# Structurally damped $\sigma-$evolution equations with power-law memory

N. Faustino  Department of Mathematics, University of Aveiro, Campus
Universitário de Santiago, 3810-193 Aveiro, Portugal<EMAIL_ADDRESS>Center for
R&D in Mathematics and Applications (CIDMA), Department of Mathematics,
University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro,
Portugal<EMAIL_ADDRESS>and  J. Marques  Faculty of Economics and
CeBER, University of Coimbra, Av. Dias da Silva, 165, 3004-512 Coimbra,
Portugal<EMAIL_ADDRESS>
###### Abstract.

We consider an integro-differential counterpart of the $\sigma-$evolution
equation of the type

$\partial_{t}^{2}u(t,x)+\mu(-\Delta)^{\frac{\sigma}{2}}\partial_{t}u(t,x)+(-\Delta)^{\sigma}u(t,x)=f(t,x),$

with $\sigma>0$ and $\mu>0$, that encodes memory of power-law type. To do so,
we replace the time derivatives $\partial_{t}$ and $\partial_{t}^{2}$ by the
so-called Caputo-Djrbashian derivatives $\partial_{t}^{\gamma}$ of order
$\gamma=\alpha$ and $\gamma=2\alpha$, respectively, and the inhomogeneous term
$f(t,x)$ by the Riemann-Liouville integral $I^{\beta-2\alpha}_{0^{+}}f(t,x)$,
whereby $0<\alpha\leq 1$ and $2\alpha\leq\beta<2\alpha+1$. For the solution
representation of the underlying Cauchy problems on the space-time
$[0,T]\times\mathbb{R}^{n}$ we then consider a wide class of pseudo-
differential operators
$\displaystyle(-\Delta)^{\frac{\eta}{2}}E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)$,
endowed by the fractional Laplacian $-(-\Delta)^{\frac{\sigma}{2}}$ and the
two-parameter Mittag-Leffler functions $E_{\alpha,\beta}$. On our approach we
are also able to provide dispersive and Strichartz estimates for the solutions
with the aid of decay properties of $E_{\alpha,\beta}(-z)$ ($z\in\mathbb{C}$)
and the boundedness properties of the Hankel transform.

###### Key words and phrases:

Fractional differential equations, Mittag-Leffler functions, structural
damping

###### 2020 Mathematics Subject Classification:

26A33, 33C10, 33E12, 35R11, 42B37, 44A20

N. Faustino was supported by The Center for Research and Development in
Mathematics and Applications (CIDMA) through the Portuguese Foundation for
Science and Technology (FCT), references UIDB/04106/2020 and UIDP/04106/2020.

J. Marques was supported by Centre for Business and Economics Research (CeBER)
through the Portuguese Foundation for Science and Technology (FCT), reference
UIDB/05037/2020

###### Contents

  1. 1 Introduction
  2. 2 Discussion
    1. 2.1 Model Problems
    2. 2.2 Solution Representation
    3. 2.3 Dispersive and Strichartz estimates
  3. 3 Organization of the Paper
  4. 4 Preliminaries
    1. 4.1 Laplace Transform and Mittag-Leffler functions
    2. 4.2 Fourier analysis, function spaces and associated operators
  5. 5 Dispersive Estimates
    1. 5.1 Proof Strategy
    2. 5.2 $L^{p}-L^{q}$ and $L^{p}-\dot{W}^{\sigma,q}$ decay estimates
    3. 5.3 $\dot{W}^{\gamma,p}-L^{q}$ and $\dot{W}^{\gamma,p}-\dot{W}^{\sigma,q}$ decay estimates
    4. 5.4 An amalgamation of dispersive estimates
  6. 6 Main Results
    1. 6.1 Notations
    2. 6.2 Solution of the Cauchy Problem 2.1
    3. 6.3 Solution of the Cauchy Problem 2.2
    4. 6.4 Strichartz estimates
  7. 7 Conclusions
    1. 7.1 Summary of results
    2. 7.2 Further remarks and open problems
  8. 8 Acknowledgments
  9. A Proof of Lemma 5.2
  10. B Proof of Lemma 5.3
  11. C Proof of Corollary 5.13

##  1\. Introduction

Let us begin with a few facts concerning the theory of structurally damped
equations. After Russell’s fundamental paper [42] on mathematical models of
structural mechanics describing elastic beams much more attention has been
paid rightly to the strongly damped plate/wave problems and to its further
applications on the crossroads of control theory (cf. [3]), dynamical systems
(cf. [47]) and transmition problems (cf. [38]). The current trends on
struturally damped equations, exhibited on the contribution of Pham et al
[40], and more recently on D’Abbicco-Ebert’s series of papers [12, 14, 15],
paved the way for considering space-fractional model problems such as the
inhomogeneous $\sigma-$evolution equation

(1.1)
$\partial_{t}^{2}(t,x)+\mu(-\Delta)^{\frac{\sigma}{2}}\partial_{t}u(t,x)+(-\Delta)^{\sigma}u(t,x)=f(t,x).$

We refer the reader to the works of Carvalho et al (cf. [6]) and Denk-
Schnaubelt (cf. [18]) just to mention a few papers on analysis of PDEs towards
struturally damped plate equations (case of $\sigma=2$) and damped wave
equations (case of $\sigma=1$). Also, we refer the seminal work of Karch (cf.
[26]) for a general account on the asymptotic behaviour of structurally damped
models.

We notice that the $\sigma-$evolution equation (1.1) describes the scenario
where the higher frequencies exhibit a strongly damped behavior in comparison
with the lower frequencies. Here, the fractional differential operator
$(-\Delta)^{\frac{\gamma}{2}}$ (that will be defined later on Subsection 4.2)
is used to model the damping term $(\gamma=\sigma)$ and the elastic term as
well $(\gamma=2\sigma)$. Noteworthy, in the view of Carvalho et al (cf. [6])
and Chill-Srivastava (cf. [7]), the model problem (1.1) brings also the
possibility to study the well-posedness of problems carrying $L^{p}$ data
($1<p<\infty$), in stark constrast with the weaker damping case so that
$\mu(-\Delta)^{\frac{\sigma}{2}}\partial_{t}u(t,x)$ $(\mu>0)$ resembles to a
regularization of the weak damping term $\mu\partial_{t}u(t,x)$, in the limit
$\sigma\rightarrow 0^{+}$.

Our aim here is to investigate a space-time-fractional counterpart of (1.1)
that encodes memory effects, ubiquitous in natural phenomena (cf. [20, 45]).
In our setting it will be replaced the time derivatives $\partial_{t}$ resp.
$\partial_{t}^{2}$ by the so-called Caputo-Djrbashian derivatives
$\partial^{\gamma}_{t}$ of order $\gamma$, defined as

(1.5)
$\displaystyle\partial^{\gamma}_{t}u(t,x):=\left\\{\begin{array}[]{lll}\displaystyle\int_{0}^{t}\dfrac{(t-\tau)^{m-\gamma-1}}{\Gamma(m-\gamma)}\frac{\partial^{m}u(\tau,x)}{\partial\tau^{m}}d\tau&,m-1<\gamma<m\\\
\\\ \displaystyle\frac{\partial^{m}u(t,x)}{\partial
t^{m}}&,\gamma=m,\end{array}\right.$

where $\Gamma(\cdot)$ denotes the Euler’s Gamma function (see eq. (4.7) of
Subsection 4.1) and $m=\lfloor\gamma\rfloor+1$ ($\lfloor\gamma\rfloor$ stands
for the integer part of $\gamma$).

For the time-fractional counterpart of the inhomogeneous term $f(t,x)$, it
will be adopted the Riemann-Liouville integral

(1.9) $\displaystyle
I^{\gamma}_{0^{+}}f(t,x):=\left\\{\begin{array}[]{lll}\displaystyle\int_{0}^{t}\dfrac{(t-\tau)^{\gamma-1}}{\Gamma(\gamma)}f(\tau,x)d\tau&,0<\gamma<1\\\
\\\ \displaystyle f(t,x)&,\gamma=0.\end{array}\right.$

We observe that for values of $\gamma\not\in{\mathbb{N}}_{0}$ resp.
$\gamma\neq 0$, the eqs. (1.5) and (1.9) always define operators with power-
law memory. The underlying memory function, given by

(1.10) $\displaystyle\displaystyle
g_{\nu}(t):=\dfrac{t^{\nu-1}}{\Gamma(\nu)}~{}~{}(0<\nu<1),$

is a probability density function over the interval $(0,\infty)$, converging
to the delta function $\delta(t)$ in the limit $\nu\rightarrow 0^{+}$.

Some of the physical reasons for the choice of the Caputo-Djrbashian
derivative (1.5) instead of the Riemann-Liouville derivative
${~{}}^{RL}\partial_{t}^{\gamma}:=\partial_{t}^{m}I^{m-\gamma}_{0^{+}}$ stems
on its ubiquity in the generalized Langevin equations (cf. [32]) and on
certain limiting processes (cf. [25]). Despite their usefulness on real-world
applications (cf. [1, 2]), the Caputo-Djrbashian derivative is suitable for
modelling Cauchy problems for the two amongst many reasons:

  1. (1)

they have similar properties to the time-derivatives;

  2. (2)

they remove the singularities at $t=0$.

We refer to the book of Samko et al. (cf. [43]) for further details on the
theory of Caputo-Djrbashian derivatives and Riemann-Liouville operators and
Stinga’s preprint [45] for an abridged overview of it. For its applications on
Cauchy problems, we refer to the books of Podlubny (cf. [41]) and Kilbas et
al. (cf. [30]). We also refer to [28, 29, 13, 31, 16] for recent applications
on fractional diffusion equations, closely related to our approach.

##  2\. Discussion

###  2.1. Model Problems

The main focus in this paper are the following two Cauchy problems, carrying
the initial condition(s) $u_{0}(x)$ [and $u_{1}(x)$] and the inhomogeneous
term $I^{\beta-2\alpha}_{0^{+}}f(t,x)$, fulfilling certain regularity
conditions to be presented a posteriori. Here and elsewhere
$I^{\beta-2\alpha}_{0^{+}}$ stands for the Riemann-Liouville integral, defined
through eq. (1.9) for $\gamma=\beta-2\alpha$.

######  Cauchy Problem 2.1.

Given $0<\alpha\leq\frac{1}{2}$, $2\alpha\leq\beta<2\alpha+1$, $\sigma>0$ and
$\mu>0$, the function $u:[0,T]\times{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$
is a (weak) solution of

$\begin{cases}\displaystyle\partial^{2\alpha}_{t}u(t,x)+\mu(-\Delta)^{\frac{\sigma}{2}}\partial^{\alpha}_{t}u(t,x)+(-\Delta)^{\sigma}u(t,x)=I^{\beta-2\alpha}_{0^{+}}f(t,x)&,\mbox{in}~{}(0,T]\times\mathbb{R}^{n}\\\
\ \\\ u(0,x)=u_{0}(x)~{}&,\mbox{in}~{}\mathbb{R}^{n}.\end{cases}$

######  Cauchy Problem 2.2.

Given $\frac{1}{2}<\alpha\leq 1$, $2\alpha\leq\beta<2\alpha+1$, $\sigma>0$ and
$\mu>0$, the function $u:[0,T]\times{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$
is a (weak) solution of

$\begin{cases}\displaystyle\partial^{2\alpha}_{t}u(t,x)+\mu(-\Delta)^{\frac{\sigma}{2}}\partial^{\alpha}_{t}u(t,x)+(-\Delta)^{\sigma}u(t,x)=I^{\beta-2\alpha}_{0^{+}}f(t,x)&,\mbox{in}~{}(0,T]\times\mathbb{R}^{n}\\\
\ \\\ u(0,x)=u_{0}(x)&,\mbox{in}~{}\mathbb{R}^{n}\\\ \ \\\
\partial_{t}u(0,x)=u_{1}(x)&,\mbox{in}~{}\mathbb{R}^{n}.\end{cases}$

The programme of studying Cauchy problems similar to Cauchy Problem 2.1 &
Cauchy Problem 2.2 has been started by Fujita in the former papers [22, 23],
where it has been shown that fractional diffusion and wave equations can be
reformulated as integro-differential equations of D’Alembert type. In our
case, the proposed model problems amalgamate the following properties,
tactically investigated on several works (see e.g. [6, 18, 40, 28, 29]):

  1. (1)

The damping part has ‘half of the order’ of the leading elastic term, which is
in accordance with Russell’s formulation [42];

  2. (2)

For values of $0<\alpha<1$, the time-fractional derivatives
$\partial^{2\alpha}_{t}$ and $\partial^{\alpha}_{t}$ endow a fractional
diffusion process;

  3. (3)

In the limit $\mu\rightarrow 0^{+}$, the Cauchy Problem 2.1 & Cauchy Problem
2.2 approaches the undamped case;

  4. (4)

For $\alpha=1$ and $1<\sigma<2$, our model problem possess mixed properties of
the plate equation (case of $\sigma=2$) resp. the wave equation (case of
$\sigma=1$) with structural damping;

  5. (5)

For $\alpha=1$, the asymptotic profile of the homogeneous part of the Cauchy
Problem 2.2 changes accordingly to the size of the damping parameter $\mu>0$.

From the point of view of potential applications, our model problem brings
also the possibility to model anomalous diffusions and memory effects in a
twofold way. Namely, the power law proportional to $t^{2\alpha}$ measures if
our diffusive process is subdiffusive $(0<2\alpha<1)$ or superdiffusive
($1<2\alpha<2$). On both cases, the fractional derivatives
$\partial_{t}^{\alpha}$ and $\partial_{t}^{2\alpha}$ are described in terms of
the memory functions $g_{1-\alpha}(t)$ and $g_{2-2\alpha}(t)$, respectively
(see eq. (1.10)).

On the other hand, the choice of the parameter $\beta$, appearing on the
Riemann-Liouville integral $I^{\beta-2\alpha}_{0^{+}}f(t,x)$, reflects if our
inhomogeneous term exhibits a memoryless pattern or not: The limit case
$\beta\rightarrow(2\alpha)^{+}$ guarantees that the shape of the memory
function $g_{\beta-2\alpha}(t)$ is close to the shape of the delta function
$\delta(t)$ so that the inhomogeneous term approaches ’lack of memory’ for
values of $\beta$ close to $2\alpha$. For values of $\beta$ a bit far from
$2\alpha$, the ubiquity of memory effects is manifested by the polynomial
shape of the graph of $g_{\beta-2\alpha}(t)$. Interesting to see, the
particular choice $\beta=1+\alpha$ ($\beta-2\alpha=1-\alpha$), with
$0<\alpha<1$, lead us always to diffusive processes with memory.

Such subtle details encoded on our model may be seen as surprising and
substantial from the vast majority of papers available on the literature. Up
to our knowledge, the overlap between the memory effects of the fractional
derivatives and the inhomogeneous term are not so often considered.

###  2.2. Solution Representation

From an operational calculus perpective, the Cauchy Problem 2.1 & Cauchy
Problem 2.2 share the same features of Cauchy problems underlying to the time-
fractional telegraph equations so that one can exploit, as in [31], Orsingher-
Beghin’s approach [39] to obtain the underlying solution representations (see
also ref. [33]). Namely, the closed-form representation for the solutions of
both Cauchy problems – to be obtained in Subsection 6.2 and Subsection 6.3,
respectively – follows from the fact that Fourier-Laplace multiplier of
$\displaystyle\partial^{2\alpha}_{t}+\mu(-\Delta)^{\frac{\sigma}{2}}\partial^{\alpha}_{t}+(-\Delta)^{\sigma}$,

$(s^{\alpha},|\xi|^{\sigma})\mapsto
s^{2\alpha}+\mu|\xi|^{\sigma}s^{\alpha}+|\xi|^{2\sigma},$

is a quadratic form on the variables $u=s^{\alpha}$ and $v=|\xi|^{\sigma}$.

Then, with the aid operational properties of the generalized Mittag-Leffler
functions $E_{\alpha,\beta}^{\gamma}$ and
$E_{\alpha,\beta}=E_{\alpha,\beta}^{1}$, highlighted on Section 5 – see, for
instance, eq. (4.8) and Theorem 4.3 – we are able to recast closed
representation formulae in terms of the wide class of pseudo-differential
operators
$\displaystyle(-\Delta)^{\frac{\eta}{2}}E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)$.
From a semigroup perspective, such class provides us fractional analogues of
the resolvent operator considered e.g. on [6, 7, 18] for the case where the
damping operator is a multiple square root of the elastic operator (cf. [21,
Chapter VI]).

By the properties of the generalized Mittag-Leffler functions, it will be also
shown in Subsection 6.2 and Subsection 6.3 that the choice of
$I^{\beta-2\alpha}_{0^{+}}f(t,x)$ as inhomogeneous term turns out to be linked
with the Fourier multipliers $\displaystyle
t^{\beta-1}E_{\alpha,\beta}^{\gamma}(-\lambda|\xi|^{\sigma}t^{\alpha})$
($\gamma=1,2$). From feasible physical descriptions of relaxation processes,
such class of functions is somewhat expected, as highlighted on the seminal
paper [19]; for an overview, we refer to [24, Subsection 8.1.].

As will be noticed by the reader, the results enclosed on this paper can
certainly be rewritten in terms of the language of fractional resolvent
families (see e.g. [33] and the references therein) in a way that the Banach
space framework, considered e.g. in the papers [7, 6, 18] and on the book [21]
can be faithfully generalized. But we have decided to use mostly pseudo-
differential calculus, rather than operator theory, to keep the approach as
self-contained as possible.

###  2.3. Dispersive and Strichartz estimates

Apart from the solution representation of these Cauchy problems being
interesting from the point of view of spectral analysis and stochastic
processes, we want to push forward the investigation of dispersive and
Strichartz estimates, with the aim of measuring the size and the decay of the
solutions of Cauchy Problem 2.1 & Cauchy Problem 2.2. The study of such type
of estimates have become a cornerstone tool for many years now, mainly due to
the Keel-Tao’s breakthrough contribution to the topic (cf. [27]); see also
[46, Chapter 2.] for an overview.

The starting point will be the decay properties of the convolution kernel of
$\displaystyle(-\Delta)^{\frac{\eta}{2}}E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)$,
represented in terms of the Hankel transform of the Fourier multiplier
$\displaystyle|\xi|^{\eta}E_{\alpha,\beta}\left(-\lambda|\xi|^{\sigma}t^{\alpha}\right)$.
To do so, we will exploit the [optimal] decay estimates for the Mittag-Leffler
functions $E_{\alpha,\beta}(-z)$, available on the book [41] (case of
$z\in\mathbb{C}$) and on the papers [44, 4] (case of $z\geq 0$) to our
setting.

Concerning the dispersive estimates, the framework enclosed in Section 5 is
comparable to Pham et al and D’Abbicco-Ebert approaches (cf. [40, 14]), who
have investigated $L^{p}-L^{q}$ estimates for Cauchy problems similar to
Cauchy Problem 2.2. Regarding the Strichartz estimates enclosed in the end of
Section 6, we have followed up some of the ideas addressed by Tao on the
monograph [46], properly adapted to our model problems. Mainly, a version of
the so-called Christ-Kiselev lemma (cf. [46, Lemma 2.4]) is always required to
ensure the boundedness of the inhomogeneous part of the solution
representation of both Cauchy problems (see also [27, THEOREM 1.2.]).

##  3\. Organization of the Paper

We organize the remaining parts of this paper as follows:

  * •

In Section 4 we collect some results to be employed in the subsequent
sections. Namely, some of the properties involving the Laplace transform and
the generalized Mittag-Leffler functions $E_{\alpha,\beta}$ and
$E_{\alpha,\beta}^{\gamma}$ are recalled in Subsection 4.1. And in Subsection
4.2 we introduce some relevant definitions and properties to define properly
the space-fractional differential operator $(-\Delta)^{\frac{\gamma}{2}}$ and
the associated function spaces. Special emphasize will be given to the
homogeneous Sobolev spaces $\dot{W}^{\gamma,p}$.

  * •

In Section 5 we deduce dispersive estimates for the wide class of pseudo-
differential operators
$(-\Delta)^{\frac{\eta}{2}}E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)$.
The results proved in Subsection 5.2, Subsection 5.3 and Subsection 5.4
describe the decay properties of it. Here, we make use of the reformulation of
the Fourier transform for radially symmetric functions in terms of the Hankel
transform to obtain, in case of $\lambda\geq 0$, a sharp control of decay of
the underlying convolution kernel, after application of Young’s inequality.

  * •

In Section 6 we provide the main results of this paper. More precisely, we
obtain in Subsection 6.2 and Subsection 6.3 closed-form representations for
the Cauchy Problem 2.1 & Cauchy Problem 2.2, respectively. And in Subsection
6.4 we turn to the study of Strichartz estimates on the mixed-normed Lebesgue
spaces $L_{t}^{s}L_{x}^{q}([0,T]\times{\mathbb{R}}^{n})$ for both Cauchy
problems, from the dispersive estimates obtained in Section 5.

  * •

Finally, in Section 7 we address the main conclusions of our approach. Some
further remarks and open problems will also be highlighted.

##  4\. Preliminaries

###  4.1. Laplace Transform and Mittag-Leffler functions

For a real-valued function $g:[0,\infty)\rightarrow\mathbb{R}$ satisfying
$\displaystyle\sup_{t\in[0,\infty)}e^{-\omega t}|g(t)|<\infty$, the Laplace
transform of $g$ is defined as

(4.1) $\displaystyle
G(s):=\mathcal{L}[g(t)](s)=\int_{0}^{\infty}e^{-st}g(t)\,\,dt$
$\displaystyle(\Re(s)>\omega).$

Here, we notice that the function $G$ (the Laplace image of $g$) is analytic
on the right-half plane
$\displaystyle\left\\{s\in{\mathbb{C}}~{}:~{}\Re(s)>\omega\right\\},$ whereby
$\omega$ is chosen as the infimum of the values of $s$ for which the right-
hand side of (4.1) is convergent. Further details may be found e.g. in [30,
Subsection 1.2].

For the inverse of the Laplace transform, defined as
$g(t)=\mathcal{L}^{-1}[G(s)](t)$, the integration is performed along the strip
$[c-i\infty,c+i\infty]$ ($c:=\Re(s)>\omega$), and whence, the resulting
formula is independent of the choice of $c$.

Associated to the Laplace transform is the Laplace convolution formula (cf.
[30, (1.4.10), (1.4.12) of p. 19] and [41, (2.237), (2.238) of pp. 103–104]),
defined for two functions $g$ and $h$ by

(4.2)
$\displaystyle\mathcal{L}\left[\int_{0}^{t}h(t-\tau)g(\tau)d\tau\right](s)=\mathcal{L}[h](s)\cdot\mathcal{L}[g](s).$

Of foremost importance are also the following operational identities,
involving the Caputo-Djrbashian derivative (1.5) and the Riemann-Liouville
integral (1.9) respectively. Namely, the property

(4.3)
$\displaystyle\mathcal{L}[\partial_{t}^{\gamma}g(t)]=s^{\gamma}\mathcal{L}[g](s)-\sum_{k=0}^{m-1}s^{\gamma-k-1}\partial_{t}^{k}g(0)$
$\displaystyle\left(~{}m-1<\gamma\leq m~{}\right)$

is fulfilled whenever $\mathcal{L}[g(t)](s)$,
$\mathcal{L}[\partial_{t}^{m}g(t)](s)$ exist and

$\displaystyle\displaystyle\lim_{t\rightarrow\infty}\left(\partial_{t}\right)^{k}g(t)=0$
$\displaystyle\left(~{}k=0,1,\ldots,m-1~{}\right),$

holds for every $g\in C^{m}(0,\infty)$ such that
$\left(\partial_{t}\right)^{m}g\in L^{1}(0,b)$ for any $b>0$ (cf. [30, Lemma
2.24]). On the other hand, the identity

(4.4)
$\displaystyle\mathcal{L}[I_{0^{+}}^{\gamma}g(t)]=s^{-\gamma}\mathcal{L}[g](s)$
$\displaystyle(\Re(s)>\omega)$

is always satisfied in case of $g\in L^{1}(0,b)$, for any $b>0$ (cf. [30,
Lemma 2.14]).

Next, we turn our attention to some of the special functions to be considered
on the sequel. We introduce the two-parameter/three-parameter Mittag-Leffler
functions $E_{\alpha,\beta}$ resp. $E_{\alpha,\beta}^{\gamma}$, as power
series expansions of the form

(4.5) $\displaystyle
E_{\alpha,\beta}(z)=\displaystyle\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\alpha
k+\beta)}$ $\displaystyle(\Re(\alpha)>0),$ (4.6) $\displaystyle
E_{\alpha,\beta}^{\gamma}(z)=\displaystyle\sum_{k=0}^{\infty}\frac{(\gamma)_{k}}{k!}\frac{z^{k}}{\Gamma(\alpha
k+\beta)}$ $\displaystyle(\Re(\alpha)>0),$

where $\Gamma(\cdot)$ stands for the Euler’s Gamma function

(4.7) $\displaystyle\Gamma(z)=\int_{0}^{\infty}e^{-t}t^{z-1}dt$
$\displaystyle(\Re(z)>0),$

and $\displaystyle(\gamma)_{k}=\frac{\Gamma(\gamma+k)}{\Gamma(\gamma)}$
($\Re(\gamma)>-k$) for the Pochhammer symbol.

The wide class of Mittag-Leffler functions, defined viz (4.5) and (4.6),
permits us to represent several transcendental and special functions (cf. [30,
Subsection 1.8 & Subsection 1.9] and [24, Chapter 3 -Chapter 5]), such as:

  * (i)

the exponential function

$e^{z}=E_{1,1}(z);$

  * (ii)

the error function

$\displaystyle\mbox{erf}(z):=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}dt=1-e^{-z^{2}}E_{\frac{1}{2},1}(-z);$

  * (iii)

the hyperbolic functions

$\displaystyle\cosh(z)=E_{2,1}(z^{2})$ and
$\displaystyle\sinh(z)=zE_{2,2}(z^{2});$

  * (iv)

the complex exponential function

$e^{iz}:=\cos(z)+i\sin(z)=E_{2,1}(-z^{2})+izE_{2,2}(-z^{2});$

  * (v)

the Kummer confluent hypergeometric function

$\varPhi(\gamma;\beta;z):={~{}}_{1}F_{1}(\gamma;\beta,z)=\Gamma(\beta)E_{1,\beta}^{\gamma}(z);$

  * (vi)

the generalized hypergeometric function

$\displaystyle{~{}}_{1}F_{m}\left(\gamma;\frac{\beta}{m},\frac{\beta+1}{m},\ldots,\frac{\beta+m-1}{m};\frac{z^{m}}{m^{m}}\right)=\Gamma(\beta)E_{m,\beta}^{\gamma}(z)$
$\displaystyle(m\in{\mathbb{N}}).$

Here, we would like to stress that $E_{\alpha,\beta}^{\gamma}$ corresponds to
a faithful generalization of $E_{\alpha,\beta}$. Indeed, from the Pochhammer
identity $(1)_{k}=k!$ it immediately follows that $E_{\alpha,\beta}^{1}$
($\gamma=1$) coincides with $E_{\alpha,\beta}$. For these functions, we have
the following Laplace transform identity (cf. [24, eq. (5.1.26) of p. 102] &
[30, eq. (1.9.13) of p. 47])

(4.8) $\displaystyle\mathcal{L}[t^{\beta-1}E_{\alpha,\beta}^{\gamma}(\lambda
t^{\alpha})](s)=\frac{s^{\alpha\gamma-\beta}}{(s^{\alpha}-\lambda)^{\gamma}}$
$\displaystyle(\Re(s)>0;~{}\Re(\beta)>0;~{}|\lambda s^{-\alpha}|<1).$

In the remainder part of this subsection we list some of the properties
required for the sake of the rest of the paper. We start to point out the
following decay estimates, involving the two-parameter Mittag-Leffler
functions (4.5), that will be employed on the sequel of results to be proved
in Section 5.

######  Theorem 4.1 (cf. [41], Theorem 1.6).

If $\alpha\leq 2$, $\beta$ is an arbitrary real number, $\theta$ is such that
$\frac{\pi\alpha}{2}<\theta<\min\\{\pi,\pi\alpha\\}$ and $C$ is a real
constant, then

$\displaystyle|E_{\alpha,\beta}(z)|\leq\frac{C}{1+|z|}$
$\displaystyle(z\in\mathbb{C}~{}~{};~{}~{}\theta\leq|\arg(z)|\leq\pi).$

######  Theorem 4.2 (cf. [44], Theorem 4. & [4], Proposition 4.).

The following optimal estimates are fulfilled by $E_{\alpha,\beta}(-z)$:

$\displaystyle\dfrac{1}{1+\Gamma(1-\alpha)z}\leq
E_{\alpha,1}(-z)\leq\dfrac{1}{1+\frac{1}{\Gamma(1+\alpha)}z}~{}~{}(z\geq
0~{};~{}0<\alpha\leq 1);$
$\displaystyle\dfrac{1}{\left(1+\sqrt{\frac{\Gamma(1-\alpha)}{\Gamma(1+\alpha)}}z\right)^{2}}\leq\Gamma(\alpha)E_{\alpha,\alpha}(-z)\leq\dfrac{1}{\left(1+\sqrt{\frac{\Gamma(1+\alpha)}{\Gamma(1+2\alpha)}}z\right)^{2}}~{}~{}(z\geq
0~{};~{}0<\alpha\leq 1);$
$\displaystyle\dfrac{1}{1+\frac{\Gamma(\beta-\alpha)}{\Gamma(\beta)}z}\leq\Gamma(\beta)E_{\alpha,\beta}(-z)\leq\dfrac{1}{1+\frac{\Gamma(\beta)}{\Gamma(\beta+\alpha)}z}~{}~{}(z\geq
0;0<\alpha\leq 1;\beta>\alpha).$

We finish this subsection with the next lemma that will be applied, in Section
6, on the proof of the solution representation for both Cauchy problems,
Cauchy Problem 2.1 resp. Cauchy Problem 2.2.

######  Lemma 4.3 (cf. [24], p. 99).

Let $z\in\mathbb{C}$ be given.

  1. (i)

If $\alpha,\beta,\gamma\in{\mathbb{C}}$ are such that $\Re(\alpha)>0$,
$\Re(\beta)>0$, $\Re(\beta-\alpha)>0$, then

$\displaystyle{zE_{\alpha,\beta}^{\gamma}(z)=E_{\alpha,\beta-\alpha}^{\gamma}(z)-E_{\alpha,\beta-\alpha}^{\gamma-1}(z)}.$

  2. (ii)

If $\alpha,\beta,\gamma\in{\mathbb{C}}$ are such that $\Re(\alpha)>0$,
$\Re(\beta)>0$, $\alpha-\beta\notin{\mathbb{N}}_{0}$, then

$\displaystyle{zE_{\alpha,\beta}(z)=E_{\alpha,\beta-\alpha}(z)-\frac{1}{\Gamma(\beta-\alpha)}}.$

  3. (iii)

If $\alpha,\beta\in{\mathbb{C}}$ are such that $\Re(\alpha)>0$,
$\Re(\beta)>1$, then

$\displaystyle{\alpha
E_{\alpha,\beta}^{2}(z)=E_{\alpha,\beta-1}(z)-(1+\alpha-\beta)E_{\alpha,\beta}(z)}.$

###  4.2. Fourier analysis, function spaces and associated operators

Let us now denote by $\mathcal{S}({\mathbb{R}}^{n})$ the Schwartz space over
${\mathbb{R}}^{n}$ and by $\mathcal{S}^{\prime}({\mathbb{R}}^{n})$ the space
of tempered distributions (dual space of $\mathcal{S}({\mathbb{R}}^{n})$).
Throughout this paper we also denote by $L^{p}$ ($1\leq p\leq\infty$) the
standard Lebesgue spaces and by $\|\cdot\|_{p}$ its norm. We adopt the bracket
notation $\langle\cdot,\cdot\rangle$ as the inner product underlying to the
Hilbert space $L^{2}$. The Fourier transform of
$\varphi\in\mathcal{S}({\mathbb{R}}^{n})$ is defined as (cf. [46, Chapter A])

(4.9)
$\widehat{\varphi}(\xi):=(\mathcal{F}\varphi)(\xi)=\int_{{\mathbb{R}}^{n}}\varphi(x)e^{-ix\cdot\xi}\,\,dx$

and the corresponding inverse Fourier transform as

(4.10)
$\varphi(x):=(\mathcal{F}^{-1}\widehat{\varphi})(x)=\frac{1}{(2\pi)^{n}}\int_{{\mathbb{R}}^{n}}\widehat{\varphi}(\xi)e^{ix\cdot\xi}\,\,d\xi\,,$

where $x\cdot\xi$ denotes the standard Euclidean inner product between
$x,\xi\in{\mathbb{R}}^{n}$.

We note that the action of the isomorphism
$\mathcal{F}:\mathcal{S}({\mathbb{R}}^{n})\longrightarrow\mathcal{S}({\mathbb{R}}^{n})$
can be extended to $L^{p}-$spaces, using the fact that
$\mathcal{S}({\mathbb{R}}^{n})$ is a dense subspace of
$L^{p}({\mathbb{R}}^{n})$, for values of $1\leq p<\infty$, and to
$\mathcal{S}^{\prime}({\mathbb{R}}^{n})$ via the duality relation

$\displaystyle\langle\widehat{\varphi},\psi\rangle=\langle\varphi,\widehat{\psi}\rangle,$
$\displaystyle\varphi\in\mathcal{S}({\mathbb{R}}^{n}),$
$\displaystyle\psi\in\mathcal{S}^{\prime}({\mathbb{R}}^{n}).$

In particular, the Plancherel identity

(4.11)
$\displaystyle\displaystyle\frac{1}{{(2\pi)^{\frac{n}{2}}}}\|\widehat{\varphi}\|_{2}=\|\varphi\|_{2}$

follows from the fact that the automorphism
$\displaystyle\varphi\mapsto\frac{1}{{(2\pi)^{\frac{n}{2}}}}\widehat{\varphi}$
underlying to $\mathcal{S}({\mathbb{R}}^{n})$ induces an unitary operator over
$L^{2}({\mathbb{R}}^{n})$.

The Fourier transform $(\ref{FT})$ and the convolution product over
${\mathbb{R}}^{n}$:

(4.12)
$\displaystyle(\varphi\ast\psi)(x)=\int_{{\mathbb{R}}^{n}}\varphi(x-y)\psi(y)dy,$

intertwined by the Fourier convolution formula (cf. [35, Theorem 5.8])

(4.13)
$\displaystyle\mathcal{F}(\varphi\ast\psi)(\xi)=\widehat{\varphi}(\xi)\widehat{\psi}(\xi)$

share many interesting features of the $L^{p}-$spaces. Of special interest so
far is the Young’s inequality (cf. [35, Theorem 4.2])

(4.14)
$\displaystyle\|\psi\ast\varphi\|_{q}\leq\left(\frac{\kappa_{p}\kappa_{r^{\prime}}}{\kappa_{q}}\right)^{n}\|\psi\|_{r^{\prime}}\|\varphi\|_{p},$

that holds for every $1\leq p,q,r^{\prime}\leq\infty~{}$ such that
$\displaystyle~{}\frac{1}{p}+\frac{1}{r^{\prime}}=\frac{1}{q}+1.$ Here and
elsewhere,
$\displaystyle\kappa_{s}=\sqrt{{s^{\frac{1}{s}}}{(s^{\prime})^{-\frac{1}{s^{\prime}}}}}$
($\displaystyle\frac{1}{s}+\frac{1}{s^{\prime}}=1$) stands for the sharp
constant appearing on the right-hand side of (4.14).

Despite the Caputo-Djrbashian derivative (1.5), the fractional differential
operator $(-\Delta)^{\frac{\gamma}{2}}$ ($\gamma\geq 0$), defined viz the
spectral formula

(4.15)
$\displaystyle\mathcal{F}\left((-\Delta)^{\frac{\gamma}{2}}\varphi\right)(\xi)=|\xi|^{\gamma}\widehat{\varphi}(\xi),$

or equivalenty, as
$(-\Delta)^{\frac{\gamma}{2}}=\mathcal{F}^{-1}|\xi|^{\gamma}\mathcal{F}$, is
also required on the formulation of Cauchy Problem 2.1 & Cauchy Problem 2.2.
Underlying to $(-\Delta)^{\frac{\gamma}{2}}$ are the homogeneous Sobolev
spaces of order $\gamma\geq 0$,

$\displaystyle\dot{W}^{\gamma,p}({\mathbb{R}}^{n}):=\left\\{~{}\varphi\in\mathcal{S}^{\prime}({\mathbb{R}}^{n})~{}:~{}(-\Delta)^{\frac{\gamma}{2}}\varphi\in
L^{p}({\mathbb{R}}^{n})~{}\right\\},$

induced by the seminorm
$\displaystyle\varphi\mapsto\left\|~{}(-\Delta)^{\frac{\gamma}{2}}\varphi~{}\right\|_{p}$.

From now on, we will adopt mostly the $L^{p}-$notation when we are referring
to the homogeneous Sobolev space of order zero (0),
$\displaystyle\dot{W}^{0,p}({\mathbb{R}}^{n})={L}^{p}({\mathbb{R}}^{n})$.

Also, the so-called Riesz fractional integral operator of order $\gamma$,
defined for values of $0<\gamma<n$ by (cf. [43, Section 25])

(4.16)
$\displaystyle(-\Delta)^{-\frac{\gamma}{2}}\varphi(x)=(R_{\gamma}\ast\varphi)(x),~{}\mbox{with}$
$\displaystyle\displaystyle
R_{\gamma}(x)=2^{-\gamma}\pi^{-\frac{n}{2}}\frac{\Gamma\left(\frac{n-\gamma}{2}\right)}{\Gamma\left(\frac{\gamma}{2}\right)}\frac{1}{|x|^{n-\gamma}}$

can also be represented in terms of its Fourier multiplier. Namely, there is
(cf. [43, Theorem 25.1])

$\displaystyle\mathcal{F}\left((-\Delta)^{-\frac{\gamma}{2}}\varphi\right)(\xi)=|\xi|^{-\gamma}\widehat{\varphi}(\xi),$
for $\displaystyle 0<\gamma<n.$

On the sequel of main results to be proved in the end of Section 6, it will
also be necessary the notion of mixed-normed Lebesgue spaces
$L^{s}_{t}L^{q}_{x}$. For values of $1\leq s<\infty$, we define
$L^{s}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})$, with $0<T<\infty$, as the
Banach space with norm

(4.17)
$\displaystyle\|u\|_{L^{s}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})}=\left(\int_{0}^{T}\|u(t,\cdot)\|_{q}^{s}dt\right)^{\frac{1}{s}}$
$\displaystyle(1\leq q\leq\infty).$

For $s=\infty$, the space
$L^{\infty}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})$ is defined in terms of
the norm

(4.18)
$\displaystyle\|u\|_{L^{\infty}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})}=\mbox{ess}\sup_{t\in[0,T]}\|u(t,\cdot)\|_{q}$
$\displaystyle(1\leq q\leq\infty).$

##  5\. Dispersive Estimates

###  5.1. Proof Strategy

In this section we shall deduce estimates for pseudo-differential operators of
the type
$\displaystyle(-\Delta)^{\frac{\eta}{2}}E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)$,
represented through the pseudo-differential formula

$\displaystyle\displaystyle(-\Delta)^{\frac{\eta}{2}}E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)\varphi(x)=$
(5.1)
$\displaystyle=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}|\xi|^{\eta}E_{\alpha,\beta}(-\lambda|\xi|^{\sigma}t^{\alpha})\widehat{\varphi}(\xi)e^{ix\cdot\xi}d\xi,$

where $0<\alpha\leq 1$, $\beta>0$, $\sigma>0$, $\lambda\in\mathbb{C}$ and
$\eta\in\mathbb{R}$.

Recall that the Fourier convolution formula (4.13) together with eq. (4.12)
states that eq. (5.1) can be rewritten as

(5.2)
$\displaystyle(-\Delta)^{\frac{\eta}{2}}E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)\varphi(x)=\displaystyle\left(~{}{\bm{K}}_{\sigma,n}^{\eta}(t,\cdot|\alpha,\beta,\lambda)\ast\varphi~{}\right)(x),$

with

(5.3)
$\displaystyle{\bm{K}}_{\sigma,n}^{\eta}(t,x|\alpha,\beta,\lambda)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}|\xi|^{\eta}E_{\alpha,\beta}(-\lambda|\xi|^{\sigma}t^{\alpha})e^{ix\cdot\xi}d\xi.$

Then, for every $1\leq p,q,r^{\prime}\leq\infty$ satisfying
$\displaystyle\frac{1}{p}+\frac{1}{r^{\prime}}=\frac{1}{q}+1$, the condition
$E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)\varphi\in\dot{W}^{\eta,q}({\mathbb{R}}^{n})$
is satisfied for every $\varphi\in L^{p}({\mathbb{R}}^{n})$, whenever
${\bm{K}}_{\sigma,n}^{\eta}(t,\cdot|\alpha,\beta,\lambda)\in
L^{r^{\prime}}({\mathbb{R}}^{n})$.

On the other hand, due to the fact that
$\displaystyle\xi\mapsto|\xi|^{\eta}E_{\alpha,\beta}\left(-\lambda|\xi|^{\sigma}t^{\alpha}\right)$
is radially symmetric one can recast eq. (5.3) in terms of the Hankel
transform of order $\nu$, $\widetilde{\mathcal{H}}_{\nu}$ (cf. [17]), defined
for values of $\nu>-1$ by

$\displaystyle\widetilde{\mathcal{H}}_{\nu}\phi(\tau)=\int_{0}^{\infty}\phi(\rho)~{}(\tau\rho)^{-\nu}J_{\nu}(\tau\rho)\rho^{2\nu+1}d\rho$
$\displaystyle(\tau>0),$

where $J_{\nu}$ stands for the Bessel function of order $\nu$ (cf. [30,
Subsection 1.7]).

Namely, from the Fourier inversion formula for radially symmetric functions
(cf. [43, p. 485, Lemma 25.1])

$\displaystyle\int_{{\mathbb{R}}^{n}}\phi(|\xi|)e^{ix\cdot\xi}d\xi=\frac{(2\pi)^{\frac{n}{2}}}{|x|^{\frac{n}{2}-1}}\int_{0}^{\infty}\phi(\rho)J_{\frac{n}{2}-1}(\rho|x|)~{}\rho^{\frac{n}{2}}d\rho$

it follows immediately that eq. (5.3) simplifies to

(5.4) $\displaystyle{\bm{K}}_{\sigma,n}^{\eta}(t,x|\alpha,\beta,\lambda)$
$\displaystyle=$
$\displaystyle\displaystyle\frac{1}{(2\pi)^{\frac{n}{2}}|x|^{\frac{n}{2}-1}}\int_{0}^{\infty}\rho^{\eta}E_{\alpha,\beta}(-\lambda\rho^{\sigma}t^{\alpha})J_{\frac{n}{2}-1}(\rho|x|)~{}\rho^{\frac{n}{2}}d\rho$
$\displaystyle=$
$\displaystyle\displaystyle\frac{1}{(2\pi)^{\frac{n}{2}}}\widetilde{\mathcal{H}}_{\frac{n}{2}-1}\left[~{}|x|^{\eta}E_{\alpha,\beta}(-\lambda|x|^{\sigma}t^{\alpha})~{}\right].$

Thereby, the representation formula (5.4) shows in turn that the condition
${\bm{K}}_{\sigma,n}^{\eta}(t,\cdot|\alpha,\beta,\lambda)\in
L^{r^{\prime}}({\mathbb{R}}^{n})$ is assured by the boundedness of the Hankel
transform

$\widetilde{\mathcal{H}}_{\frac{n}{2}-1}:L^{r}((0,\infty),\rho^{n-1}d\rho)\rightarrow
L^{r^{\prime}}((0,\infty),\rho^{n-1}d\rho),$

for values of $1\leq r\leq 2\leq r^{\prime}\leq\infty$ satisfying
$\displaystyle\frac{1}{r}+\frac{1}{r^{\prime}}=1$ – also known as Fourier-
Bessel transform (see e.g. [8] and the references therein).

It is worth mentioning that the proof that
$\widetilde{\mathcal{H}}_{\frac{n}{2}-1}$ is an isometric operator on
$L^{2}((0,\infty),\rho^{n-1}d\rho)$ ($r=r^{\prime}=2$) can be easily reached
by the eq. (5.4) and the Plancherel identity (4.11), highlighted on Subsection
4.2. On the other hand, for $r=1$ and $r^{\prime}=\infty$ the set of
identities (cf. [30, eq. (1.7.4)]):

$\displaystyle J_{-\frac{1}{2}}(z)=\left(\frac{2}{\pi
z}\right)^{\frac{1}{2}}\cos(z)$ and $\displaystyle
J_{\frac{1}{2}}(z)=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sin(z),$

together with the Poisson integral representation of $J_{\nu}$ (cf. [30, eq.
(1.7.5)]):

$\displaystyle
J_{\nu}(z)=\frac{\left(\frac{z}{2}\right)^{\nu}}{\sqrt{\pi}\Gamma(\nu+1)}\int_{-1}^{1}(1-\rho^{2})^{\nu-\frac{1}{2}}\cos(z\rho)d\rho$
$\displaystyle\left(~{}\Re(\nu)>-\frac{1}{2}~{}\right),$

shows in turn that the supremum norm $\displaystyle
b_{\frac{n}{2}-1}:=\sup_{z\in(0,\infty)}\left|z^{1-\frac{n}{2}}J_{\frac{n}{2}-1}(z)\right|$
is bounded above, and hence

$\displaystyle\left|\widetilde{\mathcal{H}}_{\frac{n}{2}-1}\phi(\tau)\right|\leq
b_{\frac{n}{2}-1}\|\phi\|_{L^{1}((0,\infty),\rho^{n-1}d\rho)}.$

Moreover, with the aid of the celebrated Riesz-Thorin convexity theorem we are
able to obtain $L^{r}-L^{r^{\prime}}$estimates, for values of $1\leq r\leq
2\leq r^{\prime}\leq\infty$ satisfying
$\displaystyle\frac{1}{r}+\frac{1}{r^{\prime}}=1$, as an interpolation between
$L^{1}-L^{\infty}$ and $L^{2}-L^{2}$ estimates. Namely, for
$\displaystyle\frac{2}{r^{\prime}}=2-\frac{2}{r}$, it can be shown that the
norm of $\widetilde{\mathcal{H}}_{\frac{n}{2}-1}$ is bounded above by
$\displaystyle\left(~{}b_{\frac{n}{2}-1}~{}\right)^{1-\frac{2}{r^{\prime}}}$
(cf. [17, eq. (2.5)]).

We would like to stress that the norm
$\displaystyle\widetilde{h}_{\frac{n}{2}-1}(r,r^{\prime}):=\left\|~{}\widetilde{\mathcal{H}}_{\frac{n}{2}-1}~{}\right\|_{L^{r}\rightarrow
L^{r^{\prime}}}$ cannot be computed accurately for
$(r,r^{\prime})=(1,\infty)$. Nevertheless, using the groundbreaking theorems
of Beckner and Lieb (cf. [5, 34]), it can be computed with precision in case
where $1<r\leq 2$ as it was proved on De Carli’s paper [17] (cf. [17,
Proposition 2.1.]).

In brief, with the aid of the mapping properties of the Hankel transform we
are able to achieve wisely the underlying endpoint $L^{r^{\prime}}-$estimates
for the kernel function (5.3) from the endpoint $L^{r}-$estimates for the
Fourier multiplier
$|\xi|^{\eta}E_{\alpha,\beta}(-\lambda|\xi|^{\sigma}t^{\alpha})$ of (5.1).

$\rho^{-s}E_{\alpha,\beta}(-\lambda\rho^{\sigma}t^{\alpha})$ bounded in
$L^{r}((0,\infty),\rho^{n-1}d\rho)$ Endpoint $L^{r^{\prime}}-$estimates for
$\displaystyle{\bm{K}}_{\sigma,n}^{\eta}(t,\cdot|\alpha,\beta,\lambda)$
$\displaystyle\left(~{}\frac{1}{r}+\frac{1}{r^{\prime}}=1~{}\right)$
$\rho^{\sigma-s}E_{\alpha,\alpha}(-\lambda\rho^{\sigma}t^{\alpha})$ bounded in
$L^{r}((0,\infty),\rho^{n-1}d\rho)$ $L^{p}-L^{q}$ and
$L^{p}-\dot{W}^{\sigma,q}$ decay estimates + $\dot{W}^{\gamma,p}-L^{q}$ and
$\dot{W}^{\gamma,p}-\dot{W}^{\sigma,q}$ decay estimates. Convolution
representation (5.2) + Hankel identity (5.4)
$s\in\\{0,\gamma\\}$$\eta=-s$$\eta=\sigma-s$$1\leq r\leq 2$Young’s ineq.
(4.14)$\displaystyle\frac{1}{p}+\frac{1}{r^{\prime}}=\frac{1}{q}+1$ Figure 1.
Tour de force of the decay estimates to be obtained on Subsection 5.2 and
Subsection 5.3.

On the sequence of results to be proved on the subsequent subsections and
elsewhere, the constants to be adopted are listed below to avoid cluttering up
the notations.

######  Notation 5.1.

We denote by

  * (a)

$\displaystyle\omega_{n-1}=\frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)}$
the $(n-1)-$dimensional measure of the sphere $\mathbb{S}^{n-1}$;

  * (b)

$\displaystyle\kappa_{s}=\sqrt{{s^{\frac{1}{s}}}{(s^{\prime})^{-\frac{1}{s^{\prime}}}}}$
(with $\displaystyle\frac{1}{s}+\frac{1}{s^{\prime}}=1$) the sharp constant
underlying to Young’s inequality (4.14);

  * (c)

$\displaystyle\widetilde{h}_{\frac{n}{2}-1}(r,r^{\prime}):=\left\|~{}\widetilde{\mathcal{H}}_{\frac{n}{2}-1}~{}\right\|_{L^{r}\rightarrow
L^{r^{\prime}}}$ (with $1\leq r\leq 2$ and
$\displaystyle\frac{1}{r}+\frac{1}{r^{\prime}}=1$) the operator norm of

$\widetilde{\mathcal{H}}_{\frac{n}{2}-1}:L^{r}((0,\infty),\rho^{n-1}d\rho)\rightarrow
L^{r^{\prime}}((0,\infty),\rho^{n-1}d\rho);$

  * (d)

$C_{r,\sigma,n}^{(s)}(\alpha,\beta,\lambda)$ the constant obtained Lemma 5.2;

  * (e)

$D_{r,\sigma,n}^{(s)}(\alpha,\lambda)$ the constant obtained in Lemma 5.3.

###  5.2. $L^{p}-L^{q}$ and $L^{p}-\dot{W}^{\sigma,q}$ decay estimates

Our first aim is to establish under which conditions
$E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)\varphi$
belongs to $L^{q}({\mathbb{R}}^{n})$ or to
$\dot{W}^{\sigma,q}({\mathbb{R}}^{n})$. In concrete, the following Mellin’s
integral identities

(5.5)
$\displaystyle\displaystyle\int_{0}^{\infty}\frac{\rho^{n-rs-1}}{(1+b\rho^{\sigma})^{a}}d\rho=\frac{\Gamma\left(\frac{n-rs}{\sigma}\right)\Gamma\left(\frac{a\sigma+rs-n}{\sigma}\right)}{\sigma~{}\Gamma(a)}~{}b^{-\frac{n-rs}{\sigma}}$
$\displaystyle~{}\displaystyle\left(~{}0<\Re\left(\frac{n-rs}{\sigma}\right)<\Re(a)\right),~{}$
(5.6)
$\displaystyle\displaystyle\int_{0}^{\infty}\frac{\rho^{n+r(\sigma-s)-1}}{(1+b\rho^{\sigma})^{2r}}d\rho=\frac{\Gamma\left(\frac{n+r(\sigma-s)}{\sigma}\right)\Gamma\left(\frac{r(\sigma+s)-n}{\sigma}\right)}{\sigma~{}\Gamma(2r)}~{}b^{-\frac{n+r(\sigma-s)}{\sigma}}$
$\displaystyle\displaystyle\left(~{}0<\Re\left(\frac{n-rs}{\sigma}\right)<\Re(r)~{}\right),$

that yield from [30, eq. (1.4.65) of p. 24], together with Theorem 4.1 and
Theorem 4.2 will be used on the proof of Lemma 5.2 and Lemma 5.3 to ensure
afterwards, on the proof of Proposition 5.5, that the kernel
${\bm{K}}_{\sigma,n}^{\eta}(\cdot,x|\alpha,\beta,\lambda)$, represented
through eq. (5.4), belongs to $L^{r^{\prime}}({\mathbb{R}}^{n})$. The proof of
both results is enclosed on A and B.

######  Lemma 5.2 (see Appendix A).

Let $0<\alpha\leq 1$ and $\beta>0$ be given. We assume that $\sigma>0$, $r\geq
1$ and $\displaystyle s\in\mathbb{R}$ satisfy one of the following conditions:

  * (i)

$r=1$ $\wedge$ $n-\sigma<s<n$;

  * (ii)

$\displaystyle\max\left\\{1,\frac{n}{\sigma}\right\\}<r<\infty$ $\wedge$
$s=0$;

  * (iii)

$\displaystyle\max\left\\{1,\frac{n}{\sigma+s}\right\\}<r<\frac{n}{s}$
$\wedge$ $0<s<n$.

Then, one has

$\displaystyle\displaystyle\left(\int_{0}^{\infty}\left|~{}\rho^{-s}E_{\alpha,\beta}(-\lambda\rho^{\sigma}t^{\alpha})~{}\right|^{r}\rho^{n-1}d\rho\right)^{\frac{1}{r}}\leq$
(5.7)
$\displaystyle\frac{\left(~{}C_{r,\sigma,n}^{(s)}(\alpha,\beta,\lambda)\right)^{\frac{1}{r}}}{\left|~{}\Gamma(\beta)~{}\right|}~{}t^{-\frac{\alpha}{\sigma}\left(\frac{n}{r}-s\right)},$

with $C_{r,\sigma,n}^{(s)}(\alpha,\beta,\lambda)$ equals to

$\displaystyle\begin{cases}\displaystyle\frac{\Gamma\left(\frac{n-rs}{\sigma}\right)\Gamma\left(\frac{r(2\sigma+s)-n}{\sigma}\right)}{\sigma~{}\Gamma(2r)}\left(\sqrt{\frac{\Gamma(1+\alpha)}{\Gamma(1+2\alpha)}}~{}\lambda\right)^{-\frac{n-rs}{\sigma}}\displaystyle~{}~{}~{}~{},~{}\lambda\geq
0,\beta=\alpha\\\ \ \\\
\displaystyle\frac{\Gamma\left(\frac{n-rs}{\sigma}\right)\Gamma\left(\frac{r(\sigma+s)-n}{\sigma}\right)}{\sigma~{}\Gamma(r)}~{}\left(\frac{\Gamma(\beta)}{\Gamma(\beta+\alpha)}~{}\lambda\right)^{-\frac{n-rs}{\sigma}}~{}~{}~{}~{},\lambda\geq
0,\beta\in\\{1\\}\cup(\alpha,+\infty)\\\ \ \\\
\displaystyle\frac{C^{r}~{}|\Gamma(\beta)|^{r}\Gamma\left(\frac{n-rs}{\sigma}\right)\Gamma\left(\frac{r(\sigma+s)-n}{\sigma}\right)}{\sigma~{}\Gamma(r)}~{}|\lambda|^{-\frac{n-rs}{\sigma}}~{},~{}\frac{\pi\alpha}{2}<\theta<\pi\alpha~{}~{}~{}~{},\theta\leq|\arg(\lambda)|\leq\pi.\end{cases}$

Here, the constant $C>0$ – that results from Theorem 4.1 – is independent of
$\alpha,\beta,r,s,\sigma,n$ and $\theta$.

######  Lemma 5.3 (see Appendix B).

Let $0<\alpha\leq 1$ be given. We assume that $\sigma>0$, $r\geq 1$ and
$\displaystyle s<n$ satisfy one of the following conditions:

  * (i)

$r=1$ $\wedge$ $n-\sigma<s<n$;

  * (ii)

$\displaystyle\max\left\\{1,\frac{n}{\sigma}\right\\}<r<\infty$ $\wedge$
$s=0$;

  * (iii)

$\displaystyle\max\left\\{1,\frac{n}{\sigma+s}\right\\}<r<\frac{n}{s}$
$\wedge$ $0<s<n$.

Then, one has

$\displaystyle\left(\int_{0}^{\infty}\left|~{}\rho^{\sigma-s}E_{\alpha,\alpha}(-\lambda\rho^{\sigma}t^{\alpha})~{}\right|^{r}\rho^{n-1}d\rho\right)^{\frac{1}{r}}\leq$
(5.8)
$\displaystyle\frac{\left(~{}D_{r,\sigma,n}^{(s)}(\alpha,\lambda)~{}\right)^{\frac{1}{r}}}{\Gamma(\alpha)}~{}t^{-\frac{\alpha}{\sigma}\left(\frac{n}{r}+{\sigma-s}\right)},$

with

$\displaystyle
D_{r,\sigma,n}^{(s)}(\alpha,\lambda)=\displaystyle\frac{\Gamma\left(\frac{n+r(\sigma-s)}{\sigma}\right)\Gamma\left(\frac{r(\sigma+s)-n}{\sigma}\right)}{\sigma~{}\Gamma(2r)}\left(\sqrt{\frac{\Gamma(1+\alpha)}{\Gamma(1+2\alpha)}}~{}\lambda\right)^{-\frac{n+r(\sigma-s)}{\sigma}}\displaystyle~{}~{}(\lambda\geq
0).$

######  Remark 5.4.

Due to the fact that the estimates depicted in Theorem 4.2 are indeed sharp
(cf. [44, 4]), one can claim that the estimates obtained in Lemma 5.2 and
Lemma 5.3 are sharp whenever $\lambda\geq 0$. The strategy considered to
compute the best upper bound, in terms of the norm of weighted Lebesgue space
$L^{r}((0,\infty),\rho^{n-1}d\rho)$, relies essentially on a straightforward
application of Mellin’s integral identities (5.5) and (5.6), respectively.

######  Proposition 5.5.

Let $0<\alpha\leq 1$ and $\beta>0$ be given. We assume that $\sigma>0$ and
$1\leq r\leq 2$ satisfy one of the following conditions:

  * (a)

$\displaystyle\sigma>n$ $\wedge$ $r=1$;

  * (b)

$\displaystyle\sigma>\frac{n}{2}$ $\wedge$
$\displaystyle\max\left\\{1,\frac{n}{\sigma}\right\\}<r\leq 2$,

and $1\leq p,q,r^{\prime}\leq\infty$ are such that

$\displaystyle\frac{1}{p}+\frac{1}{r^{\prime}}=\frac{1}{q}+1$ and
$\displaystyle\frac{1}{r}+\frac{1}{r^{\prime}}=1$.

Then we find the following estimates to hold:

$\displaystyle\displaystyle\left\|~{}E_{\alpha,\beta}\left(-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}\right)\varphi~{}\right\|_{q}\leq$
(5.9)
$\displaystyle\leq\left(~{}\frac{\kappa_{p}\kappa_{r^{\prime}}}{\kappa_{q}}\right)^{n}~{}\frac{\widetilde{h}_{\frac{n}{2}-1}(r,r^{\prime})~{}\left(~{}\omega_{n-1}~{}\right)^{\frac{1}{r^{\prime}}}\left(~{}C_{r,\sigma,n}^{(0)}(\alpha,\beta,\lambda)~{}\right)^{\frac{1}{r}}}{(2\pi)^{\frac{n}{2}}\left|~{}\Gamma(\beta)~{}\right|}~{}t^{-\frac{\alpha}{\sigma}\cdot\frac{n}{r}}~{}\|\varphi\|_{p},$
$\displaystyle\displaystyle\left\|~{}(-\Delta)^{\frac{\sigma}{2}}E_{\alpha,\alpha}\left(-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}\right)\varphi\right\|_{q}\leq$
(5.10)
$\displaystyle\leq\left(\frac{\kappa_{p}\kappa_{r^{\prime}}}{\kappa_{q}}\right)^{n}~{}\frac{\widetilde{h}_{\frac{n}{2}-1}(r,r^{\prime})~{}\left(~{}\omega_{n-1}~{}\right)^{\frac{1}{r^{\prime}}}\left(~{}D_{r,\sigma,n}^{(0)}(\alpha,\lambda)~{}\right)^{\frac{1}{r}}}{(2\pi)^{\frac{n}{2}}\Gamma(\alpha)}~{}~{}t^{-\alpha-\frac{\alpha}{\sigma}\cdot\frac{n}{r}}~{}\|\varphi\|_{p},$

whereby
$\omega_{n-1},\kappa_{s},\widetilde{h}_{\frac{n}{2}-1}(r,r^{\prime}),C_{r,\sigma,n}^{(0)}(\alpha,\beta,\lambda)$
and $D_{r,\sigma,n}^{(0)}(\alpha,\lambda)$ stand for the constants defined in
Notation 5.1.

###### Proof.

By applying Young’s inequality (4.14) to the convolution formula (5.2), we
obtain that

$\left\|~{}(-\Delta)^{\frac{\eta}{2}}E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)\varphi~{}\right\|_{q}\leq\left(\frac{\kappa_{p}\kappa_{r^{\prime}}}{\kappa_{q}}\right)^{n}\left\|~{}{\bm{K}}_{\sigma,n}^{\eta}(t,\cdot|\alpha,\beta,\lambda)~{}\right\|_{p^{\prime}}~{}\|\varphi\|_{p}$

is satisfied for every $1\leq p,q,r^{\prime}\leq\infty$ such that
$\displaystyle\frac{1}{p}+\frac{1}{r^{\prime}}=\frac{1}{q}+1$, whereby
$\displaystyle\kappa_{s}$ stands for the sharp constant arising from Young’s
inequality (see Notation 5.1).

Here, we observe that the reformulation of
${\bm{K}}_{\sigma,n}^{\eta}(t,\cdot|\alpha,\beta,\lambda)$, obtained in eq.
(5.4) in terms of the Hankel transform
$\widetilde{\mathcal{H}}_{\frac{n}{2}-1}$, gives rise to the norm equality

$\displaystyle\left\|~{}{\bm{K}}_{\sigma,n}^{\eta}(t,\cdot|\alpha,\beta,\lambda)~{}\right\|_{r^{\prime}}=\frac{1}{(2\pi)^{\frac{n}{2}}}\left(\int_{{\mathbb{R}}^{n}}\left|\widetilde{\mathcal{H}}_{\frac{n}{2}-1}\left[~{}|x|^{\eta}E_{\alpha,\beta}\left(-\lambda|x|^{\sigma}t^{\alpha}\right)~{}\right]\right|^{r^{\prime}}dx\right)^{\frac{1}{r^{\prime}}}.$

Also, from eq. (5.4) we infer that
$x\mapsto\left|\widetilde{\mathcal{H}}_{\frac{n}{2}-1}\left[~{}|x|^{\eta}E_{\alpha,\beta}\left(-\lambda|x|^{\sigma}t^{\alpha}\right)~{}\right]\right|^{r^{\prime}}$
is radially symmetric. Thereby, the change of variables to spherical
coordinates gives rise to

$\displaystyle\left\|~{}{\bm{K}}_{\sigma,n}^{\eta}(t,\cdot|\alpha,\beta,\lambda)~{}\right\|_{r^{\prime}}=\frac{\left(\omega_{n-1}\right)^{\frac{1}{r^{\prime}}}}{(2\pi)^{\frac{n}{2}}}\left(\int_{0}^{\infty}\left|\widetilde{\mathcal{H}}_{\frac{n}{2}-1}\left[~{}\rho^{\eta}E_{\alpha,\beta}\left(-\lambda\rho^{\sigma}t^{\alpha}\right)~{}\right]\right|^{r^{\prime}}\rho^{n-1}d\rho\right)^{\frac{1}{r^{\prime}}},$

where $\displaystyle\omega_{n-1}$ denotes the $(n-1)-$dimensional measure of
the sphere $\mathbb{S}^{n-1}$ (see Notation 5.1). Then, under the assumption
that one of the conditions (a) or (b) on the statement of Proposition 5.5 is
fulfilled, the inequality

(5.11)
$\displaystyle\left\|~{}{\bm{K}}_{\sigma,n}^{\eta}(t,\cdot|\alpha,\beta,\lambda)~{}\right\|_{r^{\prime}}$
$\displaystyle\leq$
$\displaystyle\frac{\widetilde{h}_{\frac{n}{2}-1}(r,r^{\prime})\left(~{}\omega_{n-1}~{}\right)^{\frac{1}{r^{\prime}}}}{(2\pi)^{\frac{n}{2}}}\times$
$\displaystyle\times$
$\displaystyle\left(\int_{0}^{\infty}\left|~{}\rho^{\eta}E_{\alpha,\beta}(-\lambda\rho^{\sigma}t^{\alpha})~{}\right|^{r}\rho^{n-1}d\rho\right)^{\frac{1}{r}}$

results from the fact that

$\widetilde{\mathcal{H}}_{\frac{n}{2}-1}:L^{r}((0,\infty),\rho^{n-1}d\rho)\rightarrow
L^{r^{\prime}}((0,\infty),\rho^{n-1}d\rho)$

is bounded for values of $1\leq r\leq 2\leq r^{\prime}\leq\infty$ satisfying
$\displaystyle\frac{1}{r}+\frac{1}{r^{\prime}}=1$. This along with the norm
estimates provided by Lemma 5.2 & Lemma 5.3 gives rise to the inequalities
(5.5) and (5.5). Indeed, the conditions $r=1$, $\sigma>n$ or

$\displaystyle\displaystyle\max\left\\{1,\frac{n}{\sigma}\right\\}<r\leq
2~{}~{}\wedge~{}~{}~{}\displaystyle\sigma>\dfrac{n}{2}~{}$
$\displaystyle\left(~{}\displaystyle~{}\max\left\\{1,\frac{n}{\sigma}\right\\}\geq\frac{n}{\sigma}~{}\right),$

fixed by assumptions (a) and (b) of Proposition 5.5, respectively, ensures
that the inequality $\displaystyle r-\frac{n}{\sigma}>0$ is always fulfilled
so that the boundedness of the right hand side of (5.11) follows from the
estimates obtained in Lemma 5.2 (case of $s=0$ and $\eta=0$) and Lemma 5.3
(case of $s=0$ and $\eta=\sigma$). ∎

######  Remark 5.6.

The proof of Proposition 5.5 encompasses part of the $L^{p}-L^{q}$ decay
estimates obtained by Pham et al in [40, Section 2.] (case of
$\alpha=\beta=1$) and Kemppainen et al in [29, Section 3.1] (case of
$\sigma=2$). We note here that our case roughly works with the same arguments
as in Pham et al. with exception of a slightly technical condition: it is not
necessary to assume, a priori, parity arguments encoded by the dimension of
the Euclidean space ${\mathbb{R}}^{n}$ (see [40, pp. 573-576] for further
comparisons).

######  Remark 5.7.

Although our technique of proof, depicted throughout Subsection 5.2, is
complementary to Kemppainen et al technique (cf. [29, Section 5]), it allows
us to rid one of the majors stumbling blocks on the computation of
$L^{r^{\prime}}-$decay estimates for the fundamental solution (the kernel
function (5.3) in our case), whose technicality of proofs heavily relies on
the asymptotics of Fox-H functions (see [29, Section 3]).

###  5.3. $\dot{W}^{\gamma,p}-L^{q}$ and
$\dot{W}^{\gamma,p}-\dot{W}^{\sigma,q}$ decay estimates

Now, we turn our attention to the homogeneous Sobolev spaces discussed in the
end of Subsection 4.2. Although the next proposition follows mutatis mutandis
the technique of proof considered in Proposition 5.5, we provide below a self-
contained proof of it for reader’s convenience.

######  Proposition 5.8.

Let $0<\alpha\leq 1$ and $\beta>0$ be given. We assume that $\sigma>0$,
$0<\gamma<n$ and $1\leq r\leq 2$ satisfy one of the following conditions:

  * (i)

$0<\gamma<n$ $\wedge$ $\sigma>n-\gamma$ $\wedge$ $r=1$;

  * (ii)

$\displaystyle 0<\gamma<\frac{n}{2}$ $\wedge$
$\displaystyle\sigma>\frac{n}{2}-\gamma$ $\wedge$
$\displaystyle\max\left\\{1,\frac{n}{\sigma+\gamma}\right\\}<r\leq 2$;

  * (iii)

$\displaystyle\frac{n}{2}\leq\gamma<n$ $\wedge$ $\sigma>0$ $\wedge$
$\displaystyle\max\left\\{1,\frac{n}{\sigma+\gamma}\right\\}<r<\frac{n}{\gamma}$,

and $1\leq p,q,r^{\prime}\leq\infty$ are such that

$\displaystyle\frac{1}{p}+\frac{1}{r^{\prime}}=\frac{1}{q}+1$ and
$\displaystyle\frac{1}{r}+\frac{1}{r^{\prime}}=1$.

Then we find the following estimates to hold:

$\displaystyle\displaystyle\left\|~{}E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)\varphi~{}\right\|_{q}$
$\displaystyle\leq\left(\frac{\kappa_{p}\kappa_{r^{\prime}}}{\kappa_{q}}\right)^{n}~{}\frac{\widetilde{h}_{\frac{n}{2}-1}(r,r^{\prime})\left(~{}\omega_{n-1}~{}\right)^{\frac{1}{r^{\prime}}}\left(~{}C_{r,\sigma,n}^{(\gamma)}(\alpha,\beta,\lambda)~{}\right)^{\frac{1}{r}}}{(2\pi)^{\frac{n}{2}}\left|~{}\Gamma(\beta)~{}\right|}~{}\times$
(5.12)
$\displaystyle\times~{}t^{-\frac{\alpha}{\sigma}\left(\frac{n}{r}-\gamma\right)}~{}\left\|~{}(-\Delta)^{\frac{\gamma}{2}}\varphi~{}\right\|_{p},$
$\displaystyle\displaystyle\left\|~{}(-\Delta)^{\frac{\sigma}{2}}E_{\alpha,\alpha}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)\varphi~{}\right\|_{q}\leq$
$\displaystyle\leq\left(\frac{\kappa_{p}\kappa_{r^{\prime}}}{\kappa_{q}}\right)^{n}~{}\frac{\widetilde{h}_{\frac{n}{2}-1}(r,r^{\prime})\left(~{}\omega_{n-1}~{}\right)^{\frac{1}{r^{\prime}}}\left(~{}D_{r,\sigma,n}^{(\gamma)}(\alpha,\beta,\lambda)~{}\right)^{\frac{1}{r}}}{(2\pi)^{\frac{n}{2}}\Gamma(\alpha)}~{}\times$
(5.13)
$\displaystyle\times~{}t^{-\alpha-\frac{\alpha}{\sigma}\left(\frac{n}{r}-\gamma\right)}\left\|~{}(-\Delta)^{\frac{\gamma}{2}}\varphi~{}\right\|_{p}.$

Hereby
$\omega_{n-1},\kappa_{s},\widetilde{h}_{\frac{n}{2}-1}(r,r^{\prime}),C_{r,\sigma,n}^{(\gamma)}(\alpha,\beta,\lambda)$
and $D_{r,\sigma,n}^{(\gamma)}(\alpha,\lambda)$ stand for the constants
defined in Notation 5.1.

###### Proof.

For the proof of the estimates (5.8) and (5.8), we recall that the convolution
representation of (5.1), provided by eq. (5.2), can be reformulated as
follows:

$\displaystyle(-\Delta)^{\frac{\eta}{2}}E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)\varphi(x)=\displaystyle\left({\bm{K}}_{\sigma,n}^{\eta-\gamma}(t,\cdot|\alpha,\beta,\lambda)\ast(-\Delta)^{\frac{\gamma}{2}}\varphi\right)(x).$

Then, by a straightforward application Young’s inequality (4.14) there holds

$\displaystyle\left\|~{}(-\Delta)^{\frac{\eta}{2}}E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)\varphi~{}\right\|_{q}\leq$
(5.14)
$\displaystyle\leq\left(\frac{\kappa_{p}\kappa_{r^{\prime}}}{\kappa_{q}}\right)^{n}\left\|~{}{\bm{K}}_{\sigma,n}^{\eta-\gamma}(t,\cdot|\alpha,\beta,\lambda)~{}\right\|_{r^{\prime}}~{}\left\|~{}(-\Delta)^{\frac{\gamma}{2}}\varphi~{}\right\|_{p}.$

On the other hand, to prove the boundedness of
$\left\|~{}{\bm{K}}_{\sigma,n}^{\eta-\gamma}(t,\cdot|\alpha,\beta,\lambda)~{}\right\|_{r^{\prime}}$
we observe that in the view of the boundedness of the Hankel transform:

$\widetilde{\mathcal{H}}_{\frac{n}{2}-1}:L^{r}((0,\infty),\rho^{n-1}d\rho)\rightarrow
L^{r^{\prime}}((0,\infty),\rho^{n-1}d\rho),$

for values of $1\leq r\leq 2\leq r^{\prime}\leq\infty$ satisfying
$\displaystyle\frac{1}{r}+\frac{1}{r^{\prime}}=1$ (cf. [17, Proposition
2.1.]), and of Lemma 5.2 & Lemma 5.3, the inequality

$\displaystyle\left\|~{}{\bm{K}}_{\sigma,n}^{\eta-\gamma}(t,\cdot|\alpha,\beta,\lambda)~{}\right\|_{r^{\prime}}$
$\displaystyle\leq$
$\displaystyle\frac{\widetilde{h}_{\frac{n}{2}-1}(r,r^{\prime})\left(~{}\omega_{n-1}~{}\right)^{\frac{1}{r^{\prime}}}}{(2\pi)^{\frac{n}{2}}}\times$
$\displaystyle\times$
$\displaystyle\left(\int_{0}^{\infty}\left|~{}\rho^{\eta-\gamma}E_{\alpha,\beta}(-\lambda\rho^{\sigma}t^{\alpha})~{}\right|^{r}\rho^{n-1}d\rho\right)^{\frac{1}{r}},$

is achieved for $s=\gamma$, with $0<\gamma<n$, whenever one of the conditions
${\bf(i)},{\bf(ii)}$ and ${\bf(iii)}$ assumed on the statement of Proposition
5.8, hold.

Thus, the estimates (5.8) and (5.8) are then immediate from the combination of
the sharp inequality (5.3) with the estimates provided by Lemma 5.2 and Lemma
5.3, respectively. ∎

######  Remark 5.9.

Despite the estimates obtained in Lemma 5.2 and Lemma 5.3 (which are sharp in
case of $\lambda\geq 0$), we also have adopted the sharp estimates obtained by
Lieb and Loss in [35, Theorem 4.2] (Young’s inequality) and the
$L^{r}-L^{r^{\prime}}$ estimates for the Hankel transform to achieve
Proposition 5.5 and, subsequently, Proposition 5.8. Moreover, one can infer
that such estimates are optimal for values of $1<r\leq 2$, accordingly to [17,
Proposition 2.1.].

######  Remark 5.10.

Worthy of mention, from the estimates obtained in Lemma 5.2 and Lemma 5.3 it
was not necessary to investigate the Mellin integral representation of the
kernel (5.3) as a Fox-H function (see, for instance, the proof of [29, Lemma
3.3] for further comparisons). In our case, the mapping properties of the
Hankel transform allowed us to downsize the computation of the [optimal]
endpoint estimates from the decay of the generalized Mittag-Leffler functions
$E_{\alpha,\beta}$ (cf. Theorem 4.2). Such approach goes towards the optimal
bound argument considered by Kemppainen et al in [28, Remark 3.1].

###  5.4. An amalgamation of dispersive estimates

Let us now amalgamate the main conclusions provided by Proposition 5.5 and
Proposition 5.8. For a sake of readability of the dispersive estimates to be
considered on the sequel, we adopt from now on the shorthand notations

$\displaystyle\|v(t,\cdot)\|_{q}\lesssim t^{\eta}~{}\|\varphi\|_{p}$ resp.
$\displaystyle\|v(t,\cdot)\|_{q}\lesssim
t^{\eta}~{}\left\|~{}(-\Delta)^{\frac{\gamma}{2}}\varphi~{}\right\|_{p}$

to express the set of inequalities

$\displaystyle\|v(t,\cdot)\|_{q}\leq At^{\eta}~{}\|\varphi\|_{p}$ resp.
$\displaystyle\|v(t,\cdot)\|_{q}\leq
Bt^{\eta}~{}\left\|~{}(-\Delta)^{\frac{\gamma}{2}}\varphi~{}\right\|_{p},$

for some $A>0$, $B>0$ and $\eta\in\mathbb{R}$. The following regions of the
$Opq$ plane will also be adopted to abridge the exposition of the results.

######  Definition 5.11.

For fixed values of $\varepsilon>0$ and $\nu>0$, we define the regions
$\mathcal{R}_{1,2}(\varepsilon)$ and $\mathcal{R}_{3}(\nu,\varepsilon)$ of the
$Opq$ plane in the following way:

  * Region $\mathcal{R}_{1,2}(\varepsilon)$:
$(p,q)\in\mathcal{R}_{1,2}(\varepsilon)$ if, and only if,
$\displaystyle\left(\frac{n}{p},\frac{n}{q}\right)$ satisfies the set of
conditions

(5.15)
$\displaystyle\displaystyle\frac{n}{2}\leq\frac{n}{p}<\min\left\\{n,\varepsilon\right\\}~{}\wedge~{}\displaystyle\max\left\\{0,\frac{n}{p}-\min\left\\{n,\varepsilon\right\\}\right\\}<\frac{n}{q}\leq\frac{n}{p}-\frac{n}{2}.$

  * Region $\mathcal{R}_{3}(\nu,\varepsilon)$:
$(p,q)\in\mathcal{R}_{3}(\nu,\varepsilon)$ if, and only if,
$\displaystyle\left(\frac{n}{p},\frac{n}{q}\right)$ satisfies the set of
conditions

(5.16)
$\displaystyle\displaystyle\nu<\frac{n}{p}<\min\left\\{n,\varepsilon\right\\}~{}\wedge~{}\displaystyle\max\left\\{0,\frac{n}{p}-\min\left\\{n,\varepsilon\right\\}\right\\}<\frac{n}{q}\leq\frac{n}{p}-\nu.$

The following assumption will also be required on the sequel:

######  Assumption 5.12.

We assume that the constant $\lambda\in\mathbb{C}$ satisfy one of the
following conditions:

  * (a)

$\lambda>0$;

  * (b)

$\theta\leq|\arg(\lambda)|\leq\pi$ and $\frac{\pi\alpha}{2}<\theta<\pi\alpha$.

######  Corollary 5.13 (see Appendix C).

Let $0<\alpha\leq 1$, $\beta>0$ be given. Assume further that Assumption 5.12
holds for the constant $\lambda\in\mathbb{C}$. Then, for every
$\varphi\in\mathcal{S}({\mathbb{R}}^{n})$ there holds

$\displaystyle
E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)\varphi\in
L^{q}({\mathbb{R}}^{n})$ $\displaystyle\wedge$ $\displaystyle
E_{\alpha,\alpha}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)\varphi\in\dot{W}^{\sigma,q}({\mathbb{R}}^{n})$

whenever the following assertions are then true:

  * (1)

– Proposition 5.5: $\varphi\in L^{p}({\mathbb{R}}^{n})$ and

$\displaystyle\left\|~{}(-\Delta)^{\frac{\eta}{2}}E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)\varphi~{}\right\|_{q}\lesssim$
(5.17) $\displaystyle\lesssim\begin{cases}\displaystyle
t^{-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}\right)}~{}\|\varphi\|_{p}&,\eta=0\\\
\ \\\ \displaystyle
t^{-\alpha-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}\right)}~{}\|\varphi\|_{p}&,\beta=\alpha~{};~{}\eta=\sigma,\end{cases}$

for tuples $(\sigma,p,q)$ satisfying the following conditions:

    * (a)

$L^{1}-L^{\infty}$ & $L^{1}-\dot{W}^{\sigma,\infty}$ estimates (first case):

$\sigma>n$ $\wedge$ $(p,q)=(1,\infty)$;

    * (b)

$L^{p}-L^{q}$ & $L^{p}-\dot{W}^{\sigma,q}$ estimates (second case):

$\displaystyle\sigma>\frac{n}{2}$ $\wedge$ $(p,q)\in\mathcal{R}_{1,2}(\sigma)$
[ see eq. (5.15) ].

  * (2)

– Proposition 5.8: $\varphi\in\dot{W}^{\gamma,p}({\mathbb{R}}^{n})$ and

$\displaystyle\left\|~{}(-\Delta)^{\frac{\eta}{2}}E_{\alpha,\beta}\left(~{}-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}~{}\right)\varphi~{}\right\|_{q}\lesssim$
(5.18) $\displaystyle\lesssim\begin{cases}\displaystyle
t^{-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\gamma\right)}~{}\left\|~{}(-\Delta)^{\frac{\gamma}{2}}\varphi~{}\right\|_{p}&,\eta=0\\\
\ \\\ \displaystyle
t^{-\alpha-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\gamma\right)}\left\|~{}(-\Delta)^{\frac{\gamma}{2}}\varphi~{}\right\|_{p}&,\beta=\alpha~{};~{}\eta=\sigma,\end{cases}$

for tuples $(\sigma,\gamma,p,q)$ satisfying the following conditions:

    * (i)

$\dot{W}^{\gamma,1}-L^{\infty}$ & $\dot{W}^{\gamma,1}-\dot{W}^{\sigma,\infty}$
estimates (first case):

$0<\gamma<n$ $\wedge$ $\sigma>n-\gamma$ $\wedge$ $(p,q)=(1,\infty)$.

    * (ii)

$\dot{W}^{\gamma,p}-L^{p}$ & $\dot{W}^{\gamma,p}-\dot{W}^{\sigma,q}$ estimates
(second case):

$\displaystyle 0<\gamma<\frac{n}{2}$ $\wedge$
$\displaystyle\sigma>\frac{n}{2}-\gamma$ $\wedge$
$(p,q)\in\mathcal{R}_{1,2}(\sigma+\gamma)$
[ see eq. (5.15) ].

    * (iii)

$\dot{W}^{\gamma,p}-L^{p}$ & $\dot{W}^{\gamma,p}-\dot{W}^{\sigma,q}$ estimates
(third case):

$\sigma>0$ $\wedge$ $\displaystyle\frac{n}{2}\leq\gamma\leq n$ $\wedge$
$(p,q)\in\mathcal{R}_{3}(\gamma,\sigma+\gamma)$
[ see eq. (5.16) ].

$\frac{n}{p}$$\frac{n}{q}$$n$$n$$\frac{n}{2}$$\sigma+s$$\frac{n}{2}$ Figure 2.
Geometric interpretation of Corollary 5.13 under the constraints
$s\in\\{0,\gamma\\}$ and $\frac{n}{2}<\sigma+s<n$. In this case, we are unable
to obtain $\dot{W}^{s,1}-L^{\infty}$ and
$\dot{W}^{s,1}-\dot{W}^{\sigma,\infty}$ estimates.
$\frac{n}{p}$$\frac{n}{q}$$n$$n$$\frac{n}{2}$$\frac{n}{2}$$\frac{n}{2}$ Figure
3. Geometric interpretation of Corollary 5.13 under the constraints
$s\in\\{0,\gamma\\}$ and $\sigma+s\geq n$. In this case, we also got
$\dot{W}^{s,1}-L^{\infty}$ and $\dot{W}^{s,1}-\dot{W}^{\sigma,\infty}$
estimates.

##  6\. Main Results

Let us turn at last our attention to the Cauchy Problem 2.1 and Cauchy Problem
2.2.

###  6.1. Notations

To state a solution representation for both problems, the auxiliar constants

(6.1) $\displaystyle\lambda_{\pm}=\begin{cases}\displaystyle\frac{\mu}{2}\pm
i\sqrt{1-\frac{\mu^{2}}{4}}&,0<\mu<2\\\ \ \\\
\displaystyle\frac{\mu}{2}\pm\sqrt{\frac{\mu^{2}}{4}-1}&,\mu\geq 2\end{cases}$

as well as the kernel functions ${\bm{M}}_{\sigma,n}(t,x|\alpha,\beta,\mu)$,
${\bm{N}}_{\sigma,n}(t,x|\alpha,\mu)$ and
${\bm{J}}_{\sigma,n}(t,x|\alpha,\mu)$, introduced as below, will be required
on the proof of Theorem 6.3, Theorem 6.4 and elsewhere.

######  Notation 6.1 (Kernel functions underlying to Cauchy Problems 2.1 &
2.2).

$\displaystyle{\bm{M}}_{\sigma,n}(t,x|\alpha,\beta,\mu)=$
$\displaystyle\begin{cases}\displaystyle\frac{t^{\beta-1}}{(2\pi)^{n}}\int_{{\mathbb{R}}^{n}}~{}\frac{\lambda_{+}E_{\alpha,\beta}(-\lambda_{+}|\xi|^{\sigma}t^{\alpha})-\lambda_{-}E_{\alpha,\beta}(-\lambda_{-}|\xi|^{\sigma}t^{\alpha})}{\lambda_{+}-\lambda_{-}}~{}e^{ix\cdot\xi}d\xi&,~{}\mu\neq
2\\\ \ \\\
\displaystyle\frac{t^{\beta-1}}{\alpha(2\pi)^{n}}\int_{{\mathbb{R}}^{n}}\left(~{}E_{\alpha,\beta-1}(-|\xi|^{\sigma}t^{\alpha})-(1+\alpha-\beta)E_{\alpha,\beta}(-|\xi|^{\sigma}t^{\alpha})~{}\right)~{}e^{ix\cdot\xi}d\xi&,~{}\mu=2.\end{cases}$

$\displaystyle{\bm{N}}_{\sigma,n}(t,x|\alpha,\mu)=$
$\displaystyle=\begin{cases}\displaystyle\frac{1}{(2\pi)^{n}}\int_{{\mathbb{R}}^{n}}\left(\frac{E_{\alpha,1}(-\lambda_{+}|\xi|^{\sigma}t^{\alpha})}{1-(\lambda_{+})^{2}}+\frac{E_{\alpha,1}(-\lambda_{-}|\xi|^{\sigma}t^{\alpha})}{1-(\lambda_{-})^{2}}\right)e^{ix\cdot\xi}d\xi&,~{}\mu\neq
2\\\ \ \\\
\displaystyle\frac{1}{(2\pi)^{n}}\int_{{\mathbb{R}}^{n}}\left(E_{\alpha,1}(-|\xi|^{\sigma}t^{\alpha})+\frac{|\xi|^{\sigma}t^{\alpha}}{\alpha}E_{\alpha,\alpha}(-|\xi|^{\sigma}t^{\alpha})\right)e^{ix\cdot\xi}d\xi&,~{}\mu=2.\end{cases}$

######  Notation 6.2 (Kernel function underlying to Cauchy Problem 2.2).

$\displaystyle{\bm{J}}_{\sigma,n}(t,x|\alpha,\mu)=$
$\displaystyle=\begin{cases}\displaystyle\frac{t}{(2\pi)^{n}}\int_{{\mathbb{R}}^{n}}\left(~{}\frac{\lambda_{+}E_{\alpha,2}(-\lambda_{+}|\xi|^{\sigma}t^{\alpha})-\lambda_{-}E_{\alpha,2}(-\lambda_{-}|\xi|^{\sigma}t^{\alpha})}{\lambda_{+}-\lambda_{-}}~{}\right)e^{ix\cdot\xi}d\xi&,~{}\mu\neq
2\\\ \ \\\
\displaystyle\frac{t}{\alpha(2\pi)^{n}}\int_{{\mathbb{R}}^{n}}\left(~{}E_{\alpha,1}(-|\xi|^{\sigma}t^{\alpha})-(\alpha-1)E_{\alpha,2}(-|\xi|^{\sigma}t^{\alpha})~{}\right)e^{ix\cdot\xi}d\xi&,~{}\mu=2.\end{cases}$

###  6.2. Solution of the Cauchy Problem 2.1

######  Theorem 6.3.

Let $0<\alpha\leq\frac{1}{2}$, $2\alpha\leq\beta<2\alpha+1$, $\sigma>0$ and
$\mu>0$ be given. If $u_{0}\in\mathcal{S}(\mathbb{R}^{n})$ and
$f(t,\cdot)\in\mathcal{S}(\mathbb{R}^{n})$, for every $t\geq 0$, then
$u:[0,\infty)\times{\mathbb{R}}^{n}\longrightarrow{\mathbb{R}}$ defined
through

$\displaystyle
u(t,x)=\left({\bm{N}}_{\sigma,n}(t,\cdot|\alpha,\mu)*u_{0}\right)(x)+\int_{0}^{t}\left[~{}{\bm{M}}_{\sigma,n}(t-\tau,\cdot|\alpha,\beta,\mu)\ast
f(\tau,\cdot)~{}\right](x)~{}d\tau$

solves the Cauchy Problem 2.1, whereby
${\bm{M}}_{\sigma,n}(t,x|\alpha,\beta,\mu)$ and
${\bm{N}}_{\sigma,n}(t,x|\alpha,\mu)$ denote the kernel functions introduced
in Notation 6.1.

###### Proof.

First, we recall that $u(t,x)$ solves the Cauchy Problem 2.1 if and only if
$\widehat{u}(t,\xi)=(\mathcal{F}u)(t,\xi)$ solves

(6.2)
$\begin{cases}\displaystyle\partial^{2\alpha}_{t}\widehat{u}(t,\xi)+\mu|\xi|^{\sigma}\partial^{\alpha}_{t}\widehat{u}(t,\xi)+|\xi|^{2\sigma}\widehat{u}(t,\xi)=I^{\beta-2\alpha}_{0^{+}}\widehat{f}(t,\xi)&,\mbox{in}~{}(0,\infty)\times\mathbb{R}^{n}\\\
\ \\\
\widehat{u}(0,\xi)=\widehat{u_{0}}(\xi)&,\mbox{in}~{}\mathbb{R}^{n},\end{cases}$

with $\widehat{u_{0}}(\xi)=(\mathcal{F}u_{0})(\xi)$ and
$I^{\beta-2\alpha}_{0^{+}}\widehat{f}(t,\xi)=I^{\beta-2\alpha}_{0^{+}}(\mathcal{F}f)(t,\xi)$.

In order to compute the solution representation of (6.2) we adopt the Laplace
transform $\mathcal{L}$ defined through eq. (4.1). First, one obtains from
eqs. (4.3) and (4.4), and from the initial conditions underlying to the Cauchy
Problem (6.2) that

$\displaystyle\mathcal{L}[\partial^{2\alpha}_{t}\widehat{u}(t,\xi)](s)$
$\displaystyle=$ $\displaystyle
s^{2\alpha}\mathcal{L}[\widehat{u}(t,\xi)](s)-s^{2\alpha-1}\widehat{u_{0}}(\xi),$
$\displaystyle\mathcal{L}[\partial^{\alpha}_{t}\widehat{u}(t,\xi)](s)$
$\displaystyle=$ $\displaystyle
s^{\alpha}\mathcal{L}[\widehat{u}(t,\xi)](s)-s^{\alpha-1}\widehat{u_{0}}(\xi),$
$\displaystyle\mathcal{L}[I^{\beta-2\alpha}_{0^{+}}\widehat{f}(t,\xi)](s)$
$\displaystyle=$ $\displaystyle
s^{2\alpha-\beta}\mathcal{L}[\widehat{f}(t,\xi)](s),$

since $0<2\alpha\leq 1$ and $0\leq\beta-2\alpha<1$.

Thus, after few algebraic manipulations we conclude that the solution of
(6.2), $\widehat{u}(t,\xi)$, admits the following Laplace representation

(6.3)
$\displaystyle\mathcal{L}[\widehat{u}(t,\xi)](s)=U_{0}(s,\xi)~{}\widehat{u_{0}}(\xi)+M(s,\xi)~{}\mathcal{L}[\widehat{f}(t,\xi)](s),$

with

(6.4) $\displaystyle M(s,\xi)$ $\displaystyle=$
$\displaystyle\frac{s^{2\alpha-\beta}}{s^{2\alpha}+\mu|\xi|^{\sigma}s^{\alpha}+|\xi|^{2\sigma}},$
(6.5) $\displaystyle U_{0}(s,\xi)$ $\displaystyle=$
$\displaystyle\frac{s^{2\alpha-1}+\mu|\xi|^{\sigma}s^{\alpha-1}}{s^{2\alpha}+\mu|\xi|^{\sigma}s^{\alpha}+|\xi|^{2\sigma}}.$

By taking into account the constants $\lambda_{+}$ and $\lambda_{-}$ defined
viz eq. (6.1), we observe that $s=-\lambda_{-}|\xi|^{\sigma}$ and
$s=-\lambda_{+}|\xi|^{\sigma}$ correspond to the algebraic roots of the
polynomial $P(s)=s^{2}+\mu|\xi|^{\sigma}s+|\xi|^{2\sigma}$. Then

$\displaystyle
s^{2\alpha}+\mu|\xi|^{\sigma}s^{\alpha}+|\xi|^{2\sigma}=(s^{\alpha}+\lambda_{+}|\xi|^{\sigma})(s^{\alpha}+\lambda_{-}|\xi|^{\sigma})$

so that for $\lambda_{+}\neq\lambda_{-}$ (i.e. for $\mu\neq 2$) the set of
identities

(6.6)
$\displaystyle~{}\frac{s^{\alpha}}{s^{2\alpha}+\mu|\xi|^{\sigma}s^{\alpha}+|\xi|^{2\sigma}}=\frac{1}{\lambda_{+}-\lambda_{-}}\left(\frac{\lambda_{+}}{s^{\alpha}+\lambda_{+}|\xi|^{\sigma}}-\frac{\lambda_{-}}{s^{\alpha}+\lambda_{-}|\xi|^{\sigma}}\right),$
(6.7)
$\displaystyle~{}\frac{1}{s^{2\alpha}+\mu|\xi|^{\sigma}s^{\alpha}+|\xi|^{2\sigma}}=\frac{-|\xi|^{-\sigma}}{\lambda_{+}-\lambda_{-}}\left(\frac{1}{s^{\alpha}+\lambda_{+}|\xi|^{\sigma}}-\frac{1}{s^{\alpha}+\lambda_{-}|\xi|^{\sigma}}\right)$

yield straightforwardly by the method of undetermined coefficients.

Thus, from eq. (6.6) we obtain that eq. (6.4) is equivalent to

(6.8) $\displaystyle M(s,\xi)$ $\displaystyle=$
$\displaystyle\begin{cases}\displaystyle\frac{1}{\lambda_{+}-\lambda_{-}}\left(\frac{\lambda_{+}~{}s^{\alpha-\beta}}{s^{\alpha}+\lambda_{+}|\xi|^{\sigma}}-\frac{\lambda_{-}~{}s^{\alpha-\beta}}{s^{\alpha}+\lambda_{-}|\xi|^{\sigma}}\right)&,~{}~{}\mu\neq
2\\\ \ \\\
\displaystyle\frac{s^{2\alpha-\beta}}{(s^{\alpha}+|\xi|^{\sigma})^{2}}&,~{}\mu=2,\end{cases}$

whereas the equivalence between eq. (6.5) and

$\displaystyle
U_{0}(s,\xi)=\displaystyle\frac{1}{s}-\frac{|\xi|^{2\sigma}s^{-1}}{s^{2\alpha}+\mu|\xi|^{\sigma}s^{\alpha}+|\xi|^{2\sigma}}$
(6.9)
$\displaystyle=\begin{cases}\displaystyle\frac{1}{s}-\frac{-|\xi|^{\sigma}}{\lambda_{+}-\lambda_{-}}\left(\frac{s^{-1}}{s^{\alpha}+\lambda_{+}|\xi|^{\sigma}}-\frac{s^{-1}}{s^{\alpha}+\lambda_{-}|\xi|^{\sigma}}\right)&,~{}\mbox{if}~{}\mu\neq
2\\\ \ \\\
\displaystyle\frac{1}{s}-|\xi|^{2\sigma}\frac{s^{-1}}{(s^{\alpha}+|\xi|^{\sigma})^{2}}&,~{}\mbox{if}~{}\mu=2.\end{cases}$

follows from eq. (6.7).

Next, from the Laplace identity (4.8) and from Lemma 4.3 we obtain the set of
identities

$\displaystyle\mathcal{L}^{-1}[M(s,\xi)](t)=\begin{cases}\displaystyle
t^{\beta-1}~{}\frac{\lambda_{+}E_{\alpha,\beta}(-\lambda_{+}|\xi|^{\sigma}t^{\alpha})-\lambda_{-}E_{\alpha,\beta}(-\lambda_{-}|\xi|^{\sigma}t^{\alpha})}{\lambda_{+}-\lambda_{-}}&,~{}\mu\neq
2\\\ \ \\\ \displaystyle
t^{\beta-1}E_{\alpha,\beta}^{2}(-|\xi|^{\sigma}t^{\alpha})&,~{}\mu=2\end{cases}$
(6.10) $\displaystyle=\begin{cases}\displaystyle
t^{\beta-1}~{}\frac{\lambda_{+}E_{\alpha,\beta}(-\lambda_{+}|\xi|^{\sigma}t^{\alpha})-\lambda_{-}E_{\alpha,\beta}(-\lambda_{-}|\xi|^{\sigma}t^{\alpha})}{\lambda_{+}-\lambda_{-}}&,~{}\mu\neq
2\\\ \ \\\
\displaystyle\frac{t^{\beta-1}}{\alpha}\left(~{}E_{\alpha,\beta-1}(-|\xi|^{\sigma}t^{\alpha})-(1+\alpha-\beta)E_{\alpha,\beta}(-|\xi|^{\sigma}t^{\alpha})~{}\right)&,~{}\mu=2.\end{cases}$

In the same order of ideas, from (4.8) we conclude that the Laplace inverse of
(6.2), $\mathcal{L}^{-1}[U_{0}(s,\xi)](t)$, equals to

$\displaystyle\begin{cases}\displaystyle
1-\frac{-|\xi|^{\sigma}t^{\alpha}}{\lambda_{+}-\lambda_{-}}\left(E_{\alpha,\alpha+1}(-\lambda_{+}|\xi|^{\sigma}t^{\alpha})-E_{\alpha,\alpha+1}(-\lambda_{-}|\xi|^{\sigma}t^{\alpha})\right)&,~{}\mu\neq
2\\\ \ \\\ \displaystyle
1-|\xi|^{2\sigma}t^{2\alpha}E_{\alpha,2\alpha+1}^{2}(-|\xi|^{\sigma}t^{\alpha})&,~{}\mu=2.\end{cases}$

Note already that the constants $\lambda_{+}$ and $\lambda_{-}$ defined by eq.
(6.1) satisfy the set of relations

(6.11) $\displaystyle\lambda_{+}\lambda_{-}=1$ $\displaystyle\wedge$
$\displaystyle\displaystyle\frac{1}{\lambda_{+}-\lambda_{-}}\left(\frac{1}{\lambda_{-}}-\frac{1}{\lambda_{+}}\right)=1.$

Then, by a straightforward application of Lemma 4.3, one has that
$\mathcal{L}^{-1}[U_{0}(s,\xi)](t)$, computed as above, simplifies to

$\displaystyle\begin{cases}\displaystyle
1-\frac{1}{\lambda_{+}-\lambda_{-}}\left(\frac{E_{\alpha,1}(-\lambda_{+}|\xi|^{\sigma}t^{\alpha})-1}{\lambda_{+}}-\frac{E_{\alpha,1}(-\lambda_{-}|\xi|^{\sigma}t^{\alpha})-1}{\lambda_{-}}\right)&,~{}\mu\neq
2\\\ \ \\\ \displaystyle
1+|\xi|^{\sigma}t^{\alpha}\left(E_{\alpha,\alpha+1}^{2}(-|\xi|^{\sigma}t^{\alpha})-E_{\alpha,\alpha+1}(-|\xi|^{\sigma}t^{\alpha})\right)&,~{}\mu=2\end{cases}$
(6.12)
$\displaystyle=\begin{cases}\displaystyle\frac{E_{\alpha,1}(-\lambda_{+}|\xi|^{\sigma}t^{\alpha})}{1-(\lambda_{+})^{2}}+\frac{E_{\alpha,1}(-\lambda_{-}|\xi|^{\sigma}t^{\alpha})}{1-(\lambda_{-})^{2}}&,~{}\mu\neq
2\\\ \ \\\ \displaystyle
E_{\alpha,1}(-|\xi|^{\sigma}t^{\alpha})+\frac{|\xi|^{\sigma}t^{\alpha}}{\alpha}E_{\alpha,\alpha}(-|\xi|^{\sigma}t^{\alpha})&,~{}\mu=2.\end{cases}$

Thereafter, by applying the Laplace convolution identity (4.2) one has that
eq. (6.3) is equivalent to

(6.13)
$\displaystyle\widehat{u}(t,\xi)=\mathcal{L}^{-1}[U_{0}(s,\xi)](t)~{}\widehat{u_{0}}(\xi)+\int_{0}^{t}\mathcal{L}^{-1}[M(s,\xi)](t-\tau)~{}\widehat{f}(\tau,\xi)d\tau.$

The above representation formula provides a solution for the Cauchy Problem
(6.2), whereby the terms $\mathcal{L}^{-1}[M(s,\xi)](t)$ and
$\mathcal{L}^{-1}[U_{0}(s,\xi)](t)$ are given by (6.2) and (6.2),
respectively.

Finally, from the Fourier convolution formula (4.13) one can moreover say that

$\displaystyle
u(t,x)=\left({\bm{N}}_{\sigma,n}(t,\cdot|\alpha,\mu)*u_{0}\right)(x)+\int_{0}^{t}\left[~{}{\bm{M}}_{\sigma,n}(t-\tau,\cdot|\alpha,\beta,\mu)\ast
f(\tau,\cdot)~{}\right](x)~{}d\tau,$

where ${\bm{M}}_{\sigma,n}(t,x|\alpha,\beta,\mu)$ and
${\bm{N}}_{\sigma,n}(t,x|\alpha,\mu)$ stand for the kernel functions
introduced in Notation 6.1, concluding in this way the proof of Theorem 6.3. ∎

###  6.3. Solution of the Cauchy Problem 2.2

######  Theorem 6.4.

Let $\frac{1}{2}<\alpha\leq 1$, $2\alpha\leq\beta<2\alpha+1$, $\sigma>0$,
$\mu>0$ be given. If $u_{0},u_{1}\in\mathcal{S}(\mathbb{R}^{n})$ and
$f(t,\cdot)\in\mathcal{S}(\mathbb{R}^{n})$, for every $t\geq 0$, then
$u:[0,\infty)\times{\mathbb{R}}^{n}\longrightarrow{\mathbb{R}}$ defined
through

$\displaystyle u(t,x)$ $\displaystyle=$
$\displaystyle({\bm{N}}_{\sigma,n}(t,\cdot|\alpha,\mu)*u_{0})(x)+({\bm{J}}_{\sigma,n}(t,\cdot|\alpha,\mu)*u_{1})(x)$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\left[~{}{\bm{M}}_{\sigma,n}(t-\tau,\cdot|\alpha,\beta,\mu)\ast
f(\tau,\cdot)~{}\right](x)~{}d\tau$

solves the Cauchy Problem 2.2, whereby
${\bm{M}}_{\sigma,n}(t,\cdot|\alpha,\beta,\mu)$,
${\bm{N}}_{\sigma,n}(t,\cdot|\alpha,\mu)$ and
${\bm{J}}_{\sigma,n}(t,\cdot|\alpha,\mu)$ denote the kernel functions
introduced in Notation 6.1 and Notation 6.2.

###### Proof.

Following mutatis mutandis the scheme of proof adopted in Theorem 6.3, we
recall that for $\frac{1}{2}<\alpha\leq 1$ and $2\alpha\leq\beta<2\alpha+1$,
one has that $u(t,x)$ solves the Cauchy Problem 2.2 if, and only if,
$\widehat{u}(t,\xi)=(\mathcal{F}u)(t,\xi)$ is a solution of

(6.14)
$\begin{cases}\displaystyle\partial^{2\alpha}_{t}\widehat{u}(t,\xi)+\mu|\xi|^{\sigma}\partial^{\alpha}_{t}\widehat{u}(t,\xi)+|\xi|^{2\sigma}\widehat{u}(t,\xi)=I^{\beta-2\alpha}_{0^{+}}\widehat{f}(t,\xi)&,\mbox{in}~{}(0,\infty)\times\mathbb{R}^{n}\\\
\ \\\ \widehat{u}(0,\xi)=\widehat{u_{0}}(\xi)&,\mbox{in}~{}\mathbb{R}^{n}\\\ \
\\\
\partial_{t}\widehat{u}(0,\xi)=\widehat{u_{1}}(\xi)&,\mbox{in}~{}\mathbb{R}^{n},\end{cases}$

with $\widehat{u_{0}}(\xi)=(\mathcal{F}u_{0})(\xi)$,
$\widehat{u_{1}}(\xi)=(\mathcal{F}u_{1})(\xi)$ and
$I^{\beta-2\alpha}_{0^{+}}\widehat{f}(t,\xi)=I^{\beta-2\alpha}_{0^{+}}(\mathcal{F}f)(t,\xi)$.

By ensuring that $1\leq 2\alpha<2$ and $0\leq\beta-2\alpha<1$, we note already
that

$\displaystyle\mathcal{L}[\partial^{2\alpha}_{t}\widehat{u}(t,\xi)](s)$
$\displaystyle=$ $\displaystyle
s^{2\alpha}\mathcal{L}[\widehat{u}(t,\xi)](s)-s^{2\alpha-1}\widehat{u_{0}}(\xi)-s^{2\alpha-2}\widehat{u_{1}}(\xi),$
$\displaystyle\mathcal{L}[\partial^{\alpha}_{t}\widehat{u}(t,\xi)](s)$
$\displaystyle=$ $\displaystyle
s^{\alpha}\mathcal{L}[\widehat{u}(t,\xi)](s)-s^{\alpha-1}\widehat{u_{0}}(\xi),$
$\displaystyle\mathcal{L}[I^{\beta-2\alpha}_{0^{+}}\widehat{f}(t,\xi)](s)$
$\displaystyle=$ $\displaystyle
s^{2\alpha-\beta}\mathcal{L}[\widehat{f}(t,\xi)](s)$

results from eqs. (4.3) and (4.4), and from the initial conditions underlying
to the Cauchy Problem (6.14). Then, $\mathcal{L}[\widehat{u}(t,\xi)](s)$ is of
the form

(6.15)
$\displaystyle\mathcal{L}[\widehat{u}(t,\xi)](s)=U_{0}(s,\xi)\widehat{u_{0}}(\xi)+U_{1}(s,\xi)\widehat{u_{1}}(\xi)+M(s,\xi)\widehat{f}(t,\xi),$

with

(6.16) $\displaystyle U_{1}(s,\xi)$ $\displaystyle=$
$\displaystyle\frac{s^{2\alpha-2}}{s^{2\alpha}+\mu|\xi|^{\sigma}s^{\alpha}+|\xi|^{2\sigma}},$

and $M(s,\xi)$, $U_{0}(s,\xi)$ defined viz eqs. (6.4) and (6.5), respectively.

Hence, the solution of the Cauchy Problem 2.2 is of the form

$\displaystyle u(t,x)$ $\displaystyle=$
$\displaystyle({\bm{N}}_{\sigma,n}(t,\cdot|\alpha,\mu)*u_{0})(x)+(\mathcal{F}^{-1}\mathcal{L}^{-1})[U_{1}(s,\xi)](t)*u_{1}(x)$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\left({\bm{M}}_{\sigma,n}(t-\tau,\cdot|\alpha,\beta,\mu)*f(\tau,\cdot)\right)(x)~{}d\tau.$

To conclude the proof it remains us to compute, at a first glance, the
function $\mathcal{L}^{-1}[U_{1}(s,\xi)](t)$ by means of the inverse of the
Laplace transform, and afterwards, use the Fourier inversion formula, provided
by eq. (4.10), to show that
$(\mathcal{F}^{-1}\mathcal{L}^{-1})[U_{1}(s,\cdot)](t)$ coincides with the
function ${\bm{J}}_{\sigma,n}(t,x|\alpha,\mu)$ introduced on Notation 6.2.

To do so, we start to observe that $U_{1}(s,\xi)$ and $M(s,\xi)$ (see eq.
(6.4)) are interrelated by the formula $U_{1}(s,\xi)=s^{\beta-2}M(s,\xi)$.
Then,

(6.17) $\displaystyle
U_{1}(s,\xi)=\begin{cases}\displaystyle\frac{1}{\lambda_{+}-\lambda_{-}}\left(\frac{\lambda_{+}~{}s^{\alpha-2}}{s^{\alpha}+\lambda_{+}|\xi|^{\sigma}}-\frac{\lambda_{-}~{}s^{\alpha-2}}{s^{\alpha}+\lambda_{-}|\xi|^{\sigma}}\right)&,~{}~{}\mu\neq
2\\\ \ \\\
\displaystyle\frac{s^{2\alpha-2}}{(s^{\alpha}+|\xi|^{\sigma})^{2}}&,~{}\mu=2\end{cases}$

is rather immediate from eq. (6.8).

Thus, from the Laplace identity (4.8), one obtains that
$\mathcal{L}^{-1}[U_{1}(s,\xi)](t)$ equals to

$\displaystyle\begin{cases}\displaystyle
t~{}\frac{\lambda_{+}E_{\alpha,2}(-\lambda_{+}|\xi|^{\sigma}t^{\alpha})-\lambda_{-}E_{\alpha,2}(-\lambda_{-}|\xi|^{\sigma}t^{\alpha})}{\lambda_{+}-\lambda_{-}}&,~{}\mu\neq
2\\\ \ \\\ \displaystyle
tE_{\alpha,2}^{2}(-|\xi|^{\sigma}t^{\alpha})&,~{}\mu=2,\end{cases}$

Moreover, from direct application of Lemma 4.3,
$\mathcal{L}^{-1}[U_{1}(s,\xi)](t)$ simplifies to

(6.18) $\displaystyle\begin{cases}\displaystyle
t~{}\frac{\lambda_{+}E_{\alpha,2}(-\lambda_{+}|\xi|^{\sigma}t^{\alpha})-\lambda_{-}E_{\alpha,2}(-\lambda_{-}|\xi|^{\sigma}t^{\alpha})}{\lambda_{+}-\lambda_{-}}&,~{}\mu\neq
2\\\ \ \\\
\displaystyle\frac{t}{\alpha}\left(E_{\alpha,1}(-|\xi|^{\sigma}t^{\alpha})-(\alpha-1)E_{\alpha,2}(-|\xi|^{\sigma}t^{\alpha})\right)&,~{}\mu=2\end{cases}$

Finally, by employing the Fourier inversion formula (4.10) to the previous
formula we obtain the kernel function ${\bm{J}}_{\sigma,n}(t,x|\alpha,\mu)$,
as desired. ∎

###  6.4. Strichartz estimates

Having already gathered the main technical results in Section 5, we are now in
a position to investigate the size and decay of the solution of Cauchy Problem
2.1 and Cauchy Problem 2.2, from its closed-form representations, in terms of
the celebrated Strichartz estimates (cf. [46, Chapter 2.]). On the formulation
of the required assumptions, we need to consider the following regions of the
$O\sigma\gamma$ plane:

(6.19)
$\displaystyle\mathcal{S}_{0}(\alpha)=\left\\{(\sigma,\gamma)~{}:~{}0<\gamma<n~{}\wedge~{}n-\gamma<\sigma\leq\alpha\gamma~{}\right\\};$
(6.20)
$\displaystyle\mathcal{S}_{1,2}(\alpha)=\left\\{(\sigma,\gamma)~{}:~{}\displaystyle
0<\gamma<\frac{n}{2}~{}\wedge~{}\displaystyle\frac{n}{2}-\gamma<\sigma\leq\alpha\gamma~{}\right\\};$
(6.21)
$\displaystyle\mathcal{S}_{3}(\alpha)=\left\\{(\sigma,\gamma)~{}:~{}\displaystyle\frac{n}{2}\leq\gamma<n~{}\wedge~{}0<\sigma\leq\alpha\gamma~{}\right\\}.$

######  Assumption 6.5.

We assume that the $4-$tuple $(\sigma,\gamma,p,q)$ satisfy, at least, one of
the following conditions:

  * (i)

$(\sigma,\gamma)\in\mathcal{S}_{0}(\alpha)$ $\wedge$ $(p,q)=(1,\infty)$
[ see eq. (6.19) ];

  * (ii)

$(\sigma,\gamma)\in\mathcal{S}_{1,2}(\alpha)$ $\wedge$
$(p,q)\in\mathcal{R}_{1,2}\left(\sigma+\varepsilon\right)\cap\mathcal{R}_{1,2}\left(\sigma+\nu\right)$
[see eqs. (6.20) and (5.15)];

  * (iii)

$(\sigma,\gamma)\in\mathcal{S}_{3}(\alpha)$ $\wedge$
$(p,q)\in\mathcal{R}_{3}\left(\varepsilon,\sigma+\varepsilon\right)\cap\mathcal{R}_{3}\left(\nu,\sigma+\nu\right)$
[ see eqs. (6.21) and (5.16) ],

with

$\displaystyle\varepsilon:=\gamma-\frac{\sigma}{\alpha}$ and
$\displaystyle\nu:=\gamma-\frac{\sigma}{\alpha}\cdot\beta.$

######  Assumption 6.6.

We assume that the following set of conditions on the function
$f:[0,T]\times{\mathbb{R}}^{n}\longrightarrow{\mathbb{R}}$:

  * (I)

$f(\tau,\cdot)\in\dot{W}^{\gamma-\frac{\sigma}{\alpha},p}({\mathbb{R}}^{n})\cap\dot{W}^{\gamma-\frac{\sigma}{\alpha}\cdot\beta,p}({\mathbb{R}}^{n})$;

  * (II)

$f\in L^{\infty}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})$;

  * (III)

$\displaystyle\left\|~{}(-\Delta)^{\frac{\gamma}{2}-\frac{\sigma}{2\alpha}\cdot\beta}f(\tau,\cdot)~{}\right\|_{p}\lesssim\tau^{\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)-1}\|f\|_{L^{\infty}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})}$,

hold for every $\tau\in[0,T]$.

We start with the following theorem, where we prove a retarded Strichartz
estimate for $f(t,x)$, appearing on the solution representation of the Cauchy
Problem 2.1 & Cauchy Problem 2.2. That will be crucial afterwards on the proof
of Theorem 6.8 & Theorem 6.9, when it will be considered the initial
datum/data

$u_{0}\in\dot{W}^{\gamma-\frac{\sigma}{\alpha},p}({\mathbb{R}}^{n})$ resp.
$(u_{0},u_{1})\in\dot{W}^{\gamma,p}({\mathbb{R}}^{n})\times\dot{W}^{\gamma-\frac{\sigma}{\alpha},p}({\mathbb{R}}^{n})$.

######  Theorem 6.7.

Let $0<\alpha\leq 1$, $2\alpha\leq\beta<2\alpha+1$, $\mu>0$ and $\sigma>0$ be
given. Assume further that the $4-$tuple $(\sigma,\gamma,p,q)$ satisfies
Assumption 6.5 and $f:[0,T]\times{\mathbb{R}}^{n}\longrightarrow{\mathbb{R}}$
satisfies Assumption 6.6.

Then, for every $t\in[0,T]$, we find the following estimate to hold:

$\displaystyle\left\|~{}\displaystyle\int_{0}^{t}{\bm{M}}_{\sigma,n}(t-\tau,\cdot|\alpha,\beta,\mu)\ast
f(\tau,\cdot)d\tau~{}\right\|_{q}\lesssim\|f\|_{L^{\infty}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})}.$

###### Proof.

Application of Minkowski inequality for integrals (cf. [35, Theorem 2.4]),
followed by the definition of ${\bm{M}}_{\sigma,n}(t,x|\alpha,\beta,\mu)$ (see
Notation 6.1) gives, for values of $1\leq q<\infty$, the set of inequalities

$\displaystyle\left\|~{}\displaystyle\int_{0}^{t}{\bm{M}}_{\sigma,n}(t-\tau,\cdot|\alpha,\beta,\mu)\ast
f(\tau,\cdot)d\tau~{}\right\|_{q}\leq$
$\displaystyle\leq\int_{0}^{t}\left\|~{}\displaystyle{\bm{M}}_{\sigma,n}(t-\tau,\cdot|\alpha,\beta,\mu)\ast
f(\tau,\cdot)~{}\right\|_{q}d\tau$
$\displaystyle\lesssim\begin{cases}\displaystyle\int_{0}^{t}(t-\tau)^{\beta-1}\max_{\lambda\in\\{\lambda_{+},\lambda_{-}\\}}\left\|~{}E_{\alpha,\beta}(-\lambda(-\Delta)^{\frac{\sigma}{2}}(t-\tau)^{\alpha})f(\tau,\cdot)~{}\right\|_{q}d\tau&,~{}\mu\neq
2\\\ \ \\\
\displaystyle\int_{0}^{t}(t-\tau)^{\beta-1}~{}\left\|~{}E_{\alpha,\alpha}(-(-\Delta)^{\frac{\sigma}{2}}(t-\tau)^{\alpha})f(\tau,\cdot)~{}\right\|_{q}d\tau&,~{}\mu=2.\end{cases}$

For $q=\infty$, the same set of inequalities holds from the norm identity
$\displaystyle\left\|~{}\displaystyle\cdot~{}\right\|_{\infty}=\lim_{q\rightarrow\infty}\left\|~{}\displaystyle\cdot~{}\right\|_{q}$
and from a straightforward application of dominated convergence theorem (cf.
[35, Theorem 1.8]).

Thereafter, under the conditions of Assumption 6.5 one gets, by direct
application of parts (1) and (2) of Corollary 5.13 that

$\displaystyle\left\|~{}\displaystyle\int_{0}^{t}{\bm{M}}_{\sigma,n}(t-\tau,\cdot|\alpha,\beta,\mu)\ast
f(\tau,\cdot)d\tau~{}\right\|_{q}\lesssim$
$\displaystyle\lesssim\int_{0}^{t}(t-\tau)^{\beta-1-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\cdot\beta\right)\right)}~{}\left\|~{}(-\Delta)^{\frac{\gamma}{2}-\frac{\sigma}{2\alpha}\cdot\beta}f(\tau,\cdot)~{}\right\|_{p}~{}d\tau$
$\displaystyle=\int_{0}^{t}(t-\tau)^{-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)}~{}\left\|~{}(-\Delta)^{\frac{\gamma}{2}-\frac{\sigma}{2\alpha}\cdot\beta}f(\tau,\cdot)~{}\right\|_{p}~{}d\tau$

hold for every $t\in[0,T]$.

Next, from the combination of the conditions (I), (II) and (III) considered on
Assumption 6.6 for the function
$f:[0,T]\times\mathbb{R}^{n}\longrightarrow\mathbb{R}$, we conclude that

(6.22)
$\displaystyle\left\|~{}\displaystyle\int_{0}^{t}{\bm{M}}_{\sigma,n}(t-\tau,\cdot|\alpha,\beta,\mu)\ast
f(\tau,\cdot)d\tau~{}\right\|_{q}\lesssim\Lambda(t)~{}\|f\|_{L^{\infty}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})},$

with

$\displaystyle\Lambda(t):=\int_{0}^{t}(t-\tau)^{-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)}~{}\tau^{\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)-1}~{}d\tau.$

We note that $\Lambda(t)$ can be determined with the aid of the Laplace
convolution formula (4.2), whereby the existence of the Laplace transforms

$\displaystyle\mathcal{L}\left\\{t^{-\eta}\right\\}(s)$ and
$\displaystyle\mathcal{L}\left\\{t^{\eta-1}\right\\}(s)$, with
$\displaystyle\eta=\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)$,

follow from the condition

(6.23) $\displaystyle\displaystyle
0<\alpha-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)<\alpha.$

Indeed, condition (I) of Assumption 6.6 ensures that
$f(\tau,\cdot)\in\dot{W}^{\gamma-\frac{\sigma}{\alpha},p}({\mathbb{R}}^{n})$,
since

$\dot{W}^{\gamma-\frac{\sigma}{\alpha},p}({\mathbb{R}}^{n})\cap\dot{W}^{\gamma-\frac{\sigma}{\alpha}\cdot\beta,p}({\mathbb{R}}^{n})\subseteq\dot{W}^{\gamma-\frac{\sigma}{\alpha},p}({\mathbb{R}}^{n}).$

Then, by setting $\displaystyle\varepsilon:=\gamma-\frac{\sigma}{\alpha},$ we
recall that the set of parameter constraints, that results from parts (1) and
(2) of Corollary 5.13 for the substitution $\gamma\rightarrow\varepsilon$,
gives rise to the following conditions:

  * (i)

In case of
$(\sigma,\gamma)\in\mathcal{S}_{0}\left(\alpha\right)~{}\wedge~{}(p,q)=(1,\infty)$,
one has

$0<\frac{n}{p}-\frac{n}{q}-\varepsilon<\sigma.$

  * (ii)

In case of
$(\sigma,\gamma)\in\mathcal{S}_{1,2}\left(\alpha\right)~{}\wedge~{}(p,q)\in\mathcal{R}_{1,2}\left(\sigma+\varepsilon\right)$,
one has

$\frac{n}{2}-\varepsilon\leq\frac{n}{p}-\frac{n}{q}-\varepsilon<\min\left\\{n-\varepsilon,\sigma\right\\}.$

  * (iii)

In case of
$(\sigma,\gamma)\in\mathcal{S}_{3}\left(\alpha\right)~{}\wedge~{}(p,q)\in\mathcal{R}_{3}\left(\varepsilon,\sigma+\varepsilon\right)$

$0<\frac{n}{p}-\frac{n}{q}-\varepsilon<\min\left\\{n-\varepsilon,\sigma\right\\}.$

Thus, $\displaystyle
0<\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\nu\right)<\alpha,$ and
hence, condition (6.23) is always satisfied. Moreover, combination of the
Laplace convolution formula (4.2) with Laplace inversion gives

$\displaystyle\Lambda(t)$ $\displaystyle=$
$\displaystyle\Gamma\left(1-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)\right)\Gamma\left(\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)\right)$
$\displaystyle=$
$\displaystyle\dfrac{\pi}{\sin\left(\frac{\alpha\pi}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)\right)},$

which is a positive constant independent of the time variable $t$.

Thus, for every $t\in[0,T]$ the inequality (6.22) becomes

$\displaystyle\left\|~{}\displaystyle\int_{0}^{t}{\bm{M}}_{\sigma,n}(t-\tau,\cdot|\alpha,\beta,\mu)\ast
f(\tau,\cdot)d\tau~{}\right\|_{q}\lesssim\|f\|_{L^{\infty}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})},$

as desired. ∎

######  Theorem 6.8.

Let $0<\alpha\leq\frac{1}{2}$, $2\alpha\leq\beta<2\alpha+1$, $\mu>0$ and
$\sigma>0$ be given.

Assume further the following:

  * •

Assumption 5.12 holds for the constants $\lambda_{+}$ and $\lambda_{-}$
defined viz (6.1);

  * •

Assumption 6.5 holds for the $4-$tuple $(\sigma,\gamma,p,q)$;

  * •

Assumption 6.6 holds for the function
$f:[0,T]\times{\mathbb{R}}^{n}\longrightarrow{\mathbb{R}}$.

Then, for every datum
$u_{0}\in\dot{W}^{\gamma-\frac{\sigma}{\alpha},p}({\mathbb{R}}^{n})$ one has
the following Strichartz estimates for the solution of the Cauchy Problem 2.1,
$u:[0,T]\times{\mathbb{R}}^{n}\longrightarrow{\mathbb{R}}$, obtained in
Theorem 6.3:

$\displaystyle\|u\|_{L^{s}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})}$
$\displaystyle\lesssim$ $\displaystyle
T^{\frac{1}{s}-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)}~{}\left\|~{}(-\Delta)^{\frac{\gamma}{2}-\frac{\sigma}{2\alpha}}u_{0}~{}\right\|_{p}$
$\displaystyle+$ $\displaystyle
T^{\frac{1}{s}}~{}\|f\|_{L^{\infty}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})},$

only if

(6.24) $\displaystyle~{}1\leq
s<\frac{1}{\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)}.$

###### Proof.

Starting from the definition of ${\bm{N}}_{\sigma,n}(t,x|\alpha,\mu)$ (see
Notation 6.1), one has

$\displaystyle\left\|~{}\displaystyle{\bm{N}}_{\sigma,n}(t,\cdot|\alpha,\mu)\ast
u_{0}~{}\right\|_{q}\lesssim$
$\displaystyle\lesssim\begin{cases}\displaystyle\max_{\lambda\in\\{\lambda_{+},\lambda_{-}\\}}\left\|~{}E_{\alpha,1}\left(-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}\right)u_{0}~{}\right\|_{q}&,~{}\mu\neq
2\\\ \ \\\
\displaystyle~{}\left\|~{}E_{\alpha,1}\left(-(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}\right)u_{0}~{}\right\|_{q}+t^{\alpha}\left\|~{}(-\Delta)^{\frac{\sigma}{2}}E_{\alpha,\alpha}\left(-(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}\right)u_{0}~{}\right\|_{q}&,~{}\mu=2.\end{cases}$

Thereafter, under the Assumption 6.6 for the constants $\lambda_{+}$ and
$\lambda_{-}$ defined through eq. (6.1), the estimate

$\left\|~{}\displaystyle{\bm{N}}_{\sigma,n}(t,\cdot|\alpha,\mu)\ast
u_{0}~{}\right\|_{q}\lesssim
t^{-\frac{\alpha}{\sigma}\left(\frac{n}{q}+\frac{n}{2}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)}~{}\left\|~{}(-\Delta)^{\frac{\gamma}{2}-\frac{\sigma}{2\alpha}}u_{0}~{}\right\|_{p}$

follows by direct application of parts (1) and (2) of Corollary 5.13 to the
substitution $\gamma\rightarrow\gamma-\frac{\sigma}{\alpha}$. On the other
hand, Assumption 6.6 ensures that Theorem 6.7 can be applied to the function
$f:[0,T]\times{\mathbb{R}}^{n}\longrightarrow{\mathbb{R}}$.

Thus, for the solution of the Cauchy Problem 2.1, determined in Theorem 6.3,
we end up with the following dispersive estimate

$\|u(t,\cdot)\|_{q}\lesssim
t^{-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)}~{}\left\|~{}(-\Delta)^{\frac{\gamma}{2}-\frac{\sigma}{2\alpha}}u_{0}~{}\right\|_{p}+\|f\|_{L^{\infty}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})}$.

Moreover, by adopting the mixed-norm (4.17) of
$L^{s}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})$, the sequence of
inequalities

$\displaystyle\|u\|_{L^{s}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})}$
$\displaystyle\leq$
$\displaystyle\left(\int_{0}^{T}t^{-s\cdot\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)}\|~{}(-\Delta)^{\frac{\gamma}{2}-\frac{\sigma}{2\alpha}}u_{0}~{}\|_{2}^{s}~{}dt\right)^{\frac{1}{s}}$
$\displaystyle+$
$\displaystyle\left(\int_{0}^{T}\|f\|_{L^{\infty}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})^{s}}^{s}dt\right)^{\frac{1}{s}}$
$\displaystyle\lesssim$ $\displaystyle
T^{\frac{1}{s}-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)}\left\|~{}(-\Delta)^{\frac{\gamma}{2}-\frac{\sigma}{2\alpha}}u_{0}~{}\right\|_{p}+T^{\frac{1}{s}}\|f\|_{L^{\infty}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})}$

results from the Minkowski inequality and by standard integration techniques,
whereby the condition (6.24) is to ensure the monotonicity of the norm of
$L^{s}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})$ and the equality

(6.25)
$\displaystyle\displaystyle\int_{0}^{T}t^{-s\cdot\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)}dt=\frac{T^{1-s\cdot\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)}}{1-s\cdot\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)}.$

This concludes the proof of Theorem 6.8. ∎

######  Theorem 6.9.

Let $0<\alpha\leq\frac{1}{2}$, $2\alpha\leq\beta<2\alpha+1$, $\mu>0$ and
$\sigma>0$ be given.

Assume further the following:

  * •

Assumption 5.12 holds for the constants $\lambda_{+}$ and $\lambda_{-}$
defined viz (6.1);

  * •

Assumption 6.5 holds for the $4-$tuple $(\sigma,\gamma,p,q)$;

  * •

Assumption 6.6 holds for the function
$f:[0,T]\times{\mathbb{R}}^{n}\longrightarrow{\mathbb{R}}$.

Then, for every data
$(u_{0},u_{1})\in\dot{W}^{\gamma,p}({\mathbb{R}}^{n})\times\dot{W}^{\gamma-\frac{\sigma}{\alpha},p}({\mathbb{R}}^{n})$,
the solution of the Cauchy Problem 2.2,
$u:[0,T]\times{\mathbb{R}}^{n}\longrightarrow{\mathbb{R}}$, obtained in
Theorem 6.4 satisfies the Strichartz estimate

$\displaystyle\|u\|_{L^{s}_{t}L^{q}_{x}\left([0,T]\times{\mathbb{R}}^{n}\right)}$
$\displaystyle\lesssim$ $\displaystyle
T^{\frac{1}{s}-\frac{\alpha}{\sigma}\left(\frac{n}{q}+\frac{n}{2}-\gamma\right)}\left(~{}\|~{}(-\Delta)^{\frac{\gamma}{2}}u_{0}~{}\|_{p}+\|~{}(-\Delta)^{\frac{\gamma}{2}-\frac{\sigma}{2\alpha}}u_{1}~{}\|_{p}~{}\right)$
$\displaystyle+$ $\displaystyle
T^{\frac{1}{s}}~{}\|f\|_{L^{\infty}_{t}L^{q}_{x}\left([0,T]\times{\mathbb{R}}^{n}\right)}$

only if

(6.26) $\displaystyle 1\leq
s<\frac{1}{\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\gamma\right)}.$

###### Proof.

From the proof of Theorem 6.8 we can immediately infer that for the auxiliar
function $v:[0,T]\times{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$, defined viz

$\displaystyle
v(t,x):=\left({\bm{N}}_{\sigma,n}(t,\cdot|\alpha,\mu)*u_{0}\right)(x)+\int_{0}^{t}\left[~{}{\bm{M}}_{\sigma,n}(t-\tau,\cdot|\alpha,\beta,\mu)\ast
f(\tau,\cdot)~{}\right](x)~{}d\tau,$

the following Strichartz estimate

$\displaystyle\displaystyle\|v\|_{L^{s}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})}$
$\displaystyle\lesssim$ $\displaystyle
T^{\frac{1}{s}-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\gamma\right)}\|~{}(-\Delta)^{\frac{\gamma}{2}}u_{0}~{}\|_{p}+T^{\frac{1}{s}}\|f\|_{L^{\infty}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{n})}$

holds true for values of $\frac{1}{2}<\alpha\leq 1$. Thus, it remains to show
that

(6.27)
$\displaystyle\left(\int_{0}^{T}\left\|~{}\displaystyle{\bm{J}}_{\sigma,n}(t,\cdot|\alpha,\mu)\ast
u_{1}~{}\right\|_{q}^{s}dt\right)^{\frac{1}{s}}\lesssim
T^{\frac{1}{s}-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{q}-\gamma\right)}~{}\|~{}(-\Delta)^{\frac{\gamma}{2}-\frac{\sigma}{2\alpha}}u_{1}~{}\|_{p},$

holds for every
$u_{1}\in\dot{W}^{\gamma-\frac{\sigma}{\alpha},p}({\mathbb{R}}^{n})$.

Indeed, from the definition of ${\bm{J}}_{\sigma,n}(t,x|\alpha,\mu)$ provided
by Notation 6.2, one has

$\displaystyle\left\|~{}\displaystyle{\bm{J}}_{\sigma,n}(t,\cdot|\alpha,\mu)\ast
u_{1}~{}\right\|_{q}\lesssim$
$\displaystyle\lesssim\begin{cases}\displaystyle\max_{\lambda\in\\{\lambda_{+},\lambda_{-}\\}}t~{}\left\|~{}E_{\alpha,2}\left(-\lambda(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}\right)u_{1}~{}\right\|_{q}&,~{}\mu\neq
2\\\ \ \\\
\displaystyle~{}t\left\|~{}E_{\alpha,1}\left(-(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}\right)u_{1}~{}\right\|_{q}+t\left\|~{}E_{\alpha,2}\left(-(-\Delta)^{\frac{\sigma}{2}}t^{\alpha}\right)u_{1}~{}\right\|_{q}&,~{}\mu=2.\end{cases}$

Then, following mutatis mutandis the scheme of proof employed in Theorem 6.7,
the estimate

$\left\|~{}\displaystyle{\bm{J}}_{\sigma,n}(t,\cdot|\alpha,\mu)\ast
u_{1}~{}\right\|_{q}\lesssim
t^{-\frac{\alpha}{\sigma}\left(\frac{n}{p}-\frac{n}{p}-\gamma\right)}~{}\|~{}(-\Delta)^{\frac{\gamma}{2}-\frac{\sigma}{2\alpha}}u_{1}~{}\|_{2}$

results by an algebraic manipulation, based on the equality

$\displaystyle
t^{-\frac{\alpha}{\sigma}\left(\frac{n}{q}+\frac{n}{2}-\gamma\right)}=t\times
t^{-\frac{\alpha}{\sigma}\left(\frac{n}{q}+\frac{n}{2}-\left(\gamma-\frac{\sigma}{\alpha}\right)\right)}.$

Hence, the estimate (6.27) follows from the condition (6.26). ∎

######  Remark 6.10.

The well-posedness of Cauchy Problem 2.1 & Cauchy Problem 2.2, provided by
Theorem 6.8 and Theorem 6.9 respectively, is ensured by the combination
dispersive estimates obtained in Corollary 5.13 with the retarded Strichartz
estimate, obtained the Theorem 6.7.

Assumptions 5.12 & 6.5$\frac{1}{2}<\alpha\leq 1$$f$Assumption
6.6$(u_{0},u_{1})$$\dot{W}^{\gamma,p}\times\dot{W}^{\gamma-{\frac{\sigma}{\alpha}},p}$
data$0<\alpha\leq\frac{1}{2}$$f$Assumption
6.6$u_{0}$$\dot{W}^{\gamma-\frac{\sigma}{\alpha},p}$ datum Figure 4. Binary
tree diagram sketch of the set of main assumptions on required on Theorem 6.8
and Theorem 6.9.
$\sigma$$\gamma$$n\alpha$$n$$\frac{n\alpha}{1+\alpha}$$\frac{n}{1+\alpha}$$n$
Figure 5. Region $\mathcal{S}_{0}(\alpha)$ ($\frac{1}{2}<\alpha<1$) adopted on
Assumption 6.5.
$\sigma$$\gamma$$n\alpha$$n$$\frac{n}{2}$$\frac{n}{2}$$\frac{n\alpha}{2(1+\alpha)}$$\frac{n}{2(1+\alpha)}$$\frac{n\alpha}{2}$$n$
Figure 6. Regions $\mathcal{S}_{1,2}(\alpha)$ and $\mathcal{S}_{3}(\alpha)$
($\frac{1}{2}<\alpha<1$) adopted on Assumption 6.5.
$\mathcal{S}_{1,2}(\alpha)$ corresponds to the dark grey region whereas
$\mathcal{S}_{3}(\alpha)$ corresponds to union of the light grey region with
the straight line
$\displaystyle\left\\{\left(\sigma,\frac{n}{2}\right)~{}:~{}0<\sigma\leq\frac{n\alpha}{2}\right\\}.$

##  7\. Conclusions

###  7.1. Summary of results

We have investigated questions related to dispersive and Strichartz estimates
for the Cauchy Problem 2.1 and the Cauchy Problem 2.2 from their closed-form
representations.

The framework developed in Section 5 constitute a significant part of the
paper. Along this we have pinned a general scheme to achieve the optimality of
the decay estimates, independently of the dimension of the ambient space
${\mathbb{R}}^{n}$. As a consequence, the main estimates collected in
Corollary 5.13, permits us to treat $L^{p}-L^{q}$ and
$L^{p}-\dot{W}^{\sigma,q}$ decay estimates, at least, in lower dimensions with
respect to $\sigma$ (see also Proposition 5.5). That also includes
$L^{1}-L^{\infty}$ and $L^{1}-\dot{W}^{\sigma,\infty}$ decay estimates,
although we are unable to claim the sharpness of these estimates, for the
reasons explained throughout Subsection 5.1.

Note also that in case where such estimates are not possible to reach – such
as the wave equation with structural damping highlighted on the beginning of
the paper [6] – our approach offers the possibility to look instead for
$\dot{W}^{\gamma,p}-L^{q}$ and $\dot{W}^{\gamma,p}-\dot{W}^{\sigma,q}$ decay
estimates, whose datum is a function with membership on the homogeneous
Sobolev space $\dot{W}^{\gamma,p}({\mathbb{R}}^{n})$ (see also Proposition
5.8).

In Section 6, we have used the estimates obtained in Section 5 to prove first,
in Theorem 6.7, a retarded Strichartz estimate for the inhomogeneous part of
the Cauchy Problem 2.1 & Cauchy Problem 2.2. Afterwards, by combining
Corollary 5.13 and Theorem 6.7, we were able to prove, in Theorem 6.8 and
Theorem 6.9, a primer version of Strichartz estimates to both Cauchy problems.

Regarding Theorem 6.7, it is worth to observe that its proof turned out to be
rather technical. Arguably, we have adopted in Assumption 6.6 the most
strongest regularity and decay properties on $\displaystyle
f:[0,T]\times{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$ with the aim achieve a
similar version of the so-called Christ-Kiselev lemma (cf. [46, Lemma 2.4]).
Amongst the many still open questions, obtaining a well-adapted version of
[27, THEOREM 1.2.] to our case is indeed an interesting issue to be deepened
in forthcoming research papers.

In comparison with other approaches, our major contribution stems, firstly, on
the possibility of proving decay estimates for both Cauchy problems without
studying the asymptotic behaviour of the fundamental solution. In particular,
we note that the fundamental solutions encoded by higher dimensional
fractional differential equations – that is, the convolution kernels that
yield from the Fourier convolution formula – are mostly Fox-H functions (cf.
[30, Subsection 1.12]). And, as it was highlighted on the paper [29],
obtaining optimal decay estimates for it revealed to be a non-trivial task.

Also, we notice that the study of the size and the decay of the solutions of
Cauchy problems of heat type, possessing memory terms, has been extensively
studied during the last years (see e.g. [28, 11] and the references given
there). And prior to the study of dispersive estimates, there seems that the
study of Strichartz estimates for the solutions of Cauchy problems, similar to
Cauchy Problem 2.1 & Cauchy Problem 2.2, has not yet been fully addressed, up
to authors’s knowledge. The present paper aims also to fill this gap.

###  7.2. Further remarks and open problems

The wide class of space-time-fractional operators, considered on the
formulation of Cauchy Problem 2.1 & Cauchy Problem 2.2, correspond to a
borderline case of the following operator:

(7.1)
$\displaystyle\partial_{t}^{\gamma}+\mu(-\Delta)^{\theta}\partial_{t}^{\alpha}+(-\Delta)^{\sigma},$
$\displaystyle\mbox{with}~{}0<\theta\leq\sigma,~{}~{}0<\alpha\leq
1,~{}~{}\alpha<\gamma\leq 2.$

For such general class of operators, Luchko-Gorenflo’s operational approach
considered in [36] provides us a general framework to determine, in a similar
fashion, closed-form representations to the ones obtained in Theorem 6.2 and
Theorem 6.3. On this case, the convolution kernels turned out be described in
terms of multi-index Mittag-Leffler functions (see [36, Theorem 4.1] and [36,
eq. (46)]).

At this point, it should be stressed that the choice of $\gamma=2\alpha$ on
eq. (7.1), already considered e.g. in [39, 31] for the time-fractional
telegraph equation, provides us a fairly accurate time-fractional counterpart
of the $\sigma-$models investigated in depth on the series of papers [40, 12,
14, 15]. Up to our knowledge, for such class of multi-index functions the
monotonicity and decay properties are not very well-known on the literature
(cf. [24, Chapter 6]) so that one cannot embody it in our scheme of proof to
obtain dispersive estimates and to plan tangible applications towards
relaxation theory as well, as tactically explained in Section 2. Therefore,
for the general case the following interesting open questions remain:

  1. Q1:

Is it possible to achieve endpoint decay estimates for the multi-index Mittag-
Leffler functions? At least, using an asymptotic scheme similar to the one
considered in [29]?

  2. Q2:

Under which choices of $\gamma$ and $\alpha$ are possible to achieve
dispersive estimates, similar to those obtained in Section 5?

For the choice $\gamma=2\alpha$, it seems likely that our results can be
extended straightforwardly for general choices of $0<\theta\leq\sigma$, since
with the aid of eq. (4.8) we are able to say, from its solution
representation, that one needs to study, in most of the cases, the decay of
the Fourier multipliers of the type

$\displaystyle\displaystyle\frac{1}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}~{}\frac{E_{\alpha,1}\left(-|\xi|^{\sigma}\lambda_{\pm}(\xi)~{}t^{\alpha}\right)}{\lambda_{\pm}(\xi)},$
$\displaystyle\displaystyle\frac{\lambda_{\pm}(\xi)~{}t^{\beta-1}~{}E_{\alpha,\beta}\left(-|\xi|^{\sigma}\lambda_{\pm}(\xi)~{}t^{\alpha}\right)}{\lambda_{+}(\xi)-\lambda_{-}(\xi)},$

($\lambda_{+}(\xi)\neq\lambda_{-}(\xi)$) and
$t^{\beta-1}E_{\alpha,\beta}^{2}(-|\xi|^{\sigma}t^{\alpha})$
($\lambda_{+}(\xi)=\lambda_{-}(\xi)$), whereby

$\displaystyle\lambda_{\pm}(\xi)=\begin{cases}\displaystyle\frac{\mu}{2}|\xi|^{2\theta-\sigma}\pm
i\sqrt{1-\frac{\mu^{2}}{4}|\xi|^{4\theta-2\sigma}}&,0<\mu<2|\xi|^{\sigma-2\theta}\\\
\ \\\
\displaystyle\frac{\mu}{2}|\xi|^{2\theta-\sigma}\pm\sqrt{\frac{\mu^{2}}{4}|\xi|^{4\theta-2\sigma}-1}&,\mu\geq
2|\xi|^{\sigma-2\theta}.\end{cases}$

In case where $\theta=\dfrac{\sigma}{2}$ we are in conditions to claim that
our results are enough to cover the so-called critical case (i.e. when
$\lambda_{\pm}(\xi)$ are constants), but we cannot yet claim it for the
remaining cases $\displaystyle 0<\theta<\frac{\sigma}{2}$ and
$\displaystyle\frac{\sigma}{2}<\theta\leq\sigma$, already treated by Pham et
al. in [40] for values of $\alpha=\beta=1$. The main gap to be to be
circumvented is the fact that the terms

$\displaystyle\dfrac{1}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}$ and
$\displaystyle\dfrac{\lambda_{\pm}(\xi)}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}$

and unbounded for values of
$\displaystyle|\xi|^{\sigma-2\theta}\approx\frac{\mu}{2}$. However, based on
the identity $\lambda_{+}(\xi)\cdot\lambda_{-}(\xi)=1$, the following question
comes naturally:

  1. Q3:

Can we also obtain estimates, similar to the ones reached in Section 5, to
regions defined implicitly by

$\displaystyle|\lambda_{+}(\xi)-\lambda_{-}(\xi)|\geq\varepsilon\cdot\max\left\\{\mu,\mu|\xi|^{2\theta-\sigma}\right\\}$,
for some $\varepsilon>0$?

Broadly speaking, we essentially conjecture a perturbation argument,
underlying to the asymptotic behaviour of $\varepsilon:=\varepsilon(\tau)$ on
small scales $0<\tau<1$:

$\displaystyle\varepsilon\sim\tan\left(\frac{\pi\tau}{2}\right)$
$\left(~{}0<\mu<2|\xi|^{\sigma-2\theta}~{}\right)$ $\wedge$
$\displaystyle\varepsilon\sim\tanh\left(\tau\right)$
$\left(~{}\mu>2|\xi|^{\sigma-2\theta}~{}\right)$.

From the background of semigroup theory, we already know from the abstract
results obtained in [21, Section VI.3.a] that the application of perturbation
arguments, similar to the above one, are mostly feasible in case of
$\displaystyle\frac{\sigma}{2}<\theta\leq\sigma$ – that is, for the case where
the elastic operator is bounded above by the damping operator. But due to the
fact that the generalizations of the framework considered in [40], on which
the authors already treat the case
$\displaystyle\frac{\sigma}{2}<\theta\leq\sigma$, requires in our case to
handle, at most, with subordination formulae for the generalized Mittag-
Leffler functions $E_{\alpha,\beta}$ in terms of probability density functions
defined through the so-called Wright functions (cf. [37]), we decided to
postpone the treatment of the aforementioned remaining cases for a forthcoming
paper.

We conclude this section by noticing that Cordero-Zucco’s approach on
Strichartz estimates for the vibrating plate equation on Sobolev and
modulation spaces (case of $\mu\rightarrow 0^{+}$; $\sigma=2$) (see [9, 10])
also falls into the framework of the class of function spaces and pseudo-
differential operators investigated in depth in Section 5 and Section 6. This
is also an interesting issue to be deepened in a nearly future, as well as the
following broadly general questions:

  1. Q4:

Employing the well-established Littlewood-Paley’s theory, can we relax some of
the results enclosed on this paper?

  2. Q5:

Can we replace the homogeneous Sobolev spaces, adopted throughout this paper,
by homogeneous analogues of Besov spaces?

##  8\. Acknowledgments

Nelson Faustino was supported by The Center for Research and Development in
Mathematics and Applications (CIDMA) through the Portuguese Foundation for
Science and Technology (FCT), references UIDB/04106/2020 and UIDP/04106/2020;
Jorge Marques was supported by Centre for Business and Economics Research
(CeBER) through the Portuguese Foundation for Science and Technology (FCT),
reference UIDB/05037/2020.

The authors would like to thank Antero Neves for his precious advice and help
on creating TikZ code used to build some of the graphics throughout the paper.
Last but not least, the authors would like to thank the reviewers for
carefully reading the preliminary version of the manuscript. Their remarks and
suggestions allowed us to clarify several issues as well as improve some of
the main results.

##  Appendix A Proof of Lemma 5.2

###### Proof.

First, we recall that for $\beta=1$, one has
$\displaystyle\frac{\Gamma(1)}{\Gamma(1+\alpha)}=\frac{1}{\Gamma(1+\alpha)}$.
Thus, application of Theorem 4.1 & Theorem 4.2 gives rise to

$\displaystyle\int_{0}^{\infty}\left|~{}\Gamma(\beta)\rho^{\eta}E_{\alpha,\beta}(-\lambda\rho^{\sigma}t^{\alpha})~{}\right|^{r}\rho^{n-1}d\rho=\int_{0}^{\infty}\left|~{}\Gamma(\beta)E_{\alpha,\beta}(-\lambda\rho^{\sigma}t^{\alpha})~{}\right|^{r}\rho^{n+r\eta-1}d\rho$
(A.1)
$\displaystyle\leq\begin{cases}\displaystyle\int_{0}^{\infty}\frac{\rho^{n+r\eta-1}}{\left(1+\sqrt{\frac{\Gamma(1+\alpha)}{\Gamma(1+2\alpha)}}~{}\lambda\rho^{\sigma}t^{\alpha}\right)^{2r}}d\rho&,\lambda\geq
0;~{}\beta=\alpha\\\ \ \\\
\displaystyle\int_{0}^{\infty}\frac{\rho^{n+r\eta-1}}{\left(1+\frac{\Gamma(\beta)}{\Gamma(\beta+\alpha)}\lambda\rho^{\sigma}t^{\alpha}\right)^{r}}d\rho&,\lambda\geq
0;~{}\beta\in\\{1\\}\cup(\alpha,+\infty)\\\ \ \\\
\displaystyle\int_{0}^{\infty}\frac{C^{r}|\Gamma(\beta)|^{r}\rho^{n+r\eta-1}}{(1+|\lambda|\rho^{\sigma}t^{\alpha})^{r}}d\rho&,\frac{\pi\alpha}{2}<\theta<\pi\alpha;~{}\theta\leq|\arg(\lambda)|\leq\pi,\end{cases}$

where $C>0$ is the constant that results from Theorem 4.1.

For the case of $\eta=-s$ and $a=r$ resp. $a=2r$, the set of constraints (i),
(ii) and (iii), appearing on the statement of Lemma 5.2, assures that

$\displaystyle\Re(r)>\Re\left(\frac{n-rs}{\sigma}\right)>0$ resp.
$\displaystyle\Re(2r)>\Re\left(\frac{n-rs}{\sigma}\right)>0$

so that one can apply the Mellin integral identity (5.5) to simplify the
previous set of inequalities, appearing on the right-hand side of (A). Namely,
one has

$\displaystyle\left|~{}\Gamma(\beta)~{}\right|^{r}\int_{0}^{\infty}\left|~{}E_{\alpha,\beta}(-\lambda\rho^{\sigma}t^{\alpha})~{}\right|^{r}\rho^{n-rs-1}d\rho\leq
C_{r,\sigma,n}^{(s)}(\alpha,\beta,\lambda)t^{-\frac{\alpha}{\sigma}\left({n-rs}\right)}$
$\displaystyle=\begin{cases}\displaystyle\frac{\Gamma\left(\frac{n-rs}{\sigma}\right)\Gamma\left(\frac{r(2\sigma+s)-n}{\sigma}\right)}{\sigma~{}\Gamma(2r)}\left(\sqrt{\frac{\Gamma(1+\alpha)}{\Gamma(1+2\alpha)}}~{}\lambda
t^{\alpha}\right)^{-\frac{n-rs}{\sigma}}\displaystyle,~{}~{}~{}~{}\lambda\geq
0~{};~{}\beta=\alpha\\\ \ \\\
\displaystyle\frac{\Gamma\left(\frac{n-rs}{\sigma}\right)\Gamma\left(\frac{r(\sigma+s)-n}{\sigma}\right)}{\sigma~{}\Gamma(r)}~{}\left(\frac{\Gamma(\beta)}{\Gamma(\beta+\alpha)}~{}\lambda
t^{\alpha}\right)^{-\frac{n-rs}{\sigma}}~{}~{}~{}~{},\lambda\geq
0~{};~{}\beta\in\\{1\\}\cup(\alpha,+\infty)\\\ \ \\\
\displaystyle\frac{C^{r}~{}|\Gamma(\beta)|^{r}\Gamma\left(\frac{n-rs}{\sigma}\right)\Gamma\left(\frac{r(\sigma+s)-n}{\sigma}\right)}{\sigma~{}\Gamma(r)}\left(~{}|\lambda|t^{\alpha}\right)^{-\frac{n-rs}{\sigma}},~{}\frac{\pi\alpha}{2}<\theta<\pi\alpha~{}~{};~{}~{}\theta\leq|\arg(\lambda)|\leq\pi,\end{cases}$

where $C_{r,\sigma,n}^{(s)}(\alpha,\beta,\lambda)$ is the constant defined on
the statement of Lemma 5.2.

Hence, from the monotonicity of the norm underlying to the weighted Lebesgue
space $L^{r}((0,\infty),\rho^{n-1}d\rho)$, there holds that the previous
inequality is equivalent to (5.2), as desired. ∎

##  Appendix B Proof of Lemma 5.3

###### Proof.

We recall first that the set of constraints (i), (ii) and (iii), imposed on
the statement of Lemma 5.3, assures that

$\displaystyle\Re(r)>\Re\left(\frac{n-rs}{\sigma}\right)>0$

so that the Mellin integral formula (5.6) can be indeed applied.

Thereby, by setting $\eta=\sigma-s$ on both sides of eq. (A), the inequality

$\displaystyle\int_{0}^{\infty}\left|~{}\Gamma(\alpha)\rho^{\sigma-s}E_{\alpha,\alpha}(-\lambda\rho^{\sigma}t^{\alpha})~{}\right|^{r}\rho^{n-1}d\rho~{}\leq$
$\displaystyle\leq\displaystyle\int_{0}^{\infty}\frac{\rho^{n+r(\sigma-s)-1}}{\left(1+\sqrt{\frac{\Gamma(1+\alpha)}{\Gamma(1+2\alpha)}}~{}\lambda\rho^{\sigma}t^{\alpha}\right)^{2r}}d\rho~{}~{}(\lambda\geq
0).$

is thus immediate.

Furthermore, by direct application of eq. (5.6) we end up with

$\displaystyle\left(~{}\Gamma(\alpha)~{}\right)^{r}\int_{0}^{\infty}\left|~{}\rho^{\sigma-s}E_{\alpha,\alpha}(-\lambda\rho^{\sigma}t^{\alpha})~{}\right|^{r}\rho^{n-1}d\rho\leq
D_{r,\sigma,n}^{(s)}(\alpha,\lambda)~{}t^{-\frac{\alpha}{\sigma}\left({n+r(\sigma-s)}\right)}$
$\displaystyle=\displaystyle\frac{\Gamma\left(\frac{n+r(\sigma-s)}{\sigma}\right)\Gamma\left(\frac{r(\sigma+s)-n}{\sigma}\right)}{\sigma~{}\Gamma(2r)}\left(\sqrt{\frac{\Gamma(1+\alpha)}{\Gamma(1+2\alpha)}}~{}\lambda
t^{\alpha}\right)^{-\frac{n+r(\sigma-s)}{\sigma}}\displaystyle~{}~{}(\lambda\geq
0)$

and hence with (5.3), after few algebraic manipulations. ∎

##  Appendix C Proof of Corollary 5.13

Firstly, we recall that the Assumption 6.6 for $\lambda\in\mathbb{C}$ results
from the closed formulae for $C_{r,\sigma,n}^{(s)}(\alpha,\beta,\lambda)$ and
$D_{r,\sigma,n}^{(s)}(\alpha,\lambda)$ ($s=0$ or $s=\gamma$), obtained in
Lemma 5.2 and Lemma 5.3. From now on, let us assume that
$\displaystyle\frac{1}{p}+\frac{1}{r^{\prime}}=\frac{1}{q}+1$ and
$\displaystyle\frac{1}{r}+\frac{1}{r^{\prime}}=1$ hold for every $1\leq
p,q,r,r^{\prime}\leq\infty$.

#### Proof of (1)

Note that $\displaystyle\frac{1}{r}=\frac{1}{p}-\frac{1}{q}$. Then, under the
conditions of Proposition 5.5, the set of estimates ((1)) are thus immediate
from the substitution $\displaystyle\frac{n}{r}=\frac{n}{p}-\frac{n}{q}$ on
the right hand of the inequalities (5.5) and (5.5), respectively.

Moreover, the condition $(p,q)=(1,\infty)$ is fulfilled for $r=1$, whereas the
condition $(p,q)\in\mathcal{R}_{1,2}(\sigma)$ – see eq. (5.15) – results from
the set of equivalences

$\displaystyle\max\left\\{1,\frac{n}{\sigma}\right\\}<r\leq 2$
$\displaystyle\Longleftrightarrow$
$\displaystyle\displaystyle\frac{n}{2}\leq\frac{n}{r}<{\min\left\\{n,\sigma\right\\}}$
$\displaystyle\Longleftrightarrow$
$\displaystyle\displaystyle\frac{n}{2}\leq\frac{n}{p}-\frac{n}{q}<{\min\left\\{n,\sigma\right\\}}$
$\displaystyle\Longleftrightarrow$
$\displaystyle\frac{n}{p}-{\min\left\\{n,\sigma\right\\}}\displaystyle<\frac{n}{q}\leq\frac{n}{p}-\frac{n}{2}$

and from the constraints $\displaystyle 0\leq\frac{n}{p}\leq n$ and
$\displaystyle 0\leq\frac{n}{q}\leq n$.

#### Proof of (2)

The proof of the set of estimates ((2)) results from the substitution
$\displaystyle\frac{n}{r}-\gamma=\frac{n}{p}-\frac{n}{q}-\gamma$ on the right
hand of the inequalities (5.8) and (5.8), respectively, whereas the conditions
(i), (ii) and (iii) follow straightforwardly by the set of equivalences:

  * (i)

$\dot{W}^{\gamma,1}-L^{\infty}$ & $\dot{W}^{\gamma,1}-\dot{W}^{\sigma,\infty}$
estimates (first case):

$\displaystyle r=1$ $\displaystyle\Longleftrightarrow$
$\displaystyle(p,q)=(1,\infty);$
$\displaystyle\sigma+\gamma>n~{}~{}\wedge~{}~{}0<\gamma<n$
$\displaystyle\Longleftrightarrow$ $\displaystyle n-\sigma<\gamma<n.$

  * (ii)

$\dot{W}^{\gamma,p}-L^{p}$ & $\dot{W}^{\gamma,p}-\dot{W}^{\sigma,q}$ estimates
(second case):

$\displaystyle\max\left\\{1,\frac{n}{\sigma+\gamma}\right\\}<r\leq
2\Longleftrightarrow$
$\displaystyle\displaystyle\frac{n}{p}-{\min\left\\{n,\sigma+\gamma\right\\}}\displaystyle<\frac{n}{q}\leq\frac{n}{p}-\frac{n}{2}.$
$\displaystyle\sigma+\gamma>\frac{n}{2}~{}~{}\wedge~{}~{}0<\gamma<\frac{n}{2}$
$\displaystyle\Longleftrightarrow$
$\displaystyle\frac{n}{2}-\sigma<\gamma<\frac{n}{2}.$

  * (iii)

$\dot{W}^{\gamma,p}-L^{p}$ & $\dot{W}^{\gamma,p}-\dot{W}^{\sigma,q}$ estimates
(third case):

$\displaystyle\max\left\\{1,\frac{n}{\sigma+\gamma}\right\\}<r\leq\frac{n}{\gamma}\Longleftrightarrow$
$\displaystyle\displaystyle\frac{n}{p}-{\min\left\\{n,{\sigma+\gamma}\right\\}}\displaystyle<\frac{n}{q}\leq\frac{n}{p}-\gamma.$
$\displaystyle\sigma+\gamma>\frac{n}{2}~{}~{}\wedge~{}~{}\frac{n}{2}\leq\gamma<n$
$\displaystyle\Longleftrightarrow$ $\displaystyle\frac{n}{2}\leq\gamma<n.$

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