Cauchy problem for a fractional anisotropic parabolic equation in anisotropic
Hölder spaces.

Sergey Degtyarev

State Agency ”Institute of applied mathematics and mechanics”,

Donetsk, Ukraine

###### Abstract

We consider a Cauchy problem for a fractional anisotropic parabolic equation
in anisotropic Hölder spaces. The equation generalizes the heat equation to
the case of fractional power of the Laplace operator and the power of this
operator can be different with respect to different groups of space variables.
The time derivative can be either fractional Caputo - Jrbashyan derivative or
usual derivative. Under some necessary conditions on the order of the time
derivative we show that the operator of the whole problem is an isomorphism of
appropriate anisotropic Hölder spaces. Under some another conditions we prove
unique solvability of the Cauchy problem in the same spaces.

##  1 Introduction.

The present paper is devoted to correctness in anisotropic Hölder spaces of a
Cauchy problem for a fractional partial differential equation which
generalizes the classical heat equation. Let $N$ be the dimension of the space
$R^{N}$, $T>0$ is a given positive number, $R_{T}^{N}\equiv R^{N}\times(0,T)$.
Let further the set of coordinates of a point $x=(x_{1},x_{2},...,x_{N})$ is
split into $r$ groups of lengths $N_{i}$, $i=1,2,...,r$,
$N_{1}+N_{2}+...N_{r}=N$. Let, besides, $\theta>0$ be an integer or a
noninteger number and $\sigma_{k}>0$, $k=1,2,...,r$. Denote

$z_{1}=(x_{1},...,x_{N_{1}}),z_{2}=(x_{N_{1}+1},...,x_{N_{1}+N_{2}}),...,z_{r}=(x_{N_{1}+...+N_{r-1}+1},...,x_{N}).$
(1.1)

Consider the following Cauchy problem for the unknown function $u(x,t)$,
$(x,t)\in$ $R_{T}^{N}$,

$D_{\ast
t}^{\theta}u(x,t)+{\sum_{k=1}^{r}}(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u(x,t)=f(x,t),\quad(x,t)\in
R_{T}^{N},$ (1.2) $\frac{\partial^{i}u}{\partial t^{i}}(x,0)=u_{i}(x),x\in
R^{N},\quad\genfrac{\\{}{.}{0.0pt}{0}{i=0,1,...,[\theta],\quad\theta\ \text{is
a noninteger},}{i=0,1,...,\theta-1,\quad\theta\ \text{is an integer},}$ (1.3)

where $[\theta]$ is the integer part of a number $\theta$. Let’s explain the
above notations. Firstly, here $f(x,t)$ and $u_{0}(x)$ are some given
functions, defined in $\overline{R_{T}^{N}}$ and $R^{N}$ correspondingly.
Further, the symbol $D_{\ast t}^{\theta}$ (with the lower asterisk) means the
fractional Caputo - Jrbashyan derivative (often called just the Caputo
derivative) of a non-integer order $\theta>0$ with respect to $t$. Such
derivative of order $\theta\in(0,1)$ is defined by

$D_{\ast
t}^{\theta}g(t)=\frac{1}{\Gamma(1-\theta)}{\int\limits_{0}^{t}}\frac{g^{\prime}(\tau)d\tau}{(t-\tau)^{\theta}}=\frac{1}{\Gamma(1-\theta)}\frac{d}{dt}{\int\limits_{0}^{t}}\frac{\left[g(\tau)-g(0)\right]d\tau}{(t-\tau)^{\theta}},\quad
t>0,$ (1.4)

and for the case of order $\theta\in(n-1,n)$

$D_{\ast t}^{\theta}g(t)=D_{\ast t}^{\theta-n+1}g^{(n-1)}(t).$ (1.5)

The second equality in (1.4) links the Caputo - Jrbashyan derivative to the
classical Riemann - Liouville fractional derivative $D_{t}^{\theta}$ (without
the lower asterisk), and the last for $\theta\in(0,1)$ is defined by

$D_{t}^{\theta}g(t)=\frac{1}{\Gamma(1-\theta)}\frac{d}{dt}{\int\limits_{0}^{t}}\frac{g(\tau)d\tau}{(t-\tau)^{\theta}},\quad
t>0.$ (1.6)

Thus, for $\theta\in(0,1)$,

$D_{\ast t}^{\theta}g(t)=D_{t}^{\theta}\left[g(t)-g(0)\right].$ (1.7)

In general, for $\theta\in(n-1,n)$ ($n$ is a positive integer), the Riemann -
Liouville fractional derivative is defined by

$D_{t}^{\theta}g(t)=\frac{1}{\Gamma(1-\theta)}\frac{d^{n}}{dt^{n}}{\int\limits_{0}^{t}}\frac{g(\tau)d\tau}{(t-\tau)^{\theta-n+1}},\quad
t>0,$ (1.8)

and the Caputo - Jrbashyan derivative can be expressed as

$D_{\ast
t}^{\theta}g(t)=\frac{1}{\Gamma(1-\theta)}{\int\limits_{0}^{t}}\frac{g^{(n)}(\tau)d\tau}{(t-\tau)^{\theta-n+1}}=$
(1.9)
$=\frac{1}{\Gamma(1-\theta)}\frac{d}{dt}{\int\limits_{0}^{t}}\frac{\left[g^{(n-1)}(\tau)-g^{(n-1)}(0)\right]d\tau}{(t-\tau)^{\theta-n+1}}=$
$=\frac{1}{\Gamma(1-\theta)}\frac{d^{n}}{dt^{n}}{\int\limits_{0}^{t}}\frac{\left[g(\tau)-{\sum\limits_{k=0}^{n-1}}\frac{g^{(k)}(0)}{k!}\tau^{k}\right]d\tau}{(t-\tau)^{\theta-n+1}}=$
$=D_{t}^{\theta}\left[g(t)-{\sum\limits_{k=0}^{n-1}}\frac{t^{k}g^{(k)}(0)}{k!}\right],\quad
t>0.$

As for different definitions of some others fractional derivatives, we can
refer to, for example, [1], [2], [3], Ch. 3. We consider the above Cauchy
problem with the Caputo - Jrbashyan derivative with respect to time $t$
because it is well known that the Cauchy problem with initial condition (1.3)
in the case of the Riemann - Liouville derivative in the equation is incorrect
and it does not carry a proper physical meaning. The correct statement for
equations with the Riemann - Liouville derivative must include some nonlocal
condition instead of (1.3) - see [1], [3], Ch. 3. In the present paper we
consider a Cauchy problem with usual initial condition (1.3) and therefore we
use namely the Caputo - Jrbashyan derivative in equation (1.2).

At last, the summands $(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u(x,t)$ in
equation (1.2), $k=1,...,r,$ are fractional powers of the Laplace operators of
powers $\sigma_{k}/2>0$ with respect to the corresponding group of the space
variables $z_{k}$. These fractional operators can be defined in terms of the
Fourier transform as follows. Let $z_{k}=(x_{j+1},x_{j+1},...,x_{j+N_{k}})$ be
such a group of the space variables and let the corresponding group of ”dual”
(in the sense of the Fourier transform) group of variables be
$\zeta_{k}\equiv(\xi_{j+1},...,\xi_{j+N_{k}})$. Denote by $F_{k}[u]$ the
Fourier image of a function $u(x,t)$ with respect to the variables $z_{k}$
that is

$F_{k}[u](x_{1},...,x_{j},\zeta_{k},x_{j+N_{k}+1},...,x_{N},t)\equiv$
$\equiv\frac{1}{(2\pi)^{\frac{N_{k}}{2}}}{\int\limits_{z_{k}\in
R^{N_{k}}}}u(x_{1},...,x_{j},z_{k},x_{j+N_{k}+1},...,x_{N},t)e^{-i(z_{k},\zeta_{k})}dz_{k}.$
(1.10)

Then by definition

$(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u(x,t)\equiv
F_{k}^{-1}[|\zeta_{k}|^{\sigma_{k}}F_{k}[u]],$ (1.11)

where $F_{k}^{-1}$ is the inverse transform to (1.10). Besides, operator
$(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}$ from (1.11) can be defined for
$\sigma_{k}>0$ as follows (see, for example, [1], Ch.5, [3], Ch.3). Denote
finite difference of a function $u(x,t)$ with respect to a group of space
variables $z_{k}$ with a step $\eta_{k}$ by

$\delta_{\eta_{k},z_{k}}u(x,t)=u(x_{1},...,x_{j},z_{k}+\eta_{k},x_{j+N_{k}+1},...,x_{N},t)-u(x,t)$

and denote the corresponding finite difference of an order $m>1$ by

$\delta_{\eta_{k},z_{k}}^{m}u(x,t)=\delta_{\eta_{k},z_{k}}\left(\delta_{\eta_{k},z_{k}}^{m-1}u(x,t)\right).$

Then

$(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u(x,t)=C_{N_{k},\sigma_{k},m}{\int\limits_{\eta_{k}\in
R^{N_{k}}}}\frac{\delta_{\eta_{k},z_{k}}^{m}u(x,t)}{|\eta_{k}|^{N_{k}+\sigma_{k}}}d\eta_{k},$
(1.12)

where $m>\sigma_{k}$ is arbitrary, $C_{N_{k},\sigma_{k},m}$ is some constant,
which depends only on $m$, on the dimension $N_{k}$ of the group of the
variables $z_{k}$, and on $\sigma_{k}$.

The question of a possible domain of definition for the operators $D_{\ast
t}^{\theta}$ and $(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}$ will be discussed
a little bit below.

Problem (1.2), (1.3) belongs to a class of mathematical models with fractional
differentiation. Similar models arises in very many contemporary
investigations in different branches of science and technology. In particular,
such models describe different processes in fractal media and differential
operators like in (1.2) serve as generators for definite random processes,
which are actively investigated at present. It is impossible even outline here
all different applications of the model under consideration and for details we
refer the reader to the monographs [1], [2], [3], to the survey papers [4],
[5], and to a plenty of other recent monographs, devoted to investigations and
applications of fractional models.

Note that we can observe nowadays almost explosive growth of investigations on
properties of mathematical models with fractional differential operators in
view of their great importance and numerous applications. At the same time
problems for equations of the kind (1.2), that contains and a fractional time
derivative, and a fractional Laplace operator are investigated quite a little.
As for the investigations of correctness for such problems in classes of
smooth functions (especially up to the initial time moment), the author is
aware of just a few papers, devoted to such questions. Therefore we describe
shortly only the main known results related to the issues studied in this
paper. And we stress that we do not comment, for example, all the papers with
different results on fundamental solutions for more-less related equations, -
at least just because we do not use the methods of explicit fundamental
solutions.

Nevertheless, we start with the paper , which contains the fundamental
solution for the ”doubly fractional” equation (in our notations)

$D_{t}^{\theta}u(x,t)+(-\Delta_{x})^{\frac{\sigma}{2}}u(x,t)=0,\quad(x,t)\in
R^{N}\times(0,\infty)$

under the restrictions $0<\theta<2$, $0<\sigma\leq 2$ on the orders of
fractional differentiation. It was shown that in dependence on relations
between $\theta$, $\sigma$, and the space dimension $N$, the fundamental
solution can be either positive or changing it’s sign.

In the paper [7] the maximum principle is proved for an initial problem for a
similar equation with nonzero right hand side.

A Cauchy problem for the equation with the usual first derivative in time

$\frac{\partial u}{\partial
t}(x,t)+(-\Delta_{x})^{\frac{\sigma}{2}}u(x,t)=f(x,t),\quad(x,t)\in
R^{N}\times(0,\infty)$

is considered in [8]. The paper deals with two types of ”heat” potentials,
which formally give in their sum a solution to the problem according to the
Duhamel formula. These potentials have as their kernel the fundamental
solution for the equation. One of them, a volume potential, is associated with
the right hand side $f(x,t)$ of the equation, and the another, an initial
potential, - with the initial datum $u_{0}(x)$. The paper gives for these
potentials estimates in the spaces $L_{p}([0,T],F_{c}^{a,b}(R^{N}))$ and
$L_{p}([0,T],B_{c}^{a,b}(R^{N}))$, where $F_{c}^{a,b}(R^{N})$ is the Triebel -
Lizorkin space, and $B_{c}^{a,b}(R^{N})$ is the Besov - Lipschitz space with
respect to the space variables. It is known that the Besov - Lipschitz spaces
in their particular case $B_{c}^{\infty,\infty}(R^{N})$ include the Hölder
spaces, and the paper contains estimates for the volume potential (with the
density $f(x,t)$) for this particular case and for $p=\infty$. Thus the paper
gives estimates of the volume potential in the space of functions with the
bounded in time norm in the space $C^{\sigma+\alpha}(R^{N})$ with respect to
$x$ under the condition of boundedness in time of the density $f(x,t)$ in the
space $C^{\alpha}(R^{N})$, $\alpha\in(0,1)$. However, the range of the spaces,
considered in the paper for the initial potential with the density $u_{0}(x)$,
does not contain estimates in the space
$L_{\infty}([0,T],B_{\alpha}^{\infty,\infty}(R^{N}))=L_{\infty}([0,T],C^{\alpha}(R^{N}))$.
That is the classical smoothness of the initial potential (both with respect
time and space) is not considered.

An analogous Cauchy problem for a similar equation with $f(x,t)\equiv 0$ was
investigated in [9]. In this paper the initial data can have a growth at
infinity with the restriction

$|u_{0}(x)|\leq C(1+|x|)^{\sigma-\varepsilon},\quad\varepsilon>0.$

For such initial data an existence and uniqueness of the classical solution
is, in particular, proved - in the sense that the solution itself and it’s
derivatives, included in the equation, are continuous, and the solution
approaches it’s initial datum in the classical sense.

The paper [10] is devoted, in particular, to smoothness issues of the solution
to the Cauchy problem for the equation

$\frac{\partial u}{\partial
t}(x,t)+\emph{L}u(x,t)=\emph{L}f(x,t),\quad(x,t)\in R^{N}\times(0,\infty)$

with the usual derivative in time and with rather general nonlocal operator L
with respect to the space variables. This operator generalizes space operator
from equation (1.2) in the sense of (1.12) (but for the same order of
fractional differentiation in each space direction
$\sigma_{1}=...=\sigma_{r}=\sigma$). Note that the right hand side
$\emph{L}f(x,t)$ in the equation is a distribution since smoothness of the
function $f(x,t)$ from an anisotropic Hölder space is less than the order of
operator L. Therefore the paper deals with a weak solution and the initial
data are supposed from some Lebesgue space. It is shown that this solution
$u(x,t)$ for $t>0$ inherits the smoothness properties of the function
$f(x,t)$, and thus $u(x,t)$ belongs to the same anisotropic Hölder space as
$f(x,t)$ with some estimate of Hölder seminorm of $u(x,t)$ over the same
seminorm of $f(x,t)$.

The papers [11], [12], [13] are also devoted to studying of equations of the
form

$\frac{\partial u}{\partial t}(x,t)+\emph{L}u(x,t)=f(x,t),\quad(x,t)\in
R^{N}\times(0,\infty)$ (1.13)

with some nonlocal operator L, which generalizes the fractional Laplace
operator $(-\Delta_{x})^{\frac{\sigma}{2}}$ in the sense of definition (1.12).

Moreover, the paper [11] deals with a completely nonlinear operator L and the
equation has the form

$\frac{\partial u}{\partial
t}(x,t)=\inf_{a\in\mathit{A}}(\emph{L}_{a}u(x,t)+f_{a}(x,t)),$

where a parameter $a$ runs through some index set $\mathit{A}$. For a solution
to this equation sharp local (inner) Schauder and some other estimates of
smoothness are obtained.

The paper [12] is also devoted to studying of equation (1.13). For the Cauchy
problem with zero initial condition existence and uniqueness of the Hölder
smooth solution is obtained. Moreover, under the assumption that the right
hand side $f(x,t)$ has a finite Hölder seminorm with respect to the space
variables $x$ there were proved sharp partial Schauder estimates with respect
to the same space variables $x$. Under the same assumption there were also
proved some interesting estimates of the smoothness with respect to time $t$
for the solution.

The paper [13] in it’s turn contains sharp inner Schauder estimates for
equation (1.13) in some natural Hölder space and some interesting boundary
estimates.

The papers [14], [15] deals with Cauchy problem (1.2), (1.3) with a fractional
derivative in time and with the usual Laplacian as the space operator. Here
some estimates of the solution in different Sobolev spaces are obtained.

Further, the papers [16] \- [19] are devoted to studying of abstract parabolic
equation with a fractional time derivative in Banach spaces.

Some other questions of regularity and qualitative behavior of solutions to
fractional equations of the kind were considered, in particular, in [21] \-
[27].

###### Remark 1.1

Let us stress that all the above regularity results and estimates do not
contain complete sharp coercive estimates in smooth functional classes up to
the initial time moment $t=0$. And question of qualified smoothness up to the
initial time moment with a corresponding sharp estimate in the case of smooth
initial data is still open. Therefore the goal of the present paper is to find
conditions for smoothness of possible solutions to (1.2), (1.3) up to $t=0$
likewise it takes place for usual parabolic equations.

Naturally, since the present paper is not a survey, it can not give even a
brief description of all huge amount of the existing results on regularity for
fractional parabolic equations. Therefore we confine ourselves to some known
to us existing investigations that are mostly adjacent (in our opinion !) to
the issues that are the main purpose of the paper.

Nevertheless, it is very important to refer here the mostly recent papers [28]
and [29], which were published after the present paper was prepared. These
papers are closely related to the questions under our consideration. They
consider a fractional parabolic equation with a general nonlocal space
operator, which generalizes the fractional Laplacian. The investigations in
[28] and [29] are based on the methods of operator semigroups and include
properties of the corresponding semigroups. In particular, along with other
questions, the questions of the Schauder estimates for the related equations
are considered.

Besides, we would like to refer shortly one more paper [30], which also was
published after the present paper was prepared. Here the authors derive long
time $L_{p}-L_{q}$ decay estimates, in the full range $1\leq p,q\leq\infty$,
for the time-dependent Fourier multipliers

$\widetilde{m}(t,\xi)=e^{\pm i|\xi|^{\sigma}t-|\xi|^{\theta}t},$

which correspond to the Cauchy problem for the homogeneous equation

$u_{tt}+(-\Delta)^{\sigma}u+(-\Delta)^{\frac{\theta}{2}}u_{t}=0.$

The subsequent content of the paper is as follows. In the next section we
define standard anisotropic Hölder spaces and those of them that corresponds
to the anisotropy of equation (1.2). These particular spaces will be the
working spaces for our considerations of problem (1.2), (1.3). The section is
concluded by the formulation of the main results of the paper in terms of the
mentioned spaces.

In section 3 we formulate some (mainly known) results on operators of
fractional differentiation in isotropic Hölder spaces.

Sections 4 and 5 are also auxiliary and they are devoted to the actions of
fractional differentiation on Schwartz and Lizorkin spaces and also on their
dual spaces of distributions.

In sections 6 and 7 we study operators of fractional differentiation in
anisotropic Hölder spaces. The results of this section show that the operator
of problem (1.2), (1.3) is a bounded linear operator in the corresponding
anisotropic spaces.

Below, to prove the existence of the bounded inverse operator to the operator
of problem (1.2), (1.3), we use some results on Fourier multipliers in
anisotropic Hölder spaces. Therefore we formulate these results in section 8.

The fact of existence (under proper conditions) of the mentioned inverse
operator to problem (1.2), (1.3) is proved in the subsequent sections 9 \- 14.

At that we fist consider separately in sections 9 and 10 the cases of minimal
fractional and integer orders of differentiation with respect to time
$\theta\in(0,1)$ and $\theta=1$. Here the exponents of smoothness in time are
supposed to be $\theta+\theta\alpha$ and $1+\alpha$ correspondingly, where
$\theta\alpha\in(0,1)$ and $\alpha\in(0,1)$.

Then, in sections 11 and 12 we show that the smoothness of the solution to
(1.2), (1.3) rises in accordance with the rising of the data of the problem.
That is we in fact consider the case of arbitrary high smoothness of the data.

To move to the problems with an arbitrary large order of time differentiation
$\theta>1$, we consider next the question of constructing of functions from
anisotropic Hölder spaces with given initial functional values at $t=0$ up to
maximal possible order. The corresponding construction is described in section
13. This is necessary to reduce an initial problem to a problem with zero
initial data. We can not use for that the known results on this subject since
such results are absent for the case of an irrational anisotropy of Hölder
spaces.

At last, section 14 concludes the proofs of the main theorems 2.1, 2.3 and 2.5
below.

##  2 Functional spaces and formulation of the main results.

In this paper we use some natural for equation (1.2) anisotropic Hölder spaces
of functions with different smoothness with respect to different variables.
Let $\overline{l}=(l_{1},l_{2},...,l_{N})$, where $l_{i}$ are arbitrary
positive non-integer numbers. Denote by $C^{\overline{l}}(R^{N})$ the Banach
space of functions $u(x)$, $x\in R^{N}$, with the finite norm

$\left\|u\right\|_{C^{\overline{l}}(R^{N})}\equiv|u|_{R^{N}}^{(\overline{l})}=|u|_{R^{N}}^{(0)}+{\sum\limits_{i=1}^{N}}\left\langle
u\right\rangle_{x_{i},R^{N}}^{(l_{i})},$ (2.1) $|u|_{R^{N}}^{(0)}=\sup_{x\in
R^{N}}|u(x)|,$ (2.2) $\left\langle
u\right\rangle_{x_{i},R^{N}}^{(l_{i})}=\sup_{x\in
R^{N},h>0}\frac{|D_{x_{i}}^{[l_{i}]}u(x_{1},x_{2},...,x_{i}+h,...,x_{N})-D_{x_{i}}^{[l_{i}]}u(x)|}{h^{l_{i}-[l_{i}]}}.$
(2.3)

Here $[l_{i}]$ the integer part of a number $l_{i}$, $D_{x_{i}}^{[l_{i}]}u(x)$
is the derivative of a function $u(x)$ of order $[l_{i}]$ with respect to a
variable $x_{i}$. Seminorm (2.3) can be equivalently defined as ([31] \- [34])

$\left\langle u\right\rangle_{x_{i},R^{N}}^{(l_{i})}\simeq\sup_{x\in
R^{N},h>0}\frac{|\delta_{h,x_{i}}^{k}u(x)|}{h^{l_{i}}},$ (2.4)

where $\delta_{h,x_{i}}=u(x_{1},x_{2},...,x_{i}+h,...,x_{N})-u(x)$ represents
the difference of a function $u(x)$ with respect to a variable $x_{i}$ with a
step $h$,
$\delta_{h,x_{i}}^{k}u(x)=\delta_{h,x_{i}}\left(\delta_{h,x_{i}}^{k-1}u(x)\right)=\left(\delta_{h,x_{i}}\right)^{k}u(x)$
is the difference of an arbitrary fixed order $k>l_{i}$. It is known (see, for
example, [32]), that functions from the space $C^{\overline{l}}(R^{N})$ admit
also some mixed derivatives up to a definite order depending on the set of the
exponents $l_{i}$. At that all the mixed and ”pure” derivatives
$D_{x_{i}}^{[l_{i}]}$ have finite Hölder seminorms with some exponents with
respect to all the variables. Namely, let
$\overline{k}=(k_{1},k_{2},...,k_{N})$, $k_{i}\leq[l_{i}]$ and

$\omega=1-{\sum\limits_{i=1}^{N}}\frac{k_{i}}{l_{i}}>0,$ (2.5)

and $\overline{d}=(d_{1},d_{2},...,d_{N})$, where $d_{i}=\omega l_{i}$. Then

$D_{x}^{\overline{k}}u(x)\in
C^{\overline{d}}(R^{N}),\quad\left\|D_{x}^{\overline{k}}u(x)\right\|_{C^{\overline{d}}(R^{N})}\leq
C(N,\overline{l},\overline{k})\left\|u(x)\right\|_{C^{\overline{l}}(R^{N})}.$
(2.6)

In the present paper we are going to consider solutions to problem (1.2),
(1.3), that is functions $u(x,t)$ defined in the domain $R_{T}^{N}\equiv
R^{N}\times(0,T)$ (including the case $R_{\infty}^{N}=R^{N}\times(0,\infty)$).
For such domains in the space $R^{N+1}$ all definitions and properties in
(2.1) - (2.6) (for the space $R^{N}$) are also valid with respect to all
variables $(x,t)\in R^{N+1}$. In view of the character of equation (1.2) and
in view of our splitting of the whole set of the space variables into $r$
groups $z_{k}$ of length $N_{k}$ (see definition (1.1)), we introduce now some
additional notations. Let $\alpha\in(0,1)$ be chosen in a way that numbers
$\theta\alpha$ and $\theta+\theta\alpha$ are positive non-integers, where
$\theta$ is the order of the derivative in $t$ from equation (1.2). We suppose
that functions $u(x,t)$ under consideration have smoothness in $t$ of order
$\theta+\theta\alpha$ in the sense of definition (2.3) that is the value of
$\left\langle u\right\rangle_{t,\overline{R_{T}^{N}}}^{(\theta+\theta\alpha)}$
is finite. We suppose also that for each group $z_{k}$ from (1.1) the
smoothness order of $u(x,t)$ with respect to each space variable from $z_{k}$
is $\sigma_{k}(1+\alpha)$, where $\sigma_{k}\alpha$ and
$\sigma_{k}+\sigma_{k}\alpha$ are non-integers. That is for each space
variable $x_{i}$ inside $z_{k}$ the seminorm $\left\langle
u\right\rangle_{x_{i},\overline{R_{T}^{N}}}^{\sigma_{k}(1+\alpha)}$ is finite.
Denote the total Hölder seminorm with respect to the group $z_{k}$ by
$\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{\sigma_{k}(1+\alpha)}$, that is

$\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{\sigma_{k}(1+\alpha)}\equiv{\sum\limits_{x_{i}\in
z_{k}}}\left\langle
u\right\rangle_{x_{i},\overline{R_{T}^{N}}}^{\left(\sigma_{k}(1+\alpha)\right)}.$
(2.7)

Denote, besides, the set of the orders of fractional differentiation with
respect to different groups of space variables in equation (1.2) by
$\overline{\sigma}=(\sigma_{1},\sigma_{2},...,\sigma_{r})$ and the set of the
smoothness exponents with respect to different groups by
$\overline{\sigma}(1+\alpha)=(\sigma_{1}(1+\alpha),\sigma_{2}(1+\alpha),...,\sigma_{r}(1+\alpha))$.
Denote, at last, the Hölder space of functions $u(x,t)$ with described
anisotropic smoothness by
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$.
That is
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$ is
the Hölder space of bounded and continuous in the closed domain
$\overline{R_{T}^{N}}$ functions with the finite over $\overline{R_{T}^{N}}$
norm

$\left\|u\right\|_{C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})}\equiv$
$\equiv|u|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}(1+\alpha),\theta+\theta\alpha)}=|u|_{\overline{R_{T}^{N}}}^{(0)}+{\sum\limits_{k=1}^{r}}\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\sigma_{k}(1+\alpha))}+\left\langle
u\right\rangle_{t,\overline{R_{T}^{N}}}^{(\theta+\theta\alpha)},$ (2.8)

where $\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\sigma_{k}(1+\alpha))}$ are
defined in (2.7) and

$|u|_{\overline{R_{T}^{N}}}^{(0)}=\sup_{(x,t)\in\overline{R_{T}^{N}}}|u(x,t)|.$
(2.9)

Besides, we denote by $C^{\overline{\sigma}(1+\alpha)}(R^{N})$ the space of
functions $u(x)$ with dependance only on the space variables, with $R^{N}$ as
the domain of their definition, and with defined above smoothness in the space
variables that is

$\left\|u\right\|_{C^{\overline{\sigma}(1+\alpha)}(R^{N})}\equiv|u|_{R^{N}}^{(\overline{\sigma}(1+\alpha))}=|u|_{R^{N}}^{(0)}+{\sum\limits_{k=1}^{r}}\left\langle
u\right\rangle_{z_{k},R^{N}}^{(\sigma_{k}(1+\alpha))}.$ (2.10)

At the same time, for the right hand side $f(x,t)$ of equation (1.2) we use a
Hölder space with a lower smoothness - according to the orders of
differentiation in (1.2) in $t$ and $x$. Namely, we use the space
$C^{\overline{\sigma}\alpha,\theta\alpha}(\overline{R_{T}^{N}})$ with the norm

$\left\|f\right\|_{C^{\overline{\sigma}\alpha,\theta\alpha}(\overline{R_{T}^{N}})}\equiv|f|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}\alpha,\theta\alpha)}=|f|_{\overline{R_{T}^{N}}}^{(0)}+{\sum\limits_{k=1}^{r}}\left\langle
f\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\sigma_{k}\alpha)}+\left\langle
f\right\rangle_{t,\overline{R_{T}^{N}}}^{(\theta\alpha)}.$ (2.11)

Note that all the above definitions of functional spaces are preserved and in
the case of bounded domains in $R^{N}$ and $R_{T}^{N}$.

In what follows we will use also the closed subspace of the space
$C^{\overline{\sigma}\alpha,\theta\alpha}(\overline{R_{T}^{N}})$ with elements
$f(x,t)$, that are identically equal to zero at $t=0$, $f(x,0)\equiv 0$. We
denote this subspace by
$\underline{C}^{\overline{\sigma}\alpha,\theta\alpha}(\overline{R_{T}^{N}})$.
And analogously we denote by
$\underline{C}^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$
and
$\underline{C}^{\overline{\sigma}(1+\alpha),\theta;\overline{\sigma}\alpha}(\overline{R_{T}^{N}})$
(underlined) the closed subspaces of the corresponding spaces consisting of
functions that equal to zero at $t=0$ together with all their derivatives in
$t$ up to the order $[\theta]$.

Turning now to the original problem (1.2), (1.3), we consider it as a linear
operator L in the introduced functional spaces. That is for an integer
$\theta=n$

$\emph{L:\,}C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})\rightarrow$
$\rightarrow
C^{\overline{\sigma}\alpha,\theta\alpha}(\overline{R_{T}^{N}})\times
C^{\overline{\sigma}(1+\alpha)}(R^{N})\times
C^{\overline{\sigma}(1+\alpha)-\frac{1}{n}\overline{\sigma}}\times...\times
C^{\overline{\sigma}(1+\alpha)-\frac{n-1}{n}\overline{\sigma}},$ (2.12)

and for a non-integer $\theta$ (under some conditions)

$\emph{L:\,}C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})\rightarrow$
$\rightarrow
C^{\overline{\sigma}\alpha,\theta\alpha}(\overline{R_{T}^{N}})\times
C^{\overline{\sigma}(1+\alpha)}(R^{N})\times
C^{\overline{\sigma}(1+\alpha)-\frac{1}{\theta}\overline{\sigma}}(R^{N})\times...\times
C^{\overline{\sigma}(1+\alpha)-\frac{[\theta]}{\theta}\overline{\sigma}}.$
(2.13)

This operator maps a function $u(x,t)$ from the space
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$ to
it’s image under applying the whole differential operator from (1.2) (that is
the corresponding function $f(x,t)$ from the space
$C^{\overline{\sigma}\alpha,\theta\alpha}(\overline{R_{T}^{N}})$) and it’s
initial traces $u_{0}(x)=u(x,0)$, $u_{1}(x)=u_{t}(x,0)$, … ,
$u_{n-1}(x)=u_{t}^{(n-1)}(x,0)$ ( …,$u_{[\theta]}(x)=u_{t}^{([\theta])}(x,0)$
) up to the order $n-1$ ( in the case of an integer $\theta=n$) or up to the
order $[\theta]$ in the case of a non-integer $\theta$. The correctness of
this definitions is stated by the following main theorems of the present
papers.

###### Theorem 2.1

Let $\theta=n>0$ be an integer and not a number of the form $n\neq 4k+2$,
$k=0,1,...$. Let further $\alpha$ be a positive non-integer such that
$n\alpha$ is a non-integer. Let, at last, $\sigma_{k}$, $k=1,...,r$, are
positive numbers such that $\sigma_{k}\alpha$ and
$\sigma_{k}+\sigma_{k}\alpha$, $k=1,...,r$, are non-integers. Then the
operator L is a linear isomorphism from the space
$C^{\overline{\sigma}(1+\alpha),n+n\alpha}(\overline{R_{T}^{N}})$ to the space
$C^{\overline{\sigma}\alpha,n\alpha}(\overline{R_{T}^{N}})\times
C^{\overline{\sigma}(1+\alpha)}(R^{N})\times
C^{\overline{\sigma}(1+\alpha)-\frac{1}{n}\overline{\sigma}}\times...\times
C^{\overline{\sigma}(1+\alpha)-\frac{n-1}{n}\overline{\sigma}}$ as it is
indicated in (2.12).

If the function $f(x,t)$ in (1.2) is defined for all $t>0$ in the domain
$\overline{R_{\infty}^{N}}=R^{N}\times[0,\infty)$ and it’s norm in the space
$C^{\overline{\sigma}\alpha,n\alpha}(\overline{R_{\infty}^{N}})$ is finite
(that is if
$\left|f(x,t)\right|_{\overline{R_{\infty}^{N}}}^{(\overline{\sigma}\alpha,n\alpha)}<\infty$),
then Cauchy problem (1.2), (1.3) has the unique solution $u(x,t)$, which
belongs to the space
$C^{\overline{\sigma}(1+\alpha),n+n\alpha}(\overline{R_{\infty}^{N}})$ locally
in time $t$ and the following estimates are valid

$\left\langle
u\right\rangle_{\overline{R_{\infty}^{N}}}^{(\overline{\sigma}(1+\alpha),n+n\alpha)}\leq
C(\overline{\sigma},\alpha)\left(|f|_{\overline{R_{\infty}^{N}}}^{(\overline{\sigma}\alpha,n\alpha)}+{\sum\limits_{i=0}^{n-1}}|u_{i}|_{R^{N}}^{(\overline{\sigma}(1+\alpha)-\frac{i}{n}\overline{\sigma})}\right),$
(2.14) $\left|u\right|_{\overline{R_{\widetilde{T}}^{N}}}^{(0)}\leq
C(\overline{\sigma},\alpha)\left(|f|_{\overline{R_{\infty}^{N}}}^{(\overline{\sigma}\alpha,n\alpha)}+{\sum\limits_{i=0}^{n-1}}|u_{i}|_{R^{N}}^{(\overline{\sigma}(1+\alpha)-\frac{i}{n}\overline{\sigma})}\right)(1+\widetilde{T}^{n+\alpha})+|u_{0}|_{R^{N}}^{(0)},\quad\widetilde{T}\leq\infty.$
(2.15)

###### Remark 2.2

For a non-integer $\theta$ operator L not always is an isomorphism of the
indicated above spaces but only under some conditions (see Theorem 2.5 below).
For example, the function with no dependance on $x$, namely $u(x,t)=t$,
belongs in fact to $C^{\infty}(\overline{R_{T}^{N}})$ and the more $u(x,t)\in
C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$
with arbitrary large exponents. Let $\theta\in(0,1)$ and $\alpha$ is
sufficiently big so that $\theta\alpha>1$. The value of the differential
expression from (1.2) for this function is equal to $f(x,t)=D_{\ast
t}^{\theta}u=C(\theta)t^{1-\theta}$, so that the maximal smoothness of
$f(x,t)$ in $t$ up to $t=0$ is equal to $1-\theta$, which is less than needed
smoothness $\theta\alpha$, $1-\theta<\theta\alpha$. Likewise, the function
$u(x,t)=t^{\theta}$ for $\theta\in(0,1)$ is a solution to (1.2), (1.3) with
the corresponding constant function $f\equiv const$ and with zero initial
condition. And this $u(x,t)$ has up to $t=0$ the order of smoothness in $t$
exactly $\theta$ and not $\theta+\theta\alpha$. Sharp consideration of such
cases are out of the scope of the present paper. And our goal is to determine
conditions for problem (1.2), (1.3) to behave likewise Cauchy problems for
usual parabolic equations (comp. [35]) - see Remark 1.1.

However, the following theorem on solvability of problem (1.2), (1.3) is
valid.

###### Theorem 2.3

Let $\theta>0$ be a non-integer, and let $\alpha>0$ be such that
$\theta\alpha$ and $\theta+\theta\alpha$ are non-integers. Let further
$\sigma_{k}$, $k=1,...,r$, be positive non-integers such that
$\sigma_{k}\alpha$ and $\sigma_{k}+\sigma_{k}\alpha$, $k=1,...,r$, are non-
integers. Let also be fulfilled the following compatibility condition

$f(x,0)={\sum_{k=1}^{r}}(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u_{0}(x),\quad
x\in R^{N}$ (2.16)

and for $\theta\alpha>1$ the condition

$\frac{d^{i}f(x,0)}{dt^{i}}\equiv 0,\quad i=1,...,[\theta\alpha].$

If $\theta\in(0,2)$ then problem (1.2), (1.3) has the unique solution from the
space
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$
with the estimate

$|u|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}(1+\alpha),\theta+\theta\alpha)}\leq
C(\overline{\sigma},\theta,\alpha,T)\left(|f|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}\alpha,\theta\alpha)}+{\sum\limits_{i=0}^{[\theta]}}|u_{i}|_{R^{N}}^{(\overline{\sigma}(1+\alpha)-\frac{i}{\theta}\overline{\sigma})}\right).$
(2.17)

In the case $T=\infty$

$\left\langle
u\right\rangle_{\overline{R_{\infty}^{N}}}^{(\overline{\sigma}(1+\alpha),\theta+\theta\alpha)}\leq
C(\overline{\sigma},\theta,\alpha)\left(|f|_{\overline{R_{\infty}^{N}}}^{(\overline{\sigma}\alpha,\theta\alpha)}+{\sum\limits_{i=0}^{[\theta]}}|u_{i}|_{R^{N}}^{(\overline{\sigma}(1+\alpha)-\frac{i}{\theta}\overline{\sigma})}\right),$
(2.18) $\left|u\right|_{\overline{R_{\widetilde{T}}^{N}}}^{(0)}\leq
C(\overline{\sigma},\theta,\alpha)\left(|f|_{\overline{R_{\infty}^{N}}}^{(\overline{\sigma}\alpha,\theta\alpha)}+{\sum\limits_{i=0}^{[\theta]}}|u_{i}|_{R^{N}}^{(\overline{\sigma}(1+\alpha)-\frac{i}{\theta}\overline{\sigma})}\right)\times$
(2.19)
$\times\left(1+\widetilde{T}^{\theta+\theta\alpha}\right)+|u_{0}|_{R^{N}}^{(0)},\quad\widetilde{T}\leq
T.$

If now $\theta>2$ then the previous statement including the corresponding
above estimates is still valid under the additional assumption
$\\{\theta\\}+\theta\alpha>1$.

###### Remark 2.4

We stress that condition (2.16) is necessary for the smoothness up to $t=0$
since the fractional derivative $D^{\theta}_{\ast t}u(x,t)$ of a function
$u(x,t)$ from the space
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$ is
always equal identically to zero at $t=0$ \- see Proposition 3.2 below.

As a sufficient condition for the operator L in the case of non-integer
$\theta$ to be an isomorphism of the corresponding spaces is the simple
condition $\\{\theta\\}+\theta\alpha<1$. That is the following theorem is
valid.

###### Theorem 2.5

Let $\theta>0$ be a non-integer and $\alpha>0$ be such that $\theta\alpha$ and
$\theta+\theta\alpha$ are non-integers. Let further $\sigma_{k}$, $k=1,...,r$
be positive numbers such that $\sigma_{k}\alpha$ and
$\sigma_{k}+\sigma_{k}\alpha$, $k=1,...,r$, are nonintegers. If
$\\{\theta\\}+\theta\alpha<1$, then the operator L is an isomorphism of the
space
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$
and the closed subspace of the space
$C^{\overline{\sigma}\alpha,\theta\alpha}(\overline{R_{T}^{N}})\times
C^{\overline{\sigma}(1+\alpha)}(R^{N})\times
C^{\overline{\sigma}(1+\alpha)-\frac{1}{\theta}\overline{\sigma}}(R^{N})\times...\times
C^{\overline{\sigma}(1+\alpha)-\frac{[\theta]}{\theta}\overline{\sigma}}$,
which consists of the sets $(f,u_{0},u_{1},...,u_{[\theta]})$ were $f$ and
$u_{0}$ obey condition (2.16).

##  3 Some preliminaries.

This section is included for the completeness and it contains some known
properties of fractional differential operators of the present paper. All the
statements of this section are formulated in the form we need below.

Firstly, on the ground of Section 3, Ch.1, in [1] (see, in particular, Section
3.1, Corollary 1), directly from definitions (1.4), (1.5), (1.9) we infer the
following proposition.

###### Proposition 3.1

The fractional differential operator $D_{\ast t}^{\theta}$ for noninteger
$\theta+\theta\alpha$ ($\theta\alpha\in(0,1)$) and
$\\{\theta\\}+\theta\alpha<1$ is a bounded linear operator from
$C^{\theta+\theta\alpha}([0,T])$ to $C^{\theta\alpha}([0,T])$. That is if
$u(t)\in C^{\theta+\theta\alpha}([0,T])$, then

$|D_{\ast t}^{\theta}u(t)|_{[0,T]}^{(\theta\alpha)}\leq
C(\alpha,\theta,T)|u(t)|_{[0,T]}^{(\theta+\theta\alpha)},$ (3.1)

where $|u(t)|_{[0,T]}^{(\lambda)}$ is the norm in the space
$C^{\lambda}([0,T])$, $\lambda>0$.

If now $u(t)\in C^{\theta+\theta\alpha}([0,T])$ and

$u^{(i)}(0)=0,\quad i=0,1,...,[\theta+\theta\alpha],$

then (3.1) is valid without the restriction $\\{\theta\\}+\theta\alpha<1$.

Further, the Caputo - Jrbashyan derivative possesses the following property
(compare [2], Theorem 2.2).

###### Proposition 3.2

If a function $u(t)\in C^{\theta+\theta\alpha}([0,T])$ with a positive
noninteger $\theta$, then it’s derivative $D_{\ast t}^{\theta}u(t)$ vanishes
at $t=0$ that is

$D_{\ast t}^{\theta}u(t)|_{t=0}=0.$ (3.2)

In other words, if a function $u(t)$ has some higher smoothness, than a
fractional $\theta>0$, then it’s Caputo - Jrbashyan derivative of order
$\theta$ necessarily vanishes at $t=0$.

We include a short proof for the completeness.

Proof. Note first that, in view of the second equality in representation
(1.9), we can without loss of generality assume $\theta\in(0,1)$.

Further, if $\theta+\theta\alpha\geq 1$, then property (3.2) follows directly
from the first equality in representation (1.9), since in this case the
expression under the integral is integrable.

If now $\theta+\theta\alpha<1$, then after some smoothing of $u(t)$ (for
example, as it was done in Section 9.4 below) we obtain a sequence
$\\{u_{n}(t)\\}$ of the class $C^{\infty}([0,T])$ with

$u_{n}(t)\rightarrow u(t)\ \ \text{in}\ \ C^{\theta+\beta}([0,T]),\ \
\beta\in(0,\theta\alpha).$ (3.3)

Since each member $u_{n}(t)$ of the sequence has the smoothness, which is
grater, than one, then all functions $u_{n}(t)$ possess property (3.2).
Moreover, from (3.1) it follows, in particular, that the derivatives $D_{\ast
t}^{\theta}u_{n}(t)$ converge uniformly on $[0,T]$ (at least for a
subsequence) to the derivative $D_{\ast t}^{\theta}u(t)$ of the original
function. This means that the function $u(t)$ itself satisfies (3.2).

Besides, Example 3.9 in [3] contains an explicit solution to the simplest
Cauchy problem

$D_{\ast t}^{\theta}u(t)=h(t),\quad t\in[0,T],\quad h(t)\in C([0,T]),\quad
n-1<\theta<n,$ $u^{(k)}(0)=a_{k},\quad k=0,1,...,n-1,$

and the mentioned solution is expressed as

$u(t)=J^{\theta}h(t)+{\sum\limits_{k=0}^{n-1}}\frac{a_{k}}{k!}t^{k},$

where

$J^{\theta}h(t)\equiv\frac{1}{\Gamma(\theta)}{\int\limits_{0}^{t}}(t-\tau)^{-1+\theta}h(\tau)d\tau.$
(3.4)

And from this, again on the base of Corollary 1 in Section 3, Ch.1 of [1] and
on the base of Lemma 13.1 in the same monograph (with the taking into account
the Remark after this lemma), follows a statement, which is the inverse to
Proposition 3.1.

###### Proposition 3.3

If $D_{\ast t}^{\theta}u(t)\in C^{\theta\alpha}([0,T])$,
$\theta\alpha\in(0,1)$ and $D_{\ast t}^{\theta}u(t)|_{t=0}=0$, $u^{(k)}(0)=0$,
$k=0,...,[\theta]$, then for a noninteger $\theta+\theta\alpha$ the function
$u(t)\in\underline{C}^{\theta+\theta\alpha}([0,T])$ and

$|u(t)|_{[0,T]}^{(\theta+\theta\alpha)}\leq C(\alpha,\theta,T)|D_{\ast
t}^{\theta}u(t)|_{[0,T]}^{(\theta\alpha)}.$ (3.5)

And if $\theta+\theta\alpha>[\theta]+1$, then also

$u^{([\theta]+1)}(0)=0.$ (3.6)

We formulate now in the form we need below some known properties of the
fractional Laplace operator in $R^{M}$.

Consider the operator $(-\Delta)^{\frac{\sigma}{2}}$ with a noninteger
$\sigma>0$ on functions $u(x)$, $x\in R^{M}$, where $R^{M}$, $M=1,2,...$, is
Euclidian space. Representation (1.12) for this operator has the form

$(-\Delta)^{\frac{\sigma}{2}}u(x)=C_{M,\sigma,m}{\int\limits_{\eta\in
R^{M}}}\frac{\delta_{\eta,x}^{m}u(x)}{|\eta|^{M+\sigma}}d\eta,\quad m>\sigma,$
(3.7)

where, remind, $\delta_{\eta,x}u(x)=u(x+\eta)-u(x)$,
$\delta_{\eta,x}^{m}u(x)=\delta_{\eta,x}(\delta_{\eta,x}^{m-1}u(x))$ is the
difference of order $m$. Directly from this representation, on the ground of
classical estimates for elliptic potentials (see, for example, [33]) it
follows that the operator $(-\Delta)^{\frac{\sigma}{2}}$ is correctly defined
on the space $C^{\sigma+\beta}(R^{M})$, $\beta>0$, with some noninteger
$\beta$ and $\sigma+\beta$. And in addition

$|(-\Delta)^{\frac{\sigma}{2}}u|_{R^{M}}^{(\beta)}\leq
C(M,\sigma,\beta)|u|_{R^{M}}^{(\sigma+\beta)}$ (3.8)

(it is convenient to choose $m>\sigma+\beta$ in (3.7)). Note here, that the
analogous estimate for the corresponding highest Hölder seminorms of the
functions $u(x)$ and $(-\Delta)^{\frac{\sigma}{2}}u(x)$ is, in fact, a
particular case of Theorem 1 in Section 5.2.3 of [34]. This theorem is proved
for the general homogeneous Besov spaces $B_{\beta}^{a,b}(R^{M})$, and in the
particular case $B_{\beta}^{\infty,\infty}(R^{M})=C^{\beta}(R^{M})$ the
assertion of the theorem gives for the Hölder seminorms

$\left\langle(-\Delta)^{\frac{\sigma}{2}}u\right\rangle_{R^{M}}^{(\beta)}\leq
C(M,\sigma,\beta)\left\langle u\right\rangle_{R^{M}}^{(\sigma+\beta)}.$ (3.9)

Without touching on here the question about the image of
$C^{\sigma+\beta}(R^{M})$ under the action of the operator
$(-\Delta)^{\frac{\sigma}{2}}$ (and this image does coincide with the whole
$C^{\beta}(R^{M})$ even in the case of the classical Laplace operator), we
formulate an assertion, which is an inverse to inequality (3.8). Namely, the
following estimate is valid (see [36], Theorem 1.2)

$|u|_{R^{M}}^{(\sigma+\beta)}\leq
C(M,\sigma,\beta)\left(|(-\Delta)^{\frac{\sigma}{2}}u|_{R^{M}}^{(\beta)}+|u|_{R^{M}}^{(0)}\right),$
(3.10)

and also for lonely the highest seminorm ([34], Section 5.2.3, Theorem 1)

$\left\langle u\right\rangle_{R^{M}}^{(\sigma+\beta)}\leq
C(M,\sigma,\beta)\left\langle(-\Delta)^{\frac{\sigma}{2}}u\right\rangle_{R^{M}}^{(\beta)}.$
(3.11)

##  4 Operators $(-\Delta)^{\frac{\sigma}{2}}$ on the Schwartz spaces
$S(R^{M})$ and $S^{\prime}(R^{M})$

Below we need some assertion about continuity of the operator
$(-\Delta)^{\frac{\sigma}{2}}$ on the Schwartz spaces $S(R^{M})$ and
$S^{\prime}(R^{M})$. It is known that the operator
$(-\Delta)^{\frac{\sigma}{2}}$ does not map the space $S(R^{M})$ to itself.
Therefore this operator is not defined on the whole space $S^{\prime}(R^{M})$.
However, it is possible to highlight some subclasses of the space
$S^{\prime}$, where the mentioned operator is continuously defined
$S^{\prime}\rightarrow S^{\prime}$. We confine ourself only to a subclass we
need below. To a pity, the author could not find in literature some
appropriate for us assertions or proofs. Therefore, for the completeness, we
give here some assertions needed and some simple proofs.

We start with the action of $(-\Delta)^{\frac{\sigma}{2}}$ on $S(R^{M})$.
Denote a usual partial derivative of a function $u(x)\in S(R^{M})$ by

$D_{x}^{\overline{\omega}}u(x)\equiv\frac{\partial^{\omega_{1}}\partial^{\omega_{2}}...\partial^{\omega_{M}}u(x)}{\partial
x_{1}^{\omega_{1}}\partial x_{2}^{\omega_{2}}...\partial
x_{M}^{\omega_{M}}},\quad\overline{\omega}=(\omega_{1},\omega_{2},...,\omega_{M})$

and for a function $u(x)\in S(R^{M})$ denote it’s (semi)norm

$\left\langle u\right\rangle_{R^{M}}^{n,p}={\sum_{|\overline{\omega}|\leq
n}}\sup_{x\in R^{M}}|D_{x}^{\overline{\omega}}u(x)(1+|x|)^{p}|,\quad
n=0,1,2,...,\quad p>0.$ (4.1)

###### Lemma 4.1

Let $u(x)\in S(R^{M})$. Then $(-\Delta)^{\frac{\sigma}{2}}u(x)\in
C^{\infty}(R^{M})$ and for any $\varepsilon>0$

$\left|D_{x}^{\overline{\omega}}\left[(-\Delta)^{\frac{\sigma}{2}}u(x)\right]\right|\leq
C(M,\sigma,\varepsilon)\left\langle
u\right\rangle_{R^{M}}^{m+|\overline{\omega}|,p(\varepsilon)}(1+|x|)^{-M-\sigma+\varepsilon}.$
(4.2)

Proof. Note that for an arbitrary $\eta\in R^{M}$

$D_{x}^{\overline{\omega}}\left(\delta_{\eta,x}^{m}u(x)\right)=\delta_{\eta,x}^{m}\left(D_{x}^{\overline{\omega}}u(x)\right)$
(4.3)

and consequently, on the ground of the mean value theorem, for any $x,\eta\in
R^{M}$

$\left|D_{x}^{\overline{\omega}}\left(\delta_{\eta,x}^{m}u(x)\right)\right|=\left|\delta_{\eta,x}^{m}\left(D_{x}^{\overline{\omega}}u(x)\right)\right|\leq
C(M,m)|D_{x}^{m+|\overline{\omega}|}u|_{B_{m|\eta|}(x)}^{(0)}|\eta|^{m},$
(4.4)

where $B_{m|\eta|}(x)$ is the centered in $x$ ball of radius $m|\eta|$,
$\left|\overline{\omega}\right|=\omega_{1}+\omega_{2}+...+\omega_{M}$,

$|D_{x}^{m+|\overline{\omega}|}u|_{B_{m|\eta|}(x)}^{(0)}\equiv{\sum\limits_{|\rho|=m+|\overline{\omega}|}}\left|D_{x}^{\overline{\rho}}u(x)\right|_{B_{m|\eta|}(x)}^{(0)},\quad\overline{\rho}=(\rho_{1},\rho_{2},...,\rho_{M}).$

Note also that for $|\eta|\leq|x|/2m$ and for any $p>0$ the value of
$|D_{x}^{m+|\overline{\omega}|}u|_{B_{m|\eta|}(x)}^{(0)}$ in (4.4) can be
estimated as follows

$|D_{x}^{m+|\overline{\omega}|}u|_{B_{m|\eta|}(x)}^{(0)}\leq
C(M,m,p)\left\langle
u\right\rangle_{R^{M}}^{m+|\overline{\omega}|,p}(1+|x|)^{-p},\quad|\eta|\leq|x|/2m.$
(4.5)

Denote, besides, for $\delta\in(0,1/10)$ and for $x\in R^{M}$

$U_{\delta}(x)\equiv{\bigcup\limits_{j=1}^{m}}B_{|x|^{\delta}}(-\frac{x}{j}),$
(4.6)

where $B_{|x|^{\delta}}(-x/j)$ is the the centered in $-x/j$ ball of radius
$|x|^{\delta}$. Considering $x$ with $|x|>(2m)^{1/(1-\delta)}$, we see that on
the set $U_{\delta}(x)$ we have

$\eta\in U_{\delta}(x)\Rightarrow|\eta|>\frac{|x|}{2m}.$ (4.7)

Consider now a derivative $D_{x}^{\overline{\omega}}$ of $\
(-\Delta)^{\frac{\sigma}{2}}u(x)$. Use representation (3.7) and split the
integral over $R^{M}$ into the four domains

$D_{x}^{\overline{\omega}}\left[(-\Delta)^{\frac{\sigma}{2}}u(x)\right]=C_{M,\sigma,m}{\int\limits_{|\eta|\leq\frac{1}{2m}|x|}}\frac{\delta_{\eta,x}^{m}\left(D_{x}^{\overline{\omega}}u(x)\right)}{|\eta|^{M+\sigma}}d\eta+$
(4.8) $+C_{M,\sigma,m}{\int\limits_{\left\\{\frac{1}{2m}|x|<|\eta|\leq
2m|x|\right\\}\cap
U_{\delta}(x)}}\frac{\delta_{\eta,x}^{m}\left(D_{x}^{\overline{\omega}}u(x)\right)}{|\eta|^{M+\sigma}}d\eta+$
$+C_{M,\sigma,m}{\int\limits_{\left\\{\frac{1}{2m}|x|<|\eta|\leq
2m|x|\right\\}\backslash
U_{\delta}(x)}}\frac{\delta_{\eta,x}^{m}\left(D_{x}^{\overline{\omega}}u(x)\right)}{|\eta|^{M+\sigma}}d\eta+$
$+C_{M,\sigma,m}{\int\limits_{2m|x|<|\eta|}}\frac{\delta_{\eta,x}^{m}\left(D_{x}^{\overline{\omega}}u(x)\right)}{|\eta|^{M+\sigma}}d\eta\equiv
I_{1}+I_{2}+I_{3}+I_{4}.$

Taking advantage of (4.4), (4.1), estimate the integral $I_{1}$ as follows,
bearing in mind that $m>\sigma$ and that on the domain of integration
$|\eta|\leq\frac{1}{2m}|x|$,

$\left|I_{1}\right|\leq C_{M,\sigma,m,k}\left\langle
u\right\rangle_{R^{M}}^{m+|\overline{\omega}|,p}{\int\limits_{|\eta|\leq\frac{1}{2m}|x|}}\frac{|\eta|^{m}(1+|x|)^{-p}}{|\eta|^{M+\sigma}}d\eta\leq$
$\leq C_{M,\sigma,m,k}\left\langle
u\right\rangle_{R^{M}}^{m+|\overline{\omega}|,p}(1+|x|)^{-p}{\int\limits_{|\eta|\leq\frac{1}{2m}|x|}}\frac{1}{|\eta|^{M-(m-\sigma)}}d\eta\leq$
$\leq C_{M,\sigma,m,k}\left\langle
u\right\rangle_{R^{M}}^{m+|\overline{\omega}|,p}(1+|x|)^{-p+m-\sigma},$ (4.9)

where $p>m-\sigma$.

Note now that

$\delta_{\eta}^{m}\left(D_{x}^{\overline{\omega}}u(x)\right)={\sum\limits_{j=0}^{m}}C(i,m)D_{x}^{\overline{\omega}}u(x+j\eta).$
(4.10)

Taking into account (4.7) and the fact that the total measure of
$U_{\delta}(x)$ does not exceed $\left|U_{\delta}(x)\right|\leq
C|x|^{M\delta}$, we have for the integral $I_{2}$ in (4.8)

$\left|I_{2}\right|\leq C_{M,\sigma,m}\left\langle
u\right\rangle_{R^{M}}^{|\overline{\omega}|,0}{\int\limits_{U_{\delta}(x)}}\frac{1}{|\eta|^{M+\sigma}}d\eta\leq$
$\leq C_{M,\sigma,m}\left\langle
u\right\rangle_{R^{M}}^{|\overline{\omega}|,0}{\int\limits_{U_{\delta}(x)}}\frac{1}{|x|^{M+\sigma}}d\eta,$

which gives under the condition $|x|>(2m)^{1/(1-\delta)}$ the estimate

$\left|I_{2}\right|\leq C_{M,\sigma,m}\left\langle
u\right\rangle_{R^{M}}^{|\overline{\omega}|,0}|x|^{M\delta}|x|^{-M-\sigma}\leq
C_{M,\sigma,m}\left\langle
u\right\rangle_{R^{M}}^{|\overline{\omega}|,0}(1+|x|)^{-M-\sigma+M\delta}.$
(4.11)

To estimate $I_{3}$ make again use of (4.10) and note that on the domain of
integration all the arguments of all the functions in (4.10) satisfy the
condition $\left|x+j\eta\right|\geq\left|x\right|^{\delta}/m$ and thus, with
the help of (4.5),

$\left|I_{3}\right|\leq C_{M,\sigma,m,p}\left\langle
u\right\rangle_{R^{M}}^{|\overline{\omega}|,p}\left(1+\left|x\right|\right)^{-p\delta}{\int\limits_{\frac{1}{2m}|x|<|\eta|\leq
2m|x|}}\frac{1}{|\eta|^{M+\sigma}}d\eta\leq$ (4.12) $\leq
C_{M,\sigma,m,p}\left\langle
u\right\rangle_{R^{M}}^{|\overline{\omega}|,p}\left(1+\left|x\right|\right)^{-p\delta}|x|^{-M-\sigma}|x|^{M}\leq
C_{M,\sigma,m,p}\left\langle
u\right\rangle_{R^{M}}^{|\overline{\omega}|,p}\left(1+\left|x\right|\right)^{-p\delta-\sigma}.$

Turning at last to $I_{4}$, we avail ourselves of the fact that on the domain
of integration in (4.10)

$\left|x+j\eta\right|\geq\genfrac{\\{}{.}{0.0pt}{0}{|x|,\quad
j=0,}{|\eta|/2m,\quad j>0}$

and therefore

$\left|\delta_{\eta}^{m}\left(D_{x}^{\overline{\omega}}u(x)\right)\right|\leq
C_{m,p}\left\langle
u\right\rangle_{R^{M}}^{|\overline{\omega}|,p}\left(1+\left|x\right|\right)^{-p}+C_{m,p}\left\langle
u\right\rangle_{R^{M}}^{|\overline{\omega}|,p}\left(1+\left|\eta\right|\right)^{-p}.$

Consequently,

$\left|I_{4}\right|\leq C_{M,\sigma,m,p}\left\langle
u\right\rangle_{R^{M}}^{|\overline{\omega}|,p}(1+|x|)^{-p}{\int\limits_{|\eta|>2m|x|}}\frac{1}{|\eta|^{M+\sigma}}d\eta+$
$+C_{M,\sigma,m,p}\left\langle
u\right\rangle_{R^{M}}^{|\overline{\omega}|,p}{\int\limits_{|\eta|>2m|x|}}\frac{(1+|\eta|)^{-p}}{|\eta|^{M+\sigma}}d\eta\leq
C_{M,\sigma,m,k}\left\langle
u\right\rangle_{R^{M}}^{|\overline{\omega}|,k}(1+|x|)^{-p-\sigma}$ (4.13)

The above estimates for the integrals $I_{1}$ \- $I_{2}$ show that, first, we
can differentiate under the sign of the integral in (3.7) and for $u(x)\in
S(R^{M})$ the function $(-\Delta)^{\frac{\sigma}{2}}u(x)$ belongs to the class
$C^{\infty}(R^{M})$. And the second, from (4.9), (4.11), (4.12) and (4.13) it
follows that for $|x|>(2m)^{1/(1-\delta)}$ we have

$\left|D_{x}^{\overline{\omega}}\left[(-\Delta)^{\frac{\sigma}{2}}u(x)\right]\right|\leq
C_{M,\sigma,m,p}\left\langle
u\right\rangle_{R^{M}}^{m+|\overline{\omega}|,p}\times$
$\times\left[(1+|x|)^{-p+m-\sigma}+(1+|x|)^{-M-\sigma+M\delta}+\left(1+\left|x\right|\right)^{-p\delta-\sigma}+(1+|x|)^{-p-\sigma}\right].$

First choosing here sufficiently small $\delta$, $M\delta=\varepsilon$, and
then choosing $p$ sufficiently large, $p=M/\delta+m$, we arrive at the lemma
statement.

The proved lemma permits to define the operator $(-\Delta)^{\frac{\sigma}{2}}$
on a subclass $S_{\sigma}^{\prime}(R^{M})\subset S^{\prime}(R^{M})$ of the
class $S^{\prime}(R^{M})$, where

$S_{\sigma}^{\prime}(R^{M})=\left\\{u(x)\in S^{\prime}(R^{M}):|u(x)|\leq
C(1+|x|)^{b},\quad C>0,\quad b<\sigma\right\\}.$ (4.14)

###### Lemma 4.2

Operator $(-\Delta)^{\frac{\sigma}{2}}$ is correctly defined on
$S_{\sigma}^{\prime}(R^{M})\subset S^{\prime}(R^{M})$ in the usual sense of
the space $S^{\prime}(R^{M})$.

This statement follows directly from Lemma 4.1, since for $u(x)\in
S_{\sigma}^{\prime}(R^{M})$ and $\varphi(x)\in S(R^{M})$ in the sense of
duality

$\left\langle(-\Delta)^{\frac{\sigma}{2}}u(x),\varphi(x)\right\rangle\equiv\left\langle
u(x),(-\Delta)^{\frac{\sigma}{2}}\varphi(x)\right\rangle,$

where the last operation is correctly defined due to Lemma 4.1 and the
definition of $S_{\sigma}^{\prime}(R^{M})$. As an another simple consequence
of Lemma 4.1 we have the following assertion, which we present here without a
proof.

###### Lemma 4.3

Let for a sequence of measurable locally bounded functions
$\\{u_{n}(x)\\}\subset S_{\sigma}^{\prime}(R^{M})$ there exist such
independent on $n$ constants $C>0$ and $b<\sigma$ that

$|u_{n}(x)|\leq C(1+|x|)^{b},\quad n=1,2,...$

Let also there exits such a function $u(x)$, that $u_{n}(x)$ converges to
$u(x)$ on each ball $B_{R}=\\{x\in R^{M}:|x|\leq R\\}$. Then

$u(x)\in S_{\sigma}^{\prime}(R^{M});\quad
u_{n}(x)\rightarrow_{S^{\prime}(R^{M})}u(x),$
$(-\Delta)^{\frac{\sigma}{2}}u_{n}(x)\rightarrow_{S^{\prime}(R^{M})}(-\Delta)^{\frac{\sigma}{2}}u(x),\quad
n\rightarrow\infty.$ (4.15)

Define now an anisotropic analog of the class $S_{\sigma}^{\prime}(R^{M})$,
which is related to the studying of the operator

$\emph{M}u\equiv{\sum_{k=1}^{r}}(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u$

that is to the space operator in equation (1.2). Namely, we denote

$S_{\overline{\sigma}}^{\prime}(R^{N})\equiv$ $\left\\{u(x)\in
S^{\prime}(R^{N}):|u(x)|\leq
C{\prod\limits_{k=1}^{r}}(1+|z_{k}|)^{b_{k}},C>0,b_{k}<\sigma_{k},k=\overline{1,r}\right\\},$
(4.16)

where $z_{k}$ are defined in (1.1). Directly from lemmas 4.1 \- 4.3 we infer
the following assertion.

###### Proposition 4.4

Let for a sequence of measurable locally bounded functions
$\\{u_{n}(x)\\}\subset S_{\overline{\sigma}}^{\prime}(R^{N})$ there exist such
independent on $n$ constants $C>0$ and $b_{k}<\sigma_{k}$, $k=1,...,r$ that

$|u_{n}(x)|\leq C{\prod\limits_{k=1}^{r}}(1+|z_{k}|)^{b_{k}},\quad n=1,2,...$

If there exists such a function $u(x)$ that $u_{n}(x)$ uniformly converges to
$u(x)$ on each ball $B_{R}=\\{x\in R^{N}:|x|\leq R\\}$, then

$u(x)\in S_{\overline{\sigma}}^{\prime}(R^{N});\quad
u_{n}(x)\rightarrow_{S^{\prime}(R^{N})}u(x),$
${\sum_{k=1}^{r}}(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u_{n}(x)\rightarrow_{S^{\prime}(R^{N})}{\sum_{k=1}^{r}}(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u(x),\quad
n\rightarrow\infty.$ (4.17)

##  5 Operators $D_{\ast t}^{\theta}$ and $(-\Delta)^{\frac{\sigma}{2}}$ on
the Lizorkin spaces $\Phi(R^{M})$ and $\Phi^{\prime}(R^{M})$

In this section we present, for the completeness, some known facts about the
acting of fractional differential operators on the Lizorkin spaces
$\Phi(R^{M})$ and $\Phi^{\prime}(R^{M})$. These spaces are a subspace and a
quotient space correspondingly of the spaces $S(R^{M})$ and
$S^{\prime}(R^{M})$. They permit to generalize known properties of usual
differential operators on the spaces $S(R^{M})$ and $S^{\prime}(R^{M})$ to
fractional differential operators. We present here briefly some necessary for
us definitions and facts according to the corresponding presentation in [1],
section 8.2, where the reader can find more detailed information.

Define first the space $\Psi(R^{M})$, This space is the closed subspace of
$S(R^{M})$, consisting of function from $S(R^{M})$ that vanish at $x=0$
together with all their derivatives. That is

$\Psi(R^{M})\equiv\left\\{\psi(x)\in S(R^{M}):\quad
D_{x}^{\overline{\omega}}\psi(0)=0,\quad|\overline{\omega}|=0,1,2,...\right\\}.$
(5.1)

The Lizorkin space $\Phi(R^{M})$ is the closed subspace of $S(R^{M})$,
consisting of functions from $S(R^{M})$ with the Fourier images from the space
$\Psi(R^{M})$. That is

$\Phi(R^{M})\equiv\left\\{\varphi(x)\in
S(R^{M}):\quad\widehat{\varphi}(\xi)\in\Psi(R^{M})\right\\}.$ (5.2)

Besides, the space $\Phi(R^{M})$ can be equivalently defined as the closed
subspace of $S(R^{M})$, consisting of function that are orthogonal to all
polynomials. That is

$\varphi(x)\in\Phi(R^{M})\Leftrightarrow{\int\limits_{R^{M}}}\varphi(x)P(x)dx=0,\forall
P(x),$

where $P(x)$ is an arbitrary polynomial.

Since $\Psi(R^{M})$ and $\Phi(R^{M})$ closed subspaces of $S(R^{M})$, their
topologies are induced by the topology of $S(R^{M})$.

The dual to the space $\Psi(R^{M})$ is denoted by $\Psi^{\prime}(R^{M})$ and
it is the quotient space of $S^{\prime}(R^{M})$ along the closed subspace of
$S^{\prime}(R^{M})$, consisting of distributions with the supports at $x=0$.
It is well known that such distributions are exactly those that are finite
linear combinations of the Dirac function $\delta(x)$ with the support at
$x=0$ and it’s derivatives.

At the same time, the dual to $\Phi(R^{M})$ space $\Phi^{\prime}(R^{M})$, is
the quotient space of $S^{\prime}(R^{M})$ along the space of polynomials
$P(x)$ (which is a closed subspace of $S^{\prime}(R^{M})$). That is the
elements of $\Phi^{\prime}(R^{M})$ are exactly the equivalence classes of the
distributions from $S^{\prime}(R^{M})$ modulo polynomials.

The topologies in the spaces $\Psi^{\prime}(R^{M})$ and $\Phi^{\prime}(R^{M})$
are the corresponding quotient topologies.

Fractional differential operators $D_{\ast t}^{\theta}$ and
$(-\Delta)^{\frac{\sigma}{2}}$ are defined and continuous from the space
$\Phi(R^{M})$ to itself and from $\Phi^{\prime}(R^{M})$ to itself. For the
operator $(-\Delta)^{\frac{\sigma}{2}}$ this fact follows directly from the
definition of the space $\Phi(R^{M})$ and from definition (1.11). The same is
also valid for the operator $D_{\ast t}^{\theta}$, when we consider it on
functions that vanish at $t=0$ together with all their derivatives with
respect to $t$ up to the order $[\theta]$. At that one needs to take into
account relation (8.24) and Remark 8.5 below in section 8.

According to the definitions of $\Psi(R^{M})$ and $\Phi(R^{M})$, the Fourier
transform is a linear homeomorphism from $\Phi(R^{M})$ to $\Psi(R^{M})$ and,
correspondingly, from $\Phi^{\prime}(R^{M})$ to $\Psi^{\prime}(R^{M})$.

Moreover, for the distributions from $\Phi^{\prime}(R^{N+1})$ all formulas of
the Fourier transform for fractional derivatives are preserved. That is if
$u(x,t)\in\Phi^{\prime}(R^{N+1})$, then, analogously to (1.11),

$\widehat{(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u(x,t)}(\xi,\xi_{0})=|\eta_{k}|^{\sigma_{k}}\widehat{u}(\xi,\xi_{0}).$
(5.3)

And if $u(x,t)\in\Phi^{\prime}(R^{N+1})$ and it’s support along with the
supports of all it’s derivatives in $t$ up to the order $[\theta]$ are
included in the set $\\{t\geq 0\\}$, then

$\widehat{D_{\ast
t}^{\theta}u(x,t)}(\xi,\xi_{0})=(i\xi_{0})^{[\theta]}(i\xi_{0})^{\\{\theta\\}}\widehat{u}(\xi,\xi_{0}).$
(5.4)

We are going to make use of these circumstances below at the proof of the
uniqueness for problem (1.2), (1.3).

##  6 Operators $D_{\ast t}^{\theta}$ and $(-\Delta)^{\frac{\sigma}{2}}$ on
Hölder spaces
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$
and $C^{\overline{\sigma}(1+\alpha)}(R^{N})$

Turning to the Hölder spaces, introduced in (2.8) and (2.10), we note that
these spaces are defined in terms of the corresponding coordinate-wise
smoothness. Therefore on the grounds of (2.4), (3.8) and (3.10) the following
assertion can be obtained.

###### Proposition 6.1

The operator $(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}$, $k=1,...,r$ is a
bounded linear operator from $C^{\overline{\sigma}(1+\alpha)}(R^{N})$ to
$C^{\overline{\sigma}\alpha}(R^{N})$. That is for $u(x)\in
C^{\overline{\sigma}(1+\alpha)}(R^{N})$

$|(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u|_{R^{N}}^{(\overline{\sigma}\alpha)}\leq
C(N,N_{k},\overline{\sigma},\alpha)|u|_{R^{N}}^{\left(\overline{\sigma}(1+\alpha)\right)}$
(6.1)

and for the Hölder seminorm of the function
$(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u$ with respect to another group of
variables $z_{l}$, $l\neq k$, the following interpolation inequality is valid
with an arbitrary $\varepsilon>0$

$\left\langle(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u\right\rangle_{z_{l},R^{N}}^{(\sigma_{l}\alpha)}\leq
C\varepsilon^{\alpha}\left\langle
u\right\rangle_{z_{k},R^{N}}^{(\sigma_{k}+\sigma_{k}\alpha)}+\frac{C}{\varepsilon}\left\langle
u\right\rangle_{z_{l},R^{N}}^{(\sigma_{l}+\sigma_{l}\alpha)}.$ (6.2)

Proof. Firstly, as for the smoothness with respect to the group $z_{k}$, from
(3.8) it follows that uniformly in the other coordinate groups

$|(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u|_{R^{N}}^{(0)}+\left\langle(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u\right\rangle_{z_{k},R^{N}}^{(\sigma_{k}\alpha)}\leq
C(N_{k},\sigma_{k},\alpha)|u|_{R^{N}}^{\left(\overline{\sigma}(1+\alpha)\right)}.$
(6.3)

To show the estimates of the Hölder seminorms for the function
$(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u(x)$ with respect to other groups
$z_{l}$, $l\neq k$, we again use representation (3.7)

$(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u(x)=C_{N_{k},\sigma_{k},m}{\int\limits_{R^{N_{k}}}}\frac{\delta_{\eta,z_{k}}^{m}u(x)}{|\eta|^{N_{k}+\sigma_{k}}}d\eta,\quad
m>\sigma_{k}+\sigma_{k}\alpha.$ (6.4)

To estimate the seminorm
$\left\langle(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u\right\rangle_{z_{l},R^{N}}^{(\sigma_{l}\alpha)}$
according to (2.4) we carry the difference with respect to $z_{l}$ inside the
integral in (6.4),

$\left\langle(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u\right\rangle_{z_{l},R^{N}}^{(\sigma_{l}\alpha)}=$
$=\sup_{x\in R^{N},h\in R^{N_{l}},h\neq
0}\frac{C_{N_{k},\sigma_{k},m}}{|h|^{\sigma_{l}\alpha}}{\int\limits_{R^{N_{k}}}}\frac{\delta_{h,z_{l}}^{p}\delta_{\eta,z_{k}}^{m}u(x)}{|\eta|^{N_{k}+\sigma_{k}}}d\eta,\quad
p>\sigma_{l}+\sigma_{l}\alpha.$ (6.5)

Estimate the expression under the $\sup$-sign in (6.5) by splitting the
integral into the two parts

${\int\limits_{R^{N_{k}}}}\frac{\delta_{h,z_{l}}^{m}\delta_{\eta,z_{k}}^{m}u(x)}{|\eta|^{N_{k}+\sigma_{k}}|h|^{\sigma_{l}\alpha}}d\eta={\int\limits_{|\eta|\leq\varepsilon|h|^{\frac{\sigma_{l}}{\sigma_{k}}}}}\frac{\delta_{h,z_{l}}^{m}\delta_{\eta,z_{k}}^{m}u(x)}{|\eta|^{N_{k}+\sigma_{k}}|h|^{\sigma_{l}\alpha}}d\eta+$
$+{\int\limits_{|\eta|\geq\varepsilon|h|^{\frac{\sigma_{l}}{\sigma_{k}}}}}\frac{\delta_{h,z_{l}}^{m}\delta_{\eta,z_{k}}^{m}u(x)}{|\eta|^{N_{k}+\sigma_{k}}|h|^{\sigma_{l}\alpha}}d\eta\equiv
I_{1}+I_{2},$

where $\varepsilon>0$ is arbitrary but fixed. Note that according to (2.4) and
according to the definition of the finite differences

$|\delta_{h,z_{l}}^{m}\delta_{\eta,z_{k}}^{m}u(x)|\leq C\left\langle
u\right\rangle_{z_{l},R^{N}}^{(\sigma_{l}+\sigma_{l}\alpha)}|h|^{\sigma_{l}+\sigma_{l}\alpha},$
$|\delta_{h,z_{l}}^{m}\delta_{\eta,z_{k}}^{m}u(x)|\leq C\left\langle
u\right\rangle_{z_{k},R^{N}}^{(\sigma_{k}+\sigma_{k}\alpha)}|\eta|^{\sigma_{k}+\sigma_{k}\alpha}.$
(6.6)

Making use of the second of these inequalities, we estimate the integral
$I_{1}$ as follows

$|I_{1}|\leq C\left\langle
u\right\rangle_{z_{k},R^{N}}^{(\sigma_{k}+\sigma_{k}\alpha)}{\int\limits_{|\eta|\leq\varepsilon|h|^{\frac{\sigma_{l}}{\sigma_{k}}}}}\frac{|\eta|^{\sigma_{k}+\sigma_{k}\alpha}}{|\eta|^{N_{k}+\sigma_{k}}|h|^{\sigma_{l}\alpha}}d\eta=$
$=\frac{C\left\langle
u\right\rangle_{z_{k},R^{N}}^{(\sigma_{k}+\sigma_{k}\alpha)}}{|h|^{\sigma_{l}\alpha}}{\int\limits_{|\eta|\leq\varepsilon|h|^{\frac{\sigma_{l}}{\sigma_{k}}}}}\frac{d\eta}{|\eta|^{N_{k}-\sigma_{k}\alpha}}=$
$=C\left\langle
u\right\rangle_{z_{k},R^{N}}^{(\sigma_{k}+\sigma_{k}\alpha)}|h|^{-\sigma_{l}\alpha}\left(\varepsilon|h|^{\frac{\sigma_{l}}{\sigma_{k}}}\right)^{\sigma_{k}\alpha}=C\varepsilon^{\sigma_{k}\alpha}\left\langle
u\right\rangle_{z_{k},R^{N}}^{(\sigma_{k}+\sigma_{k}\alpha)}.$ (6.7)

Analogously, taking advantage of the first inequality in (6.6), we have for
the integral $I_{2}$,

$|I_{2}|\leq C\left\langle
u\right\rangle_{z_{l},R^{N}}^{(\sigma_{l}+\sigma_{l}\alpha)}{\int\limits_{|\eta|\geq\varepsilon|h|^{\frac{\sigma_{l}}{\sigma_{k}}}}}\frac{|h|^{\sigma_{l}+\sigma_{l}\alpha}}{|\eta|^{N_{k}+\sigma_{k}}|h|^{\sigma_{l}\alpha}}d\eta=$
$=C\left\langle
u\right\rangle_{z_{l},R^{N}}^{(\sigma_{l}+\sigma_{l}\alpha)}|h|^{\sigma_{l}}{\int\limits_{|\eta|\geq\varepsilon|h|^{\frac{\sigma_{l}}{\sigma_{k}}}}}\frac{d\eta}{|\eta|^{N_{k}+\sigma_{k}}}=$
$=C\left\langle
u\right\rangle_{z_{l},R^{N}}^{(\sigma_{l}+\sigma_{l}\alpha)}|h|^{\sigma_{l}}\left(\varepsilon|h|^{\frac{\sigma_{l}}{\sigma_{k}}}\right)^{-\sigma_{k}}=\frac{C}{\varepsilon^{\sigma_{k}}}\left\langle
u\right\rangle_{z_{l},R^{N}}^{(\sigma_{l}+\sigma_{l}\alpha)}.$ (6.8)

Estimate (6.2) follows now from (6.5), (6.7) and (6.8) after the change of
variables $\varepsilon^{\sigma_{k}}\rightarrow\varepsilon$. And this, together
with (6.3), completes the proof.

We formulate now some more general assertion as a theorem.

###### Theorem 6.2

Let us be given a number $\rho_{k}\in(0,\sigma_{k}+\sigma_{k}\alpha)$ for some
$k\in\\{1,2,...,r\\}$ so that it can be expressed as
$\rho_{k}=(1-\omega)\sigma_{k}(1+\alpha)$, $\omega\in(0,1)$. Denote

$\overline{\rho}\equiv(1-\omega)\overline{\sigma}(1+\alpha).$ (6.9)

The operator $(-\Delta_{z_{k}})^{\frac{\rho_{k}}{2}}$, $k=1,...,r$, is a
bounded linear operator from $C^{\overline{\sigma}(1+\alpha)}(R^{N})$ to
$C^{\omega\overline{\sigma}(1+\alpha)}(R^{N})$ that is for $u(x)\in
C^{\overline{\sigma}(1+\alpha)}(R^{N})$

$|(-\Delta_{z_{k}})^{\frac{\rho_{k}}{2}}u|_{R^{N}}^{(\omega\overline{\sigma}(1+\alpha))}\leq
C(N,N_{k},\omega,\overline{\sigma},\alpha)|u|_{R^{N}}^{\left(\overline{\sigma}(1+\alpha)\right)}.$
(6.10)

Moreover, for the Hölder seminorm of the function
$(-\Delta_{z_{k}})^{\frac{\rho_{k}}{2}}u$ with respect to some another group
of the variables $z_{l}$, $l\neq k$, we have the following interpolation
inequality with an arbitrary $\varepsilon>0$

$\left\langle(-\Delta_{z_{k}})^{\frac{\rho_{k}}{2}}u\right\rangle_{z_{l},R^{N}}^{(\omega\sigma_{l}(1+\alpha))}\leq
C\varepsilon^{\frac{\omega}{1-\omega}}\left\langle
u\right\rangle_{z_{k},R^{N}}^{(\sigma_{k}+\sigma_{k}\alpha)}+\frac{C}{\varepsilon}\left\langle
u\right\rangle_{z_{l},R^{N}}^{(\sigma_{l}+\sigma_{l}\alpha)}.$ (6.11)

Proof. The theorem follows directly from the previous proposition. It is
enough to note that

$\overline{\sigma}(1+\alpha)=\overline{\rho}(1+\beta),\quad\beta\equiv\frac{1}{1-\omega}-1=\frac{\omega}{1-\omega}>0,$

and thus
$C^{\overline{\sigma}(1+\alpha)}(R^{N})=C^{\overline{\rho}(1+\beta)}(R^{N})$.
Consequently, the proof follows from Proposition 6.1.

Move now to considering of the space
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(R_{T}^{N})$ of the
functions with the independent variable $t$ besides the independent variables
$x\in R^{N}$.

###### Proposition 6.3

The operator $(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}$, $k=1,...,r$, is a
bounded linear operator from
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$ to
$C^{\overline{\sigma}\alpha,\theta\alpha}(\overline{R_{T}^{N}})$ that is for
$u(x,t)\in C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(R_{T}^{N})$

$|(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}\alpha,\theta\alpha)}\leq
C(N,N_{k},\overline{\sigma},\theta,\alpha)|u|_{\overline{R_{T}^{N}}}^{\left(\overline{\sigma}(1+\alpha),\theta+\theta\alpha\right)}.$
(6.12)

Moreover, for the Höder seminorm of
$(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u$ with respect to some another group
of the variables $z_{l}$, $l\neq k$, we have the following interpolation
inequality with an arbitrary $\varepsilon>0$

$\left\langle(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u\right\rangle_{z_{l},\overline{R_{T}^{N}}}^{(\sigma_{l}\alpha)}\leq
C\varepsilon^{\alpha}\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\sigma_{k}+\sigma_{k}\alpha)}+\frac{C}{\varepsilon}\left\langle
u\right\rangle_{z_{l},\overline{R_{T}^{N}}}^{(\sigma_{l}+\sigma_{l}\alpha)},$
(6.13)

and for the Höder seminorm of $(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u$ with
respect to $t$ we have

$\left\langle(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u\right\rangle_{t,\overline{R_{T}^{N}}}^{(\theta\alpha)}\leq
C\varepsilon^{\alpha}\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\sigma_{k}+\sigma_{k}\alpha)}+\frac{C}{\varepsilon}\left\langle
u\right\rangle_{t,\overline{R_{T}^{N}}}^{(\theta+\theta\alpha)}.$ (6.14)

The proof of this proposition is identical to the proof of Proposition 6.1
because the proofs of inequalities (6.13) and (6.14) are identical to the
proof of (6.2).

As a direct consequence of this proposition we have the following more general
assertion, whose proof replicates the proof of Theorem 6.2.

###### Theorem 6.4

Let us be given a number $\rho_{k}\in(0,\sigma_{k}+\sigma_{k}\alpha)$ for some
$k\in\\{1,2,...,r\\}$ so that $\rho_{k}=(1-\omega)\sigma_{k}(1+\alpha)$,
$\omega\in(0,1)$. Denote

$\overline{\rho}\equiv(1-\omega)\overline{\sigma}(1+\alpha).$ (6.15)

The operator $(-\Delta_{z_{k}})^{\frac{\rho_{k}}{2}}$, $k=1,...,r$, is a
linear bounded operator from
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$ to
$C^{\omega\overline{\sigma}(1+\alpha),\omega\theta(1+\alpha)}(\overline{R_{T}^{N}})$
that is for $u(x,t)\in
C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(R_{T}^{N})$

$|(-\Delta_{z_{k}})^{\frac{\rho_{k}}{2}}u|_{\overline{R_{T}^{N}}}^{(\omega\overline{\sigma}(1+\alpha),\omega\theta(1+\alpha))}\leq
C(N,N_{k},\omega,\overline{\sigma},\theta,\alpha)|u|_{\overline{R_{T}^{N}}}^{\left(\overline{\sigma}(1+\alpha),\theta+\theta\alpha\right)}.$
(6.16)

Moreover for the Hölder seminorm of $(-\Delta_{z_{k}})^{\frac{\rho_{k}}{2}}u$
with respect to some another group op the variables $z_{l}$, $l\neq k$, the
following interpolation inequality with an arbitrary $\varepsilon>0$ is valid

$\left\langle(-\Delta_{z_{k}})^{\frac{\rho_{k}}{2}}u\right\rangle_{z_{l},\overline{R_{T}^{N}}}^{(\omega\sigma_{l}(1+\alpha))}\leq
C\varepsilon^{\frac{\omega}{1-\omega}}\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\sigma_{k}+\sigma_{k}\alpha)}+\frac{C}{\varepsilon}\left\langle
u\right\rangle_{z_{l},\overline{R_{T}^{N}}}^{(\sigma_{l}+\sigma_{l}\alpha)}.$
(6.17)

And for the Hölder seminorm of $(-\Delta_{z_{k}})^{\frac{\rho_{k}}{2}}u$ with
respect to $t$ the following inequality is valid

$\left\langle(-\Delta_{z_{k}})^{\frac{\sigma_{k}}{2}}u\right\rangle_{t,\overline{R_{T}^{N}}}^{(\omega\theta(1+\alpha))}\leq
C\varepsilon^{\frac{\omega}{1-\omega}}\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\sigma_{k}+\sigma_{k}\alpha)}+\frac{C}{\varepsilon}\left\langle
u\right\rangle_{t,\overline{R_{T}^{N}}}^{(\theta+\theta\alpha)}.$ (6.18)

In what follows we will need three more useful assertions.

###### Lemma 6.5

Let $l_{1}>0$, $l_{2}>0$ and let $\overline{\Omega}\subset R^{N_{1}+N_{2}}$ be
a domain in $R^{N_{1}+N_{2}}$ with the boundary $\partial\Omega$ of the class
$C^{\max\\{l_{1},l_{2}\\}}$. Let, further, a function $u(z_{1},z_{2})\in
C_{z_{1},z_{2}}^{l_{1},l_{2}}(\overline{\Omega})$,
$(z_{1},z_{2})\in\overline{\Omega}$, $z_{1}\in R^{N_{1}}$, $z_{2}\in
R^{N_{2}}$. Then for any integers $m>l_{1}$, $k>l_{2}$, for any positive $a$
and $b$ with

$\frac{a}{l_{1}}+\frac{b}{l_{2}}=1,$ (6.19)

and for any $\varepsilon>0$ the following estimate is valid

$\left|\delta_{\tau,z_{1}}^{m}\delta_{h,z_{2}}^{k}u(z_{1},z_{2})\right|\leq
C(\overline{\Omega},l_{1},l_{2})\times$
$\times\left[\varepsilon^{l_{1}-a}\left(\left\langle
u\right\rangle_{z_{1},\overline{\Omega}}^{(l_{1})}+|u|_{\overline{\Omega}}^{(0)}\right)+\varepsilon^{-a}\left(\left\langle
u\right\rangle_{z_{2},\overline{\Omega}}^{(l_{2})}+|u|_{\overline{\Omega}}^{(0)}\right)\right]\left|\tau\right|^{a}\left|h\right|^{b},$
(6.20)

where $\tau$ and $h$ are such that the arguments of the function
$\delta_{\tau,z_{1}}^{m}\delta_{h,z_{2}}^{k}u(z_{1},z_{2})$ stay inside
$\Omega$.

If now the domain $\Omega$ is the space $R^{N_{1}+N_{2}}$ or a half-space, or
a domain of the form

$\Omega=\\{x\in R^{N_{1}+N_{2}}:x_{i_{1}}>0,...,x_{i_{k}}>0\\},$ (6.21)

then the terms $|u|_{\overline{\Omega}}^{(0)}$ in (6.20) can be omitted.

Proof. Extend the function $u(z_{1},z_{2})$ from the domain
$\overline{\Omega}$ to the whole space $R^{N_{1}+N_{2}}$ with the preserving
(up to a multiple constant) it’s norm in
$C_{z_{1},z_{2}}^{l_{1},l_{2}}(\overline{\Omega})$ to a finite function (see.,
for example, [35], [37]). That is

$|u|_{R^{N_{1}+N_{2}}}^{(l_{1},l_{2})}\leq
C(\overline{\Omega},l_{1},l_{2})|u|_{\overline{\Omega}}^{(l_{1},l_{2})},$

but note that the corresponding seminorms satisfy (in general)

$\left\langle u\right\rangle_{z_{i},R^{N_{1}+N_{2}}}^{(l_{i})}\leq
C\left(\left\langle
u\right\rangle_{z_{i},\overline{\Omega}}^{(l_{i})}+|u|_{\overline{\Omega}}^{(0)}\right).$

However, for a domain $\Omega$ of the form (6.21) the last inequality is valid
without the term $|u|_{\overline{\Omega}}^{(0)}$ and this term does not plays
a role in the further reasonings. Overall, such an extension permits to
suppose that $\overline{\Omega}$ coincides with the whole space
$R^{N_{1}+N_{2}}$.

Consider the two possible cases of relations between $\left|\tau\right|$ and
$\left|h\right|$. Let first
$\left|\tau\right|\leq\varepsilon\left|h\right|^{l_{2}/l_{1}}$. Then, making
use of (2.4), we have

$\left|\delta_{\tau,z_{1}}^{m}\delta_{h,z_{2}}^{k}u(z_{1},z_{2})\right|\leq
C(m,k)\left\langle
u\right\rangle_{z_{1},\overline{\Omega}}^{(l_{1})}\left|\tau\right|^{l_{1}}=C(m,k)\left\langle
u\right\rangle_{z_{1},\overline{\Omega}}^{(l_{1})}\left|\tau\right|^{a}\left|\tau\right|^{l_{1}-a}\leq$
$\leq C(m,k)\left\langle
u\right\rangle_{z_{1},\overline{\Omega}}^{(l_{1})}\left|\tau\right|^{a}\left(\varepsilon\left|h\right|^{\frac{l_{2}}{l_{1}}}\right)^{l_{1}-a}=C(m,k)\varepsilon^{l_{1}-a}\left\langle
u\right\rangle_{z_{1},\overline{\Omega}}^{(l_{1})}\left|\tau\right|^{a}\left|h\right|^{b},$
(6.22)

since $b=l_{2}-al_{2}/l_{1}$ in view of (6.19).

Let now $\left|\tau\right|\geq\varepsilon\left|h\right|^{l_{2}/l_{1}}$ that is
$\left|h\right|\leq\varepsilon^{-l_{1}/l_{2}}\left|\tau\right|^{l_{1}/l_{2}}$.
Then we have, analogously to the previous case,

$\left|\delta_{\tau,z_{1}}^{m}\delta_{h,z_{2}}^{k}u(z_{1},z_{2})\right|\leq
C(m,k)\left\langle
u\right\rangle_{z_{2},\overline{\Omega}}^{(l_{2})}\left|h\right|^{l_{2}}=C(m,k)\left\langle
u\right\rangle_{z_{2},\overline{\Omega}}^{(l_{2})}\left|h\right|^{b}\left|h\right|^{l_{2}-b}\leq$
$\leq C(m,k)\left\langle
u\right\rangle_{z_{2},\overline{\Omega}}^{(l_{2})}\left|h\right|^{b}\left(\varepsilon^{-l_{1}/l_{2}}\left|\tau\right|^{\frac{l_{1}}{l_{2}}}\right)^{l_{2}-b}=C(m,k)\varepsilon^{-a}\left\langle
u\right\rangle_{z_{1},\overline{\Omega}}^{(l_{1})}\left|\tau\right|^{a}\left|h\right|^{b},$
(6.23)

since $a=l_{1}-bl_{1}/l_{2}$ in view of (6.19).

The lemma follows now from (6.22) and (6.23).

The proved lemma is valid also in the case when the function under
consideration depends only on a single variable $z$ that is when
$z_{1}=z_{2}$. The proof in this case is a simple replication of the proof of
Lemma 6.5 therefore we give the following assertion without a proof.

###### Corollary 6.6

Let $l>0$ and let $\overline{\Omega}\subset R^{N}$ be a domain in $R^{N}$ with
the boundary $\partial\Omega$ of the class $C^{l}$. Let us also be given a
function $u(z)\in C^{l}(\overline{\Omega})$. Let, finally, $m>l$ and $k>l$ be
arbitrary integers and let positive numbers $a$ and $b$ be such that

$\frac{a}{l}+\frac{b}{l}=1\Leftrightarrow a+b=l.$ (6.24)

Then

$\left|\delta_{\tau,z}^{m}\delta_{h,z}^{k}u(z)\right|\leq
C(\overline{\Omega},l)\left(\left\langle
u\right\rangle_{z,\overline{\Omega}}^{(l)}+|u|_{\overline{\Omega}}^{(0)}\right)\left|\tau\right|^{a}\left|h\right|^{b},$
(6.25)

where $\tau$ and $h$ are taken in the way that the arguments of
$\delta_{\tau,z}^{m}\delta_{h,z}^{k}u(z)$ stay in the domain $\Omega$.

If now the the domain $\Omega$ coincides with $R^{N}$ or with a half-space, or
with a domain of the form

$\Omega=\\{z\in R^{N}:z_{i_{1}}>0,...,z_{i_{k}}>0\\},$ (6.26)

then the terms $|u|_{\overline{\Omega}}^{(0)}$ in (6.25) can be omitted

From Lemma 6.5 and Corollary 6.6 one can infer a useful assertion, which
generalizes, in a sense, some analogous assertion for Hölder spaces from [38].
We mean the assertion number 5 in section 6, [38], for the one-dimensional
case.

###### Proposition 6.7

Let a function $u(x)\in C\overline{{}^{l}}(\overline{\Omega})$ in the sense of
definitions (2.1) - (2.4), $\overline{l}=(l_{1},...,l_{N})$. Let, further, a
domain $\Omega$ be the whole space $R^{N}$, a half-space or a domain of the
form (6.26). Let also $k$ be a fixed index, $k\in\\{1,...,N\\}$ and let $h>0$.
Let, finally, $a\in(0,l_{k})$ be such that $l_{k}-a$ is a noninteger. Consider
for an integer $m>l_{k}$ the function

$u_{a,h,k}(x)\equiv\frac{\delta_{h,x_{k}}^{m}u(x)}{h^{a}}.$ (6.27)

The function $u_{a,h,k}(x)$ belongs to the space
$C^{\overline{l}\omega_{a}}(\overline{\Omega})$,

$\omega_{a}\equiv(1-a/l_{k}),\quad\overline{l}\omega_{a}=(l_{1}\omega_{a},...,l_{N}\omega_{a}),$
(6.28)

and uniformly in $h$

$\left\|u_{a,h,k}(x)\right\|_{C^{\overline{l}\omega_{a}}(\overline{\Omega})}\equiv|u_{a,h,k}(x)|_{\overline{\Omega}}^{(\overline{l}\omega_{a})}\leq
C|u(x)|_{\overline{\Omega}}^{(\overline{l})}$ (6.29)

with a constant $C$, which does not depend on $h$. In particular, for an
arbitrary $\varepsilon>0$

$\left\langle
u_{a,h,k}(x)\right\rangle_{x_{i},\overline{\Omega}}^{(\omega_{a}l_{i})}\leq\genfrac{\\{}{.}{0.0pt}{0}{C(l_{i},l_{k},m)\left(\varepsilon^{l_{i}-a}\left\langle
u\right\rangle_{x_{i},\overline{\Omega}}^{(l_{i})}+\varepsilon^{-a}\left\langle
u\right\rangle_{x_{k},\overline{\Omega}}^{(l_{k})}\right),\quad i\neq
k,}{C(l_{k},m)\left\langle
u\right\rangle_{x_{k},\overline{\Omega}}^{(l_{k})},\quad i=k,}$ (6.30)
$\left|u_{a,h,k}(x)\right|_{\overline{\Omega}}^{(0)}\leq
C(m)\left(\varepsilon^{l_{k}-a}\left\langle
u\right\rangle_{x_{k},\overline{\Omega}}^{(l_{k})}+\varepsilon^{-a}\left|u\right|_{\overline{\Omega}}^{(0)}\right).$
(6.31)

Proof. Estimates (6.30) are a direct consequence of Lemma 6.5 and Corollary
6.6 with the taking into account the properties of Hölder seminorms in (2.4).
To obtain estimate (6.31) we first suppose that $h\leq\varepsilon$. Then

$\left|u_{a,h,k}(x)\right|_{\overline{\Omega}}^{(0)}=\left|\frac{\delta_{h,x_{k}}^{m}u(x)}{h^{a}}\right|_{\overline{\Omega}}^{(0)}=\left|\frac{\delta_{h,x_{k}}^{m}u(x)}{h^{l_{k}}}\right|_{\overline{\Omega}}^{(0)}h^{l_{k}-a}\leq\varepsilon^{l_{k}-a}\left\langle
u\right\rangle_{x_{k},\overline{\Omega}}^{(l_{k})}.$

If now $h>\varepsilon$, then

$\left|u_{a,h,k}(x)\right|_{\overline{\Omega}}^{(0)}=\left|\frac{\delta_{h,x_{k}}^{m}u(x)}{h^{a}}\right|_{\overline{\Omega}}^{(0)}\leq\varepsilon^{-a}\left|\delta_{h,x_{k}}^{m}u(x)\right|_{\overline{\Omega}}^{(0)}\leq
C(m)\varepsilon^{-a}\left|u\right|_{\overline{\Omega}}^{(0)},$

which completes the proof of the proposition.

Consider, further, the acting of the Caputo - Jrbashyan derivative $D_{\ast
t}^{\theta}$ in the introduced anisotropic Hölder spaces.

###### Proposition 6.8

Let nonintegers $\theta>0$ and $\alpha>0$ be such that

$[\theta+\theta\alpha]=[\theta],\text{ that is }\theta\alpha<1-\\{\theta\\}.$
(6.32)

Then the operator of the Caputo - Jrbashyan derivative $D_{\ast t}^{\theta}$
is a bounded linear operator from
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$ to
$C^{\overline{\sigma}\alpha,\theta\alpha}(\overline{R_{T}^{N}})$ that is for a
function $u(x,t)\in
C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$ the
following estimate is valid

$|D_{\ast
t}^{\theta}u|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}\alpha,\theta\alpha)}\leq
C(N,\overline{\sigma},\theta,\alpha)|u|_{\overline{R_{T}^{N}}}^{\left(\overline{\sigma}(1+\alpha),\theta+\theta\alpha\right)}.$
(6.33)

Proof. Firstly, from (3.1) it follows that

$|D_{\ast t}^{\theta}u|_{\overline{R_{T}^{N}}}^{(0)}+\left\langle D_{\ast
t}^{\theta}u\right\rangle_{t,\overline{R_{T}^{N}}}^{(\theta\alpha)}\leq
C(\alpha,\theta,T)|u|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}(1+\alpha),\theta+\theta\alpha)}.$
(6.34)

Further, let $p-1<\theta<p$ with some positive integer $p$. Represent the
fractional derivative $D_{\ast t}^{\theta}u$ in the form (see (1.9),
$\theta-p+1=\\{\theta\\}$, $p-1=[\theta]$) by applying integration by parts

$D_{\ast t}^{\theta}u(x,t)=$
$=C(\theta){\int\limits_{0}^{t}}\frac{u_{\tau}^{(p)}(x,\tau)d\tau}{(t-\tau)^{\\{\theta\\}}}=C(\theta){\int\limits_{0}^{t}}\frac{\left[u_{\tau}^{(p-1)}(x,\tau)-u_{t}^{(p-1)}(x,t)\right]_{\tau}^{\prime}d\tau}{(t-\tau)^{\\{\theta\\}}}=$
$=C(\theta)\frac{u_{t}^{([\theta])}(x,t)-u_{\tau}^{([\theta])}(x,0)}{(t-0)^{\\{\theta\\}}}+$
$+C(\theta){\int\limits_{0}^{t}}\frac{\left[u_{\tau}^{([\theta])}(x,\tau)-u_{t}^{([\theta])}(x,t)\right]d\tau}{(t-\tau)^{1+\\{\theta\\}}}\equiv
I_{1}+I_{2}.$ (6.35)

Note further that on the ground of (2.6) $u_{t}^{([\theta])}(x,t)\in
C^{\overline{\sigma}(1+\alpha)(1-\frac{[\theta]}{\theta+\theta\alpha}),\\{\theta\\}+\theta\alpha}(\overline{R_{T}^{N}})$
and

$\left\langle
u_{t}^{([\theta])}\right\rangle_{t,\overline{R_{T}^{N}}}^{(\\{\theta\\}+\theta\alpha)}+{\sum\limits_{k=1}^{r}}\left\langle
u_{t}^{([\theta])}\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\sigma_{k}(1+\alpha)(1-\frac{[\theta]}{\theta+\theta\alpha}))}\leq
C\left\|u\right\|_{C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})}.$
(6.36)

Making use of the last inequality we estimate Hölder seminorms of the
expressions $I_{1}$ and $I_{2}$ with respect to some group of the space
variables $z_{n}$. Consider first the ratio $I_{1}$. For this we make use of
Lemma 6.5 as applied to $u_{t}^{([\theta])}(x,t)$ and to the variables $t$ and
$z_{n}$. Since $\\{\theta\\}+\theta\alpha<1$, we can take in (6.20) the order
of the difference in $t$ to be one, $m=1$. And we fix some integer
$k>\sigma_{n}(1+\alpha)(1-\frac{[\theta]}{\theta+\theta\alpha})$ as the order
of the difference in the space variables $z_{n}$. To apply the mentioned lemma
we choose the exponent $a$ as $a=\\{\theta\\}$ and then the exponent $b$ is
defined from the relation

$\frac{\\{\theta\\}}{\\{\theta\\}+\theta\alpha}+\frac{b}{\sigma_{n}(1+\alpha)(1-\frac{[\theta]}{\theta+\theta\alpha})}=1,$

that is

$b=\sigma_{n}(1+\alpha)\frac{\\{\theta\\}+\theta\alpha}{\theta+\theta\alpha}\frac{\theta\alpha}{\\{\theta\\}+\theta\alpha}=\sigma_{n}\alpha.$

Thus we have on the ground of (6.20) ($h\in R^{N_{n}}$)

$\left|\delta_{h,z_{n}}^{k}\left[u_{t}^{([\theta])}(x,t)-u_{\tau}^{([\theta])}(x,0)\right]\right|\leq$
$\leq C\left(\left\langle
u_{t}^{([\theta])}\right\rangle_{t,\overline{R_{T}^{N}}}^{(\\{\theta\\}+\theta\alpha)}+\left\langle
u_{t}^{([\theta])}\right\rangle_{z_{n},\overline{R_{T}^{N}}}^{(\sigma_{n}(1+\alpha)(1-\frac{[\theta]}{\theta+\theta\alpha}))}\right)t^{\\{\theta\\}}|h|^{\sigma_{n}\alpha}$

or, dividing both parts by $t^{\\{\theta\\}}$,

$\left|\delta_{h,z_{n}}^{k}I_{1}\right|\leq C\left(\left\langle
u_{t}^{([\theta])}\right\rangle_{t,\overline{R_{T}^{N}}}^{(\\{\theta\\}+\theta\alpha)}+\left\langle
u_{t}^{([\theta])}\right\rangle_{z_{n},\overline{R_{T}^{N}}}^{(\sigma_{n}(1+\alpha)(1-\frac{[\theta]}{\theta+\theta\alpha}))}\right)|h|^{\sigma_{n}\alpha}$

that is

$\left\langle
I_{1}\right\rangle_{z_{n},\overline{R_{T}^{N}}}^{(\sigma_{n}\alpha)}\leq
C\left(\left\langle
u_{t}^{([\theta])}\right\rangle_{t,\overline{R_{T}^{N}}}^{(\\{\theta\\}+\theta\alpha)}+\left\langle
u_{t}^{([\theta])}\right\rangle_{z_{n},\overline{R_{T}^{N}}}^{(\sigma_{n}(1+\alpha)(1-\frac{[\theta]}{\theta+\theta\alpha}))}\right).$
(6.37)

Estimate now the seminorm of the integral $I_{2}$ in (6.35). Let $h\in
R^{N_{n}}$ be fixed. Suppose first that $t>|h|^{\sigma_{n}/\theta}$ and
decompose the integral in $I_{2}$ into two parts as follows

$I_{2}={\int\limits_{t-|h|^{\sigma_{n}/\theta}}^{t}}\frac{\left[u_{\tau}^{([\theta])}(x,\tau)-u_{t}^{([\theta])}(x,t)\right]d\tau}{(t-\tau)^{1+\\{\theta\\}}}+$
$+{\int\limits_{0}^{t-|h|^{\sigma_{n}/\theta}}}\frac{\left[u_{\tau}^{([\theta])}(x,\tau)-u_{t}^{([\theta])}(x,t)\right]d\tau}{(t-\tau)^{1+\\{\theta\\}}}\equiv
J_{1}+J_{2}.$

Estimate separately the finite differences $\delta_{h,z_{n}}^{k}J_{1}$ and
$\delta_{h,z_{n}}^{k}J_{2}$,
$k>\sigma_{n}(1+\alpha)(1-\frac{[\theta]}{\theta+\theta\alpha})$. On the
ground of the Hölder property of $u_{t}^{([\theta])}(x,t)$ in $t$ we have for
$J_{1}$

$|\delta_{h,z_{n}}^{k}J_{1}|\leq C\left\langle
u_{t}^{([\theta])}\right\rangle_{t,\overline{R_{T}^{N}}}^{(\\{\theta\\}+\theta\alpha)}{\int\limits_{t-|h|^{\sigma_{n}/\theta}}^{t}}\frac{(t-\tau)^{\\{\theta\\}+\theta\alpha}d\tau}{(t-\tau)^{1+\\{\theta\\}}}=$
$=C\left\langle
u_{t}^{([\theta])}\right\rangle_{t,\overline{R_{T}^{N}}}^{(\\{\theta\\}+\theta\alpha)}\left(|h|^{\sigma_{n}/\theta}\right)^{\theta\alpha}=C\left\langle
u_{t}^{([\theta])}\right\rangle_{t,\overline{R_{T}^{N}}}^{(\\{\theta\\}+\theta\alpha)}|h|^{\sigma_{n}\alpha}.$
(6.38)

Use now the Hölder property of $u_{t}^{([\theta])}(x,t)$ in $z_{n}$ to obtain

$|\delta_{h,z_{n}}^{k}J_{2}|\leq C\left\langle
u_{t}^{([\theta])}\right\rangle_{z_{n},\overline{R_{T}^{N}}}^{(\sigma_{n}(1+\alpha)(1-\frac{[\theta]}{\theta+\theta\alpha}))}|h|^{\sigma_{n}(1+\alpha)(1-\frac{[\theta]}{\theta+\theta\alpha})}{\int\limits_{0}^{t-|h|^{\sigma_{n}/\theta}}}\frac{d\tau}{(t-\tau)^{1+\\{\theta\\}}}=$
$=C\left\langle
u_{t}^{([\theta])}\right\rangle_{z_{n},\overline{R_{T}^{N}}}^{(\sigma_{n}(1+\alpha)(1-\frac{[\theta]}{\theta+\theta\alpha}))}|h|^{\sigma_{n}(1+\alpha)(1-\frac{[\theta]}{\theta+\theta\alpha})}\left(|h|^{\frac{\sigma_{n}}{\theta}}\right)^{-\\{\theta\\}}=$
$=C\left\langle
u_{t}^{([\theta])}\right\rangle_{z_{n},\overline{R_{T}^{N}}}^{(\sigma_{n}(1+\alpha)(1-\frac{[\theta]}{\theta+\theta\alpha}))}|h|^{\sigma_{n}\alpha}.$
(6.39)

If now $t\leq|h|^{\sigma_{n}/\theta}$, then analogously to the estimate for
$\delta_{h,z_{n}}^{k}J_{1}$ we have for $\delta_{h,z_{n}}^{k}I_{2}$

$|\delta_{h,z_{n}}^{k}I_{2}|\leq C\left\langle
u_{t}^{([\theta])}\right\rangle_{t,\overline{R_{T}^{N}}}^{(\\{\theta\\}+\theta\alpha)}{\int\limits_{0}^{|h|^{\sigma_{n}/\theta}}}\frac{(t-\tau)^{\\{\theta\\}+\theta\alpha}d\tau}{(t-\tau)^{1+\\{\theta\\}}}=C\left\langle
u_{t}^{([\theta])}\right\rangle_{t,\overline{R_{T}^{N}}}^{(\\{\theta\\}+\theta\alpha)}|h|^{\sigma_{n}\alpha}.$
(6.40)

Collecting estimates (6.38) - (6.40), we see that

$\left\langle
I_{2}\right\rangle_{z_{n},\overline{R_{T}^{N}}}^{(\sigma_{n}\alpha)}\leq
C\left(\left\langle
u_{t}^{([\theta])}\right\rangle_{t,\overline{R_{T}^{N}}}^{(\\{\theta\\}+\theta\alpha)}+\left\langle
u_{t}^{([\theta])}\right\rangle_{z_{n},\overline{R_{T}^{N}}}^{(\sigma_{n}(1+\alpha)(1-\frac{[\theta]}{\theta+\theta\alpha}))}\right).$
(6.41)

The assertion of the proposition and estimate (6.33) follow now from (6.34),
(6.36), (6.37) and (6.41).

We have also some more general assertion.

###### Theorem 6.9

Let $\\{\theta\\}+\theta\alpha<1$ and let a noninteger
$\mu\in(0,\theta+\theta\alpha)$ so that it can be represented as
$\mu=(1-\omega)\theta(1+\alpha)$, $\omega\in(0,1)$.

The operator $D_{\ast t}^{\mu}$ is a bonded linear operator from
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$ to
$C^{\omega\overline{\sigma}(1+\alpha),\omega\theta(1+\alpha)}(\overline{R_{T}^{N}})$
that is for $u(x,t)\in
C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(R_{T}^{N})$

$|D_{\ast
t}^{\mu}u|_{\overline{R_{T}^{N}}}^{(\omega\overline{\sigma}(1+\alpha),\omega\theta(1+\alpha))}\leq
C(N,\omega,\overline{\sigma},\theta,\alpha)|u|_{\overline{R_{T}^{N}}}^{\left(\overline{\sigma}(1+\alpha),\theta+\theta\alpha\right)}.$
(6.42)

Proof. The theorem is a direct consequence of the previous Proposition. It is
enough to note that

$\theta(1+\alpha)=\mu(1+\beta),\quad\overline{\sigma}(1+\alpha)=\overline{\rho}(1+\beta),\quad\beta\equiv\frac{1}{1-\omega}-1=\frac{\omega}{1-\omega}>0,$

where

$\beta\equiv\frac{1}{1-\omega}-1=\frac{\omega}{1-\omega}>0,\quad\rho_{k}\equiv(1-\omega)\sigma_{k}(1+\alpha).$

Thus
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})=C^{\overline{\rho}(1+\beta),\mu+\mu\beta}(\overline{R_{T}^{N}})$.
Consequently, the assertion of the theorem is obtained by the application of
Proposition 6.8.

It was shown in the example after Theorem 2.1 that the restriction
$\\{\theta\\}+\theta\alpha<1$ can not be omitted in general. However this
restriction is a possible sufficient condition for the assertions of
Proposition 6.8 and Theorem 6.9 to be valid. As a conclusion of this section
we present here some another sufficient condition. It is that all the
permitted by a class derivatives in $t$ vanish at $t=0$.

But first we prove an auxiliary lemma.

###### Lemma 6.10

Let a function $f(t)\in C^{\theta+\beta}([0,\infty))$, where $\theta$ and
$\theta+\beta$ are nonintegers and let at $t=0$ the function $f(t)$ satisfy
the condition

$f_{t}^{(k)}(0)=0,\quad k=0,...,[\theta].$ (6.43)

Then it’s fractional Caputo - Jrbashyan derivative $D_{\ast t}^{\theta}f(t)$,
which was defined in (1.4), (1.9), coincides with the Marshaud derivative(see
[1], section 5.4) that is it has also the representation

$D_{\ast
t}^{\theta}f(t)=C(\theta,m){\int\limits_{0}^{\infty}}\frac{\delta_{-\tau,t}^{m}f(t)}{\tau^{1+\theta}}d\tau\equiv
C(\theta,m)T_{\theta,m}(f).$ (6.44)

Here $\delta_{-\tau,t}f(t)=f(t)-f(t-\tau)$, $\delta_{-\tau,t}^{n}f(t)=$
$\delta_{-\tau,t}\left(\delta_{-\tau,t}^{m-1}f(t)\right)$ is a backward
difference of an arbitrary but fixed order $m>\theta$ and the function $f(t)$
is extended by identical zero in the domain $t<0$.

Proof. The proof reproduces the schema of reasonings from [1], section 25.4,
Lemma 25.3 for the fractional Riess differentiation as it is applied in [1] to
obtain representation (1.12). Namely, we compare the Fourier transforms of the
left and of the right hand sides of (6.44).

Note that zero extension of the function $f(t)$ in the domain $t<0$ belongs to
the space $C^{\theta+\beta}$ in the case $\\{\theta\\}+\beta<1$
($\\{\theta\\}$ is the fractional part of $\theta$) and in the case
$\\{\theta\\}+\beta>1$ this extension belongs to the space
$C^{\theta+\beta^{\prime}}$ with an arbitrary $\beta^{\prime}<1-\\{\theta\\}$.
Because of this the following estimate is valid

$|\delta_{-\tau,t}^{m}f(t)|\leq\genfrac{\\{}{.}{0.0pt}{0}{C(f)\tau^{\theta+\gamma},\quad
t<1}{C(f),\quad t>1,},\quad\gamma=\min\\{\beta,\beta^{\prime}\\},$ (6.45)

where we preserve the same notation $f(t)$ for the extension.

Suppose first that the function $f(t)$ is more smooth in the sense that it
possess the continuous derivative of order $n=[\theta]+1$ ($n-1<\theta<n$ ).
Besides, we first suppose that $f(t)$ is integrable and even finite (for
simplicity) for $t\rightarrow+\infty$. Then the Fourier transform of the
derivative $D_{\ast t}^{\theta}f(t)$ from (1.9) is calculated in, for example,
[1], section 7 and it is equal to (see Lemma 8.4 below and Remark 8.5 after
it)

$F(D_{\ast t}^{\theta}f)(\xi)\equiv\widehat{D_{\ast
t}^{\theta}f}(\xi)=(i\xi)^{[\theta]}(i\xi)^{\\{\theta\\}}\widehat{f}(\xi).$
(6.46)

Here $\widehat{f}(\xi)$ the Fourier transform of $f(t)$ and
$(i\xi)^{\\{\theta\\}}$ is the following analytic extension of the function
$z^{\\{\theta\\}}$ from the positive real axis to the right half-plane
$\operatorname{Re}z>0$

$(i\xi)^{\\{\theta\\}}=|\xi|^{\\{\theta\\}}e^{i\theta\frac{\pi}{2}sign\xi}$
(6.47)

Calculate now the Fourier image of the integral $T_{\theta,m}(f)(t)$ in the
right hand side of (6.44). We have (comp. [1], section 25.4, Lemma 25.3)

$\widehat{T_{\theta,m}(f)}(\xi)={\int\limits_{0}^{\infty}}\frac{\widehat{\delta_{-\tau,t}^{m}f(t)}}{\tau^{1+\theta}}d\tau=\widehat{f}(\xi){\int\limits_{0}^{\infty}}\frac{(1-e^{-i\tau\xi})^{m}}{\tau^{1+[\theta]+\\{\theta\\}}}d\tau.$
(6.48)

Make in the last integral the change of variables $z=i\tau\xi$. After such
change the last integral for $\xi>0$ is transformed into the integral along
positive imaginary axis in the positive direction that is

$\widehat{T_{\theta,m}(f)}(\xi)=(i\xi)^{[\theta]}(i\xi)^{\\{\theta\\}}\widehat{f}(\xi){\int\limits_{0i}^{\infty
i}}\frac{(1-e^{-z})^{m}}{z^{1+[\theta]+\\{\theta\\}}}dz.$ (6.49)

Consider for $R>0$ the integral of the last integrand along the circular
contour

$C_{R}^{+}=[0i,Ri]+[R\cdot e^{i\varphi},\varphi\in(\pi/2,0)]-[0,R]$

This contour consists of the interval of the imaginary axis $[0i,Ri]$, of the
clock-wise oriented quarter of circle $[R\cdot
e^{i\varphi},\varphi\in(\pi/2,0)]$, and of the negatively oriented interval
$[0,R]$ of the real axis. Since the integrand is analytic and continuous in
the right half-plane, the integral along this contour $C_{R}^{+}$ is equal to
zero. Besides,

$\left|{\int\limits_{[R\cdot
e^{i\varphi},\varphi\in(\pi/2,0)]}}\frac{(1-e^{-z})^{m}}{z^{1+[\theta]+\\{\theta\\}}}dz\right|\leq\frac{2^{m}}{R^{1+[\theta]+\\{\theta\\}}}\frac{\pi}{2}R\rightarrow
0,\quad R\rightarrow\infty.$

Consequently, letting $R\rightarrow\infty$, we obtain un view of the
directions of integration

${\int\limits_{0i}^{\infty
i}}\frac{(1-e^{-z})^{m}}{z^{1+[\theta]+\\{\theta\\}}}dz={\int\limits_{0}^{\infty}}\frac{(1-e^{-x})^{m}}{x^{1+\theta}}dx\equiv
1/C(m,\theta)>0.$

If now $\xi<0$, then the last integral in (6.48) after the change $z=i\tau\xi$
goes to an analogous to (6.49) integral, but along the negatively oriented
imaginary axis. The analogous considerations for this case give the same
result that is

${\int\limits_{0i}^{-\infty
i}}\frac{(1-e^{-z})^{m}}{z^{1+[\theta]+\\{\theta\\}}}dz={\int\limits_{0}^{\infty}}\frac{(1-e^{-x})^{m}}{x^{1+\theta}}dx\equiv
1/C(m,\theta)>0$

with the same constant $C(m,\theta)$. Thus for any sign of $\xi$ we get

$\widehat{T_{\theta,m}(f)}(\xi)=C(m,\theta)(i\xi)^{[\theta]}(i\xi)^{\\{\theta\\}}\widehat{f}(\xi).$
(6.50)

From (6.46) and (6.50) it follows that $\widehat{D_{\ast
t}^{\theta}f}(\xi)=C(m,\theta)\widehat{T_{\theta,m}(f)}(\xi)$,
$\xi\in(-\infty,\infty)$, which proves assertion (6.44) of the lemma for
finite functions $f(t)$ of the class $C^{[\theta]+1}([0,\infty))$.

General case is obtained now by a mollifying and by a cutting off the function
$f(t)$ with a subsequent limiting process in the final equality (6.44), which
does not depend on extra smoothness.

On the ground of Lemma 6.10, by replication of the proof of Proposition 6.3,
we obtain the following proposition.

###### Proposition 6.11

For a function $u(x,t)\in
C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$
with

$u_{t}^{(k)}(x,0)=0,\quad k=0,...,[\theta+\theta\alpha]$

the following estimate is valid

$|D_{\ast
t}^{\theta}u|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}\alpha,\theta\alpha)}\leq
C(N,\overline{\sigma},\theta,\alpha)|u|_{\overline{R_{T}^{N}}}^{\left(\overline{\sigma}(1+\alpha),\theta+\theta\alpha\right)}.$
(6.51)

Moreover, for the Hölder seminorm of the derivative $D_{\ast t}^{\theta}u$
with respect to a group $z_{k}$ of the space variables we have the following
interpolation inequality with an arbitrary $\varepsilon>0$

$\left\langle D_{\ast
t}^{\theta}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\theta\alpha)}\leq
C\varepsilon^{\alpha}\left\langle
u\right\rangle_{t,\overline{R_{T}^{N}}}^{(\theta+\theta\alpha)}+\frac{C}{\varepsilon}\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\sigma_{k}+\sigma_{k}\alpha)}.$
(6.52)

Presenting this Proposition without a proof, we nevertheless note the
following. First of all, to make use of Lemma 6.10 and representation (6.44),
we need to extend the function $u(x,t)$ to a finite in $t$ function from the
domain $\overline{R_{T}^{N}}$ to the domain $t>T$ that is to whole subspace
$\overline{R_{\infty}^{N}}=R^{N}\times[0,\infty)$. This can be done with a
controlled preservation of the norm in
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$ in
the way, which is described in, for example, [35], [37]. In the second place,
extended further by zero in the domain $t<0$ function $u(x,t)$ preserves it’s
norm in $C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(R^{N}\times
R^{1})$. It is precisely this place, where we need all the derivatives of
$u(x,t)$ in $t$ up to the order $[\theta+\theta\alpha]$ (possibly bigger than
$[\theta]$ as it is required in Lemma 6.10) to vanish at $t=0$. After this the
proof of the above Proposition coincides with the proof of Proposition 6.1
with the making use of the representation from Lemma 6.10 for $D_{\ast
t}^{\theta}u$ (for obtaining the smoothness in the space variables) and the
proof of Proposition 3.1 (the smoothness in $t$). The only difference is that
the interval of integration $(0,\infty)$ in representation (6.44) for $D_{\ast
t}^{\theta}u$ is naturally split not into a centered at zero ball and it’s
outer part (as in the proof of Proposition 6.1), but into the intervals
$(0,\varepsilon|h|^{\frac{\sigma_{k}}{\theta}})$ and
$(\varepsilon|h|^{\frac{\sigma_{k}}{\theta}},\infty)$.

##  7 Some additional technical assertions on the properties of fractional
differentiation in anisotropic Hölder spaces.

In the present section we prove several useful for applications assertions of
the interpolation type on the properties of fractional differentiation in
anisotropic Hölder spaces.

###### Lemma 7.1

Let a function $u(x,t)$ is defined in $\overline{R_{T}^{N}}$ and possesses
bounded in $\overline{R_{T}^{N}}$ fractional derivative $D_{\ast
t}^{\mu}u(x,t)$ of a noninteger order $\mu>0$. Let also this derivative
possesses the smoothness of a noninteger order $\beta>0$ with respect to a
group $z_{k}\in R^{N_{k}}$ of the space variables that is

$\left\langle D_{\ast
t}^{\mu}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}=\sup_{(x,t)\in\overline{R_{T}^{N}},\overline{h}\in
R^{N_{k}},\overline{h}\neq 0}\frac{|\delta_{\overline{h},z_{k}}^{m}D_{\ast
t}^{\mu}u(x,t)|}{|\overline{h}|^{\beta}}<\infty,\quad m>\beta.$ (7.1)

Here $\delta_{\overline{h},z_{k}}^{m}u(x,t)$ is the finite difference of order
$m$ from the function $D_{\ast t}^{\mu}u(x,t)$ in the variables $z_{k}$ with
the step $\overline{h}\in R^{N_{k}}$.

Then

$\left|\delta_{\tau,t}^{p}\delta_{\overline{h},z_{k}}^{m}u(x,t)\right|\leq
C\left\langle D_{\ast
t}^{\mu}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}\tau^{\mu}|\overline{h}|^{\beta},\quad
p\geq[\mu]+1,m>\beta,$ (7.2)

where $\delta_{\tau,t}^{p}v(x,t)$ is the finite difference in $t$ of step
$\tau>0$ from the function $v(x,t)$.

If a function $u(x,t)$ possesses bounded in $\overline{R_{T}^{N}}$ derivative
$u_{t}^{(n)}(x,t)$ in $t$ of an integer order $n>0$, which possesses the
smoothness in a space group $z_{k}\in R^{N_{k}}$ of a noninteger order
$\beta>0$ that is

$\left\langle
u_{t}^{(n)}\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}=\sup_{(x,t)\in\overline{R_{T}^{N}},\overline{h}\in
R^{N_{k}},\overline{h}\neq
0}\frac{|\delta_{\overline{h},z_{k}}^{m}u_{t}^{(n)}(x,t)|}{|\overline{h}|^{\beta}}<\infty,\quad
m>\beta,$ (7.3)

then also

$\left|\delta_{\tau,t}^{p}\delta_{\overline{h},z_{k}}^{m}u(x,t)\right|\leq
C\left\langle
u_{t}^{(n)}\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}\tau^{n}|\overline{h}|^{\beta},\quad
m>\beta,\,p\geq n.$ (7.4)

Proof. First of all, the case of an integer order $n>0$ follows directly from
the mean value theorem and from condition (7.3) - see formulas (7.14), (7.15)
below. Therefore we consider only the case of a noninteger derivative.

Further, we can assume without loss of generality that the function $u(x,t)$
possesses the property

$\frac{\partial^{k}u(x,0)}{\partial t^{k}}\equiv 0,\quad k=0,...,[\mu].$ (7.5)

In the opposite case we can replace $u(x,t)$ with the function

$\overline{u}(x,t)\equiv
u(x,t)-{\sum\limits_{k=0}^{[\mu]}}\frac{t^{k}}{k!}\frac{\partial^{k}u(x,0)}{\partial
t^{k}}$

since
$\delta_{\tau,t}^{[\mu]+1}\delta_{\overline{h},z_{k}}^{m}u(x,t)=\delta_{\tau,t}^{[\mu]+1}\delta_{\overline{h},z_{k}}^{m}\overline{u}(x,t)$
and $D_{\ast t}^{\mu}u=D_{\ast t}^{\mu}\overline{u}$. But for a function
$u(x,t)$ with (7.5) the following representation is valid (see [3], section
3.5)

$D_{\ast t}^{\mu}u(x,t)=D_{\ast
t}^{\\{\mu\\}}\left[u_{t}^{([\mu])}(x,t)\right],$ (7.6)
$u(x,t)=J_{\\{\mu\\}}\left\\{D_{\ast
t}^{\\{\mu\\}}\left[u_{t}^{([\mu])}(x,t)\right]\right\\}.$

Here $J_{\mu}$ is the operator of fractional integration of order $\mu$ that
is

$u(x,t)=\frac{1}{\Gamma(\mu)}{\int\limits_{0}^{t}}(t-\tau)^{-1+\mu}D_{\ast\tau}^{\mu}u(x,\tau)d\tau=$
(7.7)
$=\frac{1}{\Gamma(\\{\mu\\})}{\int\limits_{0}^{t}}(t-\tau)^{-1+\\{\mu\\}}D_{\ast\tau}^{\mu}u_{\tau}^{([\mu])}(x,\tau)d\tau.$

Note also that we can assume $p=[\mu]+1$.

Let first $\mu\in(0,1)$ that is $\mu=\\{\mu\\}$, $[\mu]=0$. Then with
$m>\beta$

$\delta_{\tau,t}\delta_{\overline{h},z_{k}}^{m}u(x,t)=\frac{1}{\Gamma(\mu)}\delta_{\tau,t}\left\\{{\int\limits_{0}^{t}}(t-\omega)^{-1+\mu}\left[\delta_{\overline{h},z_{k}}^{m}D_{\ast\omega}^{\mu}u(x,\omega)\right]d\omega\right\\}.$
(7.8)

Denote

$v(x,t)=\frac{1}{\Gamma(\mu)}\left[\delta_{\overline{h},z_{k}}^{m}D_{\ast\tau}^{\mu}u(x,t)\right],$
(7.9)

and remark that according to (7.1),

$\left|v(x,t)\right|\leq C(\mu)\left\langle D_{\ast
t}^{\mu}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}|\overline{h}|^{\beta}.$
(7.10)

Assuming that $t>2\tau$, represent the difference
$\delta_{\tau,t}\delta_{\overline{h},z_{k}}^{m}u(x,t)$ in the form

$\delta_{\tau,t}\delta_{\overline{h},z_{k}}^{m}u(x,t)={\int\limits_{t-2\tau}^{t+\tau}}(t+\tau-\omega)^{-1+\mu}v(x,\omega)d\omega-$
$-{\int\limits_{t-2\tau}^{t}}(t-\omega)^{-1+\mu}v(x,\omega)d\omega+$
$+{\int\limits_{0}^{t-2\tau}}\left[(t+\tau-\omega)^{-1+\mu}-(t-\omega)^{-1+\mu}\right]v(x,\omega)d\omega\equiv
I_{1}+I_{2}+I_{3}.$ (7.11)

Taking into account (7.10) we have for $I_{1}$

$|I_{1}|\leq C(\mu)\left\langle D_{\ast
t}^{\mu}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}|\overline{h}|^{\beta}{\int\limits_{t-2\tau}^{t+\tau}}(t+\tau-\omega)^{-1+\mu}d\omega=$
$=C(\mu)\left\langle D_{\ast
t}^{\mu}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}|\overline{h}|^{\beta}\tau^{\mu}.$

The integral $I_{2}$ is estimated analogously, which gives

$|I_{1}|+|I_{2}|\leq C(\mu)\left\langle D_{\ast
t}^{\mu}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}|\overline{h}|^{\beta}\tau^{\mu}.$
(7.12)

To estimate $I_{3}$ we make use of the mean value theorem

$I_{3}=C(\mu)\tau{\int\limits_{0}^{t-2\tau}}(t+\varkappa(t-\omega)\tau-\omega)^{-2+\mu}v(x,\omega)d\omega,\quad\varkappa(t-\omega)\in(0,1),$

and note that since $\omega<t-2\tau$ and consequently $\tau<(t-\omega)/2$,
then

$|t-\omega|\leq|t+\varkappa(t-\omega)\tau-\omega|\leq\frac{3}{2}|t-\omega|.$

Therefore

$|I_{2}|\leq C\left\langle D_{\ast
t}^{\mu}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}|\overline{h}|^{\beta}\tau{\int\limits_{0}^{t-2\tau}}(t-\omega)^{-2+\mu}d\omega\leq$
$\leq C\left\langle D_{\ast
t}^{\mu}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}|\overline{h}|^{\beta}\tau{\int\limits_{-\infty}^{t-2\tau}}(t-\omega)^{-2+\mu}d\omega=C\left\langle
D_{\ast
t}^{\mu}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}|\overline{h}|^{\beta}\tau^{\mu}.$
(7.13)

If now $t<2\tau$, then we can represent the difference in the form
$\delta_{\tau,t}\delta_{\overline{h},z_{k}}^{m}u(x,t)=\delta_{\overline{h},z_{k}}^{m}u(x,t+\tau)-\delta_{\overline{h},z_{k}}^{m}u(x,t)$,
estimate modulo each term of the difference separately analogously to the
estimates of the integrals $I_{1}$ and $I_{2}$ in (7.12), and, taking into
account (7.12), (7.13), arrive at estimate (7.2) in the case $\mu\in(0,1)$.

Let now $\mu>1$. We make use of the integral mean value theorem to represent
the difference in $t$ of order $[\mu]$ as follows

$\delta_{\tau,t}^{[\mu]}u(x,t)=\tau^{[\mu]}{\int\limits_{0}^{1}}d\xi_{1}...{\int\limits_{0}^{1}}d\xi_{[\mu]}u_{t}^{[\mu]}(x,t+\xi_{1}\tau+...\xi_{1}\tau)=$
$=\tau^{[\mu]}{\int\limits_{P_{[\mu]}}}u_{t}^{[\mu]}\left(x,t+\tau{\sum\limits_{i=1}^{[\mu]}}\xi_{i}\right)d\xi,$
(7.14)

where $\xi=(\xi_{1},...,\xi_{[\mu]})\in P_{[\mu]}=\\{\xi:0<\xi_{i}<1\\}$. Then
the double difference
$\delta_{\tau,t}^{[\mu]+1}\delta_{\overline{h},z_{k}}^{m}u(x,t)$ is equal to

$\delta_{\tau,t}^{[\mu]+1}\delta_{\overline{h},z_{k}}^{m}u(x,t)=\delta_{\tau,t}\delta_{\overline{h},z_{k}}^{m}u(x,t)=\tau^{[\mu]}{\int\limits_{P_{[\mu]}}}\delta_{\tau,t}\delta_{\overline{h},z_{k}}^{m}u_{t}^{[\mu]}\left(x,t+\tau{\sum\limits_{i=1}^{[\mu]}}\xi_{i}\right)d\xi.$
(7.15)

At the same time, according to the conditions of the lemma, the function
$u(x,t)$ possesses the fractional derivative $D_{\ast t}^{\mu}u(x,t)$, which
satisfies (7.1). Therefore at a fixed $\tau$ and $\xi$ the function
$v(x,t)=u_{t}^{[\mu]}\left(x,t+\tau{\sum\limits_{i=1}^{[\mu]}}\xi_{i}\right)$
admits, by virtue of (7.6), the fractional derivative $D_{\ast
t}^{\\{\mu\\}}v(x,t)$ with (7.1). Therefore by what was proved above

$\left|\delta_{\tau,t}^{[\mu]+1}\delta_{\overline{h},z_{k}}^{m}u(x,t)\right|\leq\tau^{[\mu]}{\int\limits_{P_{[\mu]}}}\left|\delta_{\tau,t}\delta_{\overline{h},z_{k}}^{m}u_{t}^{[\mu]}\left(x,t+\tau{\sum\limits_{i=1}^{[\mu]}}\xi_{i}\right)\right|d\xi\leq$
$\leq C\tau^{[\mu]}\left\langle D_{\ast
t}^{\mu}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}|\overline{h}|^{\beta}\tau^{\\{\mu\\}}=C\left\langle
D_{\ast
t}^{\mu}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}|\overline{h}|^{\beta}\tau^{\mu},$

which finishes the proof of the lemma.

###### Theorem 7.2

Let a function $u(x,t)$ is defined in $\overline{R_{T}^{N}}$ and possesses
bounded in $\overline{R_{T}^{N}}$ derivative $D_{t}^{\theta}u(x,t)$ of an
integer or a noninteger order $\theta>0$ and in the case of a noninteger order
we mean the Caputo - Jrbashyan derivative $D_{*t}^{\theta}u(x,t)$. Suppose
that the derivative $D_{t}^{\theta}u(x,t)$ possesses the smoothness of a
noninteger order $\beta>0$ with respect to a spacial group $z_{k}\in
R^{N_{k}}$ that is

$\left\langle
D_{t}^{\theta}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}=\sup_{(x,t)\in\overline{R_{T}^{N}},\overline{h}\in
R^{N_{k}},\overline{h}\neq 0}\frac{|\delta_{\overline{h},z_{k}}^{m}D_{\ast
t}^{\theta}u(x,t)|}{|\overline{h}|^{\beta}}<\infty,\quad m>\beta.$ (7.16)

Besides, suppose that the function $u(x,t)$ itself possesses the smoothness of
a noninteger order $\gamma>\max\\{1,\beta\\}$ with respect to the same
variables $z_{k}\in R^{N_{k}}$ that is

$\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\gamma)}=\sup_{(x,t)\in\overline{R_{T}^{N}},\overline{h}\in
R^{N_{k}},\overline{h}\neq
0}\frac{|\delta_{\overline{h},z_{k}}^{m}u(x,t)|}{|\overline{h}|^{\gamma}}<\infty,\quad
m>\gamma.$ (7.17)

Then a derivative of $u(x,t)$ of an integer order $\beta<n<\gamma$ with
respect to $z_{k}\in R^{N_{k}}$

$D_{z_{k}}^{\overline{\rho}}u(x,t)=\frac{\partial^{\rho_{1}}...\partial^{\rho_{N_{k}}}u(x,t)}{\partial
x_{i_{1}}^{\rho_{1}}...\partial
x_{i_{N_{k}}}^{\rho_{N_{k}}}},\,\overline{\rho}=(\rho_{1},...,\rho_{N_{k}}),\,\rho=\rho_{1}+...+\rho_{N_{k}}=n,$

possesses the smoothness with respect to $t$ of order
$\theta(\gamma-n)/(\gamma-\beta)$ that is

$\left\langle
D_{z_{k}}^{\overline{\rho}}u\right\rangle_{t,\overline{R_{T}^{N}}}^{(\theta(\gamma-n)/(\gamma-\beta))}=\sup_{(x,t)\in\overline{R_{T}^{N}},\tau>0}\frac{|\delta_{\tau,t}^{m}D_{z_{k}}^{\overline{\rho}}u(x,t)|}{\tau^{\theta(\gamma-n)/(\gamma-\beta)}}<\infty,\quad
m>\theta(\gamma-n)/(\gamma-\beta).$

Moreover,

$\left\langle
D_{z_{k}}^{\overline{\rho}}u\right\rangle_{t,\overline{R_{T}^{N}}}^{(\theta(\gamma-n)/(\gamma-\beta))}\leq
C(\theta,\beta,\gamma,n)\left(\varepsilon\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\gamma)}+\frac{1}{\varepsilon^{c}}\left\langle
D_{t}^{\theta}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}\right),$
(7.18)

where $\varepsilon>0$ is arbitrary.

Note that in the case $n<\beta$ the derivative
$D_{z_{k}}^{\overline{\rho}}u(x,t)$ just admits the derivative
$D_{t}^{\theta}D_{z_{k}}^{\overline{\rho}}u(x,t)$ in $t$ as it follows from
the first condition of the theorem.

Proof. Let first $\beta\in(0,1)$. We prove now the assertion of the theorem
for $n=1$ that is for the derivative of the first order with respect to a
single spacial variable $x_{l}$ from the group $z_{k}$. For that we use the
schema of reasonings from [32], section 3, Lemma 2, when obtaining formula
(2.6) there. Let for the simplicity of notations $l=1$ that is we consider the
derivative $u_{x_{1}}(x,t)$. Consider the following representation for the
finite difference from $u_{x_{1}}(x,t)$ of order $m>\gamma$ with a step
$\xi>0$ with respect to $x_{1}$

$\delta_{\xi,x_{1}}^{m}u_{x_{1}}(x,t)={\sum\limits_{j=0}^{m}}(-1)^{m-j}C_{m}^{j}u_{x_{1}}(x_{1}+j\xi,x_{2},...x_{N},t).$

Expressing from this the term without a shift $u_{x_{1}}(x,t)$ over the others
terms of the identity and integrating over $\xi$ in the range from zero to
$\varepsilon\tau^{a}$, $\tau>0$, $\varepsilon>0$, $a=\theta/(\gamma-\beta)$,
we obtain

$u_{x_{1}}(x,t)=\frac{1}{\varepsilon\tau^{a}}{\sum\limits_{j=1}^{m}}(-1)^{j+1}\frac{C_{m}^{j}}{j}{\int\limits_{0}^{\varepsilon\tau^{a}}}u_{\xi}(x_{1}+j\xi,x_{2},...x_{N},t)d\xi+$
$+\frac{(-1)^{m}}{\varepsilon\tau^{a}}{\int\limits_{0}^{\varepsilon\tau^{a}}}\delta_{\xi,x_{1}}^{m}u_{x_{1}}(x,t)d\xi=$
$={\sum\limits_{j=1}^{m}}\frac{1}{\varepsilon\tau^{a}}B_{m,j}\left[u(x_{1}+j\varepsilon\tau^{a},x_{2},...x_{N},t)-u(x_{1},x_{2},...x_{N},t)\right]+$
$+\frac{(-1)^{m}}{\varepsilon\tau^{a}}{\int\limits_{0}^{\varepsilon\tau^{a}}}\delta_{\xi,x_{1}}^{m}u_{x_{1}}(x,t)d\xi\equiv{\sum\limits_{j=1}^{m}}I_{j}+I_{0}.$
(7.19)

To estimate the smoothness of $u_{x_{1}}(x,t)$ in $t$ we estimate a finite
difference along this variable of a sufficiently high order
$\delta_{\tau,t}^{p}u_{x_{1}}(x,t)$, $p>\theta$. And for this we estimate the
finite differences of the terms $I_{j}$ and $I_{0}$ in (7.19). For the terms
$I_{j}$ we have

$\left|\delta_{\tau,t}^{p}I_{j}\right|=\frac{C}{\varepsilon\tau^{a}}\left|\delta_{\tau,t}^{p}\delta_{j\varepsilon\tau^{a},x_{1}}u(x,t)\right|\leq\frac{C}{\varepsilon\tau^{a}}\left\langle
D_{t}^{\theta}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}\tau^{\theta}(\varepsilon\tau^{a})^{\beta},$

where we made use of Lemma 7.1. Thus, since $a=\theta/(\gamma-\beta)$,

$\left|\delta_{\tau,t}^{p}I_{j}\right|\leq\frac{C}{\varepsilon^{1-\beta}}\left\langle
D_{t}^{\theta}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}\tau^{\theta(\gamma-1)/(\gamma-\beta)}.$
(7.20)

Further, since for each fixed $t>0$ the function $u_{x_{1}}(x,t)$ has the
order of smoothness $\gamma-1$ in $x_{1}$, then

$|I_{0}|\leq\frac{C}{\varepsilon\tau^{a}}\left\langle
u_{x_{1}}\right\rangle_{x_{1},\overline{R_{T}^{N}}}^{(\gamma-1)}{\int\limits_{0}^{\varepsilon\tau^{a}}}\xi^{\gamma-1}d\xi\leq\frac{C}{\varepsilon\tau^{a}}\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\gamma)}(\varepsilon\tau^{a})^{\gamma}=$
$=C\varepsilon^{\gamma-1}\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\gamma)}\tau^{\theta(\gamma-1)/(\gamma-\beta)}.$

Consequently,

$\left|\delta_{\tau,t}^{p}I_{0}\right|\leq
C(p)\varepsilon^{\gamma-1}\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\gamma)}\tau^{\theta(\gamma-1)/(\gamma-\beta)}.$
(7.21)

The assertion of the theorem for $n=1$ under the condition $\beta\in(0,1)$
follows now from (7.20) and (7.21) with taking into account (7.19). Thus, the
derivatives of the first order in $x_{i}$ from $z_{k}$ belongs, under fixed
spacial variables from other groups, to the space $C_{z_{k},\,\quad
t}^{\gamma-1,\theta(\gamma-1)/(\gamma-\beta)}(R_{T}^{N_{k}})$ and estimate
(7.18) is valid for $n=1$. But since the derivatives in $z_{k}$ of order $n>1$
are the derivatives of order $n-1$ of the first derivatives, then estimate
(7.18) for $n>1$ follows now directly from (2.6). Consequently, the theorem is
proved for $\beta\in(0,1)$.

If $\beta>1$, then it is enough instead of the function $u(x,t)$ itself to
consider it’s derivative $D_{z_{k}}^{\overline{\rho}}u(x,t)$ of order
$\rho=[\beta]$ and denote this derivative by $v(x,t)=D_{z_{k}}^{\rho}u(x,t)$.
This function fully meets the conditions of the theorem with the same
$\theta$, with $\\{\beta\\}\in(0,1)$ instead of $\beta$, and with
$\gamma-[\beta]$ instead of $\gamma$. The application of what was proved above
to this function $v(x,t)$ with the subsequent recalculation of the exponents
of smoothness leads to the proof of the theorem in the general case.

The analogous assertion is valid and in the case, when instead of the usual
derivatives in a spacial group $z_{k}$ one considers the fractional Laplace
operator with respect to $z_{k}$.

###### Theorem 7.3

Let a function $u(x,t)$ be defined in $\overline{R_{T}^{N}}$ and let it’s
derivative $D_{t}^{\theta}u(x,t)$ of order $\theta>0$ be bounded in
$\overline{R_{T}^{N}}$, where $\theta>0$ can be either an integer or a
noninteger (in the case of a noninteger $\theta>0$ we mean the Caputo -
Jrbashyan derivative). Suppose that the derivative $D_{t}^{\theta}u(x,t)$
possesses the smoothness in $z_{k}\in R^{N_{k}}$ of a noninteger order
$\beta>0$ in the sense of (7.16). Suppose also that the function $u(x,t)$
itself possesses the smoothness in $z_{k}\in R^{N_{k}}$ of a noninteger order
$\gamma>\beta$ in the sense of (7.17). Then the fractional Laplace operator of
this function $(-\Delta_{z_{k}})^{\frac{\mu}{2}}u(x,t)$ in $z_{k}\in
R^{N_{k}}$ of order $\mu\in(\beta,\gamma)$ possesses the smoothness in $t$ of
order $\theta(\gamma-\mu)/(\gamma-\beta)$ that is

$\left\langle(-\Delta_{z_{k}})^{{}^{\frac{\mu}{2}}}u\right\rangle_{t,\overline{R_{T}^{N}}}^{(\theta(\gamma-\mu)/(\gamma-\beta))}=\sup_{(x,t)\in\overline{R_{T}^{N}},\tau>0}\frac{|\delta_{\tau,t}^{m}(-\Delta_{z_{k}})^{{}^{\frac{\mu}{2}}}u(x,t)|}{\tau^{\theta(\gamma-\mu)/(\gamma-\beta)}}<\infty,m>\frac{\theta(\gamma-\mu)}{(\gamma-\beta)}.$

Moreover,

$\left\langle(-\Delta_{z_{k}})^{{}^{\frac{\mu}{2}}}u\right\rangle_{t,\overline{R_{T}^{N}}}^{(\theta(\gamma-\mu)/(\gamma-\beta))}\leq$
$\leq C(\theta,\beta,\gamma,\mu)\left(\varepsilon^{\gamma-\mu}\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\gamma)}+\frac{1}{\varepsilon^{\left(\mu-\beta\right)}}\left\langle
D_{t}^{\theta}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}\right),$
(7.22)

where $\varepsilon>0$ is arbitrary.

Note that for $\mu<\beta$ the Laplace operator
$(-\Delta_{z_{k}})^{{}^{\frac{\mu}{2}}}u(x,t)$ just has the derivative
$D_{t}^{\theta}(-\Delta_{z_{k}})^{\mu}u(x,t)$, as it follows from the first
assumption of the theorem.

Proof. We use representation (1.12)

$(-\Delta_{z_{k}})^{{}^{\frac{\mu}{2}}}u(x,t)=C_{N_{k},\mu,m}{\int\limits_{\eta_{k}\in
R^{N_{k}}}}\frac{\delta_{\eta_{k},z_{k}}^{m}u(x,t)}{|\eta_{k}|^{N_{k}+\mu}}d\eta_{k},$
(7.23)

where we choose $m>\gamma$. Consider, as in the previous theorem, the
difference in $t$ of an order $p>\theta$ with a step $\tau$

$D(\tau)\equiv\delta_{\tau,t}^{p}(-\Delta_{z_{k}})^{{}^{\frac{\mu}{2}}}u(x,t)=C_{N_{k},\mu,m}{\int\limits_{\eta_{k}\in
R^{N_{k}}}}\frac{\delta_{\tau,t}^{p}\delta_{\eta_{k},z_{k}}^{m}u(x,t)}{|\eta_{k}|^{N_{k}+\mu}}d\eta_{k}.$
(7.24)

Split the integral in (7.24) into the two ones

$D(\tau)=C_{N_{k},\mu,m}{\int\limits_{\left|\eta_{k}\right|\leq\varepsilon\tau^{a}}}\frac{\delta_{\tau,t}^{p}\delta_{\eta_{k},z_{k}}^{m}u(x,t)}{|\eta_{k}|^{N_{k}+\mu}}d\eta_{k}+$
(7.25)
$+C_{N_{k},\mu,m}\int\limits_{\left|\eta_{k}\right|>\varepsilon\tau^{a}}\frac{\delta_{\tau,t}^{p}\delta_{\eta_{k},z_{k}}^{m}u(x,t)}{|\eta_{k}|^{N_{k}+\mu}}d\eta_{k}\equiv
I_{1}+I_{2},$

where $\varepsilon>0$, $a=\theta/(\gamma-\beta)$. Bearing in mind (7.17), we
can obtain for $I_{1}$ the estimate

$\left|I_{1}\right|\leq C_{N_{k},\mu,m}\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\gamma)}{\int\limits_{\left|\eta_{k}\right|\leq\varepsilon\tau^{a}}}\frac{|\eta_{k}|^{\gamma}}{|\eta_{k}|^{N_{k}+\mu}}d\eta_{k}=$
$=C_{N_{k},\mu,m}\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\gamma)}\left(\varepsilon\tau^{a}\right)^{\gamma-\mu}=C_{N_{k},\mu,m}\varepsilon^{\gamma-\mu}\left\langle
u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\gamma)}\tau^{\theta(\gamma-\mu)/(\gamma-\beta)}.$
(7.26)

To estimate $I_{2}$ we make use of (7.16) together with Lemma 7.1, which gives

$\left|I_{2}\right|\leq C_{N_{k},\mu,m}\left\langle
D_{t}^{\theta}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}\tau^{\theta}{\int\limits_{\left|\eta_{k}\right|>\varepsilon\tau^{a}}}\frac{|\eta_{k}|^{\beta}}{|\eta_{k}|^{N_{k}+\mu}}d\eta_{k}=$
$=C_{N_{k},\mu,m}\left\langle
D_{t}^{\theta}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}\tau^{\theta}\left(\varepsilon\tau^{a}\right)^{-\left(\mu-\beta\right)}=$
$=C_{N_{k},\mu,m}\left\langle
D_{t}^{\theta}u\right\rangle_{z_{k},\overline{R_{T}^{N}}}^{(\beta)}\varepsilon^{-\left(\mu-\beta\right)}\tau^{\theta(\gamma-\mu)/(\gamma-\beta)}.$

The assertion of the theorem follows now from (7.24) - (7.26) in view of
(2.4).

##  8 Theorems on Fourier multipliers in Hölder spaces

In this section we present some theorems from [39] and [40] on the Fourier
multipliers in Hölder spaces. We consider the multipliers, that act either in
spaces with finite Hölder seminorm with respect to all independent variables
or in spaces with finite Hölder seminorm with respect to a part of independent
variables. These theorems will be applied further for the proofs of theorems
2.1 \- 2.5.

We follow [40] to give some necessary definitions.

Let $K>0$ be an integer and let

$\gamma\in(0,1),\quad\beta=(\beta_{1},...,\beta_{K}),\quad\beta_{1}=1,\quad\beta_{i}\in(0,1],i=2,...,K.$
(8.1)

Consider the Hölder space $C^{\gamma\beta}(R^{K})$ with the norm

$\left\|u\right\|_{C^{\gamma\beta}(R^{K})}\equiv\left|u\right|_{R^{K}}^{(\gamma\beta)}\equiv|u|_{R^{K}}^{(0)}+{\sum\limits_{i=1}^{K}}\left\langle
u\right\rangle_{x_{i},R^{K}}^{(\gamma\beta_{i})}.$ (8.2)

Along with this space we consider the more narrow space
$H^{\gamma\beta}(R^{K})$ with the norm

$\left\|u\right\|_{H^{\gamma\beta}(R^{K})}\equiv\left\|u\right\|_{L_{2}(R^{K})}+{\sum\limits_{i=1}^{K}}\left\langle
u\right\rangle_{x_{i},R^{K}}^{(\gamma\beta_{i})},$ (8.3)

and it was shown in [39] that

$\left|u\right|_{R^{K}}^{(\gamma\beta)}\leq
C(\gamma,\beta)\left\|u\right\|_{H^{\gamma\beta}(R^{K})}.$ (8.4)

Let a measurable and bounded function $\widetilde{m}(\xi)$, $\xi\in R^{K}$, be
defined in $R^{K}$. Define the operator $M:H^{\gamma\beta}(R^{K})\rightarrow
L_{2}(R^{K})$ as follows

$Mu\equiv F^{-1}\left[\widetilde{m}(\xi)F(u)(\xi)\right]\equiv
F^{-1}\left[\widetilde{m}(\xi)\widetilde{u}(\xi)\right].$ (8.5)

Here $F(u)(\xi)\equiv\widetilde{u}(\xi)$ is the Fourier transform of $u(x)$
extended on the space $L_{2}(R^{K})$, $F^{-1}$ is the inverse Fourier
transform. Since $u(x)\in H^{\gamma\beta}(R^{K})\subset L_{2}(R^{K})$, and the
function $\widetilde{m}(\xi)$ is bounded, the operator $M$ is correctly
defined. We call the function $\widetilde{m}(\xi)$ a Fourier multiplier.

Let the whole set of the variables $(\xi_{1},...,\xi_{K})=\xi$ be split into
$r$ subsets of length $K_{i}$, $i=1,...,r$, $K=K_{1}+...+K_{r}$ so that

$\xi=(y_{1},...,y_{r}),\quad
y_{1}=(\xi_{1},...,\xi_{K_{1}}),...,y_{r}=(\xi_{K_{1}+...K_{r-1}+1},...,\xi_{K}).$
(8.6)

Let, further, $\omega_{i}$, $i=1,...,r$ be multi-indexes each of length
$K_{i}$

$\omega_{1}=(\omega_{1,1},...,\omega_{1,K_{1}}),...,\omega_{r}=(\omega_{r,1},...,\omega_{r,K_{r}}),\quad\omega_{i,j}\in\mathbf{N\cup\\{0\\}.}$
(8.7)

Denote by $D_{y_{i}}^{\omega_{i}}\widetilde{u}(\xi)$ the derivative of the
function $\widetilde{u}(\xi)$ in the group $y_{i}$ of order
$|\omega_{i}|=\omega_{i,1}+...+\omega_{i,K_{i}}$ that is
$D_{y_{i}}^{\omega_{i}}\widetilde{u}(\xi)=D_{\xi_{j_{1}}}^{\omega_{i,1}}...D_{\xi_{j_{N_{i}}}}^{\omega_{i,N_{i}}}\widetilde{u}(\xi)$.
Let also $p\in(1,2]$ and positive numbers $s_{i}$, $i=1,...,r$, satisfy the
conditions

$s_{i}>\frac{N_{i}}{p},\quad i=1,...,r.$ (8.8)

Denote for $\nu>0$

$B_{\nu}=\\{\xi\in R^{K}:\nu\leq|\xi|\leq\nu^{-1}\\}.$ (8.9)

Suppose that for some $\nu>0$ the function $\widetilde{m}(\xi)$ satisfies with
a certain $\mu>0$ and uniformly in $\lambda>0$ the condition

${\sum\limits_{|\omega_{i}|\leq
s_{i}}}\left\|D_{y_{1}}^{\omega_{1}}D_{y_{2}}^{\omega_{2}}...D_{y_{r}}^{\omega_{r}}\widetilde{m}(\lambda^{\frac{1}{\beta_{1}}}\xi_{1},...,\lambda^{\frac{1}{\beta_{K}}}\xi_{K})\right\|_{L_{p}(B_{\nu})}\leq\mu,$
(8.10)

where $\beta_{i}$ are defined in (8.1).

###### Theorem 8.1

([39]: T.2.1, L.2.1, L.2.2, T.2.2, T.2.3)

If a function $\widetilde{m}(\xi)$ satisfies conditions (8.10), then the
defined in (8.5) operator $M$, is a bounded linear operator from the space
$H^{\gamma\beta}(R^{K})$ to itself and

$\left\|Mu\right\|_{H^{\gamma\beta}(R^{K})}\leq
C(K,\gamma,\beta,p,\nu,\\{s_{i}\\})\mu\left\|u\right\|_{H^{\gamma\beta}(R^{K})},$
(8.11) ${\sum\limits_{i=1}^{K}}\left\langle
Mu\right\rangle_{x_{i},R^{K}}^{(\gamma\beta_{i})}\leq
C(K,\gamma,\beta,p,\nu,\\{s_{i}\\})\mu{\sum\limits_{i=1}^{K}}\left\langle
u\right\rangle_{x_{i},R^{K}}^{(\gamma\beta_{i})}.$ (8.12)

Condition (8.10) can be especially easily verified in the cases, when the
function $\widetilde{m}(\xi)$ has the homogeneity of degree zero that is when
$\widetilde{m}(\lambda^{\frac{1}{\beta_{1}}}\xi_{1},...,\lambda^{\frac{1}{\beta_{K}}}\xi_{K})=\widetilde{m}(\xi)$.
Note also that condition (8.10) includes the derivatives of
$\widetilde{m}(\xi)$ in $y_{i}$ only up to the orders $s_{i}$. The case $r=1$,
$K_{1}=K$, $p=2$ is considered in Lemma 2.1 in [39] and Lemma 2.2 of the same
paper contains the case $r=K$, $K_{i}=1$, $s_{i}=1$. The general case is
analogous - see lemmas 2.2 - 2.4 in [40].

Now we formulate a generalization of Theorem 8.1 to the case of Fourier
multipliers in the spaces of functions with the Hölder condition only with
respect to a part of the variables. For that we need to split the whole set of
variables $x\in R^{K}$ and the corresponding dual (in the sense of the Fourier
transform) set of variables $\xi\in R^{K}$, besides splitting (8.6) and
regardless this splitting, also as follows.

Let $x=(x^{(1)},x^{(2)})$, $x^{(1)}=(x_{1},...,x_{S})\in R^{S}$,
$x^{(2)}=(x_{S+1},...,x_{K})\in R^{K-S}$ and correspondingly
$\xi=(\xi^{(1)},\xi^{(2)})$, $\xi^{(1)}=(\xi_{1},...,\xi_{S})\in R^{S}$,
$\xi^{(2)}=(\xi_{S+1},...,\xi_{K})\in R^{K-S}$. Let, further, analogously to
(8.1),

$\gamma\in(0,1),\quad\beta=(\beta_{1},...,\beta_{S}),\quad\beta_{1}=1,\quad\beta_{i}\in(0,1],i=2,...,S,$
$\varkappa=(\varkappa_{S+1},...,\varkappa_{K}),\quad\varkappa_{i}>0,\quad
i=S+1,...,K.$ (8.13)

Note that $\varkappa_{i}$ not necessarily belongs to $(0,1]$. Analogously to
(8.3) define the space
$H_{x^{(1)},x^{(2)}}^{\gamma\beta,\quad\gamma\varkappa}(R^{K})=C_{x^{(1)},x^{(2)}}^{\gamma\beta,\quad\gamma\varkappa}(R^{K})\cap
L_{2}(R^{K})$ as the Banach space of functions with the finite norm

$\left\|u\right\|_{H_{x^{(1)},x^{(2)}}^{\gamma\beta,\quad\gamma\varkappa}(R^{K})}\equiv\left\|u\right\|_{L_{2}(R^{K})}+{\sum\limits_{i=1}^{S}}\left\langle
u\right\rangle_{x_{i},R^{K}}^{(\gamma\beta_{i})}+{\sum\limits_{i=S+1}^{K}}\left\langle
u\right\rangle_{x_{i},R^{K}}^{(\gamma\varkappa_{i})}$ (8.14)

and analogously to (8.4)

$\left|u\right|_{x^{(1)},x^{(2)},R^{K}}^{(\gamma\beta,\gamma\varkappa)}\leq
C(\gamma,\beta)\left\|u\right\|_{H_{x^{(1)},x^{(2)}}^{\gamma\beta,\quad\gamma\varkappa}(R^{K})}.$
(8.15)

Besides, define the Banach space $H_{x^{(1)}}^{\gamma\beta}(R^{K})\supset
C_{x^{(1)}}^{\gamma\beta}(R^{K})\cap L_{2}(R^{K})$ with the finite norm

$\left\|u\right\|_{H_{x^{(1)}}^{\gamma\beta}(R^{K})}\equiv\left\|u\right\|_{L_{2}(R^{K})}+{\sum\limits_{i=1}^{S}}\left\langle
u\right\rangle_{x_{i},R^{K}}^{(\gamma\beta_{i})},$ (8.16)

and we stress that the functions from $H_{x^{(1)}}^{\gamma\beta}(R^{K})$ have
bounded Hölder seminorms with respect to the variables from the group
$x^{(1)}$ only. In particular, the functions from this space are not
necessarily bounded - see an example before Theorem 2.7 in [40].

###### Theorem 8.2

([40]: T.2.7) Let a function

$\widetilde{m}(\xi)\in C(R^{K}\backslash\\{0\\})\text{ be continuous and
bounded in }R^{K}\backslash\\{0\\}$ (8.17)

and let it satisfy the condition

$\widetilde{m}(\xi)|_{\xi^{(1)}=0}=\widetilde{m}(0,\xi^{(2)})\equiv
0,\quad\xi^{(2)}\in R^{K-S}\backslash\\{0\\}.$ (8.18)

Let, further, $p\in(1,2]$ and let positive numbers $s_{i}$, $i=1,...,r$,
satisfy the condition (comp. (8.8))

$s_{i}>\frac{N_{i}}{p}+\gamma,\quad i=1,...,r.$ (8.19)

Let, besides, the following condition be satisfied (comp. (8.10))

${\sum\limits_{|\omega_{i}|\leq
s_{i}}}\left\|D_{y_{1}}^{\omega_{1}}D_{y_{2}}^{\omega_{2}}...D_{y_{r}}^{\omega_{r}}\widetilde{m}(\lambda^{\frac{1}{\beta_{1}}}\xi_{1},...,\lambda^{\frac{1}{\beta_{S}}}\xi_{S},\lambda^{\frac{1}{\varkappa_{S+1}}}\xi_{1},...,\lambda^{\frac{1}{\varkappa_{K}}}\xi_{S})\right\|_{L_{p}(B_{\nu})}\leq\mu,$
(8.20)

where $B_{\nu}$ is defined in (8.9) and $\mu$ is a positive number.

Then the operator $M$ from (8.5) is a bounded linear operator from the space
$H_{x^{(1)}}^{\gamma\beta}(R^{K})$ to the space
$H_{x^{(1)},x^{(2)}}^{\gamma\beta,\quad\gamma\varkappa}(R^{K})$, and

$\left\|Mu\right\|_{H_{x^{(1)},x^{(2)}}^{\gamma\beta,\quad\gamma\varkappa}(R^{K})}\leq
C(K,\gamma,\beta,\varkappa,p,\nu,\\{s_{i}\\})\mu\left\|u\right\|_{H_{x^{(1)}}^{\gamma\beta}(R^{K})},$
(4.22) ${\sum\limits_{i=1}^{S}}\left\langle
Mu\right\rangle_{x_{i},R^{K}}^{(\gamma\beta_{i})}+{\sum\limits_{i=S+1}^{K}}\left\langle
Mu\right\rangle_{x_{i},R^{K}}^{(\gamma\varkappa_{i})}\leq
C(K,\gamma,\beta,\varkappa,p,\nu,\\{s_{i}\\})\mu{\sum\limits_{i=1}^{S}}\left\langle
u\right\rangle_{x_{i},R^{K}}^{(\gamma\beta_{i})}.$ (4.23)

As the conclusion of the section we present two auxiliary statements we need
in what follows.

###### Lemma 8.3

([40]: L.2.8) Let a function $\widetilde{f}(i\xi_{0},\xi)$ ($i$ is the
imaginary unit, $\xi_{0}\in R^{1}$, $\xi\in R^{N}$) be defined in $R^{N+1}$
and can be extended to a function $\widetilde{f}(i\xi_{0}+a,\xi)$ in the
domain $a\geq 0$ in the way that the extension $\widetilde{f}(i\xi_{0}+a,\xi)$
possesses the properties:

1) $\widetilde{f}(i\xi_{0}+a,\xi)$ is continuous in the domain $a\geq 0$;

2) the function $\widetilde{f}(i\xi_{0}+a,\xi)$ is analytic in the domain
$a>0$ with respect to the variable $p=i\xi_{0}+a$;

3) this function satisfies in the domain $a\geq 0$ the following inequality
with some constants $C>0$, $m_{1}$, and $m_{2}$

$\left|\widetilde{f}(p,\xi)\right|\leq C(1+|p|)^{m_{1}}(1+|\xi|)^{m_{2}}.$
(8.21)

Then the inverse Fourier transform of this function ( including in the sense
of distributions )

$f(t,x)=(2\pi)^{-(N+1)}{\int\limits_{R^{N+1}}}e^{it\xi_{0}+ix\xi}\widetilde{f}(i\xi_{0},\xi)d\xi
d\xi_{0}$

vanishes for $t<0$

$f(t,x)\equiv 0,\quad t<0.$ (8.22)

###### Lemma 8.4

Let $f(t)\in C^{n}([0,\infty))$, $n\geq 1$, $f^{(k)}(t)\in L_{1}([0,\infty))$,
$k=0,1,...,n$ and $f^{(k)}(0)=0$, $k=0,1,...,n-1$ and let $\theta\in(n-1,n)$.
Then the fractional derivative $D_{\ast t}^{\theta}f(t)$, extended along with
the function $f(t)$ itself, in the domain $\\{t<0\\}$ by zero, has the
following Fourier transform

$\widehat{D_{\ast
t}^{\theta}f(t)}(\xi_{0})\equiv\frac{1}{\sqrt{2\pi}}{\int\limits_{-\infty}^{\infty}}D_{\ast
t}^{\theta}f(t)e^{-i\xi_{0}t}dt,$ (8.23)

where the integral is understood as an improper one. Moreover,

$\widehat{D_{\ast
t}^{\theta}f(t)}(\xi_{0})=(i\xi_{0})^{[\theta]}(i\xi_{0})^{\\{\theta\\}}\widehat{f}(\xi_{0}),$
(8.24)

where $\widehat{f}(\xi_{0})$ is the Fourier transform of $f(t)$ and

$(i\xi_{0})^{\\{\theta\\}}\equiv|\xi_{0}|^{\\{\theta\\}}e^{i\\{\theta\\}\frac{\pi}{2}sign(\xi_{0})}$
(8.25)

is an analytic extension of the function $z^{\\{\theta\\}}$ from the positive
real half-axis to the right half-plane $\operatorname{Re}z>0$.

###### Remark 8.5

We do not present here a detailed proof of this statement because it, by
essence, is contained in, for example, [1], Section 7. We note only that at
applications of [1], Section 7, one should take into account that the
monograph [1] considers the direct Fourier transform with $e^{ix\xi_{0}}$ and
the present paper makes use in (1.10) of the kernel $e^{-ix\xi_{0}}$. Besides,
it should be taken into account that, since $f(t)\equiv 0$ for $t<0$, then

$D_{\ast
t}^{\theta}f(t)=\frac{1}{\Gamma(1-\theta)}{\int\limits_{0}^{t}}\frac{f^{(n)}(\tau)d\tau}{(t-\tau)^{\theta-n+1}}=\frac{1}{\Gamma(1-\theta)}{\int\limits_{-\infty}^{t}}\frac{f^{(n)}(\tau)d\tau}{(t-\tau)^{\theta-n+1}}.$

##  9 Cauchy problem for equation (1.2) in the case of a noninteger
$\theta\in(0,1)$ ,$\theta\alpha\in(0,1)$.

In the present section we study problem (1.2), (1.3) with $\theta\in(0,1)$ in
the anisotropic Hölder spaces
$C^{\overline{\sigma}(1+\alpha),\theta+\theta\alpha}(\overline{R_{T}^{N}})$