\\
b_{\mu}(q)={\mathsf{b}}_{\mu}(q)=\hat{\tau}(q)<\tau(q)={\mathsf{B}}_{\mu}(q)=B_{\mu}(q),\;\;\text{for}\;\;q<0\;\text{or}\;q>1.\end{array}\right.$

Then we have the following result,

######  Theorem 10.

Let $\alpha\geq 0$.

  1. (1)

For $\alpha\in\Big{(}-\log_{2}(1-\hat{p}),\;-\log_{2}(\hat{p})\Big{)}$, then
we have

${f}_{\mu}(\alpha)=\mathsf{f}_{\mu}(\alpha)=b_{\mu}^{*}(\alpha)={\mathsf{b}}_{\mu}^{*}(\alpha).$

  2. (2)

For
$\alpha\in\Big{(}-\log_{2}(1-\hat{p}),\;-\log_{2}(\hat{p})\Big{)}\setminus\Big{(}\Big{[}-B_{\mu_{+}}^{\prime}(0),-B_{\mu_{-}}^{\prime}(0)\Big{]}\bigcup\Big{[}-B_{\mu_{+}}^{\prime}(1),-B_{\mu_{-}}^{\prime}(1)\Big{]}\Big{)}$,
we have

${F}_{\mu}(\alpha)=\mathsf{F}_{\mu}(\alpha)={\mathsf{B}}_{\mu}^{*}(\alpha)=B_{\mu}^{*}(\alpha).$

###### Proof.

All of the ideas needed to prove this theorem can be found in [11, Proposition
9], Propositions 2, 3 and 4 and Theorem 8. ∎

###  4.4. Example 4

Given a class of exact dimensional measures (inhomogeneous multinomial
measures) whose support is the whole interval $[0,1]$, the multifractal
functions $b_{\mu}$, $\mathsf{b}_{\mu}$, $\mathsf{B}_{\mu}$ and $B_{\mu}$ are
real analytic and agree at two points only $0$ and $1$ (for more details, see
[56]). These measures satisfy our multifractal formalism in the sense that,
for $\alpha$ in some interval, the Hausdorff dimension of the level sets
$E(\alpha)$ is given by the Legendre transform of lower multifractal Hewitt-
Stromberg function and their packing dimension by the Legendre transform of
the upper multifractal Hewitt-Stromberg function. More specifically,

$b_{\mu}(q)={\mathsf{b}}_{\mu}(q)<B_{\mu}(q)={\mathsf{B}}_{\mu}(q)\quad\text{for
all}\;\;q\notin\\{0,1\\},$

${f}_{\mu}(\alpha)=\mathsf{f}_{\mu}(\alpha)=b_{\mu}^{*}(\alpha)={\mathsf{b}}_{\mu}^{*}(\alpha)$

and

$\qquad\qquad\qquad\mathsf{F}_{\mu}(\alpha)={F}_{\mu}(\alpha)=\mathsf{B}_{\mu}^{*}(\alpha)={B}_{\mu}^{*}(\alpha)\;\;\text{for
some}\;\;\alpha.$

##  5\. Open problems

Motivated by some results and examples developed in [42, 43, 44, 45, 47, 65],
we therefore ask the following questions.

  1. (1)

Let $\mu\in\mathcal{P}_{D}(\mathbb{R}^{n})$,
$E\subseteq\operatorname{supp}\mu$, $p,q\in\mathbb{R}$ and $\alpha\in[0,1].$
Then, the following problem remains open:

${\mathsf{b}}_{\mu}^{\alpha
p+(1-\alpha)q}(E)\leq\alpha{\mathsf{B}}_{\mu}^{p}(E)+(1-\alpha){\mathsf{b}}_{\mu}^{q}(E)?$

  2. (2)

Let $q\in\mathbb{R}$ and assume that
${\mathsf{B}}_{\mu}(q)={\mathsf{b}}_{\mu}(q)$. Are the measures
${\mathsf{H}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}$ and
${\mathsf{P}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}$ proportional,
i.e. does there exists a constant $c_{q}>0$ such that

${\mathsf{P}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}=c_{q}{\mathsf{H}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}?$

Even though it seems rather unlikely that the lower and upper multifractal
Hewitt-Stromberg measures are proportional in general, the ratio of the
measures ${\mathsf{H}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}$ and
${\mathsf{P}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}$ might still be
bounded. We therefore ask the following question: Does there exists a number
$0<c_{q}<+\infty$ such that

${\mathsf{H}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}\leq{\mathsf{P}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}\leq
c_{q}{\mathsf{H}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}?$

  3. (3)

Let $q\in\mathbb{R}$ and assume that ${b}_{\mu}(q)={\mathsf{b}}_{\mu}(q)$. Are
the measures ${\mathsf{H}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}$
and ${{\mathcal{H}}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}$
proportional, i.e. does there exists a constant $C_{q}>0$ such that

${\mathsf{H}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}=C_{q}{{\mathcal{H}}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}?$

Even though it seems rather unlikely that the multifractal Hausdorff measure
and the lower multifractal Hewitt-Stromberg measure are proportional in
general, the ratio of the measures
${{\mathcal{H}}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}$ and
${\mathsf{H}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}$ might still be
bounded. We therefore ask the following question: Does there exists a number
$0<C_{q}<+\infty$ such that

${{\mathcal{H}}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}\leq{\mathsf{H}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}\leq
C_{q}{{\mathcal{H}}^{q,t}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}?$

  4. (4)

Let $p,q\in\mathbb{R}$ and assume that ${\mathsf{b}}_{\mu}(q)$ is
differentiable at $p$ and $q$ with
${\mathsf{b}}_{\mu}^{\prime}(p)\neq{\mathsf{b}}_{\mu}^{\prime}(q)$. Then, the
following problem remains open:

${\mathsf{H}^{p,{\mathsf{b}}_{\mu}(p)}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}\bot\;\;{\mathsf{H}^{q,{\mathsf{b}}_{\mu}(q)}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}?$

  5. (5)

Let $p,q\in\mathbb{R}$ and assume that ${\mathsf{B}}_{\mu}(q)$ is
differentiable at $p$ and $q$ with
${\mathsf{B}}_{\mu}^{\prime}(p)\neq{\mathsf{B}}_{\mu}^{\prime}(q)$. Then, the
following problem remains open:

${\mathsf{P}^{p,{\mathsf{B}}_{\mu}(p)}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}\bot\;\;{\mathsf{P}^{q,{\mathsf{B}}_{\mu}(q)}_{\mu}}_{\llcorner_{\operatorname{supp}\mu}}?$

  6. (6)

Is it true that the weaker condition ${b}_{\mu}(q)={\mathsf{b}}_{\mu}(q)$ is
sufficient to obtain the conclusion of Theorem 8?

  7. (7)

Let $\mu\in\mathcal{P}_{D}(\mathbb{R}^{n})$,
$\nu\in\mathcal{P}_{D}(\mathbb{R}^{m})$ and $q,s,t\in\mathbb{R}$. Assume that
$c>0$, $E\subseteq\mathbb{R}^{n}$, $F\subseteq\mathbb{R}^{m}$,
$H\subseteq\mathbb{R}^{n+m}$ and $H(y)=\big{\\{}x;\;(x,y)\in H\big{\\}}$.
Then, the following problem remains open:

$\displaystyle\int{{\mathsf{H}}}^{q,s}_{\mu}(H(y))~{}~{}d{{\mathsf{H}}}^{q,t}_{\nu}(y)$
$\displaystyle\leq$ $\displaystyle
c~{}~{}{{\mathsf{H}}}^{q,s+t}_{\mu\times\nu}(H)$
$\displaystyle{{\mathsf{H}}}^{q,s+t}_{\mu\times\nu}(E\times F)$
$\displaystyle\leq$ $\displaystyle
c~{}~{}{{\mathsf{H}}}^{q,s}_{\mu}(E)~{}~{}{{\mathsf{P}}}^{q,t}_{\nu}(F)$
$\displaystyle\int{{\mathsf{H}}}^{q,s}_{\mu}(H(y))~{}~{}d{{\mathsf{P}}}^{q,t}_{\nu}(y)$
$\displaystyle\leq$ $\displaystyle
c~{}~{}{{\mathsf{P}}}^{q,s+t}_{\mu\times\nu}(H)$
$\displaystyle{{\mathsf{P}}}^{q,s+t}_{\mu\times\nu}(E\times F)$
$\displaystyle\leq$ $\displaystyle
c~{}~{}{{\mathsf{P}}}^{q,s}_{\mu}(E)~{}~{}{{\mathsf{P}}}^{q,t}_{\nu}(F)$

and

${\mathsf{b}}_{\mu}^{q}(E)+{\mathsf{b}}_{\nu}^{q}(F)\leq{\mathsf{b}}_{\mu\times\nu}^{q}(E\times
F)\leq{\mathsf{b}}_{\mu}^{q}(E)+{\mathsf{B}}_{\nu}^{q}(F)\leq{\mathsf{B}}_{\mu\times\nu}^{q}(E\times
F)\leq{\mathsf{B}}_{\mu}^{q}(E)+{\mathsf{B}}_{\nu}^{q}(F)?$

  8. (8)

The multifractal Hausdorff dimension function $b_{\mu}$ and the lower
multifractal Hewitt-Stromberg function ${\mathsf{b}}_{\mu}$ do not necessarily
coincide. Motivated by the results developed in [40], we conjecture that there
exist Borel probability measures $\mu$ on $\mathbb{R}^{n}$ such that

$b_{\mu}\neq{\mathsf{b}}_{\mu}={\mathsf{B}}_{\mu}={B}_{\mu}\quad\text{or}\quad
b_{\mu}<{\mathsf{b}}_{\mu}<{\mathsf{B}}_{\mu}={B}_{\mu}<{\Lambda}_{\mu}\quad\text{for
all}\quad q\neq 1.$

In particular, this will imply that

${f}_{\mu}(\alpha)=b_{\mu}^{*}(\alpha)<\mathsf{f}_{\mu}(\alpha)=\mathsf{F}_{\mu}(\alpha)={F}_{\mu}(\alpha)=\mathsf{b}_{\mu}^{*}(\alpha)=\mathsf{B}_{\mu}^{*}(\alpha)={B}_{\mu}^{*}(\alpha)\;\;\text{for
some}\;\;\alpha\geq 0.$

## Acknowledgments

The authors would like to thank Professor Lars Olsen for his first reading of
this work, the interest he gave to it and his valuable comment which improves
the presentation of the paper. And they thank the anonymous referees for their
valuable comments and suggestions that led to the improvement of the
manuscript.

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