††thanks: Department of Mathematics, Faculty of Science, University of Tabuk,
Saudi Arabia.

# Mixed multifractal densities for quasi-Ahlfors vector-valued measures

Adel Farhat  Labo. of Algebra, Number Theory and Nonlinear Analysis LR18ES50,
Department of Mathematics, Faculty of Sciences, Monastir 5000, Tunisia.
<EMAIL_ADDRESS>Anouar Ben Mabrouk  Department of Mathematics, Higher
Institute of Applied Mathematics and Computer Science, Street of Assad Ibn
Alfourat, 3100 Kairouan, Tunisia.
Lab. Algebra, Number Theory and Nonlinear Analysis, UR11ES50, Department of
Mathematics, Faculty of Sciences, 5000 Monastir, Tunisia.
<EMAIL_ADDRESS>Corresponding author.

###### Abstract

In the present work, some density estimations associated to vector-valued
quasi-Ahlfors measures are developed within the mixed multifractal analysis
framework. The principle idea reposes on the fact that being quasi-Ahlfors is
sufficient to conduct a mixed multifractal analysis for vector-valued
measures. In the present work, we introduced a multifractal density for
finitely many measures, and showed that such density may be estimated well by
means of the mixed multifractal measures. Such estimation induces an exact
computation of multifractal spectrum of the vector-valued quasi-Ahlfors
measure.

###### keywords:

Hausdorff measure and dimension, packing measure and dimension, Multifractal
densities, Multifractal formalism, Hölderian Measures, Ahlfors measures, Mixed
cases.

###### PACS:

: 28A78, 28A80.

##  1 Introduction

The purpose of the present paper is to develop a multifractal analysis for a
special type of densities due to a special class of measures known as quasi-
Ahlfors, within the framework of mixed multifractal analysis. The latter is a
natural extension of multifractal analysis of single objects such as measures,
functions, statistical data, distributions… It is developed quite recently
(since 2014) in the pure mathematical point of view. In physics and
statistics, it was appearing on different forms but not really and strongly
linked to the mathematical theory. See for example [23], [30]. In many
applications such as clustering topics, each attribute in a data sample may be
described by more than one type of measure. This leads researchers to apply
measures well adopted for mixed-type data. See for example [23].

Mathematically speaking, the idea consists in establishing some control of the
multifractal density of some vector-valued measures instead of a single
measure as in the classic or original multifractal analysis of measures and to
introduce a simultaneous density characteristic of such measures relatively to
one of them which is characterized by a quasi-Ahlfors property. This is
important as it permits for example to characterize fractal or irregular sets
such as Moran ones. The present work will provide a natural extension of [2],
[4], [5], [8], [9], [13], [33].

The assumption of being Ahlfors for one of the measures is the essential
motivation behind the present work, where we aim at a first step to review
some existing works that have forgotten such assumption and developed some
version of multifractal densities, even-though their constructions seems
questionable ([2] and [8]). The present work may form with [2], [5], [6], [8],
[35], [36] a quite full study of the concepts of multifractal densities of
measures.

However, we noticed that there has been some lack in hypothesis in the last
recent works [2] and [8]. Although the developments in [2] are in some parts
based on [8] which also refers to [6], the authors did not pay attention to
the fact that general probability measures (even-though being doubling) may
not lead to multifractal dimensions. Indeed, it is already mentioned in [6]
that the lower bound in the definition of the $\mu$-Hausdorff measure due to a
Borel probability measure $\mu$, (respectively, the upper bound in the
definition of the $\mu$-packing measure) extends naturally on all the
$\mu$-$\rho$-coverings of the analyzed sets (respectively, all the
$\mu$-$\rho$-packings). Recall that, in [6], it is already mentioned that a
$\mu$-$\rho$-covering imposes a control of the measure $\mu(C)$ of a cylinder
by means of its radius $\rho$, such as $\mu(C)<\rho$. In [6], it is also
excluded all atomic measures, as in this case we may not be able to find any
$\mu$-$\rho$-coverings. Omitting these assumptions may lead to empty coverings
(packings) suitable for inducing a multifractal analysis.

To overcome these lacks, some weak hypothesis is assumed in the present work
stating that one of the components of the vector-value measure should be
quasi-Ahlfors. It consists of a weak form of the so-called Alhfors measures.
See [18], [32], [38] for backgrounds on these measures and their properties.
In the sequel we will denote $\mathcal{P}_{B}(\mathbb{R}^{d})$ the set of all
Borel probability measures on $\mathbb{R}^{d}$, $n\geq 1$.

###### Definition 1.1

A measure $\mu\in\mathcal{P}_{B}(\mathbb{R}^{d})$ is said to be quasi-Ahlfors
regular with index $\alpha>0$ if

$\displaystyle\limsup_{|U|\longrightarrow
0}\displaystyle\frac{\mu(U)}{|U|^{\alpha}}<+\infty.$

It is in fact easy to check that with this assumption, the extensions
developed in [2] and [8] become possible, and that an associated multifractal
analysis may be addressed.

For backgrounds and details on multifractal measures, dimensions, densities,
examples, and couter-examples, the readers are asked to review [35], [36],
[41], [47], [48], [49], [51], [51], [50], [53], [54], [55]).

The next section is devoted to some review on mixed multifractal analysis in
general, and mixed multifractal densities in particular. Interactions with
other fields, link with our results in the present work are addressed. Section
3 is devoted to the presentation of general mathematical settings and results
that will be applied in the present study. In section 4, our main results in
the present paper are provided without proofs. In section 5, proofs of main
results are developed. Section 6 is a concluding part. Finally, section 7 is a
brief appendix in which we provided the proof of a result applied in our
study, which extends the well-known Billingsley result on dimension of
measures.

##  2 Recent related works and motivations

In a simple description, mixed multifractal analysis may be defined as
mathematical glasses that permit to capture and/or to quantify the transient
higher-order dependence beyond correlation of many measures, functions, time
series, distributions, etc.

Mixed multifractal densities are applied in several contexts. In [25], a mixed
analysis of Markov maps has been conducted with at most countable number of
branches. A mixed spectrum has been introduced known as mixed Birkhoff spectra
combining many characteristics depending simultaneously on different sets of
observations.

In [46], a mixed analysis based idea has been applied in the context of
multivariate time series with time-varying joint distribution. Based on the
presence of possibly self-affine structure for the time series, the authors
proposed a multivariate estimation of the distribution. Combrexelle in [10]
showed a potential benefit of mixed multifractal analysis in remote sensing
application. More precisely, the author addressed in [10] a multivariate
analysis for images by developing a hierarchical models for distributions
generalizing the Bayesian one.

Hong et al ([27]) developed a multifractal analysis of measures for the so-
called mixed logical dynamical models to the classification of signals
especially network traffic. These models are widely applied in the control of
hybrid systems such as multi-server ones.

Mixed multifractal analysis of measures has been applied in [11] to extract
the properties of stock index series and in exploiting the eventual inner
relationship that relates them.

The last closest work to our’s is developed in [28]. However, the main
difference is that the authors there considered the multifractal formalism for
a measure relatively to another one controlled by the diameter of the covering
elements. Instead of considering usual coverings, they applied the so-called
$\nu$-$\delta$-coverings relatively to a suitable doubling measure $\nu$ to
assure the control of both the measure of the covering and the diameters
simultaneously, corresponding to the essential measure $\mu$ subject of the
multifractal formalism. In the present work, we do not need such double
covering assumption as quasi-Ahlfors assumption permits the multifractal
analysis of the vector-valued measures and their densities.

Besides, in [14], the authors considered to revisit the well-known Besicovitch
covering theorem in the framework of relative multifractal analysis. Results
on multifractal densities in Tricot’s sense have been established. The main
idea consists in the use of doubling measures, which is not the case in the
present work. In [43], density results have been established in the single
case of multifractal analysis based on some necessary condition on strong
regularity of measures. In our knowledge, the results developed in the present
work are the first in the case of mixed multifractal analysis, for vector-
valued measures and where weak assumption has been assumed on the vector.

More about the use of mixed multifractal analysis of measures as well as
functions and time series or images may be found in [1, 29] and the references
therein.

##  3 General settings

In this section, general settings will be reviewed especially those developed
in [20]. We introduce the mixed multifractal generalisations of densities
associated to vector-valued measures. Consider a vector
$\mu=(\mu_{1},\mu_{2},...,\mu_{k})\in\left(\mathcal{P}_{B}(\mathbb{R}^{d})\right)^{k}$
and denote

$\mu(B(x,r))=(\mu_{1}(B(x,r)),...,\mu_{k}(B(x,r))).$

For $q=(q_{1},q_{2},...,q_{k})\in\mathbb{R}^{d}$, we write

$\mu(B(x,r))^{q}=\left[\mu_{1}(B(x,r)\right]^{q_{1}}\times...\times\left[\mu_{k}(B(x,r)\right]^{q_{k}}.$

Let finally $\nu\in\mathcal{P}(\mathbb{R}^{d})$. The mixed multifractal
generalisations of the Hausdorff measure with respect to $(\mu,\nu)$ is
introduced in [20] as follows. For $E\subset\mathbb{R}^{d}$,

${\overline{\mathcal{H}}}_{\mu,\nu,\delta}^{q,t}(E)=\inf\\{\underset{i}{\sum}(\mu(B(x_{i},r_{i})))^{q}(\nu(B(x_{i},r_{i})))^{t}\\}.$

The inf above is taken over the set of all centred $\delta$-coverings
$(B(x_{i},r_{i})))_{i}$ of $E$. Next, we consider

${\overline{\mathcal{H}}}_{\mu,\nu}^{q,t}(E)=\underset{\delta\downarrow
0}{\text{lim}}{\overline{\mathcal{H}}}_{\mu,\nu,\delta}^{q,t}(E)=\underset{\delta>0}{\text{sup}}{\overline{\mathcal{H}}}_{\mu,\nu,\delta}^{q,t}(E),$

which induces finally,

$\mathcal{H}_{\mu,\nu}^{q,t}(E)=\underset{F\subseteq
E}{\sup}{\overline{\mathcal{H}}}_{\mu,\nu}^{q,t}(F).$

Similarly to the mixed multifractal generalisation of Hausdorrf measure, the
mixed generalized multifractal packing measure with respect to $(\mu,\nu)$ is
introduced in [20] as follows. For $E\subset\mathbb{R}^{d}$,

${\overline{\mathcal{P}}}_{\mu,\nu,\delta}^{q,t}(E)=\sup\\{\;\underset{i}{\sum}(\mu(B(x_{i},r_{i})))^{q}(\nu(B(x_{i},r_{i}))^{t}\\}.$

The sup above is taken over the set of all centred $\delta$-packings
$(B(x_{i},r_{i})))_{i}$ of $E$. Let next

${\overline{\mathcal{P}}}_{\mu,\nu}^{q,t}(E)=\underset{\delta\downarrow
0}{\lim}{\overline{\mathcal{P}}}_{\mu,\nu,\delta}^{q,t}(E)=\underset{\delta>0}{\inf}{\overline{\mathcal{P}}}_{\mu,\nu,\delta}^{q,t}(E)$

and

$\mathcal{P}_{\mu,\nu}^{q,t}(E)=\underset{E\subseteq\;\underset{i}{\cup}E_{i}}{\inf}\underset{i}{\sum}{\overline{\mathcal{P}}}_{\mu,\nu}^{q,t}(E_{i}).$

The following Theorem proved in [20] resumes the characteristics of the mixed
multifractal generalisations $\mathcal{H}_{\mu,\nu}^{q,t}$ and
$\mathcal{P}_{\mu,\nu}^{q,t}$.

###### Theorem 3.1

[[20]]

  * •

The functions $\mathcal{H}_{\mu,\nu}^{q,t}$ and $\mathcal{P}_{\mu,\nu}^{q,t}$
are metric, outer measure and thus measures on Borel sets in $\mathbb{R}^{d}$.

  * •

For all $E\subset\mathbb{R}^{d}$, there exists unique extended reel numbers
denoted $dim_{\mu,\nu}^{q}({E})$,
$Dim_{\mu,\nu}^{q}({E}),\;\Delta_{\mu,\nu}^{q}({E})\in\left[-\infty,+\infty\right]$
satisfying respectively

$\mathcal{H}_{\mu,\nu}^{q,t}(E)=\left\\{\begin{array}[]{lll}\infty&\text{if}&t<dim_{\mu,\nu}^{q}({E}),\\\
0&\text{if}&t>dim_{\mu,\nu}^{q}({E}),\end{array}\right.$
$\mathcal{P}_{\mu,\nu}^{q,t}(E)=\left\\{\begin{array}[]{lll}\infty&\text{if}&t<Dim_{\mu,\nu}^{q}({E}),\\\
0&\text{if}&t>Dim_{\mu,\nu}^{q}({E}),\end{array}\right.$
${\overline{\mathcal{P}}}_{\mu,\nu}^{q,t}(E)=\left\\{\begin{array}[]{lll}\infty&\text{if}&t<\Delta_{\mu,\nu}^{q}({E}),\\\
0&\text{if}&t>\Delta_{\mu,\nu}^{q}({E}).\end{array}\right.$

For the convenience and to prove the necessity of the assumption of quasi-
Ahlfors regularity mentioned above we recall in brief the proof of the first
cutt-off value due to $\mathcal{H}_{\mu,\nu}^{q,t}$ in Theorem 3.1. We claim
firstly the following lemma.

###### Lemma 3.1

$\forall\,E\subseteq\mathbb{R}^{d}$ and $\forall q\in\mathbb{R}^{k}$, the set

$\Gamma_{q}=\left\\{t\;\mathcal{H}_{\mu,\nu}^{q,t}(E)<+\infty\right\\}\neq\emptyset.$

###### Proof 3.1

Let $M\in\mathbb{R}_{+}^{\ast}$ be such that

$\underset{\left|U\right|\rightarrow
0}{{\overline{\lim}}}\frac{\nu(U)}{\left|U\right|^{\alpha}}<M.$

This implies that for some $\delta>0$, and $\forall r$; $0<r<\delta$, we have

$\nu(U)\leq M\left|U\right|^{\alpha};\forall\,U\hbox{ s.t. }\left|U\right|<r.$
(1)

Let next $(B(x_{i},r_{i}))_{i}$ be a $\delta$-covering of $E$ and consider the
$N_{B}$ collections defined in the Besicovitch covering theorem. We get

$\underset{i}{\sum}\mu(B(x_{i},r_{i}))^{q}\nu(B(x_{i},r_{i}))^{t}\leq\underset{i=1}{\overset{N_{B}}{\sum}}\underset{j}{\sum}\mu(B(x_{ij},r_{ij}))^{q}\nu(B(x_{ij},r_{ij}))^{t}.$
(2)

Whenever $q\geq 0,$ the right hand term is bounded by

$\underset{i=1}{\overset{N_{B}}{\sum}}\underset{j}{\sum}\nu(B(x_{ij},r_{ij}))^{t}.$

For $t=1,$ this becomes

$\underset{i=1}{\overset{N_{B}}{\sum}}\underset{j}{\sum}\nu(B(x_{ij},r_{ij})).$

As the $(B(x_{ij},r_{ij}))_{j}$ are disjoint, the last quantity will be
bounded by

$\underset{i=1}{\overset{N_{B}}{\sum}}\nu\left(\underset{j}{\cup}B(x_{ij},r_{ij})\right)\leq
N_{B}\nu(\mathbb{R}^{d})=N_{B}.$

Consequently

$\mathcal{H}_{\mu,\nu}^{q,1}(E)<+\infty.$

Assume now that $q_{i}\leq 0$ for some $i$, $1\leq i\leq k$. We get

$\underset{i}{\sum}\mu(B(x_{i},r_{i}))^{q}\nu(B(x_{i},r_{i}))^{t}\leq
2^{-\alpha}M^{t}\underset{i}{\sum}\mu(B(x_{i},r_{i}))^{q}(2r_{i})^{\alpha t}.$

Let next $t>\frac{1}{\alpha}\left[\max\left(1,dim_{\mu}^{q}(E)\right)\right].$
We obtain

$\mathcal{H}_{\mu,\nu}^{q,t}(E)\leq 2^{-\alpha
t}M^{t}\mathcal{H}_{\mu}^{q,\alpha t}(E)<+\infty.$

###### Lemma 3.2

i) $\mathcal{H}_{\mu,\nu}^{q,t}(E)<+\infty$
$\Rightarrow\mathcal{H}_{\mu,\nu}^{q,s}(E)=0,$ $\forall$ $s>t.$
ii) $\mathcal{H}_{\mu,\nu}^{q,t}(E)>0$
$\Rightarrow\mathcal{H}_{\mu,\nu}^{q,s}(E)=+\infty,$ $\forall$ $s<t.$

###### Proof 3.2

i) Let $(B(x_{i},r_{i}))_{i}$ be a $\delta$-covering of $E$. We may write that

$\begin{array}[]{lll}\displaystyle\sum_{i}\mu(B(x_{i},r_{i}))^{q}\nu(B(x_{i},r_{i}))^{s}&=&\displaystyle\sum_{i}\mu(B(x_{i},r_{i}))^{q}\nu(B(x_{i},r_{i}))^{t}\nu(B(x_{i},r_{i}))^{s-t}\\\
&\leq&M^{s-t}\delta^{s-t}\displaystyle\sum_{i}\mu(B(x_{i},r_{i}))^{q}\nu(B(x_{i},r_{i}))^{t}.\end{array}$

Consequently,

${\overline{\mathcal{H}}}_{\mu,\nu,\delta}^{q,s}(E)\leq
M^{s-t}\delta^{s-t}{\overline{\mathcal{H}}}_{\mu,\nu,\delta}^{q,t}(E).$ (3)

Hence,

$\mathcal{H}_{\mu,\nu}^{q,s}(E)=0.$

ii) Using the same arguments as in assertion i, we get

${\overline{\mathcal{H}}}_{\mu,\nu,\delta}^{q,s}(E)\geq
M^{s-t}\delta^{s-t}{\overline{\mathcal{H}}}_{\mu,\nu,\delta}^{q,t}(E)\hbox{
(as }s-t<0).$

Consequently,

$\mathcal{H}_{\mu,\nu}^{q,s}(E)=+\infty.$

The two Lemmas above permit to introduce the generalised mixed multifractal
Hausdorff dimension as

$dim_{\mu,\nu}^{q}(E)=\inf\left\\{t,\,\mathcal{H}_{\mu,\nu}^{q,t}(E)=0\right\\}=\sup\left\\{t,\,\mathcal{H}_{\mu,\nu}^{q,t}(E)=+\infty\right\\}.$

We now develop our idea about the mixed density of measures. The original
definition are introduced in [8] and [9]). It is next re-studied by several
authors, especially in [2] and [35]. In the present work, we conduct an
extension of these works to the case introduced in [33] and their most recent
generalizations in [20].

###### Definition 3.1

Let $\theta\in\mathcal{P}(\mathbb{R}^{d})$, $x\in\,{Support}(\theta)$,
$q=(q_{1},q_{2},...,q_{k})\in\mathbb{R}^{k}$, and $t\in\mathbb{R}$. The upper,
respectively, lower, $(q,t)$-density of $\theta$ at the point $x$ relatively
to $(\mu,\nu)$ is defined by

${\overline{d}}_{\mu,\nu}^{q,t}(x,\theta)=\underset{r\rightarrow
0}{\lim\sup}\frac{\theta(B(x,r))}{(\mu(B(x,r)))^{q}(\nu(B(x,r))^{t}},$

respectively,

${\underline{d}}_{\mu,\nu}^{q,t}(x,\theta)=\underset{r\rightarrow
0}{\liminf}\frac{\theta(B(x,r))}{(\mu(B(x,r)))^{q}(\nu(B(x,r))^{t}}.$

Whenever
${\overline{d}}_{\mu,\nu}^{q,t}(x,\theta)={\underline{d}}_{\mu,\nu}^{q,t}(x,\theta)$,
we denote by ${{d}}_{\mu,\nu}^{q,t}(x,\theta)$ their common value, which will
be called the $(q,t)$-density of $\theta$ relatively to $(\mu,\nu)$ at the
point $x$.

Next, for $a>1$, and a single measure $\mu\in\mathcal{P}(\mathbb{R}^{d})$, let

$P_{a}(\mu)=\underset{r\downarrow 0}{\limsup}\left(\underset{x\in
S_{\mu}}{\sup}\frac{\mu(B(x,ar))}{\mu(B(x,r))}\right),$

and for a vector-valued measure
$\mu=(\mu_{1},\mu_{2},...,\mu_{k})\in\bigl{(}\mathcal{P}(\mathbb{R}^{d})\bigr{)}^{k}$,

$P_{a}(\mu)=\displaystyle\prod_{i=1}^{k}P_{a}(\mu_{i}).$

Finally, define the set of the so-called doubling vector-valued measures on
$\mathbb{R}^{d},$ by

$P_{D}(\mathbb{R}^{d})=\displaystyle\bigcup_{a>1}\left\\{\mu\in\mathcal{P}(\mathbb{R}^{d});\;P_{a}(\mu)<\infty\right\\}.$

##  4 Main results

The first result of the present work deals with the establishment of lower and
upper bounds for the mixed multifractal density introduced above. We will see
that such bounds permit to obtain the multifractal formalism already
introduced in [35] and re-considered next in [2], [9], [20], [33] and [34].

###### Theorem 4.1

There exists $C_{1},C_{2}>0$ constants satisfying for all Borel set
$E\subseteq\,S_{\mu}$,

$C_{1}\mathcal{H}_{\mu,\nu}^{q,t}(E)\underset{x\in
E}{\inf}{\overline{d}}_{\mu,\nu}^{q,t}(x,\theta)\leq\theta(E)\leq
C_{2}\mathcal{H}_{\mu,\nu}^{q,t}(E)\underset{x\in
E}{\sup}{\overline{d}}_{\mu,\nu}^{q,t}(x,\theta),$ (4)

whenever $\mathcal{H}_{\mu,\nu}^{q,t}(E)<\infty$, and

$C_{1}\mathcal{P}_{\mu,\nu}^{q,t}(E)\underset{x\in
E}{\inf}{\underline{d}}_{\mu,\nu}^{q,t}(x,\theta)\leq\theta(E)\leq
C_{2}\mathcal{P}_{\mu,\nu}^{q,t}(E)\underset{x\in
E}{\sup}{\underline{d}}_{\mu,\nu}^{q,t}(x,\theta),$ (5)

whenever $\mathcal{P}_{\mu,\nu}^{q,t}(E)<\infty$.

###### Remark 4.1

Whenever $\mu,\nu\in P_{D}(\mathbb{R}^{d})$, we may choose $C_{1}=C_{2}=1$ in
Theorem 4.1.

As a result of the estimations above of the new mixed multifractal densities,
we aim in the next step to show that such estimations permit in some special
cases to compute the mixed multifractal spectrum for some suitable vector-
valued measures in the mixed framework. For a Borel set
$E\subset\mathbb{R}^{d}$, define

${\overline{D}}_{\mu,\nu}^{q,t}(x,E)={\overline{d}}_{\mu,\nu}^{q,t}(x,\mathcal{H}_{\mu,\nu}^{q,s}(E))$

and

${\underline{D}}_{\mu,\nu}^{q,t}(x,E)={\underline{d}}_{\mu,\nu}^{q,t}(x,\mathcal{H}_{\mu,\nu}^{q,s}(E)).$

Define similarly

${\overline{\Delta}}_{\mu,\nu}^{q,t}(x,E)={\overline{d}}_{\mu,\nu}^{q,t}(x,\mathcal{P}_{\mu,\nu}^{q,t}(E))$

and

${\underline{\Delta}}_{\mu,\nu}^{q,t}(x,E)={\underline{d}}_{\mu,\nu}^{q,t}(x,\mathcal{P}_{\mu,\nu}^{q,t}(E)).$

As usually, whenever

${\overline{D}}_{\mu,\nu}^{q,t}(x,E)={\underline{D}}_{\mu,\nu}^{q,t}(x,E)$

we denote their common value by ${D}_{\mu,\nu}^{q,t}(x,E)$. Similarly,
whenever

${\overline{\Delta}}_{\mu,\nu}^{q,t}(x,E)={\underline{\Delta}}_{\mu,\nu}^{q,t}(x,E)$

we denote their common value by ${\Delta}_{\mu,\nu}^{q,t}(x,E)$. Denote next

$\underline{K}{=}\\{\,x\in\,E,\;{\underline{D}}_{\mu,\nu}^{q,t}(x,E)=1\\},\;\;{\overline{K}}{=}\\{\,x\in\,E,\;{\overline{D}}_{\mu,\nu}^{q,t}(x,E)=1\\},$
${\underline{T}}{=}\\{\,x\in\,E,\;{\underline{\Delta}}_{\mu,\nu}^{q,t}(x,E)=1\\},\;\;{\overline{T}}{=}\\{\,x\in\,E,\;{\overline{\Delta}}_{\mu,\nu}^{q,t}(x,E)=1\\},$
$K={\underline{K}\cap\overline{K}}\;\text{\ and
}\;T={\overline{T}\cap\underline{T}}.$

The multifractal dimensions of these sets are provided in the following
theorem.

###### Theorem 4.2

Let $E\subset{S_{\mu}\cap S_{\nu}}$ be a Borel set.

  1. 1.

If $\mathcal{H}_{\mu,\nu}^{q,t}(E)<\infty$ and $\mu,\nu\in
P_{D}(\mathbb{R}^{d})$, then $dim_{\mu,\nu}^{q}({\overline{K}})=t$.

  2. 2.

If $\mathcal{P}_{\mu,\nu}^{q,t}(E)<\infty$, then
$Dim_{\mu,\nu}^{q}({\underline{T}})=t$.

  3. 3.

If $\mathcal{P}_{\mu,\nu}^{q,t}(E)<\infty$ and $\mu,\nu\in
P_{D}(\mathbb{R}^{d})$, we have the following equivalences.

a.

$\mathcal{H}_{\mu,\nu}^{q,t}=\mathcal{P}_{\mu,\nu}^{q,t}$.

b.

${\overline{D}}_{\mu,\nu}^{q,t}(x,E)={\underline{D}}_{\mu,\nu}^{q,t}(x,E)=1$,
for $\mathcal{P}_{\mu,\nu}^{q,t}-a.a.x\in E$.

c.

${\overline{\Delta}}_{\mu,\nu}^{q,t}(x,E)={\underline{\Delta}}_{\mu,\nu}^{q,t}(x,E)=1$,
for $\mathcal{P}_{\mu,\nu}^{q,t}-a.a$.$x\in E$.

  4. 4.

If $\mathcal{H}_{\mu,\nu}^{q,t}=\mathcal{P}_{\mu,\nu}^{q,t}<\infty$ and
$\mu,\nu\in P_{D}(\mathbb{R}^{d})$, then
$dim_{\mu,\nu}^{q}({K})=dim_{\mu,\nu}^{q}({T})=t$.

This result is important as it constitutes a first information leading to the
computation of the multifractal spectrum due to the densities introduced.
Indeed, related to the origins of the multifractal spectrum, such as in [4],
[5], [8], [9], [20], [33], [35], a starting point in the classical case is to
establish an estimation of the form

$\theta(B(x,r))\sim(\mu(B(x,r))^{q}(2r)^{t\pm\varepsilon},\;\;r\rightarrow 0,$

which by considering a somehow Hölder probability measure

$\nu(B(x,r))\sim(2r)^{t\pm\varepsilon},\;\;r\rightarrow 0$

means that the densities considered above are all equal to 1. These last
assumptions permit to compute the multifractal spectrum (evaluated as the
Hausdorff dimension of the density level sets) by means of a Legendre
transformation of a convex function issued from the multifractal generalized
dimensions $b_{\mu,\nu}$, $B_{\mu,\nu}$ and $\Delta_{\mu,\nu}$.

In the following part, we aim to provide in a preparatory step a
characterization of the sets of points in the support(s) of the relative
measure(s) with the same density.

###### Theorem 4.3

Let $\mu$,$\nu$ $\in P_{D}(\mathbb{R}^{d})$ and $E\subset S_{\mu}\cap S_{\nu}$
be a Borel set. Let also

$E_{1}=\\{\ x\in
E,{\overline{D}}_{\mu,\nu}^{q,t}(x,E)={\underline{D}}_{\mu,\nu}^{q,t}(x,E)\\}$

and

$E_{2}=\\{\ x\in
E,{\overline{\Delta}}_{\mu,\nu}^{q,t}(x,E)={\underline{\delta}}_{\mu,\nu}^{q,t}(x,E)\\}.$

1) If $\mathcal{H}_{\mu,\nu}^{q,t}(E)<\infty$ then
a)
${\overline{D}}_{\mu,\nu}^{q,t}(x,E_{1})={\underline{D}}_{\mu,\nu}^{q,t}(x,E_{1}),$
for $\mathcal{H}_{\mu,\nu}^{q,t}-a.$ $e$ on $E_{1}.$
b) ${\overline{D}}_{\mu,\nu}^{q,t}(x,E\diagdown
E_{1})={\underline{D}}_{\mu,\nu}^{q,t}(x,E\diagdown E_{1}),$ for
$\mathcal{H}_{\mu,\nu}^{q,t}-a.$ $e$ on $E\diagdown E_{1}.$
2) If $\mathcal{P}_{\mu,\nu}^{q,t}(E)<\infty$ then
a)
${\overline{\Delta}}_{\mu,\nu}^{q,t}(x,E_{2})={\underline{\delta}}_{\mu,\nu}^{q,t}(x,E_{2}),$
for $\mathcal{P}_{\mu,\nu}^{q,t}-a.$ $e$ on $E_{2}.$
b) ${\overline{\delta}}_{\mu,\nu}^{q,t}(x,E\diagdown
E_{2})={\underline{\delta}}_{\mu,\nu}^{q,t}(x,E\diagdown E_{2}),$ for
$\mathcal{P}_{\mu,\nu}^{q,t}-a.$ $e$ on $E\diagdown E_{2}.$

##  5 Proof of main results

###  5.1 Proof of Theorem 4.1.

We firstly show the left-hand side inequality of (4). So, denote
$\overline{d}=\underset{x\in
E}{\inf}{\overline{d}}_{\mu,\nu}^{q,t}(x,\theta)$. Whenever $\overline{d}=0$,
the inequality is obvious. So, assume that $\overline{d}>0$, and let $\eta$ be
such that $0<\eta<\overline{d}$, and denote
$\overline{d}_{\eta}=\overline{d}-\eta$. Let finally $F\subset E$ be closed,
and $H\subset F$. Finally, for $\delta>0$, let

$B_{\delta}(F)=\\{\ x\in\mathbb{R}^{d},\;dist(F,x)\leq\delta\\}.$

It is straightforward that $B_{\delta}(F)\downarrow F$ whenever
$\delta\downarrow 0$. Therefore, for all $\varepsilon>0$, there is
$\delta_{0}>0$ satisfying

$\theta(B_{\delta}(F))\leq\theta(F)+\varepsilon,\;\forall\,\delta,\;0<\delta<\delta_{0}.$

Besides, as $\mathcal{H}_{\mu,\nu}^{q,t}(H)<\infty$, we may also write

${\overline{\mathcal{H}}}_{\mu,\nu}^{q,t}(H)-\varepsilon\leq{\overline{\mathcal{H}}}_{\mu,\nu,\delta}^{q,t}(H),\;\forall\,\delta,\;0<\delta<\delta_{0}.$

Denote next

$\Gamma_{\mu,\nu}^{q,t}(B(x,r))=[\mu(B(x,r))]^{q}[\nu(B(x,r))]^{t}$

and

$\mathcal{B}_{\delta}=\left\\{B(x,r);\,x\in
H,\,0<r<\delta,\;\;\theta(B(x,r))\geq\overline{d}_{\eta}\Gamma_{\mu,\nu}^{q,t}(B(x,r))\right\\}.$

From the definition of $\overline{d}$ and $\overline{d}_{\eta}$, there exists
$\delta_{0}>0$ such that

$\displaystyle\frac{\left[\theta(B(x,r)\right]}{(\mu(B(x,r)))^{q}(\nu(B(x,r))^{t}}\geq\overline{d}_{\eta};\;\;\forall
r,\,0<r<\delta_{0}.$

Or equivalently,

$\theta(B(x,r))\geq\overline{d}_{\eta}\Gamma_{\mu,\nu}^{q,t}(B(x,r));\;\;\forall
r,\,0<r<\delta_{0}.$

Henceforth, $\mathcal{B}_{\delta}\not=\emptyset$. So, let next $N_{B}$ be the
number of at most countable collections $(\mathcal{B}_{i})_{1\leq i\leq
N_{B}}=(B(x_{ij},r_{ij}))_{j,1\leq i\leq N_{B}}$ of $\mathcal{B}_{\delta}$
obtained from the Besicovitch covering theorem, where, for all $i$,
$\mathcal{B}_{i}$ is composed of pairwise disjoint balls $B(x_{ij},r_{ij})$.
We have,

$H\subset\bigcup_{1\leq i\leq N_{B}}\bigcup_{j}B(x_{ij},r_{ij}).$

It follows that

$\begin{array}[]{lll}\displaystyle{\overline{\mathcal{H}}}_{\mu,\nu,\delta}^{q,t}(H)&\leq&{\overline{\mathcal{H}}}_{\mu,\nu,\delta}^{q,t}\,\bigl{(}\displaystyle\bigcup_{i=1}^{N_{B}}\bigcup_{j}B(x_{ij},r_{ij})\bigr{)}\\\
&\leq&\displaystyle\sum_{i=1}^{N_{B}}{\overline{\mathcal{H}}}_{\mu,\nu,\delta}^{q,t}\,\bigl{(}\displaystyle\bigcup_{j}B(x_{ij},r_{ij})\bigr{)}\\\
&\leq&\displaystyle\sum_{i=1}^{N_{B}}\sum_{j}[\mu(B(x_{ij},r_{ij}))]^{q}[\nu(B(x_{ij},r_{ij}))]^{q}\\\
&\leq&\displaystyle\frac{1}{\overline{d}_{\eta}}\sum_{i=1}^{N_{B}}\sum_{j}\theta(B(x_{ij},r_{ij}))\\\
&\leq&\displaystyle\frac{1}{\overline{d}_{\eta}}\sum_{i=1}^{N_{B}}\theta(B_{\delta}(F))\\\
&\leq&\displaystyle\frac{N_{B}}{\overline{d}_{\eta}}\theta(B_{\delta}(F)).\end{array}$

Consequently, we obtain

$\mathcal{H}_{\mu,\nu}^{q,t}(H)\leq{\overline{\mathcal{H}}}_{\mu,\nu,\delta}^{q,t}(H)+\varepsilon\leq\displaystyle\frac{N_{B}}{\overline{d}_{\eta}}\theta(B_{\delta}(F))+\varepsilon.$

As $\varepsilon\rightarrow 0$, and observing that
$\overline{d}_{\eta}=\overline{d}-\eta$, we get

$(\overline{d}-\eta)\mathcal{H}_{\mu,\nu}^{q,t}(H)\leq\,N\theta(B_{\delta}(F));\;\;\forall\eta>0,\;0<\eta<\overline{d}.$

By letting $\eta\downarrow 0$, we get

$\overline{d}\mathcal{H}_{\mu,\nu}^{q,t}(H)\leq\,N_{B}\theta(B_{\delta}(F)).$

Now, for $\delta\downarrow 0$, and taking the sup on $H\subset F$, we obtain

$\overline{d}\mathcal{H}_{\mu,\nu}^{q,t}(F)\leq\,N_{B}\theta(F).$

This is valid for all closed $F\subset E$. As a result, taking the sup on $F$,
and replacing $\overline{d}$ by its exact form, we obtain

$\mathcal{H}_{\mu,\nu}^{q,t}(E)\underset{x\in
E}{\inf}{\overline{d}}_{\mu,\nu}^{q,t}(x,\theta)\leq\,N_{B}\theta(E).$

We now proceed to show the right-hand side part of inequality (4). As
previously, let $\overline{D}=\displaystyle\sup_{x\in
E}\overline{d}_{\mu,\nu}^{q,t}(x,\theta)$, $\eta>0$ and denote
$\overline{D}+\eta$. For $\delta>0$, consider the set

$E_{\delta}=\\{x\in
E;\;\overline{D}_{\eta}\Gamma_{\mu,\nu}^{q,t}(B(x,r))\geq\theta(B(x,r),\;0<r<\delta\\}.$

It is straightforward that for all $\delta>0$,

${\overline{\mathcal{H}}}_{\mu,\nu}^{q,t}(E_{\delta})\leq\mathcal{H}_{\mu,\nu}^{q,t}(E_{\delta})<\infty.$

On the other hand, for all $\varepsilon>0$, let $(B(x_{i},r_{i}))_{i}$ be a
$\delta$-covering of $E_{\delta}$ satisfying

$\displaystyle\sum_{i}[\mu(B(x_{i},r_{i}))]^{q}[\nu(B(x_{i},r))]^{t}\leq{\overline{\mathcal{H}}}_{\mu,\nu,\delta}^{q,t}(E_{\delta})+\varepsilon.$

We get

$\begin{array}[]{lll}\theta(E_{\delta})&\leq&\theta(\bigcup_{i}B(x_{i},r_{i}))\\\
&\leq&\displaystyle\sum_{i}\theta(B(x_{i},r_{i}))\\\
&\leq&\overline{D}_{\eta}\displaystyle\sum_{i}[\mu(B(x_{i},r_{i}))]^{q}[\nu(B(x_{i},r))]^{t}\\\
&\leq&\overline{D}_{\eta}\left[{\overline{\mathcal{H}}}_{\mu,\nu,\delta}^{q,t}(E_{\delta})+\varepsilon\right].\end{array}$

As $E_{\delta}\subset E$, we get

$\theta(E_{\delta})\leq\overline{D}_{\eta}\left[\mathcal{H}_{\mu,\nu}^{q,t}(E)+\varepsilon\right].$

Whenever $\delta\downarrow 0$, we obtain

$\theta(E)\leq\overline{D}_{\eta}\left[\mathcal{H}_{\mu,\nu}^{q,t}(E)+\varepsilon\right].$

This is true for all $\varepsilon,\eta>0$. Consequently,

$\theta(E)\leq\overline{D}\mathcal{H}_{\mu,\nu}^{q,t}(E),$

or equivalently

$\theta(E)\leq\displaystyle\sup_{x\in
E}\overline{d}_{\mu,\nu}^{q,t}(x,\theta)\mathcal{H}_{\mu,\nu}^{q,t}(E).$

Now, we will prove the left-hand side of (5). Denote similarly to the previous
case $\underline{d}=\displaystyle\inf_{x\in
E}{\underline{d}}_{\mu,\nu}^{q,t}(x,\theta)$, and assume here-also that
$\underline{d}>0$. Let next $\eta$ be such that $0<\eta<\underline{d}$ and
denote $\underline{d}_{\eta}=\underline{d}-\eta$. Let also $F\subset E$ be
closed, and write for $\delta>0$,

$B_{\delta}(F)=\\{x\in\mathbb{R}^{d};\;dist(F,x),\,\leq\delta\\}.$

Let next for $\varepsilon>0$, $\delta_{0}>0$ be such that

$\theta(B_{\delta}(F))\leq\theta(F)+\varepsilon;\;\forall\delta,\,0<\delta<\delta_{0}.$

Denote next

$F_{\delta}=\left\\{x\in
F;\;\theta(B(x,r))\leq\underline{d}\Gamma_{\mu,\nu}^{q,t}(B(x,r)),\,0<r<\delta\right\\}.$

Consider a $\delta$-packing $(B(x_{i},r_{i}))_{i}$ of $F_{\delta}$. It holds
that

$\begin{array}[]{lll}\underline{d}_{\eta}\displaystyle\sum_{i}[\mu(B(x_{i},r_{i}))]^{q}[\nu(B(x_{i},r_{i}))]^{t}&\leq&\displaystyle\sum_{i}\theta(B(x_{i},r_{i}))\\\
&\leq&\displaystyle\sum_{i}\theta(B(x_{i},r_{i}))\\\
&\leq&\theta(\displaystyle\bigcup_{i}B(x_{i},r_{i}))\\\
&\leq&\theta(B_{\delta}(F))\\\ &\leq&\theta(F)+\varepsilon\\\
&\leq&\theta(E)+\varepsilon,\end{array}$

which yields that

$\underline{d}_{\eta}\mathcal{P}_{\mu,\nu}^{q,t}(F_{\delta})\leq\underline{d}_{\eta}{\overline{\mathcal{P}}}_{\mu,\nu}^{q,t}(F_{\delta})\leq\underline{d}_{\eta}{\overline{\mathcal{P}}}_{\mu,\nu,\delta}^{q,t}(F_{\delta})\leq\theta(E)+\varepsilon.$

Finally, letting $\delta$, $\varepsilon$ and $\eta$ $\rightarrow 0$, we get

$\underline{d}\mathcal{P}_{\mu,\nu}^{q,t}(F)\leq\theta(E).$

It remains to check the inequality at the right-hand side of (5). So, denote
$\overline{D}=\displaystyle\sup_{x\in
E}{\underline{d}}_{\mu,\nu}^{q,t}(x,\theta)$, $F\subset E$, and let
$\varepsilon,\eta,\delta>0$ be such that

${\overline{\mathcal{P}}}_{\mu,\nu,\delta}^{q,t}(F)\leq{\overline{\mathcal{P}}}_{\mu,\nu}^{q,t}(F)+\varepsilon.$

Denote next $\overline{D}_{\eta}=\overline{D}+\eta$, and

$\mathcal{B}_{\delta}=\left\\{\,B(x,r);\;\overline{D}_{\eta}\Gamma_{\mu,\nu}^{q,t}(B(x,r))\leq\theta(B(x,r)),\,x\in
F,\;0<r<\delta\right\\}.$

Vitali’s Theorem ([19], [31]) implies that

$\theta\left(F\setminus\left(\displaystyle\bigcup_{i}B(x_{i},r_{i})\right)\right)=0,$

for some $\delta$-packing $(B(x_{i},r_{i}))_{i}\subset\mathcal{B}_{\delta}$ of
$F$. Furthermore,

$\begin{array}[]{lll}\theta(F)&=&\theta(\displaystyle\bigcup_{i}(F\cap
B(x_{i},r_{i}))\\\ &=&\theta((F\cap\displaystyle\bigcup_{i}B(x_{i},r_{i}))\\\
&\leq&\displaystyle\sum_{i}\theta(F\cap B(x_{i},r_{i}))\\\
&\leq&\displaystyle\sum_{i}\theta(B(x_{i},r_{i}))\\\
&\leq&\overline{D}_{\eta}\displaystyle\sum_{i}[\mu(B(x_{i},r_{i}))]^{q}[\nu(B(x_{i},r_{i}))]^{t}\\\
&\leq&\overline{D}_{\eta}{\overline{\mathcal{P}}}_{\mu,\nu,\delta}^{q,t}(F)\\\
&\leq&\overline{D}_{\eta}\left[{\overline{\mathcal{P}}}_{\mu,\nu}^{q,t}(F)+\varepsilon\right].\end{array}$

For $\varepsilon\rightarrow 0$, and $\eta\rightarrow 0$, we obtain

$\theta(F)\leq\overline{D}\,{\overline{\mathcal{P}}}_{\mu,\nu}^{q,t}(F),\;\;\forall
F\subset E.$

And thus, the desired inequality follows.

###  5.2 Proof of Theorem 4.2.

1. We shall prove that

$0<\mathcal{H}_{\mu,\nu}^{q,t}(\overline{K})<\infty.$ (6)

Consider, in a first step, the set

$F=\\{\ x\in E;{\overline{D}}_{\mu,\nu}^{q,t}(x,E)>1\\},$

It may be seen as a limit of an increasing sequence of nested sets

$F_{m}=\\{\ x\in
E;{\overline{D}}_{\mu,\nu}^{q,t}(x,E)>1+\frac{1}{m}\\},\;\;m\in\mathbb{N},$

as $F=$ $\underset{m}{\cup}F_{m}$. Hence, Theorem 4.1 (combined with Remark
4.1) yields that

$\mathcal{H}_{\mu,\nu}^{q,t}(F_{m})(1+\frac{1}{m})\leq\mathcal{H}_{\mu,\nu}^{q,t}(F_{m}),\;\forall\,m.$

As a result, we obtain

$\mathcal{H}_{\mu,\nu}^{q,t}(F_{m})=0,\text{ }\forall\text{ }m.$

Consequently,

$\mathcal{H}_{\mu,\nu}^{q,t}(F)=0,$

and thus

${\overline{D}}_{\mu,\nu}^{q,t}(x,E)\leq 1,\;\text{ for
}\mathcal{H}_{\mu,\nu}^{q,t}-a.\text{ }a.\text{ }x\in E\ .$ (7)

Next, consider similarly the set

$G\ =\\{\ x\in E;{\overline{D}}_{\mu,\nu}^{q,t}(x,E)<1\ \\},$

and analogously the sequence of nested sets

$G_{m}=\\{x\in E;{\overline{D}}_{\mu,\nu}^{q,t}(x,E)\leq 1-\frac{1}{m}\\}.$

Again Theorem 4.1 (combined with Remark 4.1) yields that

$\mathcal{H}_{\mu,\nu}^{q,t}(G_{m})(1-\frac{1}{m})\geq\mathcal{H}_{\mu,\nu}^{q,t}(G_{m}),\text{
}\forall\text{ }m.$

Consequently,

$\mathcal{H}_{\mu,\nu}^{q,t}(G_{m})=0,\text{ }\forall\text{ }m.$

Since $G=\underset{m}{\cup}G_{m}$, we obtain

$1\leq{\overline{D}}_{\mu,\nu}^{q,t}(x,E),\text{ for
}\mathcal{H}_{\mu,\nu}^{q,t}-a.\text{ }a.\text{ }x\in E\ .$ (8)

Equations (7) and (8) yield that

${\overline{D}}_{\mu,\nu}^{q,t}(x,E)=1,\text{ for
}\mathcal{H}_{\mu,\nu}^{q,t}-a.\text{ }a.\text{ }x\in E.$

As a result,

$\mathcal{H}_{\mu,\nu}^{q,t}(\\{\ x\in E,\text{
}{\overline{D}}_{\mu,\nu}^{q,t}(x,E)=1\\})>0.$

Observing now that

$\mathcal{H}_{\mu,\nu}^{q,t}(\\{\ x\in E,\text{
}{\overline{D}}_{\mu,\nu}^{q,t}(x,E)=1\\})\leq\mathcal{H}_{\mu,\nu}^{q,t}(E)<\infty,$

it follows that

$0<\mathcal{H}_{\mu,\nu}^{q,t}(\\{\ x\in E,\text{
}{\overline{D}}_{\mu,\nu}^{q,t}(x,E)=1\\})<\infty.$

Hence (6) holds, and

$dim_{\mu,\nu}^{q}({\overline{K}})=t.$

Assertion 2. may be checked by similar techniques.3. We shall prove that
$a)\Rightarrow b)\Rightarrow c)\Rightarrow a)$.
$a)\Rightarrow b).$ From assertion 1. above, it follows that

${\overline{D}}_{\mu,\nu}^{q,t}(x,E)=1,\text{ for
}\mathcal{H}_{\mu,\nu}^{q,t}-a.\text{ }a.\text{ }x\in E.$

So, as $\mathcal{H}_{\mu,\nu}^{q,t}=\mathcal{P}_{\mu,\nu}^{q,t}$, we get

${\overline{D}}_{\mu,\nu}^{q,t}(x,E)=1,\text{ for
}\mathcal{P}_{\mu,\nu}^{q,t}-a.\text{ }a.\text{ }x\in E.$ (9)

Now, proceeding as in the proof of equations (7) and (8) in assertion 1.
above, we get

${\underline{D}}_{\mu,\nu}^{q,t}(x,E)=1,\text{ for
}\mathcal{P}_{\mu,\nu}^{q,t}-a.a.x\in E,$

which by the hypothesis
$\mathcal{H}_{\mu,\nu}^{q,t}=\mathcal{P}_{\mu,\nu}^{q,t}$ yields that

${\underline{D}}_{\mu,\nu}^{q,t}(x,E)=1,\text{ for
}\mathcal{H}_{\mu,\nu}^{q,t}-a.a.x\in E.$

So, assertion b. is proved.
$b)\Rightarrow c).$ Using assertion b., Theorem 4.1 (and Remark 4.1), it
holds, for any ball $B(x,r)$, that

$\mathcal{H}_{\mu,\nu}^{q,t}(B(x,r))=\mathcal{P}_{\mu,\nu}^{q,t}(B(x,r)).$

Again, using assertion b. above, we get

$\mathcal{H}_{\mu,\nu}^{q,t}(B(x,r))\sim\Gamma_{\mu,\nu}^{q,t}(B(x,r)),\;\;\hbox{for}\;\;\mathcal{H}_{\mu,\nu}^{q,t}-a.a.x\in
E.$

Consequently,

$\mathcal{P}_{\mu,\nu}^{q,t}(B(x,r))\sim\Gamma_{\mu,\nu}^{q,t}(B(x,r)),\;\;\hbox{for}\;\;\mathcal{P}_{\mu,\nu}^{q,t}-a.a.x\in
E.$

Hence,

${\underline{\Delta}}_{\mu,\nu}^{q,t}(x,E)={\overline{\Delta}}_{\mu,\nu}^{q,t}(x,E)=1,\text{
for }\mathcal{P}_{\mu,\nu}^{q,t}-a.a.x\in E.$

So as assertion c.
$c)\Rightarrow a).$ Applying Theorem 4.1 and Remark 4.1, we get for all Borel
set $F\subset E$,

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

Consequently,

$\mathcal{P}_{\mu,\nu}^{q,t}=\mathcal{H}_{\mu,\nu}^{q,t}.$

4) is an immediate consequence of assertions 1., 2. and 3.

###  5.3 Proof of Theorem 4.3

We develop here-after the proof of assertion 1.a. The remaining assertions may
be proved by following similar techniques.
1.a. We claim that for all $\mathcal{H}_{\mu,\nu}^{q,t}$-measurable set
$F\subset E$, we have

${\overline{D}}_{\mu,\nu}^{q,t}(x,E)={\overline{D}}_{\mu,\nu}^{q,t}(x,F)\text{
and
}{\underline{D}}_{\mu,\nu}^{q,t}(x,E)={\underline{D}}_{\mu,\nu}^{q,t}(x,F);\,\mathcal{H}_{\mu,\nu}^{q,t}-a.e\text{
on }F.$ (10)

Indeed, denote as in [2], [9], for $\theta\in P(\mathbb{R}^{d})$,

$\theta_{E}(A)=\theta(E\cap A)\;\mbox{and}\;\lambda_{E}(A)=\theta(A\cap
E^{c}),\;\text{ for all Borel set }A.$

It is straightforward that

${\overline{d}}_{\mu,\nu}^{q,t}(x,\theta_{E})\leq{\overline{d}}_{\mu,\nu}^{q,t}(x,\theta)$

and

${\underline{d}}_{\mu,\nu}^{q,t}(x,\theta_{E})\leq{\underline{d}}_{\mu,\nu}^{q,t}(x,\theta).$

On the other hand, we have

${\underline{d}}_{\mu,\nu}^{q,t}(x,\theta)\leq{\underline{d}}_{\mu,\nu}^{q,t}(x,\theta_{E})+{\overline{d}}_{\mu,\nu}^{q,t}(x,\lambda_{E})$

and

${\overline{d}}_{\mu,\nu}^{q,t}(x,\theta)\leq{\overline{d}}_{\mu,\nu}^{q,t}(x,\theta_{E})+{\overline{d}}_{\mu,\nu}^{q,t}(x,\lambda_{E}).$

We now claim that

${\overline{d}}_{\mu,\nu}^{q,t}(x,\lambda_{E})=0.$ (11)

Indeed, the set
$G=\\{x;\;{\overline{d}}_{\mu,\nu}^{q,t}(x,\lambda_{E})\not=0\\}$ is a
countable union of

$G_{k}=\\{\ x\in E;\text{
}{\overline{d}}_{\mu,\nu}^{q,t}(x,\lambda)\geq\frac{1}{k}\\},\;\;k\geq 1.$

From Theorem 4.1 (and Remark 4.1), we get

$\lambda_{E}(G_{k})\geq\frac{1}{k}\mathcal{H}_{\mu,\nu}^{q,t}(G_{k}).$

Consequently,

$\mathcal{H}_{\mu,\nu}^{q,t}(G_{k})=0,\;\forall k,$

and thus

$\mathcal{H}_{\mu,\nu}^{q,t}(G)=0.$

Therefore,

${\overline{d}}_{\mu,\nu}^{q,t}(x,\lambda_{E})=0\text{ for
}\mathcal{H}_{\mu,\nu}^{q,t}-a.\text{ }a\text{ on }E,$

which leads to (11). As a result, we get

${\underline{d}}_{\mu,\nu}^{q,t}(x,\theta)\leq{\underline{d}}_{\mu,\nu}^{q,t}(x,\theta_{E})$

and

${\overline{d}}_{\mu,\nu}^{q,t}(x,\theta)\leq{\overline{d}}_{\mu,\nu}^{q,t}(x,\theta_{E}).$

These estimations together yield claim (10).

##  6 Conclusion

In the present paper, a class of quasi Ahlfors vector-valued measures has been
applied for the estimation of mixed multifractal densities. We served of a
mixed multifractal analysis developed relatively to such a class to show that
the computation of the fractal dimension is always possible when estimating
well the mixed multifractal densities. Even-though the present work is more
theoretical, however, many practical cases may be addressed. Indeed, such
densities have been applied for many time in the understanding of many
phenomena, such as climate factors distributions in different regions, crashes
propagation and the effect on markets, etc. See [1, 7, 22, 24, 39, 42]. These
applications have been tackled by using the single multifractal analysis and
densities. However, in the recent decades there has been a ’proof’ of the fact
that taking simultaneously many phenomena and/or many regions for the same
task may induce well understanding. This may be investigated by means of the
mixed analysis.

In a pure mathematical point of view, a new framework of mixed multifractal
analysis for vector-valued measures has been developed in [34], where a
general context has been considered, in which the measures considered are not
necessary Gibbs, nor doubling, nor quasi-Ahlfors, but controlled by an extra
gauge function to prove the validity of the multifractal formalism. An
eventual extension of the present work is to combine with [34] for more
general classes of multifractal densities.

##  7 Appendix - The quasi-Ahlfors regularity

Remark that whenever the measure $\nu$ is quasi-Ahlfors regular with index
$\alpha$, we get for $t\geq 0$,

$\mathcal{H}_{\mu,\nu}^{q,t}\leq M^{t}\mathcal{H}_{\mu}^{q,\alpha t}.$

Henceforth, for $\alpha t>max(0,dim_{\mu}^{q})$, we get

$t\geq dim_{\mu,\nu}^{q},$

which yields that

$\left(dim_{\mu}^{q}\right)_{+}\geq\alpha\,dim_{\mu,\nu}^{q}.$

Similarly, we get

$\left(dim_{\mu}^{q}\right)_{-}\leq\alpha\,dim_{\mu,\nu}^{q}.$

As a result, we obtain an analogous of Billingsley theorem as

$\left(dim_{\mu}^{q}\right)_{-}\leq\alpha\,dim_{\mu,\nu}^{q}\leq\left(dim_{\mu}^{q}\right)_{+}.$
(12)

Whenever $\nu$ is $\alpha$-Ahlfors regular we get a mixed version of
Billingsley theorem

$dim_{\mu}^{q}=\alpha\,dim_{\mu,\nu}^{q}.$ (13)

## References

  * [1] P. Abry, H. Wendt and G. Didier, Detecting and estimating multivariate self-similar sources in high-dimensional noisy mixtures. 2018 IEEE Workshop on Statistical Signal Processing (SSP), 2017, pp. 688-692.
  * [2] N. Attia, B. Selmi and C. Souissi, Some density results of relative multifractal analysis. Chaos, Solitons and Fractals 103 (2017), pp. 1–11.
  * [3] N. Attia and B. Selmi, Relative Multifractal Box-Dimensions. Filomat 33:9 (2019), 2841–2859. DOI: 10.2298/FIL1909841A.
  * [4] A. Ben Mabrouk, A note on Hausdorff and packing measures, Interna. J. Math. Sci., 8(3-4) (2009), pp. 135-142.
  * [5] A. Ben Mabrouk, A higher order multifractal formalism, Stat. Prob. Lett. 78 (2008), pp. 1412-1421.
  * [6] P. Billingsley, Ergodic Theory and Information. John Wiley & Sons, Inc., New York, London, Sydney 1965.
  * [7] G. Chen and Q. Cheng, Fractal density modeling of crustal heterogeneity from the KTB deep hole, J. Geophys. Res. Solid Earth, 122, (2017), 1919–1933, doi:10.1002/2016JB013684.
  * [8] J. Cole, Relative multifractal analysis. Choas Solitons Fractals 11 (2000), pp. 2233–2250.
  * [9] J. Cole and L. Olsen, Multifractal variation measures and multifractal density theorems. Real Anal Exch 28 (2003), pp. 501–514.
  * [10] S. Combrexelle, Multifractal analysis for multivariate data with application to remote sensing. Thèse de doctorat de l’université de Toulouse, Spécialité: signal, image, acoustique et optimisation, 2016.
  * [11] M. Dai, J. Hou, J. Gao, W. Su, L. Xi and D. Ye, Mixed multifractal analysis of China and US stock index series. Chaos, Solitons and Fractals 87 (2016) 268–275.
  * [12] M. Das, Hausdorff measures, dimensions and mutual singularity. Trans Am Math Soc 357 (2005), pp. 4249–4268.
  * [13] Z. Douzi and B. Selmi, Multifractal variation for projections of measures. Chaos Solitons Fractals 91 (2016), pp. 414–420 .
  * [14] Z. Douzi and B. Selmi, Regularities of general Hausdorffand packing functions. Chaos, Solitons and Fractals 123 (2019) 240–243.
  * [15] Z. Douzi and B. Selmi, On the Projections of the Mutual Multifractal Rényi Dimensions. Anal. Theory Appl., Vol. 36, No. 2 (2020), pp. 1-20. DOI: 10.4208/ata.OA-2017-0036.
  * [16] Z. Douzi and B. Selmi, On the mutual singularity of multifractal measures. Electronic Research Archive. DOI: 10.3934/era.2020024 volume 28 number 1, march 2020 pages 423-432.
  * [17] Z. Douzi, A. Samti and B. Selmi, Another example of the mutual singularity of multifractal measures. Proyeccinones (Antofagasta. On line) 40(1) (2021), pp. 17-23.
  * [18] G. A. Edgar, Centered densities and fractal measures. New York J Math 13 (2007), pp. 33–87.
  * [19] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, No 85, Cambridge Univ. Press, New York/London, 1985.
  * [20] A. Farhat and A. Ben Mabrouk, Mixed Generalized Multifractal Densities for vector-valued Quasi-Ahlfors Measures, (2018), viXra:1804.0405.
  * [21] A. Farhat and A. Ben Mabrouk, Mixed Generalized Multifractal Densities for vector-valued Quasi-Ahlfors Measures, (2018), viXra:1804.0405.
  * [22] B. C. L. de Figueiredo, G. R. Moreira, B. Stosic and T. Stosic, multifractal analysis of hourly wind speed records in petrolina, northeast brazil. Rev. Bras. Biom., Sao Paulo, 32(4) (2014), p.599-608, .
  * [23] G. Gan, C. Ma and J. Wu, Data Clustering Theory, Algorithms, and Applications. ASA-SIAM Series on Statistics and Applied Probability, SIAM, Philadelphia, ASA, Alexandria, VA, 2007.
  * [24] A. P. Garcia-Marin, J. Estévez, F. J. Jiménez-Hornero, and J. L. Ayuso-Munoz, Multifractal analysis of validated wind speed time series. Chaos: An Interdisciplinary Journal of Nonlinear Science 23, 013133 (2013); doi: 10.1063/1.4793781
  * [25] J. Jaerisch and H. Takahasi, Mixed multifractal spectra of Birkhoff averages for non-uniformly expanding one-dimensional Markov maps with countably many branches. arXiv:2004.04347v2 [math.DS] 29 Jun 2020. 38 pages.
  * [26] E. Jarvenpaa, M. Jarvenpaa, A. Kaenmaki, T. Rajala, S. Rogovin and V. Suomala, Packing Dimension and Ahlfors Regularity of Porus Sets in Metric Spaces. arXiv:1701.08593v1 [math.CA] 30 Jan 2017
  * [27] L. Hong, Y. Tie and W. Lanlan, Network Traffic Prediction Based on Multifractal MLD Model. 2010 International Workshop on Chaos-Fractal Theory and its Applications. IEEE Computer Society, 2010, pp. 466-470. DOI: 10.1109/IWCFTA.2010.109.
  * [28] M. Khelifi, H. Lotfi, A. Samti and B. Selmi, A relative multifractal analysis. Chaos, Solitons and Fractals 140 (2020) 110091.
  * [29] A. Kinnison and P. Mörters, Simultaneous Multifractal Analysis of the Branching and Visibility Measure on a Galton-Watson Tree. Advances in Applied Probability, 42(1) (2010), 226-245. DOI:10.1239/aap/1269611151.
  * [30] C. Meneveau, K. R. Sreenivasan, P. Kailasnath, and M. S. Fan, Joint multifractal measures: Theory and applications to turbulence. Physical Review A 41(2) (1990), pp. 894–913.
  * [31] P. Mattila, Geometry of sets and measures in euclidian spaces: fractals and rectifiability. Cambridge University Press; 1995 .
  * [32] P. Mattila and P. Saaramen, Ahlfors-David regular sets and bilipschitz maps. arXiv:0809.4877v1 [math.MG] 29 Sep 2008.
  * [33] M. Menceur, A. Ben Mabrouk and K. Betina, The Multifractal Formalism For Measures, Review and Extension to Mixed Cases. Anal. Theory Appl., Vol. 32, No. 1 (2016), pp. 77-106.
  * [34] M. Menceur and A. Ben Mabrouk, A mixed multifractal formalism for finitely many non Gibbs Frostman-like measures, (2018), 19 pages, ArXiv:1804.09034v1.
  * [35] L. Olsen, A multifractal formalism, Adv. Math., 116 (1995), pp. 82–196.
  * [36] L. Olsen, Multifractal dimensions of product measures. Math Proc Camb Phil Soc 120 (1996), pp. 709–34.
  * [37] L. Olsen, Mixed generalized dimensions of self-similar measures, J. Math. Anal. and Appl., 306 (2005), pp. 516-539.
  * [38] H. Pajot, Sous-ensembles de courbes Ahlfors-régulières et nombres de Jones. Publicacions Matematiques, 40 (1996), pp. 497–526.
  * [39] P. Pavon-ominguez, F. J. Jimenez-Hornero, E. Gutierrez-de-Rave, Multifractal analysis of ground–level ozone concentrations at urban, suburban and rural background monitoring sites in Southwestern Iberian Peninsula. Atmospheric Pollution Research 4 (2013), 229-237.
  * [40] Y. B. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications (Chicago Lectures in Mathematics) (Chicago, IL: University of Chicago Press), 1997.
  * [41] C. Qu, The Lower Densities of Symmetric Perfect Sets, Anal. Theory Appl., 28 (2012), pp. 377-384.
  * [42] I. Scheuring and R. H. Riedi, Application of multifractals to the analysis of vegetation pattern.Journal of Vegetation Science 5: 489-496, 1994.
  * [43] B. Selmi, On the strong regularity with the multifractal measures in a probability space. Analysis and Mathematical Physics (2019) 9:1525–1534. doi: 10.1007/s13324-018-0261-5.
  * [44] B. Selmi, Remarks on the mutual singularity of multifractal measures. Proyeccinones (Antofagasta. On line) 40(1) (2021), pp. 73-84.
  * [45] B. Selmi, The relative multifractal analysis, review and examples. Acta Sci. Math. (Szeged) 86 (2020), 635–666. DOI: 10.14232/actasm-020-801-8.
  * [46] C. M. Taylor and A. Salhi, On Partitioning Multivariate Self-Affine Time Series. IEEE Transactions on Evolutionary Computation, vol. 21, no. 6, pp. 845-862, Dec. 2017. DOI: 10.1109/TEVC.2017.2688521.
  * [47] M. Xu and S. Wang, The Boundedness of Bilinear Singular Integral Operators on Sierpinski Gaskets, Anal. Theory Appl., 27 (2011), pp. 92-100.
  * [48] S. Xu and W. Xu, Note on the Paper ”An Negative Answer to a Conjecture on the Self-similar Sets Satisfying the Open Set Condition”, Anal. Theory Appl., 28 (2012), pp. 49-57.
  * [49] S. Xu, W. Xu and D. Zhong, Some New Iterated Function Systems Consisting of Generalized Contractive Mappings, Anal. Theory Appl., 28 (2012), pp. 269-277.
  * [50] Y. Yuan, Spectral Self-Affine Measures on the Generalized Three Sierpinski Gasket, Anal. Theory Appl., 31 (2015), pp. 394-406.
  * [51] Y.-L. Ye, Self-similar vector-valued measures Adv. Appl. Math. 38 (2007): 71..96.
  * [52] Y.-L. Ye, Multifractal analysis of non-uniformly contracting iterated function systems. Nonlinearity 30 (2017) 1708–1733.
  * [53] C. Zeng, D. Yuan and S. Xui, The Hausdorff Measure of Sierpinski Carpets Basing on Regular Pentagon, Anal. Theory Appl., 28 (2012), pp. 27-37.
  * [54] Z. Zhou and L. Feng, A Theoretical Framework for the Calculation of Hausdorff Measure Self-similal Set Satisfying OSC, Anal. Theory Appl., 27 (2011), pp. 387-398.
  * [55] Z. Zhu and Z. Zhou, A Local Property of Hausdorff Centered Measure of Self-Similar Sets, Anal. Theory Appl., 30 (2014), pp. 164-172.