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

# A mixed multifractal analysis 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

The multifractal formalism for measures in its original formulation is checked
for special classes of measures such as doubling, self similar, and Gibbs-like
ones. Out of these classes, suitable conditions should be taken into account
to prove the validity of the multifractal formalism. In the present work, a
large class of measures satisfying a weak condition known as quasi Ahlfors is
considered in the framework of mixed multifractal analysis. A joint
multifractal analysis of finitely many quasi Ahlfors probability measures is
developed. Mixed variants of multifractal generalizations of Hausdorff and
packing measures, and corresponding dimensions are introduced. By applying
convexity arguments, some properties of these measures and dimensions are
established. Finally, an associated multifractal formalism is introduced and
proved to hold for the class of quasi Ahlfors measures.

###### keywords:

Hausdorff measure and dimension, packing measure and dimensions, Multifractal
formalism, mixed cases, Ahlfors measures.

###### PACS:

: 28A78, 28A80.

##  1 Introduction

The present work is devoted to the topic of multifractal analysis of measures
and the validity of multifractal formalism. We precisely focus on the
simultaneous behaviors of finitely many measures instead of a single measure
as in the classical or original multifractal analysis of measures. We call
such a study mixed multifractal analysis. Such a mixed analysis has been
generating a great attention recently and thus proved to be powerful in
describing the local behavior of measures especially fractal ones. (See [9],
[10], [11], [49], [50], [52], [53], [54], [60], [69], [70], [71], [73], [74],
[75], [76]).

In this paper, multi purposes will be done. Firstly we introduce the general
context in which the new joint multifractal analysis will be developed. We
introduce firstly some new variants of Hausdorff and packing measures based on
some vector-valued measures controlled by a common measure assumed to be
quasi-Ahlfors. Next, in the same section, associated multifractal dimensions
due to the introduced measures are defined. Eventual relations between joint
Hausdorff measures, joint packing measures, joint Hausdorff dimensions, joint
packing dimensions as well as joint variants of Renyi dimensions are used. A
second aim is to introduce a corresponding multifractal spectrum/formalism
permitting to investigate the behavior of finitely many measures
simultaneously.

As it is noticed from the literature on multifractal analysis of measures,
this latter always considered a single measure and studies its scaling
behavior as well as the multifractal formalism associated. Recently, many
works have been focused on the study of simultaneous behaviors of finitely
many measures. In [55], a mixed multifractal analysis has been developed
dealing with a generalization of Rényi dimensions for a finite set of self
similar measures, constituting thus a first motivation to our present paper.
We intend to combine the generalized Hausdorff and packing measures, and the
corresponding dimensions with Olsen’s results in [54] to define and develop a
more general multifractal analysis for finitely many measures by studying
their simultaneous regularity, spectrum and to define a mixed multifractal
formalism which may describe better the geometry of the singularities’s sets
of these measures. We apply the techniques of L. Olsen especially in [49] and
[54] with the necessary modifications to study in details general mixed
dimensions of simultaneously finite number of measures one of them at least is
characterized by a quasi-Ahlfors property. Our results May be next shown to
valid easily for a single measure, which confirms their generecity.

The assumption of being Ahlfors for one of the measures is essential
contrarily to some existing works that have omitted such assumption and
developed some questionable version of multifractal densities ([2, 20, 24]).
Indeed, in such references the authors referred to [13] to show that a real-
valued dimension may exist without a link between the measure of a ball and
its diameter. However, the authors did not pay attention to the fact that
general probability measures (although being doubling) may not lead to
multifractal dimensions. Indeed, it is assumed in [13] (but not in [2] and
[20]) that

  * •

both the lower bound taken in the definition of the $\mu$-Hausdorff measure
and the upper bound taken in the $\mu$-packing measure, may extend on all
$\mu$-$\rho$-coverings and packings whenever $\mu$ is a Borel probability
measure. Besides, a $\mu$-$\rho$-covering is already a covering by cylinders
$C$ satisfying $\mu(C)<\rho$, where $\rho$ is the diameter or the radius.

  * •

The measure $\mu$ is assumed to be nonatomic. Otherwise, the set of
$\mu$-$\rho$-coverings may be empty.

Consequently, one may check for [2] and [20] the validity of the multifractal
formalism in a general case without any assumption on the measure of balls and
their diameters. Relatively to [49], where the measure $\nu$ is equivalent to
a Lebesgue’s one, the results obtained in [2] and [20] are different. We
proposed in the present work to relax hypothesis on the applied measures and
assume a weak form of the so-called Alhfors measures. Backgrounds on such
measures may be found in [30], [37], [45], [47], [58], [59].

###### Definition 1.1

Let $\nu$ be a Borel probability measure on $\mathbb{R}^{d}$. We say that
$\nu$ is a quasi-Ahlfors measure with regularity index $\alpha>0$ if it
satisfies

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

We say that $\nu$ is an Ahlfors measure with regularity index $\alpha>0$ if it
satisfies

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

In the rest of the paper, we will write $\mathcal{QAHP}(\mathbb{R}^{d})$
(respectively, $\mathcal{AHP}(\mathbb{R}^{d})$) to designate the set of quasi-
Ahlfors (respectively, Ahlfors) probability measures on $\mathbb{R}^{d}$.

With these assumptions, it becomes natural to extend the multifractal analysis
to the cases introduced in [2] and [20]. The readers may refer to [13], [21],
[48], [49], [50], [60], [69], [70], [71], [72], [72], [73], [74], [75], [76]
for more understanding.

Resuming, mixed multifractal analysis 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 [34], [43]. 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 [34].

In section 2, we address a brief review on single/mixed multifractal analysis.
We precisely focus on eventual applications, existing works and possible
future direction, already with enlightening the link to our work in the
present paper and the novelty relatively to existing works in the same topic.
In section 3, we introduce the new context of the joint multifractal analysis.
We introduce the new joint extensions of Hausdorff and packing measures and we
show that they associate as for the classical case some multifractal
dimensions to sets. We prove some properties of convexity, monotony, upper and
lower bounds for these dimensions. Next, section 4 is devoted to the
development of an associated multifractal spectrum relative to the new
dimensions and measures developed previously. An upper bound of such a
spectrum is provided. In section 5, we develop an associated multifractal
formalism issued from the previous framework. Results on the validity of such
formalism are established as well as an application to the case of projections
of measures. Section 6 is a conclusion, in which we resume the results
obtained, and raise eventual extensions and applications.

##  2 On the utility of mixed multifractal analysis 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, and distributions.

Multifractal analysis has now applications and/or links to many cases in AI.
It is indeed used in smart grids such as fractal ones, smart cities,
buildings, nano-materials such as nano-circuits, patterns, complexity in
nature, Patterns, biosignals, bioimages, …etc. The bibliography in such topics
is growing up rapidly. Interested readers may refer to the further reading
section ([77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88]) for more
information.

Mixed multifractal analysis (MXMA) has been applied in explaining joint
movements in volatility for asset markets such as joint multifractal Markov-
switching models. Besides, mixed multifractal analysis is not really new in
financial series processing. It has been in contrast merged under the name of
multivariate multifractal analysis (MVMA), where many situations in financial
markets, and their volatility have been described. Multivariate models have
been also applied for long memory with mixture distributions. See for instance
[16, 29, 33, 41, 42].

Mixed multifractal analysis of measures has been applied in [22] to extract
the properties of stock index series, and in exploiting the eventual inner
relationship that relates them.

Mixed multifractal analysis has been also applied in the last decade to
generalize the so-called detrended fluctuation analysis, which is applied
widely for modeling time series. Mixed multifractal analysis has been emerged
in such a field to model the nonlinear correlation in time series and/or their
volatility to understand better the dependence between markets’ indices. See
[12, 14, 38, 44].

Recently, joint multifractal analysis, another face of mixed multifractal
analysis, has been applied in economic-environmental linkage of indices. In
[15], the dynamics of atmospheric carbon emissions and industrial production
index have been investigated by means of joint multifractal analysis for both
short and long terms. Already in the study of environmental indices and/or
climate factors, a mixed multifractal study has been developed in [19] to
understand the thermal structure of the mixed layer in sea surface. Recall
that temperature changes are basic causes of the well-known NINO phenomenon
detected on sea surfaces.

In [68], multivariate multifractal analysis has been applied to texture
characterization of natural images. The MVMA shows that image intensity may be
subject to fluctuated regularity. In [66], MVMA has been applied for
characterizing volatility and cross-correlation for agricultural futures’
markets. Several statistical measures such as autocorrelation, coupling
correlation as well as cross-correlation for some Chicago board of trade
series have been estimated via the MVMA to decide about the state of markets.

In relation to AI applications of fractals and multifractals, Castiglioni and
Faini addressed in [17] a multivariate multifractal analysis for biomedical
applications by developing a fast DFA algorithm on maximally overlapped blocks
suitable for short and long time series due to physiological applications. See
also [67].

From a theoretical point of view, in [25], a mixed multifractal analysis has
been developed to detect mutual singularities of the multifractal Hausdorff
measure $\mathcal{H}_{\mu}^{q,t}$ and the packing measure
$\mathcal{P}_{\mu}^{q,t}$. This differs from our work as we seek here the
mutual singularities for the measure $\mu$ included in the definition of the
Hausdorff/packing measure and not the singularities of
$\mathcal{H}_{\mu}^{q,t}$ and $\mathcal{P}_{\mu}^{q,t}$.

In [3], a mixed multifractal analysis has been developed for box dimension of
measures. Recall that box dimensions may coincide with the Hausdorff and
packing ones under suitable conditions such as the compactness of the metric
space. In our work, we get such coincidence without restricting to compact
metric spaces. See also [28, 64]. In [27], the authors applied the mixed
multifractal analysis of measures to estimate the Rényi dimension relative to
a couple of measures where the product of such measures is controlled by the
diameter/radius of the coverings. In [63], some known concepts about the
multifractal analysis are revisited in the framework of a relative
multifractal analysis. New simplified proofs and examples have been provided.

More about the use of mixed multifractal analysis of measures as well as
functions and time series or images may be found in [1, 5, 23, 40, 57] and the
references therein.

##  3 New joint multifractal measures and associated dimensions

In this section, we propose to develop the general framework of the joint
multifractal analysis. General joint variants of both Hausdorff and packing
measures will be introduced as well as the associated dimensions of sets.

Let $\mathcal{P}(\mathbb{R}^{d})$ be the set of all probability Borel measures
on $\mathbb{R}^{d}$. For a single or vector-valued measure $\mu$ denote
$S_{\mu}$ its topological support. Let $k\in\mathbb{N}$ be fixed and
$\mu=(\mu_{1},\mu_{2},...,\mu_{k})\in\mathcal{P}(\mathbb{R}^{d})^{k}$. Let
also $q=(q_{1},q_{2},...,q_{k})\in\mathbb{R}^{k}$, $x\in\mathbb{R}^{d}$ and
$r>0$. We denote

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

where we designate by $B(x,r)$ the ball of center $x$ and radius $r$. Next,
given $E\subseteq\mathbb{R}^{d}$ and $\varepsilon>0$, we call an
$\varepsilon$-covering of $E$ any finite or countable set ($U_{i})_{i}$ of
non-empty subsets $U_{i}\subseteq\mathbb{R}^{d}$ satisfying

$E\subseteq{\cup}_{i}U_{i}\hbox{ and }|U_{i}|<\varepsilon,$ (1)

where $|.|$ is the diameter.

The last assumption is already assumed in [13], [10], [11], [48], [49], [50],
[51], [52], [53], [55] but unfortunately not in [2] and [20]. When assuming
that the measure is quasi-Ahlfors, this assumption is not necessary and may be
replaced by the original one in [13] on $\mu$-$\varepsilon$-coverings.

###### Definition 3.1

Given $E\subseteq\mathbb{R}^{d}$ and $\varepsilon>0$, we call an
$\mu$-$\varepsilon$-covering of $E$ any at most countable set $(U_{i})_{i}$ of
non-empty subsets $U_{i}\subseteq\mathbb{R}^{d}$ satisfying

$E\subseteq{\cup}_{i}U_{i}\hbox{ and }\mu(U_{i})<\varepsilon.$

For a subset $E\subseteq\mathbb{R}^{d}$ and $\varepsilon>0$, a countable set
$(B(x_{i},r_{i}))_{i}$ of balls in $\mathbb{R}^{d}$ is called a centered
$\varepsilon$-$\mu$-covering of $E$ if

$E\subseteq\underset{i}{\cup}B(x_{i},r_{i}),\text{ }x_{i}\in E\text{ and
}0<\mu(B(x_{i},r_{i}))<\varepsilon\text{ for all }i.$

###### Definition 3.2

Let $\mu=(\mu_{1},\mu_{2},...,\mu_{k})$ be a positive vector-valued measure on
$E=E_{1}\times E_{2}\times...\times E_{k}$. Then the support of $\mu$ is the
complement of the largest open subset $U$ of $E$ such that
$\mu(U)=(0,0,...,0)$.

###### Theorem 3.1

(Besicovitch covering theorem) There exists a fixed
$\mathcal{N}_{B}\in\mathbb{N}$ satisfying the following assertion: For any
$A\subset\mathbb{R}^{d}$ and any bounded set $(r_{x})_{x\in
A}\subset(0,\infty)$, there exists ${N}_{B}$ subsets
$C_{1},C_{2},...,C_{N_{B}}$ of $\left\\{B(x,r_{x}),\,x\in A\right\\}$, such
that

  1. 1.

Each set $C_{i}$ is finite or countable, for all $i=1,...,N_{B}$.

  2. 2.

Each $C_{i}$ is composed of disjoint sets, for all $i=1,...,{N}_{B}$.

  3. 3.

$A\subseteq\displaystyle\bigcup_{i=1}^{N_{B}}\displaystyle\bigcup_{B\in
C_{i}}B$.

We now proceed in introducing the construction of the new variants of
multifractal Hausdorff and packing measures and the associated dimensions. We
will see later that being quasi-Ahlfors is necessary for at least one measure.

Let
$\xi=(\mu,\nu)=(\mu_{1},\mu_{2},...,\mu_{k},\nu)\in\mathcal{P}(\mathbb{R}^{d})^{k}\times\mathcal{QAHP}(\mathbb{R}^{d})$
and $(q,t)=(q_{1},q_{2},...,q_{k},t)\in\mathbb{R}^{k+1}$. For
$E\subset\mathbb{R}^{d}$ and $\varepsilon>0$, let

${\overline{\mathcal{H}}}_{\xi,\varepsilon}^{q,t}(E)=\inf\biggl{\\{}\sum_{i}(\mu(B(x_{i},r_{i})))^{q}(\nu(B(x_{i},r_{i})))^{t}\biggr{\\}}$

and

${\overline{\mathcal{H}}}_{\xi}^{q,t}(E)=\lim_{\varepsilon\downarrow
0}{\overline{\mathcal{H}}}_{\xi,\varepsilon}^{q,t}(E),$

where the inf is over the set of centred $\varepsilon$-coverings of $E$.
Similarly, let

${\overline{\mathcal{P}}}_{\xi,\varepsilon}^{q,t}(E)=\sup\\{\sum_{i}(\mu(B(x_{i},r_{i})))^{q}(\nu(B(x_{i},r_{i})))^{t}\\}$

and

${\overline{\mathcal{P}}}_{\xi}^{q,t}(E)=\lim_{\varepsilon\downarrow
0}{\overline{\mathcal{P}}}_{\xi,\varepsilon}^{q,t}(E),$

where the sup is over the set of centred $\varepsilon$-packings of $E$.

###### Definition 3.3

The mixed generalized Hausdorff measure relatively to $\xi$ is stated as

$\mathcal{H}_{\xi}^{q,t}(E)=\displaystyle\sup_{F\subseteq
E}{\overline{\mathcal{H}}}_{\xi}^{q,t}(F).$

The mixed generalized packing measure relatively to $\xi$ is expressed by

$\mathcal{P}_{\xi}^{q,t}(E)=\inf_{E\subseteq\cup_{i}E_{i}}{\sum_{i}}{\overline{\mathcal{P}}}_{\xi}^{q,t}(E_{i}).$

It is straightforward that $\mathcal{H}_{\xi}^{q,t}$ and
$\mathcal{P}_{\xi}^{q,t}$ are outer metric and regular measures on
$\mathbb{R}^{d}$. Borel sets are thus measurable relatively to them.
Furthermore, we may prove using the well known Besicovitch covering theorem
that

$\mathcal{H}_{\xi}^{q,t}(E)\leq\mathcal{N}_{B}\mathcal{P}_{\xi}^{q,t}\leq\mathcal{N}_{B}{\overline{\mathcal{P}}}_{\xi}^{q,t}(E),\;\forall(q,t)\in\mathbb{R}^{k+1},\;\forall
E\subseteq\mathbb{R}^{d}.$ (2)

$\mathcal{N}_{B}$ is the constant number related to the Besicovitch covering
theorem.

The original idea in [49] considers for $\mu\in\mathcal{P}(\mathbb{R}^{d})$
and $(q,t)\in\mathbb{R}^{2}$ the dimension function

$h_{q,t}(r)=\mu(B(x,r))^{q}r^{t},\;\;r>0,$

and associates variants $\mathcal{H}_{h_{q,t}}$ for Hausdorff and
$\mathcal{P}_{h_{q,t}}$ for packing measures relatively to $h_{q,t}$. These
measures have special supports

$X_{\mu}(\alpha)=\Biggl{\\{}x\in\mathcal{S}_{\mu},\;\displaystyle\lim_{r\downarrow
0}\displaystyle\frac{\log(\mu(B(x,r)))}{\log\,r}=\alpha\Biggr{\\}}$

known as multifractal decomposition sets, for suitable parameters $\alpha$ and
$\mu$ as in doubling, Hölderian, and Gibbs cases. To show the validity of the
multifractal formalism in some special cases $X_{\nu}(f(\alpha))$ for suitable
functions $f$, a large deviation formalism has been applied to obtain Gibbs
measures supported by these sets. Billingsley theorem known in ergodic theory
has been next applied to deduce the dimension of $X_{\mu}(\alpha)$ from the
dimension of $X_{\nu}(f(\alpha))$.

In the present situation, a more general case is investigated for vector-
valued measures using the cross-correlation dimension function

$H_{q_{1},\dots,q_{k},t}(r)=\mu_{1}(B(x,r))^{q_{1}}\dots\mu_{k}(B(x,r))^{q_{k}}\nu(B(x,r))^{t},\;\;r>0,$

for $q_{1},\dots,q_{k},t\in\mathbb{R}$, and where $\mu_{1},\dots,\mu_{k},\nu$
are Borel probability measures on $\mathbb{R}^{d}$. $\nu$ plays the role of a
gauge function used to control the simultaneous behavior of the measures
$\mu_{i}$, $i=1,\dots,k$. Variants of Hausdorff and packing measures
relatively to the multi-variables dimension function $H_{q_{1},\dots,q_{k},t}$
are introduced. For example, in the single case $k=1$ and the choice
$\nu(B(x,r))\simeq r$, we immediately obtain

$H_{q,t}(r)=h_{q,t}(r)$

for all $q,t\in\mathbb{R}$, which means that the variants of Hausdorff and
packing measures relative to $H_{q_{1},\dots,q_{k},t}$ are indeed extensions
of those introduced in [49].

The results developed in the present work lie in the whole topic of dimension
theory, especially Carathéodory dimension, which in turns constitutes a
general form of Hausdorff one. Indeed, in [59], a link between dynamical
systems and dimension theory has been pointed out, which makes a motivation to
our work here. The idea in [59] consists in interpreting the concept of
dimension as a C-structure. Mathematically speaking, let $\mathcal{C}$ be any
collection of subsets of a space $X$ known as the physical space, and consider
some non-negative functions $\varphi$, $\Lambda$ and $\Theta$ defined on
$\mathcal{C}$, which play respectively the role of the diameter and some
statistical physics measures. In this framework, the dimension function
$\mathcal{H}_{\xi}^{q,t}$ applied in the present work may be interpreted as a
multivariate form of the Carathéodory one. Under suitable conditions on these
functions, we may introduce a free energy analogue to the fractal measures by
setting

$\mathcal{H}_{\alpha}(X)=\inf\\{\displaystyle\sum_{U\in\mathcal{G}}\varphi(U)\Lambda(U)^{\alpha}\\},$

where $\mathcal{G}$ depends eventually on $X$ and $\Lambda$. This which yields
as usual a dimension called the capacity of $X$. One of the related frameworks
in multifractal analysis is the adoption of the continuous analysis to
discrete spaces such as $\mathbb{Z}^{d}$. Some essays have been already
started the last decades by comparing the log-measure of balls to some
quantities that are not log-powers of the diameter. As we know, the Hausdorff
dimension of discrete sets is not important in the framework of classical
definitions. Therefore, different variants of function dimensions should be
applied appropriately for discrete sets. These set functions are no longer
power-laws of the diameter. Interested readers may refer to [4], [7], [35] and
the references therein for the discrete framework.

In another parallel direction, J. Cole in [20] proposed to control the
analyzed measure $\mu$ by another suitable measure $\nu$ via a relative
multifractal analysis of the relative singularity sets

$X_{\xi}(\alpha,\beta)=\Biggl{\\{}x\in\mathcal{S}_{\xi},\;\displaystyle\lim_{r\downarrow
0}\displaystyle\frac{\log(\mu(B(x,r)))}{\log\,r}=\alpha\,\mbox{and}\,\displaystyle\lim_{r\downarrow
0}\displaystyle\frac{\log(\mu(B(x,r)))}{\log(\nu(B(x,r)))}=\beta\Biggr{\\}}.$

We now introduce the associated mixed dimensions relative to the generalized
Hausdorff and packing measures $\mathcal{H}_{\xi}^{q,t}$ and
$\mathcal{P}_{\xi}^{q,t}$. We will notice the necessity of the quasi-Ahlfors
assumption. We have the following result which prepares to introduce the new
mixed dimensions and shows the necessity of being Ahlfors for some measures
composing the $(k+1)$-tuple measure $\xi$.

###### Lemma 3.1

Let
$\xi=(\mu,\nu)\in\mathcal{P}(\mathbb{R}^{d})^{k}\times\mathcal{QAHP}(\mathbb{R}^{d})$
and $E\subseteq\mathbb{R}^{d}$. $\forall q\in\mathbb{R}^{k}$, The set
$\Gamma_{q}:=\left\\{t\in\mathbb{R},\;\mathcal{H}_{\xi}^{q,t}(E)<+\infty\right\\}$
is nonempty.

###### Proof 3.1

Let $\alpha,M\in\mathbb{R}_{+}$ be such that

$\displaystyle\limsup_{|U|\rightarrow
0}\displaystyle\frac{\nu(U)}{|U|^{\alpha}}<M.$

We obtain, for some $\delta>0$ and $\forall r$, $0<r<\delta$,

$\nu(U)\leq M|U|^{\alpha};\;\forall U,\;|U|<r.$

Consider next a $\delta$-covering $(B(x_{i},r_{i}))_{i}$ of $E$, and the
$\mathcal{N}_{B}$ Besicovitch collections. We obtain

$\sum_{i}\mu(B(x_{i},r_{i}))^{q}\nu(B(x_{i},r_{i}))^{t}\leq\sum_{i=1}^{\mathcal{N}_{B}}\sum_{j}\mu(B(x_{ij},r_{ij}))^{q}\nu(B(x_{ij},r_{ij}))^{t}.$

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

$\sum_{i=1}^{\mathcal{N}_{B}}\sum_{j}\nu(B(x_{ij},r_{ij}))^{t}.$

For $t=1,$ this becomes

$\sum_{i=1}^{\mathcal{N}_{B}}\sum_{j}\nu(B(x_{ij},r_{ij})).$

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

$\sum_{i=1}^{\mathcal{N}_{B}}\nu\left(\cup_{j}B(x_{ij},r_{ij})\right)\leq\mathcal{N}_{B}\nu(\mathbb{R}^{d})=\mathcal{N}_{B}.$

Consequently, we obtain

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

Assume now that there exist $i_{j},$ $1\leq j\leq k$ such that $q_{i_{j}}\leq
0$. For $t>0,$ we get

$\nu(B(x_{i_{j}},r_{i_{j}}))^{t}\leq M^{t}r_{i_{j}}^{\alpha t},\forall j.$

As a result, we get

$\sum_{j}\mu(B(x_{i_{j}},r_{i_{j}}))^{q}\nu(B(x_{i_{j}},r_{i_{j}}))^{t}\leq
2^{-\alpha t}M^{t}\sum_{j}\mu(B(x_{i_{j}},r_{i_{j}}))^{q}(2r_{i_{j}})^{\alpha
t}.$

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

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

As a consequence of Lemma 3.1, we get

$\mathcal{H}_{\xi}^{q,t}(E)<+\infty\;\;\Longrightarrow\;\;\mathcal{H}_{\xi}^{q,s}(E)=0,\;\;\forall\,s>t,$
(3)

and

$\mathcal{H}_{\xi}^{q,t}(E)>0\;\;\Longrightarrow\;\;\mathcal{H}_{\xi}^{q,s}(E)=+\infty,\;\;\forall\,s<t.$
(4)

This permits to introduce now the generalised mixed multifractal dimensions
due to the variants $\mathcal{H}_{\xi}^{q,t}$ and $\mathcal{P}_{\xi}^{q,t}$.

###### Proposition 3.1

For any set $E\subseteq\mathbb{R}^{d}$, there exists unique values denoted by
$dim_{\xi}^{q}({E})$, $\Delta_{\xi}^{q}({E})$ and $Dim_{\xi}^{q}({E})$ in
$\left[-\infty,+\infty\right]$, and satisfying respectively,

  1. 1.

$\mathcal{H}_{\xi}^{q,t}(E)=\left\\{\begin{array}[]{c}\infty\hbox{ for
}t<dim_{\xi}^{q}({E}),\\\ 0\hbox{ for
}t>dim_{\xi}^{q}({E}).\end{array}\right.$

  2. 2.

${\overline{\mathcal{P}}}_{\xi}^{q,t}(E)=\left\\{\begin{array}[]{c}\infty\hbox{
for }t<\Delta_{\xi}^{q}({E}),\\\ 0\hbox{ for
}t>\Delta_{\xi}^{q}({E}).\end{array}\right.$

  3. 3.

$\mathcal{P}_{\xi}^{q,t}(E)=\left\\{\begin{array}[]{c}\infty\hbox{ for
}t<Dim_{\xi}^{q}({E}),\\\ 0\hbox{ for
}t>Dim_{\xi}^{q}({E}).\end{array}\right.$

###### Proof 3.2

Item 1 is a consequence of equations (3) and (4) by setting

$dim_{\xi}^{q}({E})=\inf\\{\,t\in\mathbb{R}\,;\;\mathcal{H}_{\xi}^{q,t}(E)=0\\}.$

Item 2 is a consequence of the following assertion, stating that for all
$E\subseteq\mathbb{R}^{d}$ and $t>0$,

${\overline{\mathcal{P}}}_{\xi}^{q,t}(E)<+\infty\;\;\Longrightarrow\;\;{\overline{\mathcal{P}}}_{\xi}^{q,s}(E)=0,\;\;\forall\,s>t,$

and

${\overline{\mathcal{P}}}_{\xi}^{q,t}(E)>0\;\;\Longrightarrow\;\;{\overline{\mathcal{P}}}_{\xi}^{q,s}(E)=+\infty,\;\;\forall\,s<t,$

by setting

$\Delta_{\xi}^{q}({E})=\inf\\{\,t\in\mathbb{R}\,;\;{\overline{\mathcal{P}}}_{\xi}^{q,t}(E)=0\\}.$

Item 3 is a consequence of the following assertion, stating that for all
$E\subseteq\mathbb{R}^{d}$ and $t>0$,

${{\mathcal{P}}}_{\xi}^{q,t}(E)<+\infty\;\;\Longrightarrow\;\;{{\mathcal{P}}}_{\xi}^{q,s}(E)=0,\;\;\forall\,s>t,$

and

${{\mathcal{P}}}_{\xi}^{q,t}(E)>0\;\;\Longrightarrow\;\;{{\mathcal{P}}}_{\xi}^{q,s}(E)=+\infty,\;\;\forall\,s<t,$

by setting

$Dim_{\xi}^{q}({E})=\inf\\{\,t\in\mathbb{R}\,;\;\mathcal{P}_{\xi}^{q,t}(E)=0\\}.$

###### Definition 3.4

  * •

$dim_{\xi}^{q}({E})$ is called the mixed multifractal generalization of the
Hausdorff dimension of the set $E$.

  * •

$Dim_{\xi}^{q}({E})$ is called the mixed multifractal generalization of the
packing dimension of the set $E$.

  * •

$\Delta_{\xi}^{q}({E})$ is called the mixed multifractal generalization of the
logarithmic index of the set $E$.

Remark easily that the original definitions of the single Hausdorff and
packing measures and dimensions are obtained for $k=1$ and $q=0$. Besides, the
multifractal generalizations due to Olsen are obtained for $k=1$ and
$q\in\mathbb{R}$. We have precisely,

$dim_{\xi}^{Q_{i}}({E})=dim_{\mu_{i},\nu}^{q_{i}}({E}),\;Dim_{\xi}^{Q_{i}}({E})=Dim_{\mu_{i},\nu}^{q_{i}}({E}),\;\Delta_{\xi}^{Q_{i}}({E})=\Delta_{\mu_{i},\nu}^{q_{i}}({E}),$

and

$dim_{\xi}^{0}({E})=dim_{\nu}({E}),\;Dim_{\xi}^{0}({E})=Dim_{\nu}({E}),\;\Delta_{\xi}^{0}({E})=\Delta_{\nu}({E}).$

From now on, we will denote for $E\subseteq\mathbb{R}^{d}$,
$q=(q_{1},q_{2},...,q_{k})\in\mathbb{R}^{k}$, $t\in\mathbb{R}$,
$\mu\in\mathcal{P}(\mathbb{R}^{d})$ and
$\nu\in\mathcal{QAHP}(\mathbb{R}^{d})$,

$b_{\xi}(E,q)=dim_{\xi}^{q}({E}),\;B_{\xi}(E,q)=Dim_{\xi}^{q}({E}),\;\Delta_{\xi}(E,q)=\Lambda_{\xi}^{q}({E}).$

For $E=S_{(\mu,\nu)}$, we denote

$b_{\xi}(q)=dim_{\xi}^{q}(S_{(\mu,\nu)}),\;B_{\xi}(q)=Dim_{\xi}^{q}(S_{(\mu,\nu)}),\;\Delta_{\xi}(q)=\Lambda_{\xi}^{q}(S_{(\mu,\nu)}).$

For $x=(x_{1},x_{2},\dots,x_{k})$ and $q=(q_{1},q_{2},\dots,q_{k})$ in
$\mathbb{R}^{k}$ we denote

$|x|=x_{1}+x_{2}+\dots+x_{k}\;\;\mbox{and}\;\;x^{q}=x_{1}^{q_{1}}x_{2}^{q_{2}}\dots
x_{k}^{q_{k}}.$

###### Theorem 3.2

The following assertions are true.

a. $b_{\xi}(.,q)$ and $B_{\xi}(.,q)$ and $\Delta_{\xi}(.,q)$ are non-
decreasing with respect to the inclusion proprerty in $\mathbb{R}^{d}$.

b. $b_{\xi}(.,q)$ and $B_{\xi}(.,q)$ are $\sigma$-stable.

c. $B_{\xi}(q)$ and $\Lambda_{\xi}(q)$ are convex.

d. For $\widehat{q_{i}}=(q_{1},...,q_{i-1},q_{i+1},...,q_{k})$ fixed, the
functions $q_{i}\mapsto b_{\xi}(q)$, $q_{i}\mapsto B_{\xi}(q)$ and
$q_{i}\mapsto\Lambda_{\xi}(q)$ are non-increasing, $\forall\,i=1,2,...,k$.

###### Proof 3.3

a. follows from the non decreasing property of $\mathcal{H}_{\xi}^{q,t}$,
$\mathcal{P}_{\xi}^{q,t}$ and $\overline{\mathcal{P}}_{\xi}^{q,t}$ with
respect to the inclusion in $\mathbb{R}^{d}$.
b. follows from the sub-additivity property of $\mathcal{H}_{\xi}^{q,t}$ and
$\mathcal{P}_{\xi}^{q,t}$ in $\mathbb{R}^{d}$.
c. We start by proving the convexity of $\Lambda_{\xi}(E,.)$. Consider
$p,q\in\mathbb{R}^{k}$, $\alpha\in]0,1[$, and $s,t\in\mathbb{R}$, such that

$s>\Lambda_{\xi}(E,p)\mbox{ and }t>\Lambda_{\xi}(E,q).$

Let $\varepsilon>0$ be fixed arbitrary, and $(B_{i}=B(x_{i},r_{i}))_{i}$ be a
centered $\varepsilon$-packing of $E$. We have

$\displaystyle{\sum_{i}}(\mu(B_{i}))^{\alpha
q+(1-\alpha)p}(\nu(B_{i}))^{\alpha t+(1-\alpha)s}$ $\displaystyle\leq$
$\displaystyle\left[{\sum_{i}}(\mu(B_{i}))^{q}(\nu(B_{i}))^{t}\right]^{\alpha}\left[{\sum_{i}}(\mu(B_{i}))^{p}(\nu(B_{i}))^{s}\right]^{1-\alpha}.$

Hence,

${\overline{\mathcal{P}}}_{\xi,\varepsilon}^{{}^{\alpha q+(1-\alpha)p,\alpha
t+(1-\alpha)s}}(E)\leq({\overline{\mathcal{P}}}_{\xi,\varepsilon}^{q,t}(E))^{\alpha}({\overline{\mathcal{P}}}_{\xi,\varepsilon}^{p,s}(E))^{1-\alpha}.$

The limit as $\varepsilon\downarrow 0$ gives

${\overline{\mathcal{P}}}_{\xi}^{{}^{\alpha q+(1-\alpha)p,\alpha
t+(1-\alpha)s}}(E)\leq({\overline{\mathcal{P}}}_{\xi}^{q,t}(E))^{\alpha}({\overline{\mathcal{P}}}_{\xi}^{p,s}(E))^{1-\alpha}.$

Consequently,

${\overline{\mathcal{P}}}_{\xi}^{{}^{\alpha q+(1-\alpha)p,\alpha
t+(1-\alpha)s}}(E)=0,\;\forall\,s>\Lambda_{\xi}(E,p)\hbox{ and
}t>\Lambda_{\xi}(E,q).$

It results that

$\Lambda_{\xi}(\alpha
q+(1-\alpha)p,E)\leq\alpha\Lambda_{\xi}(E,q)+(1-\alpha)\Lambda_{\xi}(E,p).$

We now prove the convexity of $B_{\xi}(E,.)$. We set in this case

$t=B_{\xi}(E,q)\hbox{ and }s=B_{\xi}(E,p).$

We have

$\mathcal{P}_{\xi}^{q,t+\varepsilon}(E)=\mathcal{P}_{\xi}^{q,s+\varepsilon}(E)=0.$

Therefore, there exists $(H_{i})_{i}$ and $(K_{i})_{i}$ coverings of the set
$E$ for which

${\sum_{i}}{\overline{\mathcal{P}}}_{\xi}^{q,t+\varepsilon}(H_{i})\leq
C<+\infty\hbox{ and
}{\sum_{i}}{\overline{\mathcal{P}}}_{\xi}^{p,s+\varepsilon}(K_{i})\leq
C<+\infty.$

$C$ being a positive constant. Then, the sequence
$\bigl{(}E_{n}={\cup_{i,j=1}^{n}}(H_{i}\cap K_{j}))\bigr{)}_{n\in\mathbb{N}}$
is a covering of $E$. So that,

$\begin{array}[]{lll}&&\mathcal{P}_{\xi}^{{}^{\alpha q+(1-\alpha)p,\alpha
t+(1-\alpha)s}}(E_{n})\\\
&\leq&\displaystyle{\sum_{i,j=1}^{n}}\mathcal{P}_{\xi}^{{}^{\alpha
q+(1-\alpha)p,\alpha t+(1-\alpha)s}}(H_{i}\cap K_{j})\\\
&\leq&\displaystyle{\sum_{i,j=1}^{n}}{\overline{\mathcal{P}}}_{\xi}^{{}^{\alpha
q+(1-\alpha)p,\alpha t+(1-\alpha)s}}(H_{i}\cap K_{j})\\\
&\leq&\displaystyle\left({\sum_{i,j=1}^{n}}{\overline{\mathcal{P}}}_{\xi}^{q,t+\varepsilon}(H_{i}\cap
K_{j})\right)^{\alpha}\left({\sum_{i,j=1}^{n}}{\overline{\mathcal{P}}}_{\xi}^{p,s+\varepsilon}(H_{i}\cap
K_{j})\right)^{1-\alpha}\\\ &\leq&nC<\infty.\end{array}$

Consequently,

$B_{\xi}(E_{n},\alpha q+(1-\alpha)p)\leq\alpha
t+(1-\alpha)s+\varepsilon,\;\forall\,\varepsilon>0.$

Hence,

$B_{\xi}(E,\alpha q+(1-\alpha)p)\leq\alpha
B_{\xi}(E,q)+(1-\alpha)B_{\xi}(E,p).$

d. For $i=1,2,...,n$ and
$\widehat{q_{i}}=(q_{1},...,q_{i-1},q_{i+1},...,q_{k})$ fixed and $p_{i}\leq
q_{i}$ denote

$q=(q_{1},...,q_{i-1},q_{i},q_{i+1},...,q_{k})\hbox{ and
}p=(q_{1},...,q_{i-1},p_{i},q_{i+1},...,q_{k}).$

For any $A\subseteq E$ and any centered $\varepsilon$-covering
$(B(x_{i},r_{i}))_{i}$ of $A$ we have

$(\mu(B(x_{i},r_{i})))^{q}(\nu(B(x_{i},r_{i})))^{t}\leq(\mu(B(x_{i},r_{i})))^{p}(\nu(B(x_{i},r_{i})))^{t},\;\forall\,t\in\mathbb{R}.$

Hence,

${\overline{\mathcal{H}}}_{\xi,\varepsilon}^{q,t}(A)\leq{\overline{\mathcal{H}}}_{\xi,\varepsilon}^{p,t}(A),\;\forall\,A\subseteq
E.$

When $\varepsilon\downarrow 0$, we get

${\overline{\mathcal{H}}}_{\xi}^{q,t}(A)\leq{\overline{\mathcal{H}}}_{\xi}^{p,t}(A),\;\forall\,A\subseteq
E.$

Therefore,

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

As a result,

$\mathcal{H}_{\xi}^{q,t}(E)=0,\;\forall t>b_{\xi}(E,p).$

Consequently

$b_{\xi}(E,q)<t,\;\forall t>b_{\xi}(E,p).$

Which means that

$b_{\xi}(E,q)\leq b_{\xi}(E,p).$

The proof of the monotonicity of $B_{\xi}(E,.)$ and $\Lambda_{\xi}(E,.)$ is
similar.

###### Theorem 3.3

Let $\mu=(\mu_{1},\mu_{2},...,\mu_{k})\in\mathcal{P}(\mathbb{R}^{d})^{k}$ and
$\nu\in\mathcal{QAHP}(\mathbb{R}^{d})$. We have
1. $0\leq b_{\xi}(q)\leq B_{\xi}(q)\leq\Lambda_{\xi}(q)$, $\forall\,q_{i}<1$, $\forall\,1\leq i\leq k$.
2. $b_{\xi}(e_{i})=B_{\xi}(e_{i})=\Lambda_{\xi}(e_{i})=0$ with $e_{i}=(0,0,0...,1,0,0...,0).$
3. $b_{\xi}(q)\leq B_{\xi}(q)\leq\Lambda_{\xi}(q)\leq 0$, $\forall\,q_{i}>1$, $\forall\,1\leq i\leq k$.

###### Proof 3.4

Using (2) we get

$b_{\xi}(E,q)\leq B_{\xi}(E,q)\leq\Lambda_{\xi}(E,q),\;\forall
q\in\mathbb{R}^{k}.$

We are going to prove now that $b_{\xi}(e_{i})\geq 0$ and
$\Lambda_{\xi}(e_{i})\leq 0$ with $e_{i}=(0,0,...,0,1,0,...,0)$. Indeed if
$t<0,$ $0<\varepsilon<\frac{1}{2}$ and $(B(x_{i},r_{i}))_{i}$ is an
$\varepsilon$-covering of $E$, then

${\sum_{i}}\mu(B(x_{i},r_{i}))^{e_{i}}\nu(B(x_{i},r_{i}))^{t}\geq
1\;\;\Longrightarrow\;\;{\overline{\mathcal{H}}}_{\xi,\varepsilon}^{e_{i},t}(E)\geq
1,\,\forall t>0.$

Therefore,

$t\leq b_{\xi}(e_{i}),\forall t<0.$

Consequently,

$b_{\xi}(e_{i})\geq 0.$

Consider now $t>0,$ $0<\delta<\frac{1}{2}$ and $(B(x_{i},r_{i}))_{i}{}$ is a
centered $\varepsilon$-packing of $E$, then

${\overline{\mathcal{P}}}_{\xi,\varepsilon}^{e_{i},t}(E)\leq\underset{i}{\hbox{
}\sum}\mu(B(x_{i},r_{i}))^{e_{i}}\nu(B(x_{i},r_{i}))^{t}\leq 1.$

Consequently,

${\overline{\mathcal{P}}}_{\xi,\varepsilon}^{e_{i},t}(E)\leq 1,\hbox{
}\forall\hbox{ }t>0,$

which implies that

$\Lambda_{\xi}(e_{i})\leq t,\hbox{ }\forall t>0.$

Finally, we get

$\Lambda_{\xi}(e_{i})\leq 0.$

As a conclusion, whenever $q_{i}>1,\;\forall i=1,2,...,n$, we get
$\Lambda_{\xi}(q)<\Lambda_{\xi}(e_{i})\leq 0$ and then

$b_{\xi}(q)\leq B_{\xi}(q)\leq\Lambda_{\xi}(q)\leq\Lambda_{\xi}(e_{i})\leq 0.$

Similarly, for $q_{i}<1,\;\forall i=1,2,...,n$, we obtain
$b_{\xi}(q)>b_{\xi}(e_{i})\geq 0$ and thus

$0\leq b_{\xi}(e_{i})\leq b_{\xi}(q)\leq B_{\xi}(q)\leq\Lambda_{\xi}(q).$

Furthermore, we have

$\forall q\in\mathbb{R}^{k},\hbox{ }b_{\xi}(q)\leq
B_{\xi}(q)\leq\Lambda_{\xi}(q).$

Then,

$0\leq b_{\xi}(e_{i})\leq B_{\xi}(e_{i})\leq\Lambda_{\xi}(e_{i})\leq 0,$

which implies that

$b_{\xi}(e_{i})=B_{\xi}(e_{i})=\Lambda_{\xi}(e_{i})=0.$

Next, we need to introduce the following quantities which will be useful
later. Let $\mu=(\mu_{1},\mu_{2},...,\mu_{k})$ be a vector-valued probability
measure on $\mathbb{R}^{d}$. For $E\subseteq S_{\mu}$, and $a>1$ we denote

$T_{a}^{j}(\mu)=\displaystyle\lim\sup_{r\downarrow
0}\left[\displaystyle\sup_{x\in
S_{\mu}}\frac{\mu_{j}(B(x,ar))}{\mu_{j}(B(x,r))}\right],\;1\leq j\leq k,$

and for $x\in S_{\mu}$, $T_{a}^{j}(x)=T_{a}^{j}(\left\\{x\right\\})$. We
define the set $P_{D}(\mathbb{R}^{n})$ of doubling probability measures on
$\mathbb{R}^{n}$ by

$P_{D}(\mathbb{R}^{d})=\left\\{\mu\in P(\mathbb{R}^{d});\hbox{
}T_{a}^{j}(\mu)<\infty\hbox{ for some }a,\hbox{ }\forall\hbox{ }j\right\\}.$

We denote also

$\mathcal{QAHP}_{D}(\mathbb{R}^{d})=\mathcal{QAHP}(\mathbb{R}^{d})\cap
P_{D}(\mathbb{R}^{d}).$

Obviously, these sets are independent of $a$.

###### Proposition 3.2

Let $\mu=(\mu_{1},\mu_{2},...,\mu_{k})\in\mathcal{P}(\mathbb{R}^{d})^{k}$ and
$\nu\in\mathcal{QAHP}_{D}(\mathbb{R}^{d})$, $E\subset\mathbb{R}^{d}$,
$p,q\in\mathbb{R}^{k}$ and $\alpha\in[0,1]$. Then, we have

$b_{\xi}(E,\alpha p+(1-\alpha)q)\leq\alpha
B_{\xi}(E,p)+(1-\alpha)b_{\xi}(E,q).$

###### Proof 3.5

Let $t=B_{\xi}(E,p)$ and $s=b_{\xi}(E,q)$. We will prove that

$b_{\xi}(E,\alpha p+(1-\alpha)q)\leq\alpha t+(1-\alpha)s,\hbox{ }\forall\hbox{
}\varepsilon>0.$

Let $\varepsilon>0$, $m\in\mathbb{N}^{\ast}$, and denote

$E_{m}=\\{x\in E;\;\frac{\mu_{j}(B(x,5r))}{\mu_{j}(B(x,r))}<m,\,\forall
j,\,\frac{\nu(B(x,5r))}{\nu(B(x,r))}<m,\,0<r<\frac{1}{m}\\}.$

As $E={\cup_{m}}E_{m}$, we shall prove that

$\mathcal{H}_{\xi}^{\alpha p+(1-\alpha)q,\alpha
t+(1-\alpha)s+\varepsilon}(E_{m})<\infty\hbox{ , }\forall\hbox{
}m\in\mathbb{N}^{\ast}.$

So, let $F\subset E_{m}$ and $(F_{i})_{i}$ be an arbitrary covering of $F$,
and $\delta>0$. Let next $\varepsilon>0$, $i\in\mathbb{N}$, and $\delta_{i}>0$
be such that

${\overline{\mathcal{P}}}_{\xi,\delta_{i}}^{p+\varepsilon,t}(F_{i})\leq{\overline{\mathcal{P}}}_{\xi}^{p+\varepsilon,t}(F_{i})+\frac{1}{2^{i}}.$

Since $F_{i}\cap F\subset F\subset E$, it holds that

$b_{\xi}(F_{i}\cap F,q)\leq b_{\xi}(E,q)=s<s+\varepsilon.$

Consequently

$b_{\xi}(F_{i}\cap F,q)<s+\varepsilon,$

which yields that

${\overline{\mathcal{H}}}_{\xi}^{q,s+\varepsilon}(F_{i}\cap F)=0.$

There exists consequently a centered
$(\frac{\delta}{5}\wedge\frac{1}{m}\wedge\delta_{i})$-covering
$(B(x_{ij},r_{ij}))_{j\in I_{i}}$ of $F_{i}\cap F$ satisfying

$\displaystyle\sum_{j\in
I_{i}}\mu(B(x_{ij},r_{ij}))^{q}\nu(B(x_{ij},r_{ij}))^{s+\varepsilon}\leq\frac{1}{2^{i}}.$

Let now $J_{i}\subset I_{i}$ composed of disjoint balls such that

$\displaystyle\bigcup_{j\in
I_{i}}B(x_{ij},r_{ij})\subset\displaystyle\bigcup_{j\in
J_{i}}B(x_{ij},5r_{ij}).$

Since $(B(x_{ij},5r_{ij}))_{j\in J_{i}}$ is a centered $\delta$-covering of
$F_{i}\cap F$ and $(B(x_{ij},r_{ij}))_{j\in J_{i}}$ is a centered
$\delta_{i}$-packing of $F_{i}$, we obtain

$\begin{array}[]{lll}\displaystyle{\overline{\mathcal{H}}}_{\xi,\delta}^{\alpha(p,q),\alpha_{\varepsilon}(t,s)}(F)&\leq&\displaystyle{\overline{\mathcal{H}}}_{\xi,\delta}^{\alpha(p,q),\alpha_{\varepsilon}(t,s)}\bigl{(}\displaystyle\bigcup_{i}\displaystyle\bigcup_{j\in
J_{i}}B(x_{ij},5r_{ij})\bigr{)}\\\
&\leq&\displaystyle\sum_{i}\displaystyle\sum_{j\in
J_{i}}\left[\mu(B(x_{ij},5r_{ij}))\right]^{\alpha(p,q)}\left[\nu(B(x_{ij},5r_{ij}))\right]^{\alpha_{\varepsilon}(t,s)},\end{array}$
(5)

where $\alpha(p,q)=\alpha p+(1-\alpha)q$ and $\alpha_{\varepsilon}(t,s)=\alpha
t+(1-\alpha)s+\varepsilon$. Consequently, whenever
$\alpha(p,q)\in(0,+\infty)^{k}$ and $\alpha_{\varepsilon}(t,s)\in(0,+\infty)$,
we get

$\left[\mu(B(x_{ij},5r_{ij}))\right]^{\alpha(p,q)}\leq\,m^{|\alpha(p,q)|}\bigl{[}\mu(B(x_{ij},r_{ij}))\bigr{]}^{\alpha(p,q)},$

and

$\left[\nu(B(x_{ij},5r_{ij}))\right]^{\alpha_{\varepsilon}(t,s)}\leq\,m^{\alpha_{\varepsilon}(t,s)}\bigl{[}\nu(B(x_{ij},r_{ij}))\bigr{]}^{\alpha_{\varepsilon}(t,s)}.$

Consequently, using (5), we get

${\overline{\mathcal{H}}}_{\xi,\delta}^{\alpha(p,q),\alpha_{\varepsilon}(t,s)}(F)\leq
m^{|\alpha(p,q)|+\alpha_{\varepsilon}(t,s)}({\sum_{i}}({\overline{\mathcal{P}}}_{\xi}^{p,t+\varepsilon}(F_{i})+\frac{1}{2^{i}}))^{\alpha},$

which yields that

${\overline{\mathcal{H}}}_{\xi}^{\alpha(p,q),\alpha_{\varepsilon}(t,s)}(F)\leq
m^{|\alpha(p,q)|+\alpha_{\varepsilon}(t,s)}({\sum_{i}}{\overline{\mathcal{P}}}_{\xi}^{p,t+\varepsilon}(F_{i})+1)^{\alpha}.$

Hence, $\forall\hbox{ }F\subseteq E_{m}$, we get

${\overline{\mathcal{H}}}_{\xi}^{\alpha(p,q),\alpha_{\varepsilon}(t,s)}(F)\leq
m^{|\alpha(p,q)|+\alpha_{\varepsilon}(t,s)}({{\mathcal{P}}}_{\xi}^{p,t+\varepsilon}(F)+1)^{\alpha},$

which implies that

${\overline{\mathcal{H}}}_{\xi}^{\alpha(p,q),\alpha_{\varepsilon}(t,s)}(E_{m})\leq
m^{|\alpha(p,q)|+\alpha_{\varepsilon}(t,s)}({{\mathcal{P}}}_{\xi}^{p,t+\varepsilon}(E_{m})+1)^{\alpha}.$

Consequently,

${\overline{\mathcal{H}}}_{\xi}^{\alpha(p,q),\alpha_{\varepsilon}(t,s)}(E_{m})<\infty,\forall
m.$

Therefore,

$b_{\xi}(E_{m},\alpha(p,q))\leq\alpha_{\varepsilon}(t,s),\forall\varepsilon>0,\,\forall
m,$

which yields finally that

$b_{\xi}(E,\alpha p+(1-\alpha)q)\leq\alpha t+(1-\alpha)s=\alpha
B_{\xi}(E,p)+(1-\alpha)b_{\xi}(E,q).$

###### Theorem 3.4

Let $\mu=(\mu_{1},\mu_{2},...,\mu_{k})\in\mathcal{P}(\mathbb{R}^{d})^{k}$ and
$\nu\in\mathcal{QAHP}_{D}(\mathbb{R}^{d})$, $q\in\mathbb{R}^{k}$ and
$E\subseteq S_{\mu}\cap S_{\nu}$. The following assertions hold.
1\. Whenever $q_{i}\leq 0,\,\forall k$, we have

$b_{\xi}^{q}(E)\geq\dim_{\nu}(E)\left(1-\frac{|q|}{k}\right).$ (6)

2\. Whenever $0\leq q_{i}\leq 1,\,\forall k$, we have

$b_{\xi}^{q}(E)\leq\dim_{\nu}(E)\left(1-\frac{|q|}{k}\right)\leq\frac{n}{k}(k-|q|).$
(7)

3\. Whenever $q_{i}\geq 1,\,\forall k$, we have

$b_{\xi}^{q}(E)\geq\frac{\beta}{\beta-1}\dim_{\nu}(E)\geq\frac{\beta}{\beta-1}n,\mbox{
\ with }\beta={\max_{i}}(1-\frac{1}{q_{i}}).$ (8)

###### Proof 3.6

1. For $q=(q_{1},...,q_{k})\in\mathbb{R}^{k}$ take in Proposition 3.2, $p=e_{i}$, $\widetilde{q}_{i}=q_{i}e_{i}$ and $\alpha=\frac{-q_{i}}{1-q_{i}}$. As $q_{i}\leq 0,\,\forall i$, we get in one hand

$b_{\xi}(0)\leq\alpha
B_{\xi}(e_{i})+(1+\frac{q_{i}}{1-q_{i}})b_{\xi}(\widetilde{q}_{i}).$

Recall now that $B_{\xi}(e_{i})=0$. Therefore,

$(1-q_{i})b_{\xi}(0)\leq b_{\xi}(\widetilde{q}_{i})\leq b_{\xi}(q),$

which implies that

$(1-q_{i})\dim_{\nu}(E)\leq b_{\xi}^{q}(E).$

The summation on $i=1,2,...,k$ gives

$b_{\xi}^{q}(E)\geq\dim_{\nu}(E)\bigl{(}1-\frac{|q|}{k}\bigr{)}.$

2. For $q=(q_{1},...,q_{k})\in\mathbb{R}^{k}$ take in Proposition 3.2, $p=e_{i}$, $\widetilde{q}_{i}=q_{i}e_{i}$, and $\alpha=q_{i}$, and follow similar techniques as in assertion 1\.
3. For $q=(q_{1},...,q_{k})\in\mathbb{R}^{k}$ take in Proposition 3.2, $p=0$, and $\alpha=\beta$, and follow as usual similar techniques as previously.

###### Theorem 3.5

Let
$\xi=(\mu,\nu)=(\mu_{1},\mu_{2},...,\mu_{k},\nu)\in\mathcal{P}(\mathbb{R}^{d})^{k}\times\in\mathcal{QAHP}_{D}(\mathbb{R}^{d})$,
$q\in\mathbb{R}^{k}$, and $E\subseteq S_{\mu}\cap S_{\nu}$. The following
assertions hold.
1\. Whenever $q_{i}\leq 0,\,\forall k$, we have

$B_{\xi}^{q}(E)\geq\,Dim_{\nu}(E)\left(1-\frac{|q|}{k}\right).$ (9)

2\. Whenever $0\leq q_{i}\leq 1,\,\forall k$, we have

$B_{\xi}^{q}(E)\leq\,Dim_{\nu}(E)\left(1-\frac{|q|}{k}\right)\leq\frac{n}{k}(k-|q|).$
(10)

3\. Whenever $q_{i}\geq 1,\,\forall k$, we have

$B_{\xi}^{q}(E)\geq\frac{\beta}{\beta-1}\,Dim_{\nu}(E)\geq\frac{\beta}{\beta-1}n,\mbox{
\ with }\beta=\hbox{ }\underset{i}{\max}(1-\frac{1}{q_{i}}).$ (11)

The proof follows similar techniques as in Theorem 3.4.

###### Proposition 3.3

Let
$\xi=(\mu,\nu)=(\mu_{1},\mu_{2},...,\mu_{k},\nu)\in\mathcal{P}(\mathbb{R}^{d})^{k}\times\mathcal{AHP}_{D}(\mathbb{R}^{d})$
be Radon vector-valued measure on $\mathbb{R}^{d}$ with compact support, and
such that $S_{\mu}\subseteq S_{\nu}$. Assume further that $\mu$ is absolutely
continuous relatively to the Lebesgue measure on $S_{\mu_{i}}$. Then, for all
Borel set $E\subset S_{\mu}$ such that $\mu_{i}(E)>0$, $\forall i$ and
$q\in\mathbb{R}^{k}$ with $|q|\in[0,1]$, we have

$\alpha b_{\xi}(E,q)\geq(1-|q|),$

with $|q|=q_{1}+q_{2}+...+q_{k}$ and $\alpha$ is the Ahlfors regularity index
of $\nu$.

###### Proof 3.7

Since $\mu$ is absolutely continuous relatively to the Lebesgue measure on
$S_{\mu_{i}}$, we may find for each $i$ a function $g_{i}\geq 0$ such that
$\mu_{i}=g_{i}\lambda_{|_{S_{\mu_{i}}}}^{n}$. Therefore, as $\mu_{i}(E)>0$,
there exists a $B_{i}\subset E$ (being Borel) with Lebesgue measure
$\lambda^{n}(B_{i})>0$, and a constant $\gamma_{i}>0$ satisfying

$g_{i}(x)\geq\gamma_{i},\,\forall x\in B_{i}.$ (12)

Let next $\varepsilon>0$ and $(B(x_{i},r_{i}))_{i}$ be a centred
$\varepsilon$-covering of $E$. For $t>0$, we have

$\underset{i}{\sum}(\mu(B(x_{i},r_{i})))^{q}(\nu(B(x_{i},r_{i})))^{t}\geq
C_{\nu}\underset{i}{\sum}(\mu(B(x_{i},r_{i})))^{q}(\lambda^{n}(B(x_{i},r_{i})))^{\alpha
t},$

where $C_{\nu}$ is a constant due to Ahlfors regularity of $\nu$. Denote next
$B=\underset{i}{\cup}B_{i}\subset E$. It holds from (12) that

$(\mu(B(x_{i},r_{i})))^{q}\geq\gamma^{q}\Bigl{(}\lambda^{n}(B(x_{i},r_{i})\cap
B\Bigr{)}^{q}.$

As a result,

$\underset{i}{\sum}(\mu(B(x_{i},r_{i})))^{q}(\nu(B(x_{i},r_{i})))^{t}\geq\,C_{\nu}\gamma^{q}\underset{i}{\sum}\Bigl{(}\lambda^{n}(B(x_{i},r_{i})\cap
B)\Bigr{)}^{|q|+\alpha t}.$

For $\alpha t<1-|q|$, it yields that

$\underset{i}{\sum}(\mu(B(x_{i},r_{i})))^{q}(\nu(B(x_{i},r_{i})))^{t}\geq\,C_{\nu}\gamma^{q}\lambda^{n}(B)>0.$

Consequently, $\forall\,t$ such that $0<\alpha t<1-|q|$, we get

$\mathcal{H}_{\xi}^{q,t}(E)>0,$

which implies that

$b_{\xi}(E,q)\geq t.$

By letting $t\rightarrow\displaystyle\frac{1-|q|}{\alpha}$, we obtain

$b_{\xi}^{q}(E)\geq\displaystyle\frac{1-|q|}{\alpha}.$

###### Proposition 3.4

Let $p>1$,
$\xi=(\mu,\nu)=(\mu_{1},\mu_{2},...,\mu_{k},\nu)\in\mathcal{P}(\mathbb{R}^{d})^{k}\times\mathcal{AHP}(\mathbb{R}^{d})$
be a vector-valued Radon probability measure on $\mathbb{R}^{d}$ with compact
support, and $S_{\mu}\subset S_{\nu}$. Suppose further that $\mu_{i}\in
L^{p}(\mathbb{R}^{d})$. Then for $q_{i}\geq 1,$ we have

$\alpha B_{\xi}(q)\leq\max\left\\{k-|q|,\frac{-|q|(p-1)}{p}\right\\}.$

###### Proof 3.8

Let for $i=1,2,...,k$, $g_{i}\in L^{p}(\mathbb{R}^{d})$ be such
$d\mu_{i}=g_{i}d\lambda^{n}$ on $S_{\mu_{i}}$. Of course, the $g_{i}$’s are
compactly supported functions. Assume for instance that $q_{i}\geq p>1$,
$\forall i$. Let next $\delta>0$, and consider a centred $\delta$-packing
$(B(x_{i},r_{i}))_{i}$ of $S_{\mu}$. Let finally, $g=\underset{i}{\max}g_{i}$.
Analogously to Proposition 3.3, it follows for $t>\frac{-|q|(p-1)}{p}$, that

$\underset{i}{\sum}(\mu(B(x_{i},r_{i})))^{q}(\nu(B(x_{i},r_{i})))^{t}\leq
C_{\nu}\underset{i}{\sum}\left(\lambda^{n}(B(x_{i},r_{i}))\right)^{\alpha
t+\frac{|q|(p-1)}{p}}\left(\underset{B(x_{i},r_{i})}{\int
g^{p}}d\lambda^{n}\right)^{{\frac{|q|}{p}}}.$

As $\alpha t>\frac{-|q|(p-1)}{p}$, we get

$\underset{i}{\sum}(\mu(B(x_{i},r_{i})))^{q}(\nu(B(x_{i},r_{i})))^{t}\leq
C_{\mu}\underset{i}{\sum}\left(\underset{B(x_{i},r_{i})}{\int
g^{p}}d\lambda^{n}\right).$

This yields that

$\underset{i}{\sum}(\mu(B(x_{i},r_{i})))^{q}(\nu(B(x_{i},r_{i})))^{t}\leq
C_{\mu}(\int g^{p}d\lambda^{n})^{\frac{|q|}{p}}<\infty.$

Hence,

${\overline{\mathcal{P}}}_{\xi}^{q,t}(S_{\mu})<\infty,$

and consequently,

$\mathcal{P}_{\xi}^{q,t}(S_{\mu})<\infty.$

Therefore,

$B_{\xi}(E,q)\leq t,\;\forall t>\frac{-|q|(p-1)}{\alpha p}.$

As a result

$\alpha B_{\xi}(E,q)\leq\frac{-|q|(p-1)}{p}.$

Now, assume that $1\leq q_{i}<p$, $\forall i$. Let $t>n-|q|$, $\delta>0$, and
consider a centred $\delta$-packing $(B_{i}=B(x_{i},r_{i}))_{i}$ of $S_{\mu}$.
We get

$\underset{i}{\sum}(\mu(B_{i}))^{q}(\nu(B_{i}))^{t}\leq
C_{\mu}\underset{i}{\sum}(\mu(B_{i}))^{q}((\lambda^{n}(B_{i}\cap
S_{\mu}))^{\alpha t}).$

Therefore,

$\underset{i}{\sum}(\mu(B_{i}))^{q}(\nu(B_{i}))^{t}\leq
C_{\mu}\underset{i}{\sum}\prod_{l}\left(\underset{B_{i}\cap S_{\mu}}{\int
g_{l}}d\lambda^{n}\right)^{{q_{l}}}(\lambda^{n}(B_{i}\cap S_{\mu}))^{\alpha
t}.$

Next, by Hölder’s inequality, it follows that

$\underset{i}{\sum}(\mu(B_{i}))^{q}(\nu(B_{i}))^{t}\leq
C_{\mu}\underset{i}{\sum}\prod_{l}\left(\underset{B_{i}\cap S_{\mu}}{\int
g_{l}^{q_{l}}}d\lambda^{n}\right)\left[\lambda^{n}(B_{i}\cap
S_{\mu})\right]^{q_{l}-1}\lambda^{n}(B_{i}\cap S_{\mu}))^{\alpha t}.$

This yields that

$\underset{i}{\sum}(\mu(B_{i}))^{q}(\nu(B_{i}))^{t}\leq
C_{\mu}\underset{i}{\sum}(\lambda^{n}(B_{i}\cap S_{\mu}))^{\alpha
t+|q|-k}\prod_{l}\left(\underset{B_{i}\cap S_{\mu}}{\int
g^{q_{l}}}d\lambda^{n}\right)<C<\infty.$

As a consequence, we get

${\overline{\mathcal{P}}}_{\xi}^{q,t}(S_{\mu})<\infty,$

which means that

$B_{\xi}(q)\leq t,\;\forall t>\displaystyle\frac{k-|q|}{\alpha}.$

This in turns yields that

$B_{\xi}(q)\leq\displaystyle\frac{k-|q|}{\alpha}.$

###### Remark 3.1

For $\alpha=1$, the measure $\nu$ is equivalent to the Lebesgue’s one. If
further $k=1$, Propositions 3.3 and 3.4 are the classical cases raised by
Olsen.

##  4 The associated joint multifractal spectrum

We propose in this section to introduce an associated multifractal spectrum
relatively to the joint generalisations of Hausdorff measure/dimension and
their analogues of packing measure/dimension developed in the previous
section.

Consider a vector $\xi=(\mu,\nu)=(\mu_{1},\mu_{2},...,\mu_{k},\nu)$ of Borel
probability measures on $\mathbb{R}^{d}$. We define the local upper dimension
of $\mu_{j}$ relatively to $\nu$ at $x\in\mathbb{R}^{d}$ by

${\overline{\mathcal{\gamma}}}_{\mu_{j},\nu}(x)=\underset{r\rightarrow
0}{\hbox{lim sup}}\frac{\log(\mu_{j}(B(x,r)))}{\log(\nu(B(x,r)))}.$

Similarly, the local lower dimension of $\mu_{j}$ relatively to $\nu$ at
$x\in\mathbb{R}^{d}$ is

${\underline{\gamma}}_{\mu_{j,\nu}}(x)=\underset{r\rightarrow 0}{\hbox{ lim
inf}}\frac{\log(\mu_{j}(B(x,r)))}{\log(\nu(B(x,r)))}.$

Whenever
${\overline{\mathcal{\gamma}}}_{\mu_{j},\nu}(x)={\underline{\gamma}}_{\mu_{j,\nu}}(x)$,
we call local dimension of $\mu_{j}$ relatively to $\nu$ at $x$ their common
value, which will be written ${\gamma}_{\mu_{j},\nu}(x)$. Next, for
$\gamma=(\gamma_{1},\gamma_{2},\dots,\gamma_{k})\in\mathbb{R}_{+}^{k}$, we set

${\overline{\mathcal{K}}}^{\gamma}=\left\\{x\in{S}_{\xi}\,;\;|{\overline{\mathcal{\gamma}}}_{\mu_{j},\nu}(x)\leq\gamma_{j}\,\;\forall
j=1,\dots,k\right\\},$
${\overline{\mathcal{K}}}_{\gamma}=\left\\{x\in{S}_{\xi}\,;\;\gamma_{j}\leq{\overline{\mathcal{\gamma}}}_{\mu_{j},\nu}(x),\;\forall
j=1,\dots,k\right\\},$
${\underline{K}}^{\gamma}=\left\\{x\in{S}_{\xi}\,;\;{\underline{\gamma}}_{\mu_{j},\nu}(x)\leq\gamma_{j},\;\forall
j=1,\dots,k\right\\}$

and

${\underline{K}}_{\gamma}=\left\\{x\in{S}_{\xi}\,;\;\gamma_{j}\leq{\underline{\gamma}}_{\mu_{j},\nu}(x),\;\forall
j=1,\dots,k\right\\}.$

Let also

$K(\gamma)={\underline{K}}_{\gamma}\cap{\overline{\mathcal{K}}}^{\gamma}.$

The joint spectrum of singularities associated to the vector $\xi$ is

$d(\gamma)=dimK(\gamma),$

where the symbol dim designates the Hausdorff dimension. In the present
section, we aim to establish some bounds for such a spectrum.

###### Proposition 4.1

Consider a metric space $X$, and a vector-valued Borel probability measure
$\xi=(\mu,\nu)=(\mu_{1},\mu_{2},...,\mu_{k},\nu)$ on $\mathbb{R}^{d}$. Fix
$\gamma=(\gamma_{1},\gamma_{2},...,\gamma_{k})\in\mathbb{R}_{+}^{k}$,
$q=(q_{1},q_{2},...,q_{k})\in\mathbb{R}^{k}$, $t\in\mathbb{R}$, and $\delta>0$
satisfying $0<\delta\leq\langle\gamma,q\rangle+t$. The following assertions
are true.

  1. 1.

$\mathcal{H}_{\nu}^{\langle\gamma,q\rangle+t+k\delta}({\overline{\mathcal{K}}}^{\gamma})\leq\mathcal{H}_{\xi}^{q,t}({\overline{\mathcal{K}}}^{\gamma})$
for $q\in\mathbb{R}_{+}^{k}.$

  2. 2.

$\mathcal{H}_{\nu}^{\langle\gamma,q\rangle+t+k\delta}({\underline{K}}_{\gamma})\leq\mathcal{H}_{\xi}^{q,t}({\underline{K}}_{\gamma})$
for $q\in\mathbb{R}_{-}^{k}.$

  3. 3.

$\mathcal{P}_{\nu}^{\langle\gamma,q\rangle+t+k\delta}({\overline{\mathcal{K}}}^{\gamma})\leq\mathcal{P}_{\xi}^{q,t}({\overline{\mathcal{K}}}^{\gamma})$
for $q\in\mathbb{R}_{+}^{k}.$

  4. 4.

$\mathcal{P}_{\nu}^{\langle\gamma,q\rangle+t+k\delta}({\underline{K}}_{\gamma})\leq\mathcal{P}_{\xi}^{q,t}({\underline{K}}_{\gamma})$
for $q\in\mathbb{R}_{-}^{k}.$

  5. 5.

For all Borel set $A\subseteq{\overline{\mathcal{K}}}^{\gamma}$,
$\mathcal{H}_{\xi}^{q,t}({A})\leq\mathcal{H}_{\nu}^{\langle\gamma,q\rangle+t-k\delta}({\overline{\mathcal{K}}}^{\gamma})$
if $q\in\mathbb{R}_{-}^{k}.$

  6. 6.

For all Borel set $A\subseteq{\underline{K}}_{\gamma}$,
$\mathcal{H}_{\xi}^{q,t}({A})\leq\mathcal{H}_{\nu}^{\langle\gamma,q\rangle+t-k\delta}({\overline{\mathcal{K}}}^{\gamma})$
if $q\in\mathbb{R}_{+}^{k}.$

  7. 7.

For all Borel set $A\subseteq{\overline{\mathcal{K}}}^{\gamma}$,
$\mathcal{P}_{\xi}^{q,t}({A})\leq\mathcal{P}_{\nu}^{\langle\gamma,q\rangle+t-k\delta}({\overline{\mathcal{K}}}^{\gamma})$
if $q\in\mathbb{R}_{-}^{k}.$

  8. 8.

For all Borel set $A\subseteq{\underline{K}}_{\gamma}$,
$\mathcal{P}_{\xi}^{q,t}({A})\leq\mathcal{P}_{\nu}^{\langle\gamma,q\rangle+t-k\delta}({\overline{\mathcal{K}}}^{\gamma})$
if $q\in\mathbb{R}_{+}^{k}.$

###### Proof 4.1

We will develop the proofs of assertions 1., 3., 5. and 7. The remaining
assertions may be shown by similar arguments.
1\. For $q=0$ the statement is obvious. So, let next $q_{j}>0$, for all $j$.
For $m\in\mathbb{N}$, let

${\overline{\mathcal{K}}}_{m}^{\gamma}=\left\\{x\in{\overline{\mathcal{K}}}^{\gamma};\;\frac{\log(\mu_{j}(B(x,r)))}{\log(\nu(B(x,r)))}\leq\gamma_{j}\hbox{}+\frac{\delta}{q_{j}},\;0<r<\frac{1}{m},\,\forall
j,\hbox{ s.t. }1\leq j\leq k\right\\}.$

Let $\eta$ be such that $0<\eta<\frac{1}{m}$ and $(B_{i}=B(x_{i},r_{i}))_{i}$
be an $\eta$-covering of $E\subseteq{\overline{\mathcal{K}}}_{m}^{\gamma}$. It
follows that

$\frac{\log(\mu_{j}(B(x,r)))}{\log(\nu(B(x,r)))}\leq\gamma_{j}\hbox{}+\frac{\delta}{q_{j}}\Rightarrow\mu_{j}(B(x,r))\geq\nu(B(x,r)^{\gamma_{j}\hbox{}+\frac{\delta}{q_{j}}}.$

Since $q\in\mathbb{R}_{+}^{k}$, we get

$\mu(B(x,r))^{q}\geq\nu(B(x,r)^{\langle\gamma,q\rangle+k\delta}.$

Then,

$\mathcal{H}_{\nu,\eta}^{\langle\gamma,q\rangle+t+k\delta}({E})\leq\underset{i}{\hbox{}\sum}\nu(B(x,r)^{\langle\gamma,q\rangle+t+k\delta}\leq\underset{i}{\hbox{}\sum}\mu(B(x,r))^{q}\nu(B(x,r)^{t},$

which implies that

$\mathcal{H}_{\nu,\eta}^{\langle\gamma,q\rangle+t+k\delta}({E})\leq\mathcal{H}_{\xi,\eta}^{q.t}({E}),\forall\hbox{}\eta>0.$

When $\eta\downarrow 0$, we get for all
$E\subseteq{\overline{\mathcal{K}}}_{m}^{\gamma}$,

$\mathcal{H}_{\nu}^{\langle\gamma,q\rangle+t+k\delta}({E})\leq\mathcal{H}_{\xi}^{q.t}({E})\leq\mathcal{H}_{\xi}^{q.t}({\overline{\mathcal{K}}}_{m}^{\gamma}).$

Hence,

$\mathcal{H}_{\nu}^{\langle\gamma,q\rangle+t+k\delta}({\overline{\mathcal{K}}}_{m}^{\gamma})\leq\mathcal{H}_{\xi}^{q.t}({\overline{\mathcal{K}}}_{m}^{\gamma}).$

Finally, the result follows since
${\overline{\mathcal{K}}}^{\gamma}=\underset{m}{\cup}{\overline{\mathcal{K}}}_{m}^{\gamma}.$
3\. For $m\in\mathbb{N}$, denote

${\overline{\mathcal{K}}}_{m}^{\gamma}=\left\\{x\in{\overline{\mathcal{K}}}^{\gamma};\;\frac{\log(\mu_{j}(B(x,r)))}{\log(\nu(B(x,r)))}\leq\gamma_{j}+\frac{\delta}{q_{j}},\,0<r<\frac{1}{m},\forall\,j,\hbox{
s.t. }1\leq j\leq k\right\\}.$

Let $\eta$ be such that $0<\eta<\frac{1}{m}$ and $(B_{i}=B(x_{i},r_{i}))_{i}$
be an $\eta$-packing of $E\subseteq{\overline{\mathcal{K}}}_{m}^{\gamma}$. We
have

$\frac{\log(\mu_{j}(B(x,r)))}{\log(\nu(B(x,r)))}\leq\gamma_{j}\hbox{}+\frac{\delta}{q_{j}}\Rightarrow\mu_{j}(B(x,r))\geq\nu(B(x,r))^{\gamma_{j}\hbox{}+\frac{\delta}{q_{j}}}.$

Since $q\in\mathbb{R}_{-}^{k},$ then

$\mu(B(x,r))^{q}\geq\nu(B(x,r))^{\langle\gamma,q\rangle+k\delta},$

which implies that

$\underset{i}{\hbox{}\sum}\nu(B(x,r))^{\langle\gamma,q\rangle+t+k\delta}\leq\underset{i}{\hbox{}\sum}\mu(B(x,r))^{q}\nu(B(x,r))^{t}\leq{\overline{\mathcal{P}}}_{\xi,\eta}^{q,t}(E).$

Therefore,

${\overline{\mathcal{P}}}_{\nu,\eta}^{\langle\gamma,q\rangle+t+k\delta}(E)\leq{\overline{\mathcal{P}}}_{\xi,\eta}^{q,t}(E),\;\forall\hbox{}\eta>0.$

By letting $\eta\downarrow 0$ we obtain

${\overline{\mathcal{P}}}_{\nu}^{\langle\gamma,q\rangle+t+k\delta}({E})\leq{\overline{\mathcal{P}}}_{\xi}^{q.t}({E}),\;\forall\hbox{}\eta>0\hbox{
and }\forall\hbox{}E\subseteq{\overline{\mathcal{K}}}_{m}^{\gamma}.$

Consequently, for any covering $(E_{i})_{i}$ of
${\overline{\mathcal{K}}}_{m}^{\gamma}$, we get

$\begin{array}[]{lll}\mathcal{P}_{\nu}^{\langle\gamma,q\rangle+t+k\delta}({\overline{\mathcal{K}}}_{m}^{\gamma})&=&\mathcal{P}_{\nu}^{\langle\gamma,q\rangle+t+k\delta}(\underset{i}{\cup}({\overline{\mathcal{K}}}_{m}^{\gamma}\cap
E_{i}))\\\
&\leq&\displaystyle\sum_{i}\mathcal{P}_{\nu}^{\langle\gamma,q\rangle+t+k\delta}({\overline{\mathcal{K}}}_{m}^{\gamma}\cap
E_{i})\\\
&\leq&\displaystyle\sum_{i}{\overline{\mathcal{P}}}_{\nu}^{\langle\gamma,q\rangle+t+k\delta}({\overline{\mathcal{K}}}_{m}^{\gamma}\cap
E_{i})\\\
&\leq&\displaystyle\sum_{i}{\overline{\mathcal{P}}}_{\xi}^{q.t}({E}_{i}),\end{array}$

which leads to

$\mathcal{P}_{\nu}^{\langle\gamma,q\rangle+t+k\delta}({\overline{\mathcal{K}}}_{m}^{\gamma})\leq\mathcal{P}_{\xi}^{q,t}({\overline{\mathcal{K}}}_{m}^{\gamma}),\;\forall\hbox{}m>0.$

Finally, since
${\overline{\mathcal{K}}}^{\gamma}=\underset{m}{\cup}{\overline{\mathcal{K}}}_{m}^{\gamma}$,
we obtain

$\mathcal{P}_{\nu}^{\langle\gamma,q\rangle+t+k\delta}({\overline{\mathcal{K}}}^{\gamma})\leq\mathcal{P}_{\xi}^{q,t}({\overline{\mathcal{K}}}^{\gamma}).$

5\. For $m\in\mathbb{N}$, consider

${T}_{m}=\left\\{x\in{A};\;\frac{\log(\mu_{j}(B(x,r)))}{\log(\nu(B(x,r)))}\leq\gamma_{j}-\frac{\delta}{q_{j}},\,0<r<\frac{1}{m},\;\forall\,j,\hbox{
s.t. }1\leq j\leq k\right\\}.$

Consider a subset $E\subseteq{T}_{m}$, $0<\eta<\frac{1}{m}$, and a covering
$(E_{i})_{i\in\mathbb{N}}$ of $E$ with $r_{i}=diamE_{i}<\eta$, $\forall i$.
Consider next the set $I=\left\\{i;\;E_{i}\cap E=\emptyset\right\\}$, and let
$x_{i}\in E_{i}\cap E$. We get an $\eta$-covering $(B(x_{i},r_{i}))_{i}$ of
$E$ for which we may write that

$\frac{\log(\mu_{j}(B(x_{i},r_{i})))}{\log(\nu(B(x_{i},r_{i})))}\leq\gamma_{j}\hbox{}-\frac{\delta}{q_{j}}.$

Thus,

$\mu(B(x_{i},r_{i}))^{q}\geq\nu(B(x_{i},r_{i}))^{\langle\gamma,q\rangle-k\delta}.$

Therefore,

$\begin{array}[]{lll}{\overline{\mathcal{H}}}_{\xi,\eta}^{q.t}({E})&\leq&\displaystyle\sum_{i\in{I}}\mu(B(x_{i},r_{i}))^{q}\nu(B(x_{i},r_{i}))^{t}\\\
&\leq&\displaystyle\sum_{i\in{I}}\left[\nu(B(x_{i},r_{i}))\right]^{\langle\gamma,q\rangle-k\delta+t}\\\
&\leq&\displaystyle\sum_{i\in{I}}\left[\nu(B(x_{i},r_{i}))\right]^{\langle\gamma,q\rangle-k\delta+t}.\end{array}$

Consequently,

${\overline{\mathcal{H}}}_{\xi,\eta}^{q.t}({E})\leq\mathcal{H}_{\nu,\eta}^{\langle\gamma,q\rangle-k\delta+t}({E})\hbox{
for }\eta<\frac{1}{m}.$

By letting $\eta\rightarrow 0$, we obtain

$\overline{\mathcal{H}}_{\xi}^{q,t}(E)\leq\mathcal{H}_{\nu}^{\langle\gamma,q\rangle-k\delta+t}(E)\leq\mathcal{H}_{\nu}^{\langle\gamma,q\rangle-k\delta+t}(T_{m}),\;\forall
E\subseteq T_{m}.$

Then,

$\mathcal{H}_{\xi}^{q,t}({T}_{m})\leq\mathcal{H}_{\nu}^{\langle\gamma,q\rangle-k\delta+t}({T}_{m})\hbox{},\;\forall\hbox{}m\in\mathbb{N}.$

Finally, the result follows from the equality ${A}=\underset{m}{\cup}{T}_{m}.$
7\. Denote for $m\in\mathbb{N}$,

${T}_{m}=\left\\{x\in{A};\;\frac{\log(\mu_{j}(B(x,r)))}{\log(\nu(B(x,r)))}\leq\gamma_{j}-\frac{\delta}{q_{j}},\,0<r<\frac{1}{m},\,\forall
j\hbox{ s.t. }1\leq j\leq k\right\\}.$

Given $E\subseteq{T}_{m}$, $0<\eta<\frac{1}{m}$ and an $\eta$-packing
$(B(x_{i},r_{i}))_{i}$ of $E$, we get

$\frac{\log(\mu_{j}(B(x_{i},r_{i})))}{\log(\nu(B(x_{i},r_{i})))}\leq\gamma_{j}\hbox{}-\frac{\delta}{q_{j}}\Rightarrow\mu(B(x_{i},r_{i}))^{q}\leq\nu(B(x_{i},r_{i}))^{\langle\gamma,q\rangle-k\delta}.$

Then

$\displaystyle\sum_{i}\mu(B(x_{i},r_{i}))^{q}\nu(B(x_{i},r_{i}))^{t}\leq\displaystyle\sum_{i}\left[\nu(B(x_{i},r_{i}))\right]^{\langle\gamma,q\rangle-k\delta+t}\leq{\overline{\mathcal{P}}}_{\nu,\eta}^{\langle\gamma,q\rangle-k\delta+t}(E).$

Consequently,

${\overline{\mathcal{P}}}_{\xi,\eta}^{q,t}(E)\leq{\overline{\mathcal{P}}}_{\nu,\eta}^{\langle\gamma,q\rangle-k\delta+t}(E).$

By letting $\eta\rightarrow 0$ we obtain

${\overline{\mathcal{P}}}_{\xi}^{q,t}(E)\leq{\overline{\mathcal{P}}}_{\nu}^{\langle\gamma,q\rangle-k\delta+t}(E),\hbox{}\forall\hbox{}E\subset
E_{m}.$

Now, as $(E_{i})_{i}$ is a covering of ${T}_{m}$, we get

$\begin{array}[]{lll}\mathcal{P}_{\xi}^{q,t}({T}_{m})&\leq&\mathcal{P}_{\xi}^{q,t}(\underset{i}{\cup}({T}_{m}\cap
E_{i}))\\\ &\leq&\displaystyle\sum_{i}\mathcal{P}_{\xi}^{q,t}({T}_{m}\cap
E_{i})\\\
&\leq&\displaystyle\sum_{i}{\overline{\mathcal{P}}}_{\xi}^{q,t}({T}_{m}\cap
E_{i})\\\
&\leq&\displaystyle\sum_{i}{\overline{\mathcal{P}}}_{\nu}^{\langle\gamma,q\rangle-k\delta+t}({T}_{m}\cap
E_{i}).\end{array}$

Finally, as $A=\underset{m}{\cup}{T}_{m}$, we obtain

$\mathcal{P}_{\xi}^{q,t}({A})\leq\mathcal{P}_{\nu}^{\langle\gamma,q\rangle+t-k\delta}({\overline{\mathcal{K}}}^{\gamma}),\;\hbox{
for }q\in\mathbb{R}_{-}^{k}.$

###### Corollary 4.1

Consider a metric space $X$, and a vector-valued Borel probability measure
$\xi=(\mu,\nu)=(\mu_{1},\mu_{2},...,\mu_{k},\nu)$ on $\mathbb{R}^{d}$. Fix
$\gamma=(\gamma_{1},\gamma_{2},...,\gamma_{k})\in\mathbb{R}_{+}^{k}$, and
$q=(q_{1},q_{2},...,q_{k})\in\mathbb{R}^{k}$. Then, the following assertions
hold.
1\. Whenever $\langle\gamma,q\rangle+b_{\xi}(q)\geq 0$, we have

$\left\\{\begin{array}[]{lll}\dim_{\nu}({\overline{\mathcal{K}}}^{\gamma})\leq\langle\gamma,q\rangle+b_{\xi}(q)\;\hbox{for}\;q\in\mathbb{R}_{+}^{k},\\\
\\\
\dim_{\nu}({\underline{K}}_{\gamma})\leq\langle\gamma,q\rangle+b_{\xi}(q)\;\hbox{for}\;q\in\mathbb{R}_{-}^{k}.\end{array}\right.$

In particular, $\dim_{\nu}({\overline{\mathcal{K}}}^{\gamma})\leq\gamma_{i}$,
$\forall 1\leq i\leq k.$
2\. Whenever $\langle\gamma,q\rangle+B_{\xi}(q)\geq 0$, we have

$\left\\{\begin{array}[]{lll}Dim_{\nu}({\overline{\mathcal{K}}}^{\gamma})\leq\langle\gamma,q\rangle+B_{\xi}(q)\;\hbox{for}\;q\in\mathbb{R}_{+}^{k},\\\
\\\
Dim_{\nu}({\underline{K}}_{\gamma})\leq\langle\gamma,q\rangle+B_{\xi}(q)\;\hbox{for}\;q\in\mathbb{R}_{-}^{k}.\end{array}\right.$

In particular, $Dim_{\nu}({\overline{\mathcal{K}}}^{\gamma})\leq\gamma_{i}$,
$\forall 1\leq i\leq k.$
3\. Whenever $A\subseteq{\underline{K}}_{\gamma}$ is Borel with $\mu_{i}(A)>0$
for all $i$, we have $\gamma_{i}\leq\dim_{\nu}({A}).$
4\. Whenever $A\subseteq{\underline{K}}_{\gamma}$ is Borel with $\mu_{i}(A)>0$
for all $i$, we have $\gamma_{i}\leq Dim_{\nu}({A}).$

Corollary 4.1 allows as to study the eventual link between the dimension
functions $b_{\xi}$ and $B_{\xi}$ and the joint multifractal spectrum. Indeed,
consider the following quantities,

${\underline{a}}_{\xi}=\underset{q_{i}>0}{\sup}\left(-\frac{b_{\xi}(q)}{\left|q\right|}\right),\quad{\overline{a}}_{\xi}=\underset{q_{i}<0}{\inf}\left(-\frac{b_{\xi}(q)}{\left|q\right|}\right),$
${\underline{A}}_{\xi}=\underset{q_{i}>0}{\sup}\left(-\frac{B_{\xi}(q)}{\left|q\right|}\right)\;\;\hbox{and}\;\;\overline{A}_{\xi}=\underset{q_{i}<0}{\inf}\left(-\frac{B_{\xi}(q)}{\left|q\right|}\right).$

We now establish upper bounds for $b_{\xi}$ and $B_{\xi}$.

###### Theorem 4.1

Consider a metric space $X$, a vector-valued Borel probability measure
$\xi=(\mu,\nu)=(\mu_{1},\mu_{2},...,\mu_{k},\nu)$ on $\mathbb{R}^{d}$, and
$\gamma=(\gamma_{1},\gamma_{2},...,\gamma_{k})\in\mathbb{R}^{k}_{+}$. It holds
that

$dim_{\nu}(K(\gamma))\leq\left\\{\begin{array}[]{lll}b_{\xi}^{\ast}(\gamma)=\underset{q}{\inf}(\langle\gamma,q\rangle+b_{\xi}(q))&,&\forall\gamma\in({\underline{a}}_{\xi},{\overline{a}}_{\xi})^{k},\\\
0&,&\forall\gamma\notin({\underline{a}}_{\xi},{\overline{a}}_{\xi})^{k},\end{array}\right.$

and

$Dim_{\nu}(K(\gamma))\leq\left\\{\begin{array}[]{lll}B_{\xi}^{\ast}(\gamma)=\underset{q}{\inf}(\langle\gamma,q\rangle+B_{\xi}(q))&,&\forall\gamma\in({\underline{a}}_{\xi},{\overline{a}}_{\xi})^{k},\\\
0&,&\forall\gamma\notin({\underline{a}}_{\xi},{\overline{a}}_{\xi})^{k}.\end{array}\right.$

###### Proof 4.2

We will sketch the proof of the first part. The second may be checked by
similar techniques. Since
$\gamma_{i}\in({\underline{a}}_{\xi},{\overline{a}}_{\xi})$ then
$K(\gamma)\neq\emptyset$. Consequently

$\langle\gamma,q\rangle+b_{\xi}(q)\geq 0,$

which yields that

$dim_{\nu}(K(\gamma))\leq\left\\{\begin{array}[]{c}\langle\gamma,q\rangle+b_{\xi}(q)\,;\;\forall
q\in\mathbb{R}_{+}^{k},\\\ \langle\gamma,q\rangle+b_{\xi}(q)\,;\;\forall
q\in\mathbb{R}_{-}^{k}.\end{array}\right.$

Consequently,

$dim_{\nu}(K(\gamma))\leq\underset{q}{\inf}(\langle\gamma,q\rangle+b_{\xi}(q))=b_{\xi}^{\ast}(q).$

##  5 Validity of an associated joint multifractal formalism

In the classical case of single measures, the multifractal formalism is
resumed in a mathematical formula stating that the fractal dimension of the
singularities set in the Hausdorff sense (known as the spectrum of
singularities) is evaluated by means of the Legendre transform of a free
energy evaluated by some dimension functions $b$ and $B$, the original
versions of $b_{\xi}$ and $B_{\xi}$ is our case. Such a formalism has been
proved to hold for many cases of measures such as doubling, Gibbs, self
similar, … etc. Moreover, it has been extended for more large cases such as
the multifractal case due to [49] and the mixed case due to [48].

In the present part, we propose to develop an extending multifractal formalism
relatively to the joint case introduced in the previous sections. Such a joint
multifractal formalism conjectures that

$d(\gamma)=b_{\xi}^{*}(\gamma)=B_{\xi}^{*}(\gamma)$

for suitable $\gamma$. So, let

$E_{\xi}(\gamma)=\left\\{x\in{S}_{(}\xi)\,;\;\hbox{ }\underset{r\rightarrow
0}{\lim}\frac{\log(\mu(B(x,r))}{\log(\nu(B(x,r))}=\gamma\right\\},$

where

$\dfrac{\log(\mu(B(x,r))}{\log(\nu(B(x,r))}=\Biggl{(}\dfrac{\log(\mu_{1}(B(x,r))}{\log(\nu(B(x,r))},\dfrac{\log(\mu_{2}(B(x,r))}{\log(\nu(B(x,r))},\dots,\dfrac{\log(\mu_{k}(B(x,r))}{\log(\nu(B(x,r))}\Biggr{)}.$

The following theorem provides a case of validity of the extended joint
multifractal formalism.

###### Theorem 5.1

Consider a vector-valued Borel probability measure
$\xi=(\mu,\nu)=(\mu_{1},\mu_{2},...,\mu_{k})$ on $\mathbb{R}^{d}$ with compact
support. Let $q\in\mathbb{R}^{k}$ at which $B_{\xi}$ satisfies
$B_{\xi}^{q,B_{\xi}(q)}({S}_{\xi})>0$. Then,

$Dim_{\nu}\left[E_{\xi}(-B_{\xi}^{\prime}(q))\right]=\dim_{\nu}\left[E_{\xi}(-B_{\xi}^{\prime}(q))\right]=B_{\xi}^{\ast}(-B_{\xi}^{\prime}(q)).$

where
$f^{\ast}(\alpha)=\underset{\beta}{\inf}(\left\langle\alpha,\beta\right\rangle+f(\beta)).$

The proof reposes on the following lemmas.

###### Lemma 5.1

Let $\alpha=-\nabla B_{\xi}(q).$ Then

$\mathcal{H}_{\nu}^{\left\langle\alpha,q\right\rangle+B_{\xi}(q)-k\eta}(E_{\xi}(\alpha))\geq\mathcal{H}_{\xi}^{q,B_{\xi}(q)}(E_{\xi}(\alpha)),\;\forall\,\eta>0.$

###### Lemma 5.2

We have

$\mathcal{H}_{\xi}^{q,B_{\xi}(q)}((S_{\mu}\cap S_{\nu})\diagdown
E_{\xi}(-\nabla B_{\xi}(q)))=0.$

Proof of Theorem 5.1. By Lemma 5.1 and Lemma 5.2, it holds that

$\mathcal{H}_{\nu}^{\left\langle\alpha,q\right\rangle+B_{\xi}(q)-k\eta}(E_{\xi}(\alpha))\geq\mathcal{H}_{\xi}^{q,B_{\xi}(q)}(E_{\xi}(\alpha))=0,\;\forall\,\eta>0.$

Hence,

$dim_{\nu}E_{\xi}(-\nabla B_{\xi}(q))\geq\left\langle-\nabla
B_{\xi}(q),q\right\rangle+B_{\xi}(q)-k\eta.$

Letting $\eta\rightarrow 0$, this yields that

$dim_{\nu}E_{\xi}(-\nabla B_{\xi}(q))\geq\left\langle-\nabla
B_{\xi}(q),q\right\rangle+B_{\xi}(q),$

As a consequence of this result, we finish our paper by establishing an
important result relating the multifractal formalism introduced here of the
vector-valued measures to their projections according to some suitable
subspaces of $\mathbb{R}^{d}$.

Let $m\in\mathbb{N}$ be such that $0<m<d$, and $G_{d,m}$ be the Grassamannian
manifold composed of all $m$-dimensional linear subspaces of $\mathbb{R}^{d}$.
Let also $\gamma_{d,m}$ be the invariant Haar measure on $G_{d,m}$ satisfying
$\gamma_{d,m}(G_{d,m})=1$. For $V\in G_{d,m}$, we designate by $P_{V}$ the
usual orthogonal projection onto $V$. Consider also a Borel probability
measure $\mu$ on $\mathbb{R}^{d}$, and denote $\mu_{V}$ its projection on $V$
relatively to $P_{V}$, i.e.,

$\mu_{V}(A)=\mu(P_{V}^{-1}(A)),\hbox{ }\forall A\subset V.$

The following result holds. It extends the results of [6], [8], [11], [24],
[31], [32], [36], [39], [46], [47], [56], [61], [61], [65].

###### Theorem 5.2

Let $\xi=(\mu,\nu)=(\mu_{1},\mu_{2},...,\mu_{k})$ be a vector-valued Borel
probability measure on $\mathbb{R}^{d}$ with $S_{\mu}=S_{\nu}$ being compact,
and $q\in\mathbb{R}_{-}^{k}$. Assume further that

$(H_{1})$

$\mathcal{H}_{\xi}^{q,B_{\xi}(q)}(S_{\mu})>0$.

$(H_{2})$

$B_{\xi}(q)<1$.

$(H_{3})$

$B_{\xi}(q)$ is differentiable at $q$.

It holds for all $V\in G_{n,m}$ that

$\begin{array}[]{lll}\dim_{\nu_{v}}\left(E_{\mu_{v},\nu_{v}}(-\nabla
B_{\xi}(q))\right)&=&Dim_{\nu_{v}}\left(E_{\xi_{v}}(-\nabla
B_{\xi}(q))\right)\\\ &=&\dim_{\nu}(E_{\xi_{v}}(-\nabla B_{\xi}(q)))\\\
&=&Dim_{\nu}(E_{\xi_{v}}(-\nabla B_{\xi}(q)))\\\ &=&B^{\ast}(-\nabla
B_{\xi}(q)).\end{array}$

###### Proof 5.1

Using equations (3)-(4), Proposition 3.1 and the hypotheses $(H_{1})$ and
$(H_{2})$, we get

$b_{\xi}(q)=B_{\xi}(q).$ (13)

Similarly,

$0<\mathcal{H}_{\xi}^{q,B_{\xi}(q)}(S_{\mu})\leq\mathcal{H}_{\xi_{v}}^{q,B_{\xi_{v}}(q)}(S_{\mu_{v}}),\hbox{
}\forall\hbox{ }V\in G_{n,m}.$

Next, Theorem 5.1 and equation (13) imply that

$\dim_{\nu_{v}}\left[E_{\xi_{v}}(-\nabla
B_{\xi}(q))\right]\geq\hbox{}\left\langle-\nabla
B_{\xi}(q),q\right\rangle+B_{\xi}(q).$

The other estimation is satisfied since,

$\dim_{\nu_{v}}(E_{\xi_{v}}(-\nabla B_{\xi}(q)))\geq\left\langle-\nabla
B_{\xi}(q),q\right\rangle+B_{\xi_{v}}(q)=\left\langle-\nabla
B_{\xi}(q),q\right\rangle+B_{\xi}(q).$

This achieves the proof.

##  6 Conclusion

In the present paper, a class of quasi Ahlfors vector-valued measures has been
considered for mixed multifractal analysis. It is shown that under the weak
assumption of quasi Ahlfors, the construction of mixed generalizations of
fractal measures such as Hausdorff and packing is possible. Such mixed
multifractal analysis has induced in a natural way some corresponding
multifractal dimensions, which have been shown to satisfy the multifractal
formalism in mixed multifractal framework. The results developed here are
subject of several extensions. Indeed, the developments showed that to get a
valid variant of the multifractal formalism does not necessitate to apply
radius power-laws equivalent measures. This leads to think about a general
framework where the restriction of the vector-valued measure on balls
$\mu(B(x,r))$ may be any vector-valued function $\varphi(r)$ which is not
equivalent to power-laws $r^{\alpha}$ and develop a general formulation for
mixed multifractal analysis. From the practical point of view, this is
important, as it may lead for example to a mixed multifractal fluctuation
analysis version where correlations, cross-correlations and auto-correlations
have more flexible laws. Some initial thoughts may be extracted from [1, 5,
12, 14, 16, 22, 40, 41, 42].

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## Further readings

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* [79] E. Chandrasekhar, V. P. Dimri and V. M. Gadre, Wavelets and Fractals in Earth System Sciences. Taylor & Francis Group 2014.
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* [81] B. Ghanbarian and A. G. Hunt, Fractals Concepts and Applications in Geosciences. Taylor and Francis Group, 2017.
* [82] D. Ghosh, S. Samanta and S. Chakraborty, Multifractals and Chronic Diseases of the Central Nervous System. Springer 2019.
* [83] A. K. Janahmadov and M. Y. Javadov, Synergetics and Fractals in Tribology, Materials Forming, Machining and Tribology. Springer 2016.
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* [85] S.-A. Ouadfeul, Fractal analysis and chaos in geosciences. InTech Open, 2012.
* [86] L. Seuront, Fractals and multifractals in ecology and aquatic science. Taylor and Francis Group, 2010.
* [87] M. Tolotti, Fractal and multifractal models for price changes, An attempt to manage the Black Swan. Corso di Laurea magistrale in Economia e Finanza, Universita CaFoscari Venisia, 2012/2013.
* [88] Z.-X. Zhang, Rock Fracture and Blasting Theory and Applications. Elsevier 2016.