# Is the emergent scenario in the early Universe a consequence of the
quantization of the real scalar field?

Subhayan<EMAIL_ADDRESS>Department of Mathematics, Jadavpur
University, Kolkata-700032, West Bengal, India.  Sujayita
<EMAIL_ADDRESS>Department of Physics, Diamond Harbour Women
University, West Bengal- 743388, India.

###### Abstract

The emergent scenario of cosmic evolution is a topic of great interest in
recent cosmology, especially because it describes a non-singular origin of the
Universe, unlike the Big Bang models. This cosmic evolution pattern has
already been established through the non-equilibrium thermodynamic
prescription. But those models are phenomenological and require physical
interpretation from the quantum field theory perspective. This work is an
effort to search for a quantum field theoretical reason to justify the
emergent nature of the Universe and the nature of cosmic evolution at the
early phase of cosmic expansion.

Keywords: Non-singular evolution of the Universe, Quantum field theory,
Cosmology.

Cosmology, Non-singular evolution of Universe, Quantization of real scalar
field.

##  1 Introduction

The emergent model of the Universe is becoming a very popular and interesting
cosmic evolution model nowadays. Even in an ever expanding cosmic model, the
Universe can be assumed to start from an emergent epoch [7, 6, 5, 1, 3, 2, 4]
with a non-zero constant scale factor($a_{E}$). It is a non-singular
evolutionary model, unlike the Big Bang model where there is a singularity in
the infinite past. In some previous works[9, 8], S.Maity in collaboration with
S.Chakraborty has already successfully demonstrated the existence of the
emergent Universe under diffusion mechanism. The cosmic Diffusion process is a
consequence of the non-equilibrium thermodynamic evolution of the Universe.
The dissipation of the energy-momentum tensor of cosmic fluid is compensated
by considering an interaction with a real scalar field[13, 11, 12, 10]. For
suitable phenomenological choice of the time dependent scalar field leads to
the emergent nature of the Universe. But there is no field theoretical model
to justify the diffusion mechanism.

Again the non-equilibrium thermodynamics in cosmic perspective is an outcome
of the particle creation-annihilation process which can be described by the
quantization of the Hamiltonian of the cosmic fluid.

This work is such an attempt to justify the possibility of the emergent
Universe from the quantum field theoretical point of view. Here the cosmic
fluid is assumed to be the real Klein - Gordon field (real scalar field) under
flat FLRW space-time. Then canonical quantization of the corresponding
Hamiltonian is done and the conditions for complete and smooth quantized
Hamiltonian are obtained. Then the possibility of an emergent era has been
determined from these conditions.

##  2 Dynamics of real scalar field in FLRW space-time and its quantization

In the framework of Einstein’s general relativity, the evolution of the cosmic
fluid is governed by Friedmann equations in a flat FLRW Universe
$ds^{2}=dt^{2}-a^{2}(t)(dx^{2}+dy^{2}+dz^{2})$, $a$ is the time dependent
scale factor of the Universe. Considering the cosmic fluid as a barotropic
fluid with barotropic index $W$, the trivial form of Friedmann equations are

$3H^{2}=k\rho,~{}2\dot{H}+3H^{2}=-kP,$ (1)

where $H=\frac{\dot{a}}{a}$, Hubble parameter of the Universe. $P$ and $\rho$
are the thermodynamic pressure and energy density of the cosmic fluid
respectively. Effectively the conservation equation of the fluid will be as,

$\dot{\rho}+3(1+W)H\rho=0.$ (2)

In this work, the dynamics of the cosmic fluid is aimed to explore following
the ideas of quantum field theory. Here we start from representing the cosmic
fluid by the Lagrangian of a real scalar field (Klein-Gordon field). It is the
most trivial field with no charge symmetry. It holds neither a rigid nor a
gauge symmetry. In flat space-time (Minkowski geometry), the form of the
Lagrangian density is given by,

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2},$
(3)

with $m$, the mass of the particle. $\phi^{*}=\phi$ is a real scalar field of
space-time. This Lagrangian density is Lorentz invariant. This Lagrangian
density yields the equation of motion (standard Klein-Gordon equation) as

$(\Box^{2}+m^{2})\phi=0,$ (4)

where $\Box^{2}=\partial_{\mu}\partial^{\mu}$, the D’ Alembertier operator.
The standard KG equation follows Lorentz covariance.

Under the curved space-time geometry, it can be modified in the form

$\mathcal{L}=\frac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi-\frac{1}{2}(m^{2}+\zeta
R)\phi^{2},$ (5)

$\nabla_{\mu}$ represents the covariant derivative. $R$ is the Ricci scalar
and $\zeta$ is the coupling parameter. For minimal coupling, we have chosen
$\zeta=0$ . This modified Lagrangian density is also a Lorentz invariant
quantity. The Euler- Lagrangian equation yields the K-G equation from this
Lagrangian density of equation (5),

$\left(\nabla_{\mu}\nabla^{\mu}+m^{2}\right)\phi(X).=0$ (6)

$X=x^{\mu}=(t,\vec{x})$ is the space-time four - vector. For flat FLRW metric,
equation (6) takes the form

$\ddot{\phi}+3H\dot{\phi}-\nabla^{\prime^{2}}\phi+m^{2}\phi=0.$ (7)

$H=\frac{\dot{a}}{a}$, the Hubble parameter.
$\vec{\nabla}^{\prime}=\frac{1}{a}\vec{\nabla}$. Let’s consider the formal
solution of the equation (7) in terms of the Fourier transform $\phi(X)=\int
d^{3}k\tilde{\phi}(K)e^{-i\int\tilde{K}dX}$ .
$\tilde{K}=\tilde{k}^{\mu}=(k^{0},\vec{k})$, the effective four momenta in
this case. This modified KG equation does not hold Lorentz covariance. So it
exhibits spontaneous Lorentz symmetry breaking. The term $3H\dot{\phi}$ causes
the dissipation of energy from the real scalar field like a damped oscillator.

Here we adopt the modification of the Lagrangian of the cosmic fluid field
only due to the altered metric as a manifestation of the gravity. Here we
haven’t imposed any extra term containing the signature of coupling between
the real scalar field and the gravity. This approach is justified because we
aimed to focus on the dynamics of the cosmic fluid under curved space-time.

From the formal solution, one can estimate from equation (7),

${K^{0}}^{2}+3iHK^{0}-k^{\prime 2}-m^{2}=0.$

Here we assume the slow variation of $k^{0}$ i.e.
$\frac{\dot{k}^{0}}{k^{0}}<<1$.
$\vec{x}\rightarrow\vec{x}^{\prime}=a\vec{x},\vec{k}\rightarrow\vec{k}^{\prime}=\frac{\vec{k}}{a}$.
$\vec{x}^{\prime},\vec{k}^{\prime}$ are the comoving space-coordinate and
momenta respectively.Eventually
$\vec{k}.\vec{x}=\vec{k}^{\prime}.\vec{x}^{\prime}$.

The expression of $K^{0}$ can be found as,

$K^{0}=-\frac{3}{2}iH\pm\omega(t),$

Here
$\omega(t)=\sqrt{|\omega_{0}^{2}-\frac{9}{4}H^{2}|},~{}\omega_{0}=\sqrt{k^{\prime
2}+m^{2}}$ .

The solution of this K-G equation (7) is found as,

$\phi(X)=\frac{1}{(2\pi)^{\frac{3}{2}}}\int\frac{1}{\sqrt{2\omega_{0}}}d^{3}k^{\prime}\left[\mathcal{A}(\vec{k}^{\prime},t)e^{-i\int
KdX}+\mathcal{A}^{*}(\vec{k}^{\prime},t)e^{+i\int KdX}\right],$ (8)

where $\mathcal{A}(\vec{k^{\prime}},t)=e^{-\frac{3}{2}\int
Hdt}\alpha(\vec{k^{\prime}})$ and
$\mathcal{A}^{*}(\vec{k^{\prime}},t)=e^{-\frac{3}{2}\int
Hdt}\alpha^{*}(\vec{k^{\prime}})$. $\int KdX=\int\omega(t)dt-\vec{k}.\vec{x}$.

Generally, the comoving momenta $k^{\prime}$ is dependent on time. Hence
$\omega_{0}$ is also time dependent.

The solution of K-G equation is the superposition of infinite numbers of
damped harmonic oscillators with momentum values ranging
$-\infty<k^{\prime}<\infty$.

For quantization of the Hamiltonian, this solution of the field will be
replaced by the field operator,

$\hat{\phi}(X)=\frac{1}{(2\pi)^{\frac{3}{2}}}\int\frac{1}{\sqrt{2\omega_{0}}}d^{3}k^{\prime}\left[\hat{A}(\vec{k}^{\prime},t)e^{-i\int
KdX}+\hat{A^{\dagger}}(\vec{k}^{\prime},t)e^{+i\int KdX}\right].$ (9)

The Hamiltonian in this case will be in the trivial form (similar as in the
Minkowski space-time),

$\hat{h}=\frac{1}{2}\int
d^{3}x^{\prime}\left[{\dot{{\hat{\phi}}}}^{2}+|\vec{\nabla}^{\prime}\hat{\phi}|^{2}\
+m^{2}\hat{\phi}^{2}\right]$ (10)

The form of the Hamiltonian as in the equation (10) looks very trivial in this
form but the explicit form of $\dot{\hat{\phi}}$ in FLRW metric adds some non
-trivial terms in the Hamiltonian.

The form of the normal ordered Hamiltonian in this scenario can be found as,

$:\hat{h}:=\int\omega_{0}~{}d^{3}k^{\prime}\\{\hat{A}^{\dagger}(\vec{k}^{\prime},t)\hat{A}(\vec{k}^{\prime},t)\\}+\left(\frac{9H^{2}}{2}+i3H\omega\right)\int\frac{d^{3}k^{\prime}}{2\omega_{0}}\hat{A}(\vec{k}^{\prime},t)\hat{A}(-\vec{k}^{\prime},t)\
e^{-2i\int\omega
dt}+\left(\frac{9H^{2}}{2}-i3H\omega\right)\int\frac{d^{3}k^{\prime}}{2\omega_{0}}\hat{A}^{\dagger}(\vec{k}^{\prime},t)\hat{A}^{\dagger}(-\vec{k}^{\prime},t)\
e^{+2i\int\omega dt}$ (11)

i.e.
$:\hat{h}:=\int\omega_{0}~{}d^{3}k^{\prime}\\{\hat{A}^{\dagger}(\vec{k}^{\prime},t)\hat{A}(\vec{k}^{\prime},t)\\}+\sqrt{9H^{2}\omega^{2}+\frac{81}{4}H^{4}}~{}e^{+i\theta}\int\frac{d^{3}k^{\prime}}{2\omega_{0}}\hat{A}(\vec{k}^{\prime},t)\hat{A}(-\vec{k}^{\prime},t)\
e^{-2i\int\omega
dt}+\sqrt{9H^{2}\omega^{2}+\frac{81}{4}H^{4}}~{}e^{-i\theta}\int\frac{d^{3}k^{\prime}}{2\omega_{0}}\hat{A}^{\dagger}(\vec{k}^{\prime},t)\hat{A}^{\dagger}(-\vec{k}^{\prime},t)\
e^{+2i\int\omega dt}$, with $\theta=\tan^{-1}(\frac{2w}{3H})$.

The first term in the right hand side
$\hat{h}_{0}=\int\omega_{0}d^{3}k^{\prime}\\{\hat{A}^{\dagger}(\vec{k}^{\prime},t)\hat{A}(\vec{k}^{\prime},t)\\}$
of the equation (11) is perfectly quantized with
$\hat{A}^{\dagger}(\vec{k}^{\prime},t)$ and $\hat{A}(\vec{k}^{\prime},t)$ ,
the creation and annihilation operator respectively of the boson particle with
comoving momenta $\vec{k}^{\prime}$.

$\hat{A}^{\dagger}(\vec{k}^{\prime},t)\ket{0}=\ket{\vec{k}^{\prime}},\hat{A}(\vec{k}^{\prime},t)\ket{0}=0,$
(12)

where $\ket{0}$ is vacuum state or zero particle state. The corresponding
number operator can be defined as

$\hat{N}(\vec{k}^{\prime},t)=\hat{A}^{\dagger}(\vec{k}^{\prime},t)\hat{A}(\vec{k}^{\prime},t)$
(13)

The second term in the expression of the Hamiltonian in the equation(11) can
not be quantized.

In this case, we have several alternative restrictions on the parameters to
handle the non-trivial term and get the quantized form of the Hamiltonian
operator.

### Case : $1$

For quantization of the Hamiltonian, one condition may be $H=0$ but $a\neq 0$.

Such condition is found to be satisfied at the origin of the emergent
Universe[7, 6, 5, 1, 3, 2, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32].

$H\simeq 0,a\simeq a_{E}~{}\mbox{when}~{}t\rightarrow-\infty,$ (14)

with $a_{E}$ is the scale factor of the Universe at the emergent epoch. Under
this condition, one has

$\hat{A}^{\dagger}(\vec{k}^{\prime},t)\rightarrow\left[a_{E}\right]^{-\frac{3}{2}}\alpha^{\dagger}(\vec{k}),\hat{A}^{(}\vec{k}^{\prime},t)\rightarrow\left[a_{E}\right]^{-\frac{3}{2}}\alpha^{(}\vec{k}).$
(15)

Also the non-trivial terms vanishes at this epoch. Hence one may find the
Hamiltonian in the form

$:\hat{h}:=(a_{E})^{-3}~{}:\hat{h}_{\mbox{Minkowski}}:,$ (16)

where $:\hat{h}_{\mbox{Minkowski}}:$ is the normal ordered Hamiltonian of the
real K-G field in Minkowski space-time.

Here the number operator will also be static with time,

$\hat{N}(\vec{k},t)=(a_{E})^{-3}\hat{N}(\vec{k})_{\mbox{Minkowski}}.$ (17)

### Case : $2$

The other condition for the perfect free field quantization of the Hamiltonian
is $\mbox{Re}(\omega)=\frac{3}{2}H,H\neq 0$. Hence we have
$\omega_{0}=\frac{3}{\sqrt{2}}H,K^{\prime}=\sqrt{\frac{9}{2}H^{2}-m^{2}}$.
Consequently here, the system will be quantized with one particular time
dependent energy eigenvalue $\omega_{0}=\frac{3}{\sqrt{2}}H$ and one momentum
value $K^{\prime}=\sqrt{\frac{9}{2}H^{2}-m^{2}}$.

Hence the particles created at different epochs from the vacuum state have
different energy and different momentum value. So we have to label the
spectrum also by $H$ value.

$\hat{\vec{P}}\ket{H}=\vec{k}\ket{H}=\sqrt{\frac{9}{2}H^{2}-m^{2}}~{}~{}~{}\hat{k}\ket{H},$
(18)

with $\hat{k}$ is the unit vector along the momentum vector.

$:\hat{h}:\ket{H}=\frac{3}{\sqrt{2}}H\ket{H}.$ (19)

Here the number operator $\hat{N}(\vec{k}^{\prime},t)$ will be evolving with
time as,

$\hat{N}(\vec{k}^{\prime},t)=a^{-3}\hat{N}(\vec{k}^{\prime})_{\mbox{Minkowski}}.$
(20)

### Case : $3$

If $\mbox{Re}(\omega)\neq\frac{3}{2}H$ and $H$ is non-zero, then one may
develop the quantization method analogs with the interacting field theory.
Here the explicit form of $\dot{\phi}$ can be written as

$\dot{\hat{\phi}}(X)=e^{-\frac{3}{2}\int
Hdt}\left[\dot{\hat{\phi}}_{0}-\frac{3}{2}H\hat{\phi}_{0}\right]=e^{-\frac{3}{2}\int
Hdt}\dot{\hat{\phi}}_{0}-\frac{3}{2}H\hat{\phi}.$ (21)

$\hat{\phi}=e^{\pm\frac{3}{2}\int Hdt}\hat{\phi}_{0}$ .

Consequently the form of Hamiltonian can be found as,

$\hat{h}=\int
d^{3}x\frac{1}{2}\left[\dot{\hat{\phi}}_{0}^{2}+|\vec{\nabla}^{\prime}\hat{\phi}_{0}|^{2}+m^{2}\hat{\phi}_{0}^{2}\right]+i\frac{3}{2}\omega
H\int
d^{3}x\left[\dot{\hat{\phi}}_{0}\hat{\phi}_{0}+\hat{\phi}_{0}\dot{\hat{\phi}}_{0}\right]$
(22)

Evidently, the Hamiltonian contains two parts,
$\hat{h}=\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}^{\prime}$. The first part
$\hat{\mathcal{H}}_{0}=\int
d^{3}x\frac{1}{2}\left[\dot{\hat{\phi}}_{0}^{2}+|\vec{\nabla}^{\prime}\hat{\phi}_{0}|^{2}+m^{2}\hat{\phi}_{0}^{2}\right]$
is the free field Hamiltonian and it can be quantized with energy eigenvalue
$\frac{\omega^{2}}{\omega_{0}a^{3}}$. But the second part
$\hat{\mathcal{H}}^{\prime}$ can not be quantized and here we shall treat it
as an interacting Hamiltonian. Hence we can calculate the $S$\- matrix for
this interacting Hamiltonian.

$\hat{S}=Te^{-i\int_{\mathcal{T}}^{t}\hat{\mathcal{H}}^{\prime}_{I}(t)dt},$
(23)

$T$ represents the time order product and $\hat{\mathcal{H}}^{\prime}_{I}(t)$
is the interacting Hamiltonian in the interaction picture representation.

$\hat{\mathcal{H}}^{\prime}_{I}(t)=(i\frac{3}{2}\omega H)\int
d^{3}x\left[\dot{\hat{\phi}}_{0I}\hat{\phi}_{0I}+\hat{\phi}_{0I}\dot{\hat{\phi}}_{0I}\right].$
(24)

Hence the spectrum at any epoch $t$ can be found from the relation

$\ket{\vec{k},H(t)}_{I}=\hat{S}\ket{\vec{k},H(\mathcal{T})}_{I},$ (25)

The system (the Universe) is truly free (the interacting part
$\hat{\mathcal{H}}^{\prime}=0$) when $H=0$ . Again a valid free field
Hamiltonian requires the conditions $H\simeq 0$ but $a\neq 0$. Evidently,
these conditions are found in the infinite past (the origin of the emergent
Universe), $t\rightarrow-\infty,H\simeq 0,a\rightarrow a_{E}$. The energy
eigenvalue of the free Hamiltonian leads to $\frac{\omega_{0}}{a_{E}^{3}}$. In
other epochs, when $\mbox{Re}(\omega)\neq\frac{3}{2}H$, the particle’s state
at any epoch can be related to the free field state (particle’s state at
$H\simeq 0$) through the $\hat{S}$ matrix. Let the free field state in the
interaction picture is denoted as

$\ket{free}_{I}=\lim_{t\rightarrow{-\infty}}\ket{\vec{k},H}_{I}.$ (26)

Hence the state at any arbitrary epoch $t$ with Hubble parameter $H$ can be
found as,

$\ket{\vec{k},H}_{I}=\hat{S}\ket{free}_{I},$ (27)

Here the form of $S$ matrix is as

$\hat{S}=Te^{-i\int_{-\infty}^{t}\hat{\mathcal{H}}^{\prime}_{I}(t)dt}.$ (28)

##  3  emergent scenario and cosmic evolution

The conditions for quantization of the cosmic fluid suggests that both the
free field quantization and the interacting field quantization require the
existence of an emergent scenario (when $H=0$). In both cases, the Hamiltonian
gets static at $t\rightarrow-\infty$ with $a\rightarrow
a_{E},:\hat{h}:=\frac{1}{a_{E}^{3}}:\hat{h}_{\mbox{Minkowski}}:$, and behave
like a KG field in a Minkowski space-time.

At other epochs (when $H\neq 0$), the quantization is possible in the
following ways.

$(1)$ Free field quantization is possible only for one specific time dependent
energy and momentum eigenvalue $\epsilon=\frac{3}{\sqrt{2}}H$ ,
$k^{\prime}=\sqrt{\frac{9}{2}H^{2}-m^{2}}$.

$(2)$ The interacting field quantization is possible due to the interacting
Hamiltonian ($\mathcal{H}^{\prime}$) proposed in the equation (24) except
$\epsilon=\frac{3}{\sqrt{2}}H$.

One may expect the same cosmic evolution pattern from the dynamics of the
cosmic fluid under both quantization processes . In this work, we shall follow
the free field quantization process of the fluid particle with energy
$\epsilon=\frac{3}{\sqrt{2}}H$ to obtain the cosmic evolution pattern in this
model.

In an isolated Universe, the total energy will be conserved. Hence one may
write the total energy

$E=<\hat{N}><\hat{h}>,\mbox{Constant}.$ (29)

$<\hat{h}>=\bra{H}\hat{h}\ket{H}=\epsilon(H)$ is the average energy of a
particle at any particular epoch when the value of the Hubble parameter is
$H$. $\hat{N}=\bra{H}\hat{N}\ket{H}=N(H)=N_{0}a^{-3}$. Here
$N_{0}=<\hat{\alpha}^{\dagger}(H)\hat{\alpha}(H)>$, number operator in
Minkowski space.

When $H\neq 0,\omega=\frac{3}{2}H$, we have $\epsilon(H)=\frac{3}{\sqrt{2}}H$.
Hence One may find from the equation (29),

$H\dot{N}+\dot{H}N=0.$ (30)

Incorporating the expression of $N$ in the equation(30), we get the evolution
equation

$\dot{H}-3H^{2}=0.$ (31)

The solution of the equation (31) yields

$\displaystyle H=H_{0}\left(\frac{a}{a_{0}}\right)^{3}$ (32) $\displaystyle
a=a_{0}\left[1-3H_{0}(t-t_{0})\right]^{-\frac{1}{3}},$ (33)

where $a_{0}=a(t_{0}),H_{0}=H(t_{0})$ and $t_{0}$ is a reference epoch of
time. It is already mentioned that this solution will be valid at $H\neq 0$.
Hence it will not reflect the continuous evolution from the epoch of origin
$t\rightarrow-\infty,a\simeq a_{E},H=0$ to any $H\neq$ epoch. Clearly, there
is a discontinuity immediately after the universe starts accelerating from
$H=0$.

In order to connect the evolution of $H=0$ phase and non-zero Hubble parameter
phase, we have proposed a modified form of the scale factor and Hubble
parameter as follows

$\displaystyle
a=a_{E}+\left(a_{0}-a_{E}\right)\left[1-3H_{0}(t-t_{0})\right]^{-\frac{1}{3}}$
(34) $\displaystyle
H=H_{0}\left(1-\frac{a_{E}}{a}\right)\left[\frac{a-a_{E}}{a_{0}-a_{E}}\right]^{3}.$
(35)

The proposed solutions (34),(35) satisfy the evolution equation (31) in the
limit $\frac{a_{E}}{a_{0}}<<1$. On the other hand, it satisfies the condition
of emergent scenario as equation (14).

The choice of scale factor and Hubble parameter perfectly describes the cosmic
evolution pattern under the quantization of cosmic fluid.

Also at the limit $t\rightarrow t_{0}$, the scale factor starts to increase
very rapidly and it satisfies the symptoms of the beginning of inflation or
early time acceleration.

(a)

(b)

(c)

Figure 1:  Comparison between the solution (long dash, red) of equations
(32),(33) and the proposed forms (solid, Black) (34) ,(35) of the scale factor
and the Hubble parameter . $(a)$ In the top left, variation of scale factor
with time in the range of the time epoch $(-50000,0)$. $(b)$ In the top right,
the variation of scale factor with time in the range of very early epoch of
time $(-10^{100},-10^{99})$. $(c)$ In the bottom, the variation of the Hubble
parameter with the scale factor. Here we take
$\frac{a_{E}}{a_{0}}=3*10^{-8},H_{0}=1$.

(a)

(b)

Figure 2:  (a)Variation of Scale factor : $\frac{a}{a_{E}}$ with time $t$ (b)
Variation of Hubble parameter : $\frac{H}{H_{0}}$with $\frac{a}{a_{E}}$, for
$t_{0}=0$ and three different values for the ratio $\frac{a_{E}}{a_{0}}$ for
$H_{0}=1$ .

(a)

(b)

Figure 3:  (a) Evolution of Scale factor : $\frac{a}{a_{E}}$with time $t$ and
values of $\frac{a_{E}}{a_{0}}$ (left) and (b) Variation of Hubble parameter :
$\frac{H}{H_{0}}$ with $\frac{a}{a_{E}}$ and $\frac{a_{E}}{a_{0}}$ (right) for
$t_{0}=0$ in $3$d plots.

Reasonably one may consider the reference epoch of time $t_{0}$ in the
equation (34) is the starting point of the inflationary era. Therefore the
present model is the suitable description of the pre-inflationary cosmic
evolution. From $t_{0}$, the cosmic evolution pattern will be changed and the
evolution equation will have the different form. Thus the existence of
mathematical Big-rip singularity of the solutions in equations (34) and (35)
at $t=t_{0}+\frac{1}{3H_{0}}$ has no physical relevance in this scenario.

##  4 Discussion

In this present model, we have successfully established the non-singular
origin of the Universe from the perspective of quantum field theory. Also we
have shown two different alternatives for which the cosmic fluid field may be
quantized at other epochs. As the conclusions, one gets that the Universe
consisted of the free real K-G field fluid at the emergent epoch. Then it
starts evolving with particle creation process with a specific time dependent
energy and momenta value or alternatively it may be evolved under an
interacting Lagrangian mentioned (proposed) in the article(equation (24)).

Following the first option, the evolution pattern of the Universe has been
obtained in the non-zero Hubble parameter epoch.

Also we have proposed a general evolutionary scenario from the the epoch of
the origin up to an inflationary phase. It is established that these proposed
forms of the scale factor and the Hubble parameters fit well with the
condition of quantization process and the cosmic evolution.

The deviation of the derived solutions (32), (33) from the modified solutions
(34),(35) in Fig.$1$. It is found that these solutions are identical except
the early epoch of time. They are going to coincide in the limit $a>>a_{E}$.

The proposed solutions(34),(35) are represented graphically in Fig.$2$ for
three different values of the ratio $\frac{a_{E}}{a_{0}}$. In Fig.3 , these
parameters are shown in $3$d plots for better realization of the outcomes of
this model.

This work is a demonstration of finding the effect of traditional Friedmann
cosmology from the dynamics of quantum fields.

Finally it is noteworthy that it will be aimed to set up a convenient quantum
field model to interpret the present late time acceleration along with a
complete and continuous cosmic evolutionary scenario in a series of future
works.

## Acknowledgment

The author SM acknowledges Prof. Subenoy Chakraborty, Dept. of Mathematics,
Jadavpur University, Kolkata-$700032$ for his valuable suggestion on this
topic and the author SB thanks all the faculties, Dept. of Physics, Diamond
Harbour Women University for their assistance while this work.

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# On the projections of the multifractal packing dimension for $q>1$

Bilel Selmi

###### Abstract

The aim of this article is to study the behavior of the multifractal packing
function $B_{\mu}(q)$ under projections in Euclidean space for $q>1$. We show
that $B_{\mu}(q)$ is preserved under almost every orthogonal projection. As an
application, we study the multifractal analysis of the projections of a
measure. In particular, we obtain general results for the multifractal
analysis of the orthogonal projections on $m$-dimensional linear subspaces of
a measure $\mu$ satisfying the multifractal formalism.

MSC-2010: 28A20, 28A80.

Keyword: Hausdorff dimension; Packing dimension; Projection; Multifractal
analysis.

##  1 Introduction and statement of the results

The notion of singularity exponents or spectrum and generalized dimensions are
the major components of the multifractal analysis. They were introduced with a
view of characterizing the geometry of measure and to be linked with the
multifractal spectrum which is the map which affects the Hausdorff or packing
dimension of the iso-Hölder set

$E_{\mu}(\alpha)=\left\\{x\in\operatorname{supp}\mu;\;\;\lim_{r\to
0}\frac{\log\big{(}\mu B(x,r)\big{)}}{\log r}=\alpha\right\\}$

for a given $\alpha\geq 0$ and $\operatorname{supp}\mu$ is the topological
support of probability measure $\mu$ on $\mathbb{R}^{n}$, $B(x,r)$ is the
closed ball of center $x$ and radius $r$. It unifies the multifractal spectra
to the multifractal packing function $B_{\mu}(q)$ via the Legendre transform
[5, 26], i.e.,

$\dim_{H}\Big{(}E_{\mu}(\alpha)\Big{)}=\inf_{q\in\mathbb{R}}\Big{\\{}q\alpha+B_{\mu}(q)\Big{\\}}.$

There has been a great interest in understanding the fractal dimensions of
projections of the iso-Hölder sets and measures. Recently, the projectional
behavior of dimensions and multifractal spectra of sets and measures have
generated a large interest in the mathematical literature [3, 7, 14, 15, 18,
19, 20, 29, 30]. The first significant work in this area was the result of
Marstrand [22] who showed who proved a well-known theorem according to which
the Hausdorff dimension of a planar set is preserved under orthogonal
projections. This result was later generalized to higher dimensions by Kaufman
[21], Mattila [23] and Hu and Taylor [16] and they obtain similar results for
the Hausdorff dimension of a measure.

Let us mention that Falconer and Mattila [12] and Falconer and Howroyd [11]
have proved that the packing dimension of the projected set or measure will be
the same for almost all projections. However, despite these substantial
advances for fractal sets, only very little is known about the multifractal
structure of projections of measures, except a paper by O’Neil [28]. Later, in
[2] Barral and Bhouri, studied the multifractal analysis of the orthogonal
projections on $m$-dimensional linear subspaces of singular measures on
$\mathbb{R}^{n}$ satisfying the multifractal formalism. The result of O’Neil
was later generalized by Selmi et al. in [7, 8, 9, 32, 33].

O’Neil [28] has compared the generalized Hausdorff and packing dimensions of a
set $E$ of $\mathbb{R}^{n}$ with respect to a measure $\mu$ with those of
their projections onto $m$-dimensional subspaces. More specifically, given a
compactly supported Borel probability measure $\mu$ on $\mathbb{R}^{n}$ and
$q\in\mathbb{R}$, let $B_{\mu}(q)$ the multifractal packing function of
$\operatorname{supp}\mu$. Then we have $B_{\mu_{V}}(q)\leq B_{\mu}(q)$ for all
$q\leq 1$ and all $m$-dimensional linear subspaces $V$. Then, what can be said
about the multifractal packing function and its projection onto a lower
dimensional linear subspace for $q>1$? The goal of this work is giving an
answer to this question. We are interested in knowing whether or not this
property is preserved after orthogonal projections on $\gamma_{n,m}$-almost
every linear $m$-dimensional subspaces for $q>1$, where $\gamma_{n,m}$ is the
uniform measure on $G_{n,m}$, the set of linear $m$-dimensional subspaces of
$\mathbb{R}^{n}$ endowed with its natural structure of a compact metric space
(see [24]).

In the present paper we pursue those kinds of studies and we consider the
multifractal formalism developed in [28]. The aims of this study are twofold.
First, the behavior of the packing dimensions $B_{\mu}(q)$ under projection.
In particular, we show that $B_{\mu}(q)$ is preserved under
$\gamma_{n,m}$-almost every orthogonal projection for $q>1$. We have treated
an unsolved case by O’Neil which is $q>1$ and the result that we have obtained
is optimal. Secondly, to investigate a relationship between the multifractal
spectrum and its projection onto a lower dimensional linear subspace. We also
obtain general results for the multifractal analysis of the orthogonal
projections on $m$-dimensional linear subspaces of a measure $\mu$ satisfying
the multifractal formalism.

##  2 Preliminaries

We start by recalling the multifractal formalism introduced by O’Neil in [28].
Let $\mu$ be a compactly supported probability measure on $\mathbb{R}^{n}$.
For $q,s\in\mathbb{R}$, $E\subseteq\operatorname{supp}\mu$ and $\delta>0$, we
define the multifractal packing pre-measure,

$\overline{{\mathscr{P}}}^{q,s}_{\mu,\delta}(E)=\displaystyle\sup\left\\{\sum_{i}\mu\left(B\left(x_{i},\frac{r_{i}}{3}\right)\right)^{q}r_{i}^{s}\right\\},$

where the supremum is taken over all $\delta$-packings of $E$,

$\overline{{\mathscr{P}}}^{q,s}_{\mu}(E)=\displaystyle\inf_{\delta>0}\overline{{\mathscr{P}}}^{q,s}_{\mu,\delta}(E).$

The function $\overline{{\mathscr{P}}}^{q,s}_{\mu}$ is increasing but not
$\sigma$-subadditive. That is the reason why O’Neil introduced the
modification of the multifractal packing measure ${\mathscr{P}}^{q,s}_{\mu}$:

${\mathscr{P}}^{q,s}_{\mu}(E)=\inf_{E\subseteq\bigcup_{i}E_{i}}\sum_{i}\overline{\mathscr{P}}^{q,s}_{\mu}(E_{i}).$

In a similar way, we define the Hausdorff measure,

${{\mathscr{H}}}^{{q},s}_{{\mu},\delta}(E)=\displaystyle\inf\left\\{\sum_{i}\mu\Big{(}B(x_{i},3r_{i})\Big{)}^{q}~{}r_{i}^{s};\;\;\Big{(}B(x_{i},r_{i})\Big{)}_{i}\text{is
a $\delta$-covering of}\;E\right\\},$

and

${{\mathscr{H}}}^{{q},s}_{{\mu}}(E)=\displaystyle\sup_{\delta>0}{{\mathscr{H}}}^{q,s}_{{\mu},\delta}(E).$

The functions ${\mathscr{P}}^{q,s}_{\mu}$ and ${\mathscr{H}}^{q,s}_{\mu}$ are
metric outer measures and thus measures on the family of Borel subsets of
$\mathbb{R}^{n}$. An important feature of the pre-packing, packing and
Hausdorff measure is that
${\mathscr{P}}^{q,s}_{\mu}\leq{\overline{\mathscr{P}}}^{q,s}_{\mu}$ and there
is a constant $c$ depending also on the dimension of the ambient space, such
that ${\mathscr{H}}^{q,s}_{\mu}\leq c~{}{\mathscr{P}}^{q,s}_{\mu}$ (see [28]).

The functions $\overline{{\mathscr{P}}}^{q,s}_{\mu}$,
${\mathscr{P}}^{q,s}_{\mu}$ and ${\mathscr{H}}^{q,s}_{\mu}$ assign, in the
usual way, a dimension to each subset $E$ of $\operatorname{supp}\mu$. They
are respectively denoted by $\Lambda_{\mu}^{q}(E)$, $B_{\mu}^{q}(E)$ and
$b_{\mu}^{q}(E)$.

  1. 1.

There exists a unique number $\Lambda_{\mu}^{q}(E)\in[-\infty,+\infty]$ such
that

$\overline{{\mathscr{P}}}^{q,s}_{\mu}(E)=\left\\{\begin{matrix}\infty&\text{if}&s<\Lambda_{\mu}^{q}(E),\\\
\\\ 0&\text{if}&\Lambda_{\mu}^{q}(E)<s.\end{matrix}\right.$

  2. 2.

There exists a unique number $B_{\mu}^{q}(E)\in[-\infty,+\infty]$ such that

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

  3. 3.

There exists a unique number $b_{\mu}^{q}(E)\in[-\infty,+\infty]$ such that

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

We note that for all $q\in\mathbb{R}$

$b_{\mu}^{q}(\emptyset)=B_{\mu}^{q}(\emptyset)=\Lambda_{\mu}^{q}(\emptyset)=-\infty,$

and if $\mu(E)=0$, then

$b_{\mu}^{q}(E)=B_{\mu}^{q}(E)=\Lambda_{\mu}^{q}(E)=-\infty\quad\text{for}\quad
q>0.$

Next, we define the separator functions $\Lambda_{\mu}$, $B_{\mu}$ and
$b_{\mu}$ : $\mathbb{R}\rightarrow[-\infty,+\infty]$ by,

$\Lambda_{\mu}(q)=\Lambda_{\mu}^{q}(\operatorname{supp}\mu)$,
$B_{\mu}(q)=B_{\mu}^{q}(\operatorname{supp}\mu)$ and
$b_{\mu}(q)=b_{\mu}^{q}(\operatorname{supp}\mu).$

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

###### Proposition 2.1

[28] Let $\mu$ be compactly supported probability measure on $\mathbb{R}^{n}$.
Then, we have

  1. 1.

For $q<1$, $0\leq b_{\mu}(q)\leq B_{\mu}(q)\leq\Lambda_{\mu}(q).$

  2. 2.

$b_{\mu}(1)=B_{\mu}(1)=\Lambda_{\mu}(1)=0.$

  3. 3.

For $q>1$, $b_{\mu}(q)\leq B_{\mu}(q)\leq\Lambda_{\mu}(q)\leq 0.$

###### Remark 2.1

The multifractal Hausdorff and packing measures introduced by O’Neil are
different from those developed by Olsen [26]. Although, when $\mu$ satisfies a
doubling condition, the multifractal measures are equivalent.

##  3 Main result

Let $\mu$ be a compactly supported probability measure on $\mathbb{R}^{n}$ and
$q\in\mathbb{R}$. In the following, we require an alternative characterization
of the generalized upper $L^{q}$-spectrum of $\mu$ in terms of a potential
obtained by convolving $\mu$ with a certain kernel. For this purpose let us
introduce some notations. For $1\leq s\leq n$ and $r>0$ we define

$\begin{array}[]{llll}\phi_{r}^{s}:&\mathbb{R}^{n}&\longrightarrow&\mathbb{R}\\\
&x&\longmapsto&\min\Big{\\{}1,\>r^{s}|x|^{-s}\Big{\\}},\end{array}$

and

$\mu\ast\phi_{r}^{s}(x)=\int\min\Big{\\{}1,\>r^{s}|x-y|^{-s}\Big{\\}}d\mu(y).$

Let $E$ be a compact subset of $\operatorname{supp}\mu$. For $1\leq s\leq n$
and $q>1$, write

$N_{\mu,r}^{q,s}(E)=\int_{E}\Big{(}\mu\ast\phi_{r/3}^{s}(x)\Big{)}^{q-1}d\mu(x),$

and

$\overline{\tau}_{\mu}^{q,s}(E)=\displaystyle\limsup_{r\to 0}\frac{\log
N_{\mu,r}^{q,s}(E)}{-\log
r}\quad\text{and}\quad\underline{\tau}_{\mu}^{q,s}(E)=\displaystyle\liminf_{r\to
0}\frac{\log N_{\mu,r}^{q,s}(E)}{-\log r}.$

The definition of these dimensions is, frankly, messy, indirect and
unappealing. In an attempt to make the concept more attractive, we present
here an alternative approach to the dimension $\overline{\tau}_{\mu}^{q,s}$
and his application to projections in terms of a potential obtained by
convolving $\mu$ with a certain kernel. For $E$ a compact subset of
$\operatorname{supp}\mu$ we can try to decompose $E$ into a countable number
of pieces $E_{1},E_{2},...$ in such a way that the largest piece has as small
a dimension as possible. The present approach was first used by Falconer in
[10, Section 3.3] and further developed by O’Neil in [28, Proposition 2.4].
This idea leads to the following modified dimension in terms of the
convolutions:

$\displaystyle\mathfrak{T}_{\mu}^{q,s}(E)$ $\displaystyle=$
$\displaystyle\inf\left\\{\sup_{1\leq
i<\infty}\overline{\tau}_{\mu}^{q,s}(E_{i}),\;\;E\subset\bigcup_{i}E_{i}\;\;\text{with
each $E_{i}$ compact }\right\\}$

and

$\mathfrak{T}_{\mu}^{s}(q)=\mathfrak{T}_{\mu}^{q,s}(\operatorname{supp}\mu)\quad\text{for
all}\quad s\geq 1.$

Let $m$ be an integer with $0<m\leq n$ and $G_{n,m}$ the Grassmannian manifold
of all $m$-dimensional linear subspaces of $\mathbb{R}^{n}$. Denote by
$\gamma_{n,m}$ the invariant Haar measure on $G_{n,m}$ such that
$\gamma_{n,m}(G_{n,m})=1$. For $V\in G_{n,m}$, we define the projection map
$\pi_{V}:\mathbb{R}^{n}\longrightarrow V$ as the usual orthogonal projection
onto $V$. Then, the set $\\{\pi_{V},\;V\in G_{n,m}\\}$ is compact in the space
of all linear maps from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ and the
identification of $V$ with $\pi_{V}$ induces a compact topology for $G_{n,m}$.
Also, for a Borel probability measure $\mu$ with compact support
$supp\mu\subset\mathbb{R}^{n}$ and for $V\in G_{n,m}$, we denote by $\mu_{V}$,
the projection of $\mu$ onto $V$, i.e.,

$\mu_{V}(A)=\mu(\pi_{V}^{-1}(A))\quad\forall A\subseteq V.$

Since $\mu$ is compactly supported and $supp\mu_{V}=\pi_{V}(supp\mu)$ for all
$V\in G_{n,m}$, then, for any continuous function
$f:V\longrightarrow\mathbb{R}$, we have

$\displaystyle\int_{V}fd\mu_{V}=\int f(\pi_{V}(x))d\mu(x),$

whenever these integrals exist. Then for all $V\in G_{n,m}$,
$x\in\mathbb{R}^{n}$ and $0<r<1$, we have

$\mu\ast\phi_{r}^{m}(x)=\int\mu_{V}(B(x_{V},r))dV=\int\min\Big{\\{}1,\>r^{m}|x-y|^{-m}\Big{\\}}d\mu(y).$

In [28], O’Neil has compared the generalized Hausdorff and packing dimensions
of a set $E$ of $\mathbb{R}^{n}$ with respect to a measure $\mu$ with those of
their projections onto $m$-dimensional subspaces. More specifically, he proved
the following result:

###### Theorem 3.1

Let $\mu$ be a compactly supported probability measure on $\mathbb{R}^{n}$ and
$E\subseteq\operatorname{supp}\mu$. For $q\leq 1$ and all $V\in G_{n,m}$, we
have

$B_{\mu_{V}}^{q}(\pi_{V}(E))\leq B_{\mu}^{q}(E).$

In this paper, we show that $B_{\mu}^{q}(E)$ is preserved under
$\gamma_{n,m}$-almost every orthogonal projection for $q>1$. We have treated
an unsolved case by O’Neil which is $q>1$ and the result that we have obtained
is optimal. More precisely, we have the following result.

###### Theorem 3.2

Let $E$ be a compact subset of $\operatorname{supp}\mu$ and $q>1$.

  1. 1.

If $1<q\leq 2$, one has

$B_{\mu_{V}}^{q}(\pi_{V}(E))=\mathfrak{T}_{\mu}^{q,m}(E)=\max\Big{(}m(1-q),B_{\mu}^{q}(E)\Big{)},\;\text{for
$\gamma_{n,m}$-almost every}\;\;V\in G_{n,m}.$

  2. 2.

If $q>2$ and $({E}_{i})_{i}$ is a cover of $E$ by a countable collection of
compact sets is such that $\Lambda_{\mu}^{q}(E_{i})\geq-m$ for all $i$, then

$B_{\mu_{V}}^{q}(\pi_{V}(E))=\mathfrak{T}_{\mu}^{q,m}(E)=B_{\mu}^{q}(E),\;\text{for
$\gamma_{n,m}$-almost every}\;\;V\in G_{n,m}.$

As a consequence we have, the following corollary

###### Corollary 3.1

Let $q>1$.

  1. 1.

If $1<q\leq 2$, one has

$B_{\mu_{V}}(q)=\mathfrak{T}_{\mu}^{m}(q)=\max\Big{(}m(1-q),B_{\mu}(q)\Big{)},\;\text{for
$\gamma_{n,m}$-almost every}\;\;V\in G_{n,m}.$

  2. 2.

If $q>2$ and $({E}_{i})_{i}$ is a cover of $\operatorname{supp}\mu$ by a
countable collection of compact sets is such that
$\Lambda_{\mu}^{q}(E_{i})\geq-m$ for all $i$, then

$B_{\mu_{V}}^{q}(q)=\mathfrak{T}_{\mu}^{m}(q)=B_{\mu}(q),\;\text{for
$\gamma_{n,m}$-almost every}\;\;V\in G_{n,m}.$

###### Remark 3.1

The hypothesis $\Lambda_{\mu}^{q}(E_{i}\cap E)\geq-m$ for all $i$ implies that
$\Lambda_{\mu}^{q}(E)\geq-m$. Nevertheless, we don’t know if the weaker
condition $\Lambda_{\mu}^{q}(E)\geq-m$ is sufficient to obtain the conclusion
of Theorem 3.2.

##  4 Proof of the main result

###  4.1 Preliminary results

We present the tools, as well as the intermediate results, which will be used
in the proof of our main result. Let $\mu$ be a compactly supported
probability measure on $\mathbb{R}^{n}$ and $q\in\mathbb{R}$. We define the
upper and lower $L^{q}$\- spectrum of a measure $\mu$. For a subset
$E\subset\operatorname{supp}\mu$, write

$N_{\mu,r}^{q}(E)=\sup\left\\{\displaystyle\sum_{i}\mu\left(B\left(x_{i},\frac{r}{3}\right)\right)^{q};\;\;\Big{(}B(x_{i},r)\Big{)}_{i}\quad\text{is
a packing of}\;\;E\right\\}.$

The upper respectively lower $L^{q}$\- spectrum $\overline{\tau}_{\mu}^{q}$
and $\underline{\tau}_{\mu}^{q}$ of $E$ is defined by

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

By convention, if $r\in(0,1)$: $\frac{\log 0}{-\log r}=-\infty$.

The following proposition is a consequence of the multifractal formalism
developed in [28].

###### Proposition 4.1

Let $E$ be a compact subset of $\operatorname{supp}\mu$ and $q\in\mathbb{R}$.
One has

$\displaystyle B_{\mu}^{q}(E)=\inf\left\\{\sup_{1\leq
i<\infty}\overline{\tau}_{\mu}^{q}(E_{i}),\;\;E\subset\bigcup_{i}E_{i}\;\;\text{with
each $E_{i}$ compact}\;\right\\}.$

Proposition 4.1 is a consequence from the following lemmas.

###### Lemma 4.1

Let $E$ be a subset of $\operatorname{supp}\mu$ and $q\in\mathbb{R}$. Then we
have

$B_{\mu}^{q}(E)=\inf\left\\{\sup_{1\leq
i<\infty}\overline{\tau}_{\mu}^{q}(E_{i}),\;\;E\subset\bigcup_{i}E_{i}\right\\}=\inf\left\\{\sup_{1\leq
i<\infty}\Lambda_{\mu}^{q}(E_{i}),\;\;E\subset\bigcup_{i}E_{i}\right\\}.$

Proof. The Lemma is Proposition 2.4 of [28].

###### Lemma 4.2

  1. 1.

For $q,s\in\mathbb{R}$, we have
$\overline{{\mathscr{P}}}^{q,s}_{\mu}(E)=\overline{{\mathscr{P}}}^{q,s}_{\mu}(\overline{E})$
and $\Lambda_{\mu}^{q}(E)=\Lambda_{\mu}^{q}(\overline{E})$ for all
$E\subset\operatorname{supp}\mu$.

  2. 2.

Let $E$ be a compact subset of $\operatorname{supp}\mu$ and $q\in\mathbb{R}$.
If

$\Lambda_{\mu}^{q}(E\cap V)=\Lambda_{\mu}^{q}({E})\quad\text{for all open
sets}\;\;V\;\;\text{with}\;\;E\cap V\neq\emptyset,$

then,

$B_{\mu}^{q}(E)=\Lambda_{\mu}^{q}({E}).$

Proof. The first part is Lemma 5.4.1 in [25]. We will prove the second part.
Let $E\subset\cup_{i}E_{i}$. Since $E\subset\cup_{i}\overline{E}_{i}$, Baire’s
category theorem implies that there exists an integer $k\in\mathbb{N}$ and an
open set $V$ such that $\emptyset\neq E\cap V\subset\overline{E}_{k}.$ Hence,

$\sup_{i}\Lambda_{\mu}^{q}(\overline{E}_{i})\geq\Lambda_{\mu}^{q}(\overline{E}_{k})\geq\Lambda_{\mu}^{q}(E\cap
V)=\Lambda_{\mu}^{q}({E}).$

Since the covering $(E_{i})_{i}$ of $E$ was arbitrary, the previous lemma now
implies that

$\displaystyle B_{\mu}^{q}(E)$ $\displaystyle=$
$\displaystyle\inf\left\\{\sup_{1\leq
i<\infty}\Lambda_{\mu}^{q}(E_{i}),\;\;E\subset\bigcup_{i}E_{i}\right\\}$
$\displaystyle=$ $\displaystyle\inf\left\\{\sup_{1\leq
i<\infty}\Lambda_{\mu}^{q}(\overline{E_{i}}),\;\;E\subset\bigcup_{i}E_{i}\right\\}$
$\displaystyle\geq$ $\displaystyle\Lambda_{\mu}^{q}({E}).$

The following straightforward estimates concern the behavior of
$\mu\ast\phi_{r}^{n}(x)$ as $r\to 0$.

###### Lemma 4.3

[13] Let $1\leq m\leq n$ and $\mu$ be a compactly supported probability
measure on $\mathbb{R}^{n}$. For all $x\in\mathbb{R}^{n}$, we have

$cr^{m}\leq\mu\ast\phi_{r}^{m}(x)$

for all sufficiently small $r$, where $c>0$ is independent of $r$.

###### Lemma 4.4

[13] Let $\mu$ be a compactly supported probability measure on
$\mathbb{R}^{n}$.

  1. 1.

For all $x\in\mathbb{R}^{n}$ and $r>0$

$\mu(B(x,r))\leq\mu\ast\phi_{r}^{n}(x).$

  2. 2.

Let $\varepsilon>0$. We have that for $\mu$-almost all $x$

$r^{-\varepsilon}\mu(B(x,r))\geq\mu\ast\phi_{r}^{n}(x),$

if $r$ is sufficiently small.

We use the properties of $\mu\ast\phi_{r}^{n}(x)$ to have a relationship
between the kernels and projected measures.

###### Lemma 4.5

[13] Let $1\leq m\leq n$, $\mu$ be a compactly supported probability measure
on $\mathbb{R}^{n}$, $\varepsilon>0$ and $r$ is sufficiently small.

  1. 1.

For all $V\in G_{n,m}$ and for $\mu$-almost all $x\in\mathbb{R}^{n}$

${r}^{\varepsilon}\mu\ast\phi_{r}^{m}(x)\leq\mu_{V}(B(x_{V},r)).$

  2. 2.

For $\gamma_{n,m}$-almost all $V\in G_{n,m}$ and all $x\in\mathbb{R}^{n}$

${r}^{\varepsilon}\mu\ast\phi_{r}^{m}(x)\geq\mu_{V}(B(x_{V},r)).$

The next result is essentially a restatement of [2, Proposition 4.2] and [6,
Proposition 5.1] (see also [13, Lemma 2.6 (a)] and [34]). We provide a proof
for the reader’s convenience.

###### Proposition 4.2

Let $E$ be a compact subset of $\operatorname{supp}\mu$. For $q>1$, we have

$\underline{\tau}_{\mu}^{q}(E)=\liminf_{r\to 0}\frac{1}{-\log
r}\log\int_{E}\Big{(}\mu(B(x,r/3))\Big{)}^{q-1}d\mu(x)$

and

$\overline{\tau}_{\mu}^{q}(E)=\limsup_{r\to 0}\frac{1}{-\log
r}\log\int_{E}\Big{(}\mu(B(x,r/3))\Big{)}^{q-1}d\mu(x).$

Proof. Let $r>0$ and $\Big{(}B(x_{i},r/3)\Big{)}_{i}$ be a family of disjoint
balls centered on $E$. Since $q>1$, we have

$\displaystyle\int_{E}\Big{(}\mu(B(x,r/3))\Big{)}^{q-1}d\mu(x)$
$\displaystyle\geq$
$\displaystyle\int_{\bigcup_{i}B(x_{i},r/9)}\Big{(}\mu(B(x,r/3))\Big{)}^{q-1}d\mu(x)$
$\displaystyle\geq$
$\displaystyle\sum_{i}\Big{(}\mu\left(B\left(x_{i},r/9\right)\right)\Big{)}^{q}.$

On the other hand, for every $r>0$ we can apply Besicovitch’s covering theorem
to $\Big{(}B(x,r/3)\Big{)}_{x\in E}$, to get a positive integer $\xi(n)$
depending on $n$ only, as well as
$\mathcal{B}_{1}=\Big{(}B(x_{1,j},r/3)\Big{)}_{j}$,…,$\mathcal{B}_{\xi(n)}=\Big{(}B(x_{\xi(n),j},r/3)\Big{)}_{j},$
$\xi(n)$ families of disjoint balls of radius $r/3$, such that
$E\subset\bigcup_{i=1}^{\xi(n)}\bigcup_{j}B(x_{i,j},r/3)$ and

$\displaystyle\int_{E}\Big{(}\mu(B(x,r/9))\Big{)}^{q-1}d\mu(x)$
$\displaystyle\leq$
$\displaystyle\sum_{i=}^{\xi(n)}\sum_{j}\int_{B(x_{i,j},r/3)}\Big{(}\mu(B(x,r/9))\Big{)}^{q-1}d\mu(x)$
$\displaystyle\leq$
$\displaystyle\sum_{i=}^{\xi(n)}\sum_{j}\Big{(}\mu(B(x_{i,j},r/3))\Big{)}^{q}.$

Taking the logarithms and letting $r\to 0$ yields the result.

###### Remark 4.1

Let $E$ be a compact subset of $\operatorname{supp}\mu$ and $q>1$. It is clear
that from Proposition 4.2 and Proposition 2.8 in [13],

$\overline{\tau}_{\mu}^{q}(E)=\overline{\tau}_{\mu}^{q,n}(E)\quad\text{and}\quad\underline{\tau}_{\mu}^{q}(E)=\underline{\tau}_{\mu}^{q,n}(E).$

With these definitions we have the following corollaries.

###### Corollary 4.1

Let $E$ be a compact subset of $\operatorname{supp}\mu$. For $q>1$ and $1\leq
m\leq n$, we have

$\displaystyle\underline{\tau}_{\mu}^{q,m}(E)$ $\displaystyle=$
$\displaystyle\inf\left\\{s\geq 0;\;\;\limsup_{r\to
0}r^{s}\int_{E}\Big{(}\mu\ast\phi_{r/3}^{m}(x)\Big{)}^{q-1}d\mu(x)=0\right\\}$

and

$\displaystyle\overline{\tau}_{\mu}^{q,m}(E)$ $\displaystyle=$
$\displaystyle\inf\left\\{s\geq 0;\;\;\liminf_{r\to
0}r^{s}\int_{E}\Big{(}\mu\ast\phi_{r/3}^{m}(x)\Big{)}^{q-1}d\mu(x)=0\right\\}.$

###### Corollary 4.2

Let $E$ be a compact subset of $\operatorname{supp}\mu$. For $q>1$, we have

$\displaystyle\underline{\tau}_{\mu}^{q}(E)$ $\displaystyle=$
$\displaystyle\inf\left\\{s\geq 0;\;\;\limsup_{r\to
0}r^{s}\int_{E}\Big{(}\mu(B(x,r/3))\Big{)}^{q-1}d\mu(x)=0\right\\}$

and

$\displaystyle\overline{\tau}_{\mu}^{q}(E)$ $\displaystyle=$
$\displaystyle\inf\left\\{s\geq 0;\;\;\liminf_{r\to
0}r^{s}\int_{E}\Big{(}\mu(B(x,r/3))\Big{)}^{q-1}d\mu(x)=0\right\\}.$

###### Proposition 4.3

Let $E$ be a compact subset of $\operatorname{supp}\mu$. For $q>1$, we have

$\overline{\tau}_{\mu}^{q,m}(E)=\max\Big{(}m(1-q),\overline{\tau}_{\mu}^{q}(E)\Big{)}.$

Proof. Recalling that, from Lemma 4.4

$\mu\ast\phi_{r/3}^{m}(x)\geq\mu(B(x,r/3)),$

it will be clear that for $q>1$ we have

$\int_{E}\Big{(}\mu\ast\phi_{r/3}^{m}(x)\Big{)}^{q-1}d\mu(x)\geq\int_{E}\Big{(}\mu(B(x,r/3))\Big{)}^{q-1}d\mu(x).$

Hence,

$\underline{\tau}_{\mu}^{q,m}(E)\geq\underline{\tau}_{\mu}^{q}(E)\quad\text{and}\quad\overline{\tau}_{\mu}^{q,m}(E)\geq\overline{\tau}_{\mu}^{q}(E).$

Without loss of generality, we may assume that $\mu(E)>0$. From Lemma 4.3, we
get

$\displaystyle\int_{E}\Big{(}\mu\ast\phi_{r/3}^{n}(x)\Big{)}^{q-1}d\mu(x)$
$\displaystyle\geq$ $\displaystyle c_{1}r^{m(q-1)}\mu(E).$

Therefore, we obtain

$\underline{\tau}_{\mu}^{q,m}(E)\geq m(1-q).$

Let $t>\overline{\tau}_{\mu}^{q}(E)$. Suppose that $\operatorname{supp}\mu$
have diameter $h$. Then

$\displaystyle\int_{E}\Big{(}\mu(B(x,r/3))\Big{)}^{q-1}d\mu(x)\leq
c_{1}r^{-t},\quad\forall r\leq 6h,$

where $c_{1}$ is independent of $r$, and

$\displaystyle\int_{E}\Big{(}\mu(B(x,r/3))\Big{)}^{q-1}d\mu(x)=1,\quad\forall
r\geq 6h.$

For $\varepsilon>0$, $p=q-1$ and $r$ is small enough, by using Proposition 2.5
in [13], we obtain

$\displaystyle\int_{E}\Big{(}\mu\ast\phi_{r/3}^{m}(x)\Big{)}^{p}d\mu(x)$
$\displaystyle\leq$ $\displaystyle
B\;(r/3)^{mp-\varepsilon}\int_{r/3}^{+\infty}u^{-mq-1}\int_{E}\Big{(}\mu(B(x,u))\Big{)}^{p}d\mu(x)du$
$\displaystyle=$ $\displaystyle
B\;(r/3)^{mp-\varepsilon}\int_{r/3}^{6h}u^{-mq-1}\int_{E}\Big{(}\mu(B(x,u))\Big{)}^{p}d\mu(x)du$
$\displaystyle+$ $\displaystyle
B\;(r/3)^{mp-\varepsilon}\int_{6h}^{+\infty}u^{-mp-1}\int_{E}\Big{(}\mu(B(x,u))\Big{)}^{p}d\mu(x)du$
$\displaystyle\leq$ $\displaystyle
C_{1}(r/3)^{mp-\varepsilon}\int_{r/3}^{6h}u^{-mp-1-t}du$ $\displaystyle+$
$\displaystyle C_{2}(r/3)^{mp-\varepsilon}\int_{6h}^{+\infty}u^{-mp-1}du$
$\displaystyle\leq$ $\displaystyle\left\\{\begin{array}[]{ll}\vskip 12.0pt
plus 4.0pt minus
4.0ptC_{3}~{}~{}(r/3)^{-t-\varepsilon}\quad\quad\quad\;si&\hbox{$t>-mp$,}\\\
\\\ C_{4}~{}~{}(r/3)^{mp-\varepsilon}\quad\quad\quad si&t\leq-
mp,\end{array}\right.$

where $B$ and $C_{i}$ $(i=1,...,4)$ are independent of $r$. This gives that

$\overline{\tau}_{\mu}^{q,m}(E)\leq\max(-mp,t),\quad\text{for all}\quad
t>\overline{\tau}_{\mu}^{q}(E).$

Finally, we obtain

$\overline{\tau}_{\mu}^{q,m}(E)\leq\max\Big{(}m(1-q),\overline{\tau}_{\mu}^{q}(E)\Big{)}.$

The following results present alternative expressions of the $L^{q}$-spectrum
in terms of the convolutions as well as general relations between the
$L^{q}$-spectrum of a measure and that of its orthogonal projections.

###### Theorem 4.1

Let $E$ be a compact subset of $\operatorname{supp}\mu$. Then, we have

  1. 1.

for all $q>1$ and $V\in G_{n,m},$

$\underline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E))\geq\underline{\tau}_{\mu}^{q,m}(E)\quad\text{and}\quad\overline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E))\geq\overline{\tau}_{\mu}^{q,m}(E).$

  2. 2.

For all $1<q\leq 2$ and $\gamma_{n,m}$-almost every $V\in G_{n,m},$

$\overline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E))=\overline{\tau}_{\mu}^{q,m}(E)=\max\Big{(}m(1-q),\overline{\tau}_{\mu}^{q}(E)\Big{)}$

and

$\underline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E))=\underline{\tau}_{\mu}^{q,m}(E).$

  3. 3.

For all $q>2$ and $\gamma_{n,m}$-almost every $V\in G_{n,m},$

    1.

    2. (a)

If $-m\leq\overline{\tau}_{\mu}^{q}(E)$ then
$\overline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E))=\overline{\tau}_{\mu}^{q,m}(E)=\overline{\tau}_{\mu}^{q}(E).$

    3. (b)

$\underline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E))=\max\Big{(}m(1-q),\underline{\tau}_{\mu}^{q,m}(E)\Big{)}.$

###### Remark 4.2

The assertion (2) is essentially a restatement of the main result of Hunt et
al. in [17] and Falconer et al. in [13, Theorem 3.9]. The assertion (3)
extends the result of Hunt and Kaloshin (of Falconer and O’Neil) to the case
$q>2$ untreated in their work.

Proof. The first and second parts follows from Proposition 4.3 and the
following lemma which is a consequence of Lemma 4.5.

###### Lemma 4.6

Let $E$ be a compact subset of $\operatorname{supp}\mu$. Then, we have

  1. 1.

for all $q>1$ and $V\in G_{n,m},$

$\displaystyle\limsup_{r\longrightarrow 0}\frac{1}{-\log
r}\left[\log\left(\displaystyle\frac{\displaystyle\int_{E}\Big{(}\mu\ast\phi_{r/3}^{n}(x)\Big{)}^{q-1}d\mu(x)}{\displaystyle\int_{\pi_{V}(E)}\Big{(}\mu_{V}(B(x_{V},r/3))\Big{)}^{q-1}d\mu_{V}(x_{V})}\right)\right]\leq
0,$

  2. 2.

for $1<q\leq 2$ and $\gamma_{n,m}$-almost every $V\in G_{n,m},$

$\displaystyle\lim_{r\rightarrow 0}\frac{1}{-\log
r}\left[\log\left(\displaystyle\frac{\displaystyle\int_{E}\Big{(}\mu\ast\phi_{r/3}^{n}(x)\Big{)}^{q-1}d\mu(x)}{\displaystyle\int_{\pi_{V}(E)}\Big{(}\mu_{V}(B(x_{V},r/3))\Big{)}^{q-1}d\mu_{V}(x_{V})}\right)\right]=0.$

See [1, Theorem 2.1], [6, Theorem 4.1] and [31] for the key ideas needed to
prove the third part of Theorem 4.1.

###  4.2  Proof of Theorem 3.2

Let us prove our main theorem. Let $q>1$.

  1. 1.

If $s>\mathfrak{T}_{\mu}^{q,m}(E)$ we may cover $E$ by a countable collection
of sets $E_{i}$, which we may take to be compact, such that
$\overline{\tau}_{\mu}^{q,m}(E_{i})<s$. By using Theorem 4.1 (2.), we have
$\overline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E_{i}))\leq s$ for
$\gamma_{n,m}$-almost every $V\in G_{n,m}.$ Proposition 4.1 implies that
${B}_{\mu_{V}}^{q}(\pi_{V}(E))\leq s$ for $\gamma_{n,m}$-almost every $V\in
G_{n,m}$ and so,
${B}_{\mu_{V}}^{q}(\pi_{V}(E))\leq\mathfrak{T}_{\mu}^{q,m}(E)$ for
$\gamma_{n,m}$-almost every $V\in G_{n,m}.$

Now, if $s<\mathfrak{T}_{\mu}^{q,m}(E)$. Fix $V\in G_{n,m}$ and let
$(\widetilde{E}_{i})_{i}$ be a cover of the compact set $\pi_{V}(E)$ by a
countable collection of compact sets. Put for each $i$,
$E_{i}=E\cap\pi_{V}^{-1}(\widetilde{E}_{i}),$ then
$\sup_{i}\overline{\tau}_{\mu}^{q,m}(E_{i})>s$. By using Theorem 4.1 (1.), we
have $\sup_{i}\overline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E_{i}))\geq s$ and
$\sup_{i}\overline{\tau}_{\mu_{V}}^{q}(\widetilde{E}_{i})\geq s$, this implies
that ${B}_{\mu_{V}}^{q}(\pi_{V}(E))\geq s$. Therefore, we obtain
${B}_{\mu_{V}}^{q}(\pi_{V}(E))\geq\mathfrak{T}_{\mu}^{q,m}(E)$.

Thus, the part concerning the equality between
$\max\Big{(}m(1-q),B_{\mu}^{q}(E)\Big{)}$ and $\mathfrak{T}_{\mu}^{q,m}(E)$ is
a consequence of Proposition 4.3.

  2. 2.

Let $({E}_{i})_{i}$ be a cover of $E$ by a countable collection of compact
sets is such that
$\overline{\tau}_{\mu}^{q}(E_{i})=\Lambda_{\mu}^{q}(E_{i})\geq-m$ for all $i$.
Then, by using Lemma 4.1 and Proposition 4.3, we have
$\mathfrak{T}_{\mu}^{q,m}(E)=B_{\mu}^{q}(E)$.

Now, if $s>\mathfrak{T}_{\mu}^{q,m}(E)$ we may cover $E$ by a countable
collection of sets $E_{i}$, which we may take to be compact, such that
$\overline{\tau}_{\mu}^{q,m}(E_{i})<s$. By using Theorem 4.1 (2.) and since
$-m\leq\overline{\tau}_{\mu}^{q}(E_{i})=\Lambda_{\mu}^{q}(E_{i})$ for all $i$,
we have $\overline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E_{i}))\leq s$ for
$\gamma_{n,m}$-almost every $V\in G_{n,m}.$ Proposition 4.1 implies that
${B}_{\mu_{V}}^{q}(\pi_{V}(E))\leq s$ for $\gamma_{n,m}$-almost every $V\in
G_{n,m}$ and so,
${B}_{\mu_{V}}^{q}(\pi_{V}(E))\leq\mathfrak{T}_{\mu}^{q,m}(E)$ for
$\gamma_{n,m}$-almost every $V\in G_{n,m}.$ In similar way, we prove
${B}_{\mu_{V}}^{q}(\pi_{V}(E))\geq\mathfrak{T}_{\mu}^{q,m}(E)$ for all $V\in
G_{n,m}.$

We can improve substantially the O’Neil’s result [28, Corollary 5.12] in the
following example:

###### Example 4.1

Fix $0<m\leq n$ and let $\mu$ be a self-similar measure on $\mathbb{R}^{n}$
with support equal to $K$ such that $\dim_{P}(K)=s\leq m$. Let $q\geq 0$ and
$({E}_{i})_{i}$ be a cover of $E$ by a countable collection of compact sets is
such that $\Lambda_{\mu}^{q}(E_{i})\geq-m$ for all $i$. By using Corollary 3.1
and Corollary 5.12 in [28], we have for $\gamma_{n,m}$-almost every $V\in
G_{n,m}$

$B_{{\mu}_{V}}({q})=b_{{\mu}_{V}}({q})=b_{{\mu}}({{q}})=B_{{\mu}}({q}).$

##  5 Application

When $\mu$ obeys the multifractal formalism over some interval, we are
interested in knowing whether or not this property is preserved after
orthogonal projections on $\gamma_{n,m}$-almost every linear $m$-dimensional
subspaces.

This section is devoted to study the behavior of projections of measures
obeying to the multifractal formalism. More precisely, we prove that for $q>1$
if the multifractal formalism holds for $\mu$ at
$\alpha=-B^{\prime}_{\mu}(q)$, it holds for $\mu_{V}$ for
$\gamma_{n,m}$-almost every $V\in G_{n,m}$. Before detailing our results let
us recall the multifractal formalism introduced by O’Neil. For $\alpha\geq 0$,
let

${E}_{\mu}(\alpha)=\left\\{x\in\operatorname{supp}\mu;\;\;\lim_{r\to
0}\frac{\log\Big{(}\mu B(x,3r)\Big{)}}{\log r}=\alpha\right\\}.$

We mention that in the last decade there has been a great interest for the
multifractal analysis and positive results have been written in various
situations (see for example [4, 5, 26, 27]).

The function $B_{\mu}(q)$ is related to the multifractal spectrum of the
measure $\mu$. More precisely,
$f^{*}(\alpha)=\displaystyle\inf_{\beta}\big{(}\alpha\beta+f(\beta)\big{)}$
denotes the Legendre transform of the function $f,$ it has been proved in [4,
5, 26, 27] a lower and upper bound estimate of the singularity spectrum using
the Legendre transform of the function $B_{\mu}(q)$. The following theorem is
a consequence of the multifractal formalism developed in [5].

###### Theorem 5.1

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

  1. 1.

$\mathscr{H}_{\mu}^{q,B_{\mu}(q)}(\operatorname{supp}\mu)>0,$

  2. 2.

$B_{\mu}$ is differentiable at $q$.

Then,

$\displaystyle\dim_{H}E_{\mu}\big{(}-B^{\prime}_{\mu}(q)\big{)}=\dim_{P}E_{\mu}\big{(}-B^{\prime}_{\mu}(q)\big{)}=B^{*}_{\mu}\big{(}-B^{\prime}_{\mu}(q)\big{)}=b^{*}_{\mu}\big{(}-B^{\prime}_{\mu}(q)\big{)}.$

Here $\dim_{H}$ and $\dim_{P}$ denote, respectively, the Hausdorff and the
packing dimension, see [26] for precise definitions of this.

The following proposition is established in [28].

###### Proposition 5.1

Let $\mu$ be a compactly supported Borel probability measure on
$\mathbb{R}^{n}$. For $q\geq 1$ and all $V\in G_{n,m}$, we have

$b_{\mu_{V}}(q)\geq\max\Big{(}m(1-q),b_{\mu}(q)\Big{)}.$

In the following, we study the validity of the multifractal formalism under
projection. More specifically, we obtain general result for the multifractal
analysis of the orthogonal projections on $m$-dimensional linear subspaces of
measure $\mu$ satisfying the multifractal formalism.

###### Theorem 5.2

Let $\mu$ be a compactly supported Borel probability measure on
$\mathbb{R}^{n}$ and $q>1$. Suppose that

$(\mathsf{H}_{1})$
$\mathscr{H}_{\mu}^{q,B_{\mu}(q)}(\operatorname{supp}\mu)>0,$

$(\mathsf{H}_{2})$ $B_{\mu}$ is differentiable at $q$,

$(\mathsf{H}_{3})$ $({E}_{i})_{i}$ be a cover of $\operatorname{supp}\mu$ by a
countable collection of compact sets is such that
$b_{\mu}^{q}(E_{i}\cap\operatorname{supp}\mu)\geq\max(-m,m(1-q))$ for all $i$.

Then, for $\gamma_{n,m}$-almost every $V\in G_{n,m}$,

$\displaystyle\dim_{P}{E}_{\mu_{V}}\big{(}-B^{\prime}_{\mu}(q)\big{)}$
$\displaystyle=$
$\displaystyle\dim_{H}{E}_{\mu_{V}}\big{(}-B^{\prime}_{\mu}(q)\big{)}=\dim_{H}E_{\mu}\big{(}-B^{\prime}_{\mu}(q)\big{)}$
$\displaystyle=$
$\displaystyle\dim_{P}E_{\mu}\big{(}-B^{\prime}_{\mu}(q)\big{)}=B^{*}_{\mu}\big{(}-B^{\prime}_{\mu}(q)\big{)}=b^{*}_{\mu}\big{(}-B^{\prime}_{\mu}(q)\big{)}.$

###### Remark 5.1

The results of Theorem 5.2 hold if we replace the condition

$\mathscr{H}_{\mu}^{q,B_{\mu}(q)}(\operatorname{supp}\mu)>0$

by the existence of a nontrivial (Frostman) measure $\nu_{q}$ satisfying

$\nu_{q}(B(x,r))\leq\mu(B(x,3r))^{q}~{}r^{B_{\mu}(q)}$

where $x\in\operatorname{supp}\mu$ and $0<r<1$.
For more details, the reader can see [28, Theorem 5.1].

Proof. By using Corollary 3.1, Proposition 5.1, $(\mathsf{H}_{1})$ and
$(\mathsf{H}_{3})$ we have, for $\gamma_{n,m}$-almost every $V\in G_{n,m}$,

$b_{\mu}(q)=b_{\mu_{V}}(q)=B_{\mu_{V}}(q)=B_{\mu}(q).$ (5.1)

$(\mathsf{H}_{1})$, (5.1) and the proof of Lemma 3.2 in [28] ensure that,
there exists a positive constant $c$ such that

$0<\mathscr{H}_{\mu}^{q,B_{\mu}(q)}(\operatorname{supp}\mu)\leq
c~{}\mathscr{H}_{\mu_{V}}^{q,B_{\mu_{V}}(q)}(\operatorname{supp}\mu_{V}),$ for
$\gamma_{n,m}$-almost every $V\in G_{n,m}$.

So, the hypothesis $(\mathsf{H}_{2})$, Theorem 5.1 and the equalities (5.1)
imply that

$\displaystyle\dim_{H}{E}_{\mu_{V}}\big{(}-B^{\prime}_{\mu}(q)\big{)}\geq-
qB^{\prime}_{\mu}(q)+B_{\mu}(q),\quad\text{for}\;\;\gamma_{n,m}\text{-almost
every}\;\;V\in G_{n,m}.$ (5.2)

Hence, the assumption (5.1) give that

$\displaystyle\dim_{P}{E}_{\mu_{V}}\big{(}-B^{\prime}_{\mu}(q)\big{)}$
$\displaystyle\leq$ $\displaystyle
B^{*}_{\mu_{V}}\big{(}-B^{\prime}_{\mu}(q)\big{)}$ (5.3) $\displaystyle=$
$\displaystyle B^{*}_{\mu}\big{(}-B^{\prime}_{\mu}(q)\big{)},$

for $\gamma_{n,m}$-almost every $V\in G_{n,m}$. Thus, the result is a
consequence from (5.2) and (5.3).

## Acknowledgments

The author is greatly indebted to the referee for his/her carefully reading
the first submitted version of this paper and giving elaborate comments and
valuable suggestions on revision so that the presentation can be greatly
improved.

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Bilel SELMI
Analysis, Probability and Fractals Laboratory LR18ES17
Faculty of Sciences of Monastir
Department of Mathematics
University of Monastir
5000-Monastir
Tunisia

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