Pseudo-differential operators in vector-valued spaces and applications

TL;DR

This paper investigates pseudo-differential operators with parameters in vector-valued spaces, establishing resolvent estimates, regularity properties, and applications to anisotropic and system equations, with implications for analytic semigroup generation.

Contribution

It introduces new resolvent estimates and regularity results for pseudo-differential operators with parameters, extending their analysis in vector-valued function spaces.

Findings

Operators are positive and generate analytic semigroups.

Maximal regularity properties are established for pseudo-differential parabolic equations.

Applications include anisotropic parameter-dependent and system pseudo-differential equations.

Abstract

Pseudo-differential operator equations with parameter are studied. Uniform separability properties and resolvent estimates are obtained in terms of fractional derivatives. Moreover, maximal regularity properties of the pseudo-differential abstract parabolic equation are established. Particularly, it is proven that the operators generated by these pseudo-differential equations are positive and aso are generators of analytic semigroups. As an application, the anisotropic parameter dependent pseudo-differential equations and the system of pseudo-differential equations are studied

Equations

$$P_t\left(D\right)u+Au+\sum _{|\alpha |<m}t\left(\alpha \right)A_{\alpha }\left(x\right)D^{\alpha }u+\lambda u=f\left(x\right),x\in R^n,$$

$${\Vert f\Vert }_{L_p\left(\Omega ;E\right)}={\left(\underset{\Omega }{\int }{\Vert f\left(x\right)\Vert }_E^{p}dx\right)}^{\frac{1}{p}},1\le p<\infty .$$

$$\sum _{|\alpha |\le m}t\left(\alpha \right){\left|\lambda \right|}^{1-\frac{|\alpha |}{m}}{\Vert D^{\alpha }u\Vert }_{L_p\left(R^n;E\right)}+{\Vert Au\Vert }_{L_p\left(R^n;E\right)}\le C{\Vert f\Vert }_{L_p\left(R^n;E\right)}$$

$$\frac{\partial u}{\partial y}+P_t\left(D\right)u+Au=f\left(y,x\right),u(0,x)=0,$$

$$u\in W_{\mathbf{p}}^{1,m}\left(R_{+}^{n+1};E\left(A\right),E\right)$$

$${\Vert \frac{\partial u}{\partial y}\Vert }_{L_{{}_{\mathbf{p}}}\left(R_{+}^{n+1};E\right)}+{\Vert P_t\left(D\right)u\Vert }_{L_{\mathbf{p}}\left(R_{+}^{n+1};E\right)}+$$

$${\Vert Au\Vert }_{L_{\mathbf{p}}\left(R_{+}^{n+1};E\right)}\le M{\Vert f\Vert }_{L_{\mathbf{p}}\left(R_{+}^{n+1};E\right)}.$$

$${\Vert f\Vert }_{L_p\left(\Omega ;\Sigma ,\mu ,E\right)}={\left(\underset{\Omega }{\int }{\Vert f\left(x\right)\Vert }_E^{p}d\mu \right)}^{\frac{1}{p}}.$$

$${\Vert f_0+\sum _{k=1}^N{\epsilon }_k\left(f_k-f_{k-1}\right)\Vert }_{L_p\left(\Omega ,\Sigma ,\mu ,E\right)}\le C_p{\Vert f_N\Vert }_{L_p\left(\Omega ,\Sigma ,\mu ,E\right)}.$$

$$\left(Hf\right)\left(x\right)=\underset{\epsilon \to 0}{\lim }\underset{|x-y|>\epsilon }{\int }\frac{f\left(y\right)}{x-y}dy$$

$$S_{\varphi}=\left\{\lambda;\text{ \ }\lambda\in\mathbb{C}\text{, }\left|\arg\lambda\right|\leq\varphi\right\}\cup\left\{0\right\},\text{ }0\leq\varphi<\pi.\$$

$${\Vert {\left(A+\lambda I\right)}^{-1}\Vert }_{B\left(E\right)}\le M{\left(1+\left|\lambda \right|\right)}^{-1}$$

$${\Vert u\Vert }_{E\left(A^{\theta }\right)}={\left({\Vert u\Vert }^p+{\Vert A^{\theta }u\Vert }^p\right)}^{\frac{1}{p}},1\le p<\infty ,0<\theta <\infty .$$

$$\underset{0}{\overset{1}{\int }}{\Vert \sum _{j=1}^m{r}_j\left(y\right)T_j{u}_j\Vert }_{E_2}dy\le C\underset{0}{\overset{1}{\int }}{\Vert \sum _{j=1}^m{r}_j\left(y\right)u_j\Vert }_{E_1}dy,$$

$$\underset{\Omega }{\int }{\Vert \sum _{j=1}^m{r}_j\left(y\right)T_j\left(h\right)u_j\Vert }_{E_2}dy\le C\underset{\Omega }{\int }{\Vert \sum _{j=1}^m{r}_j\left(y\right)u_j\Vert }_{E_1}dy$$

$$\underset{h\in Q}{\sup }R\left(G_h\right)\le C.$$

$$\left\{\lambda {\left(A\left(h\right)+\lambda \right)}^{-1}:\lambda \in S_{\phi }\right\}$$

$$\left(i\xi\right)^{\alpha}=\left\{\begin{array}[]{c}\left(i\xi_{1}\right)^{\alpha_{1}}...\left(i\xi_{n}\right)^{\alpha_{n}}\text{, \ \ }\xi_{1}\xi_{2},...,\xi_{n}\neq 0\\ 0\text{,\ \ \ \ \ }\xi_{1},\xi_{2},...,\xi_{n}=0,\end{array}\right.$$

$${\left(i{\xi }_k\right)}^{{\alpha }_k}=\exp \left[{\alpha }_k\left(\ln \left|{\xi }_k\right|+i\pi \text{ sgn }{\xi }_k/2\right)\right]\text{, }k=1,2,...,n.$$

$$Fu={\left(2\pi \right)}^{-\frac{n}{2}}\underset{R^n}{\int }\left[\exp \left(x,\xi \right)\right]u\left(x\right)dx,F^{-1}u={\left(2\pi \right)}^{-\frac{n}{2}}\underset{R^n}{\int }\left[\exp -\left(x,\xi \right)\right]u\left(\xi \right)d\xi ,$$

$$x=\left(x_1,x_2,...,x_n\right),\xi =\left({\xi }_1,{\xi }_2,...,{\xi }_n\right)\in R^n\text{, }\left(x,\xi \right)=\sum _{k=1}^n{x}_k{\xi }_k.$$

$$\ F\left(D_{x}^{\alpha}u\right)=\left(i\xi_{1}\right)^{\alpha_{1}}...\left(i\xi_{n}\right)^{\alpha_{n}}\hat{u},\ D_{\xi}^{\alpha}\left(F\left(u\right)\right)=F\left[\left(-ix_{n}\right)^{\alpha_{1}}...\left(-ix_{n}\right)^{\alpha_{n}}u\ \ \right]\ \ \ \ \$$

$$u\to \Lambda u=F^{-1}\Psi \left(\xi \right)Fu,u\in S\left(R^n;E_1\right)$$

$$\Lambda :L_p\left(R^n;E_1\right)\to L_p\left(R^n;E_2\right).$$

$$\left\|F^{-1}\Psi_{h}Fu\right\|_{L_{p}\left(R^{n};E_{2}\right)}\leq M\left\|u\right\|_{L_{p}\left(R^{n};E_{1}\right)}\ \ \ \ \ \$$

$$H_p^s(R^n;E_0,E)=\{uu\in S^{\prime }\left(R^n;E_0\right)\text{, }{F}^{-1}{\left(1+\left|\xi \right|\right)}^{\frac{s}{2}}Fu\in L_p\left(R^n;E\right)\text{, }$$

$${\Vert u\Vert }_{H_p^s\left(R^n;E_0,E\right)}={\Vert u\Vert }_{L_p\left(R^n;E_0\right)}+{\Vert F^{-1}{\left(1+\left|\xi \right|\right)}^{\frac{s}{2}}Fu\Vert }_{L_p\left(R^n;E\right)}<\infty \}.$$

$${\Vert u\Vert }_{H_{p,t}^s\left(R^n;E_0,E\right)}={\Vert u\Vert }_{L_p\left(R^n;E_0\right)}+{\Vert F^{-1}{\left[1+{\left(\sum _{k=1}^n{t}_k^{\frac{2}{s}}{\xi }_k^2\right)}^{\frac{1}{2}}\right]}^{s}Fu\Vert }_{L_p\left(R^n;E\right)}<\infty .$$

$${\Psi }_h\in C^n\left(R^n\backslash \left\{0\right\};B\left(E_1,E_2\right)\right).$$

$$\underset{h\in Q}{\sup }R\left(\left\{{\left|\xi \right|}^{|\beta |}{D}^{\beta }{\Psi }_h\left(\xi \right)\text{: }\xi \in R^n\backslash \left\{0\right\},{\beta }_i\in \left\{0,1\right\}\right\}\right)\le K,$$

Full text

**Pseudo-differential operators in vector-valued spaces and applications **

VELI SHAKHMUROV

Okan University, Department of Mechanical Engineering, Akfirat, Tuzla 34959 Istanbul, Turkey, E-mail: veli.sahmurov@okan.edu.tr;

ABSTRACT

Pseudo-differential operator equations with parameter are studied. Uniform $L_{p}$-separability properties and resolvent estimates are obtained in terms of fractional derivatives. Moreover, maximal regularity properties of the pseudo-differential abstract parabolic equation are established. Particularly, it is proven that the operators generated by these pseudo-differential equations are positive and aso are generators of analytic semigroups. As an application, the anisotropic parameter dependent pseudo-differential equations and the system of pseudo-differential equations are studied.

AMS: 47GXX, 35JXX, 47FXX, 47DXX, 43AXX

**Key Word: **pseudo-differential equations, Sobolev-Lions spaces, differential-operator equations, maximal $L_{p}$ regularity, abstract parabolic equations, operator-valued multipliers

**1. Introduction, notations and background **

Differential-operator equations (DOEs) have found many applications in PDEs and pseudo-differential equations (PsDEs) (see e.g. $\left[1\right],$ $\left[5\right],$ $\left[11\right],$ $\left[12\right],$ $\left[16-19\right],$ $\left[24\right]$ ). Regularity properties of PsDEs have been studied extensively by many researchers; see e.g. $\left[\text{6, 10}\right],$ \left[\text{21-22}\right]\and the references therein. The boundedness of PsDEs in Sobolev spaces have been treated e.g. in $\left[10\right],$ $\left[14\right],$ $\left[22\right].$ Moreover, the smoothness of PsDEs with bounded operator coefficients have been explored e.g. in $\left[8\right]$, $\left[15\right].$ In contrast to $\left[8\right]$, $\left[15\right],$ the PsDE considered here contain unbounded operators and parameters. In particular, the main objective of the present paper is to discuss the uniform $L_{p}\left(R^{n};E\right)-$maximal regularity of elliptic pseudo-differential operator equations (PsDOEs) with parameters

[TABLE]

where $P_{t}\left(D\right)$ is the pseudo-differential operator, $A$ and $A_{\alpha}\left(x\right)$ are linear operators in a Banach space $E$, for $\alpha_{i}\in\left[0,\infty\right)$, $\alpha=\left(\alpha_{1},\alpha_{2},...\alpha_{n}\right)$ and $D^{\alpha}=D_{1}^{\alpha_{1}}D_{2}^{\alpha_{2}}...D_{n}^{\alpha_{n}}$ are the Liouville derivatives; $m$ is a positive number, $t_{k}$ are positive, $\lambda$ is a complex parameter, $t=\left(t_{1},t_{2},...,t_{n}\right)$ and $t\left(\alpha\right)=\mathop{\displaystyle\prod}\limits_{k=1}^{n}t_{k}^{\frac{\alpha_{k}}{m}};$ $L_{p}\left(\Omega;E\right)$ denotes the space of strongly measurable $E$-valued functions that are defined on the measurable subset $\Omega\subset R^{n}$ with the norm given by

[TABLE]

We prove that problem $\left(1.1\right)$ has a maximal regular unique solution and the following uniform coercive estimate holds

[TABLE]

for $f\in L_{p}\left(R^{n};E\right)$, $\lambda\in S_{\varphi},$ where $S_{\varphi}$ is a set of complex numbers that is related with the spectrum of the operator $A.$ The estimate $\left(1.2\right)$ implies that the operator $O_{t}$ generated by $\left(1.1\right)$ has a bounded inverse from $L_{p}\left(R^{n};E\right)$ into the space $H_{p}^{m}\left(R^{n};E\left(A\right),E\right),$ which will be defined subsequently. Particularly, from the estimate $\left(1.2\right)$ we obtain that the operator $O_{t}$ is uniformly positive in $L_{p}\left(R^{n};E\right).$ By using this property we prove the uniform well posedness of the Cauchy problem for the following parabolic PsDOE with parameter

[TABLE]

in $E$-valued mixed spaces $L_{\mathbf{p}},$ $\mathbf{p=}\left(p,p_{1}\right)$. In other words, we show that problem $\left(1.3\right)$ has a unique solution

[TABLE]

for $f\in L_{\mathbf{p}}\left(R_{+}^{n+1};E\right)$ satisfying the following uniform coercive estimate

[TABLE]

[TABLE]

Note that, constants $C$, $M$ in estimates $\left(1.2\right)$ and $\left(1.4\right)$ are independent of parameters. As an application in this paper the following are established: (a) maximal regularity properties of the anisotropic elliptic PsDE in mixed $L_{\mathbf{p}},$ $\mathbf{p=}\left(p_{1},p\right)$ spaces; (b) coercive properties of the system of PsDEs of infinite order in $L_{p}$ spaces.

Let $\left(\Omega;\Sigma,\mu\right)$ be a complete probability space, $1\leq p<\infty.$ $L_{p}\left(\Omega;\Sigma,\mu,E\right)$ denotes $\mu-$measurable $E-$valued Bochner space with norm

[TABLE]

A Banach space $E$ is called UMD space (see $\left[\text{7, \S\ 5}\right]$) if $E$-valued martingale difference sequences are unconditional in $L_{p}\left(\Omega;\Sigma,\mu,E\right)$ for $p\in\left(1,\infty\right)$, i.e., there exists a positive constant $C_{p}$ such that for any martingale $\left\{f_{k}\right\}$ any choice of signs $\left\{\varepsilon_{k}\right\}\in\left\{-1,1\right\}$, $k\in\mathbb{N}$ and $N\in\mathbb{N}$

[TABLE]

It is shown (see $\left[3-4\right]$) that the Hilbert operator

[TABLE]

is bounded in the space $L_{p}\left(R,E\right),$ $p\in\left(1,\infty\right)$ for those and only those spaces $E$ which possess the property of UMD spaces. UMD spaces include e.g. $L_{p}$, $l_{p}$ and Lorentz spaces $L_{pq},$ $p,$ $q\in\left(1,\infty\right)$.

Let $\mathbb{C}$ denote the set of complex numbers and

[TABLE]

A linear operator $A$ is said to be $\varphi$-positive (or positive ) in a Banach space $E$ if $D\left(A\right)$ is dense on $E$ and

[TABLE]

for any $\lambda\in S_{\varphi},$ where $\varphi\in\left[0\right.,\left.\pi\right)$, $I$ is the identity operator in $E,$ $B\left(E\right)$ is the space of bounded linear operators in $E.$ Sometimes $A+\lambda I$ will be written $A+\lambda$ and will be denoted by $A_{\lambda}.$ It is known $\left[\text{20, \S 1.15.1}\right]$ that the powers $A^{\theta}$, $\theta\in\left(-\infty,\infty\right)$ for a positive operator $A$ exist.

The operator $A\left(h\right),$ $h\in Q\subset\mathbb{C}$ is said to be $\varphi$-positive (or positive) in $E$ uniformly with respect to $h\in Q$ if $D\left(A\left(h\right)\right)$ is independent of $h$, $D\left(A\left(h\right)\right)$ is dense in $E$ and $\left\|\left(A\left(h\right)+\lambda\right)^{-1}\right\|\leq M\left(1+\left|\lambda\right|\right)^{-1}$ for all $\lambda\in S_{\varphi},$ $0\leq\varphi<\pi$, where $M$ does not depend on $h$ and $\lambda.$ Let $E\left(A^{\theta}\right)$ denote the space $D\left(A^{\theta}\right)$ with the norm

[TABLE]

A set $W\subset B\left(E_{1},E_{2}\right)$ is called $R$-bounded (see e.g. $\left[23\right]$) if there is a constant $C>0$ such that for all $T_{1},T_{2},...,T_{m}\in W$ and $u_{1,}u_{2},...,u_{m}\in E_{1},$ $m\in\mathbb{N}$

[TABLE]

where $\left\{r_{j}\right\}$ is an arbitrary sequence of independent symmetric $\left\{-1,1\right\}$-valued random variables on $\left[0,1\right]$.

The smallest $C$ for which the above estimate holds is called a $R$-bound of the collection $W$ and is denoted by $R\left(W\right).$

A set of operators $G_{h}\subset B\left(E_{1},E_{2}\right)$ depending on parameter $h\in Q\subset\mathbb{C}$ is called uniformly $R$-bounded with respect to $h$ if there is a constant $C$ independent of $h\in Q$ such that

[TABLE]

for all $T_{1}\left(h\right),T_{2}\left(h\right),...,T_{m}\left(h\right)\in G_{h}$ and $u_{1,}u_{2},...,u_{m}\in E_{1},$ $m\in\mathbb{N}$.

It implies that

[TABLE]

The operator $A$ is said to be $R$-positive in a Banach space $E$ if the set $\left\{\lambda\left(A+\lambda\right)^{-1}\text{: }\lambda\in S_{\varphi}\right\}$ is $R$-bounded.

A positive operator $A\left(h\right)$ is said to be uniformly $R$-positive in a Banach space $E$ if there exists $\varphi\in\left[0\right.,\left.\pi\right)$ such that the set

[TABLE]

is uniformly $R$-bounded. Let $S\left(R^{n};E\right)$ denote the $E-$valued Schwartz class, i.e., the space of all $E$-valued rapidly decreasing smooth functions on $R^{n}$ equipped with its usual topology generated by seminorms. For $E=\mathbb{C}$ this space will be denoted by $S=S\left(R^{n}\right)$. $S^{\prime}\left(E\right)=S^{\prime}\left(R^{n};E\right)$ denotes the space of linear continuous mappings from $S$ into $E$ and is called $E$-valued Schwartz distributions. For any $\alpha=\left(\alpha_{1},\alpha_{2},...\alpha_{n}\right)$,  $\alpha_{i}\in\left[0,\infty\right)$ the function $\left(i\xi\right)^{\alpha}$ will be defined such that

[TABLE]

where

[TABLE]

The Liouville derivatives $D^{\alpha}u$ of an $E$-valued function $u$ are defined similarly to the case of scalar functions $\left[\text{13}\right].$

$C\left(\Omega;E\right)$ and $C^{\left(m\right)}\left(\Omega;E\right)$ will denote the spaces of $E-$valued bounded uniformly strongly continuous and $m$ times continuously differentiable functions on $\Omega$, respectively. Let $F$ and $F^{-1}$ denote the Fourier and inverse Fourier transforms defined as

[TABLE]

where

[TABLE]

Through this section, the Fourier transformation of a function $u$ will be denoted by $\hat{u}.$ It is known that

[TABLE]

for all $u\in S^{\prime}\left(R^{n};E\right)$. Let $E_{1}$ and $E_{2}$ be two Banach spaces. $B\left(E_{1},E_{2}\right)$ denotes the space of bounded linear operators from $E_{1}$ to $E_{2}$. A function $\Psi\in C\left(R^{n};B\left(E_{1},E_{2}\right)\right)$ is called a Fourier multiplier from $L_{p}\left(R^{n};E_{1}\right)$ to $L_{p}\left(R^{n};E_{2}\right)$ if the map

[TABLE]

is well defined and extends to a bounded linear operator

[TABLE]

The set of all multipliers from $L_{p}\left(R^{n};E_{1}\right)$ to $L_{p}\left(R^{n};E_{2}\right)$ will be denoted by $M_{p}^{p}\left(E_{1},E_{2}\right).$ For $E_{1}=E_{2}=E$ it is denoted by $M_{p}^{p}\left(E\right).$

Let $\Phi_{h}=\left\{\Psi_{h}\in M_{p}^{p}\left(E_{1},E_{2}\right),\text{ }h\in Q\right\}$ denote a collection of multipliers depending on the parameter $h.$

We say that $W_{h}$ is a uniform collection of multipliers if there exists a positive constant $M$ independent of $h\in Q$ such that

[TABLE]

for all $h\in Q$ and $u\in S\left(R^{n};E_{1}\right).$

Let $E_{0}$ and $E$ be two Banach spaces and $E_{0}$ be continuously and densely embedded into $E.$ Let $s\in R$ and $\xi=\left(\xi_{1},\xi_{2},...,\xi_{n}\right)\in R^{n}.$ Consider the following Liouville-Lions space

[TABLE]

[TABLE]

Let $t=\left(t_{1},t_{2},...,t_{n}\right)$ and $t_{k}$ be positive parameters. We define the following parameterized norm in $H_{p}^{s}(R^{n};E_{0},E):$

[TABLE]

Sometimes we use one and the same symbol $C$ without distinction in order to denote positive constants which may differ from each other even in a single context. When we want to specify the dependence of such a constant on a parameter, say $\alpha$, we write $C_{\alpha}$.

By using the techniques of $\left[\text{9},\text{Theorem 3.7}\right]$ and reasoning as in $\left[\text{19, Theorem A}_{0}\right]$ we obtain the following proposition.

Proposition A${}_{0}.$ Let $E_{1}$ and $E_{2}$ be two $UMD$ spaces and

[TABLE]

Suppose there is a positive constant $K$ such that

[TABLE]

for

[TABLE]

Then $\Psi_{h}$ is a uniform collection of multipliers from $L_{p}\left(R^{n};E_{1}\right)$ to $L_{p}\left(R^{n};E_{2}\right)$ for $p\in\left(1,\infty\right).$

**Proof: **Some steps (Lemma 3.1, Proposition 3.2) of proof $\left[\text{9},\text{Theorem 3.7}\right]$ trivially work for the parameter dependent case. Other steps (Theorem 3.3, Lemma 3.5) can be easily shown by replacing

[TABLE]

with

[TABLE]

and by using the uniform $R$-boundedness of the set $\Sigma_{h}$. However, the parameter dependent analog of Proposition 3.4 in $\left[\text{9}\right]$ is not straightforward. Let $M_{h}$, $M_{h,N}\in L_{1}^{loc}\left(R^{n},B\left(E_{1},E_{2}\right)\right)$ be Fourier multipliers from $L_{p}\left(R^{n};E_{1}\right)$ to $L_{p}\left(R^{n};E_{2}\right).$ Let $M_{h,N}$ converge to $M_{h}$ in $L_{1}^{loc}\left(R^{n},B\left(E_{1},E_{2}\right)\right)$ and $T_{h,N}=F^{-1}M_{h,N}F$ be uniformly bounded with respect to $h$ and $N.$ Then the operator $T_{h}=F^{-1}M_{h}F$ is uniformly bounded, so we obtain the assertion of Proposition A${}_{0}.$

The embedding theorems in vector valued spaces play a key role in the theory of DOEs. For estimating lower order derivatives in terms of interpolation spaces we use following embedding theorems from $\left[17\right]$.

Theorem A1. Suppose $E$ is an UMD space, $0<t_{k}\leq t_{0}<\infty$, $1<p\leq q<\infty$ and $A$ is an $R$-positive operator in $E.$ Then for $s$ $\in\left(0,\infty\right)$ with $\varkappa=\left|\alpha\right|+n\left(\frac{1}{p}-\frac{1}{q}\right)\leq s$, $0\leq\mu\leq 1-\varkappa$ the embedding

[TABLE]

is continuous and there exists a constant  $C_{\mu}$  $>0$, depending only on $\mu$ such that

[TABLE]

for all $u\in H_{p}^{s}\left(R^{n};E\left(A\right),E\right)$ and $0<h\leq h_{0}<\infty$ .

2. PsDOE with parameters in Banach spaces

Consider the principal part of the problem $\left(1.1\right):$

[TABLE]

where $P_{t}\left(D\right)$ is the pseudo-differential operator defined by

[TABLE]

**Condition 2.1. **Assume $P_{t}\left(\xi\right)\in$ $S^{m}$ for some positive number $m,$ i.e.,

[TABLE]

for all $\xi\in R^{n}$ and $t_{k}\in\left(0,\left.t_{0}\right].\right.$ Suppose $P_{t}\left(\xi\right)\in S_{\varphi_{1}}$ for all $\xi\in R^{n},$ $t_{k}\in\left(0,\left.t_{0}\right],\right.$ $0\leq\varphi_{1}<\pi$ and there is a constant $\gamma>0$ such that $\left|P_{t}\left(\xi\right)\right|\geq\gamma\mathop{\displaystyle\sum}\limits_{k=1}^{n}t_{k}\left|\xi_{k}\right|^{m}$.

Let

[TABLE]

In this section we prove the following

Theorem 2.1. Assume the Condition 2.1 hold. Suppose $E$ is an UMD space, $p\in\left(1,\infty\right)$ and $A$ is an $R$-positive operator in $E$ with respect to $\varphi\in\left(0\right.,\left.\pi\right].$ Then for $f\in L_{p}\left(R^{n};E\right)$, $\lambda\in S_{\varphi_{2}},$ $0\leq\varphi_{2}<\pi-\varphi_{1}$ and $\varphi_{1}+\varphi_{2}\leq\varphi$ there is a unique solution $u$ of the equation $\left(2.1\right)$ belonging to $Y$ and the following coercive uniform estimate holds

[TABLE]

**Proof. **By applying the Fourier transform to equation $\left(2.1\right)$ we obtain

[TABLE]

By construction $\lambda+P_{t}\left(\xi\right)\in S_{\varphi},$ for all $t_{k}\in\left(0\right.,\left.t_{0}\right],$ $\xi\in R^{n}$ and the operator $A+\lambda+P_{t}\left(\xi\right)$ is invertible in $E$. So, from $\left(2.4\right)$ we obtain that the solution of equation $\left(2.1\right)$ can be represented in the form

[TABLE]

By definition of the positive operator $A,$ the inverse of $A^{-1}$ is bounded in $E$. Then the operator $A$ is a closed linear operator (as an inverse of bounded linear operator $A^{-1}$). By the differential properties of the Fourier transform and by using $\left(2.5\right)$ we have

[TABLE]

[TABLE]

where $X=L_{p}\left(R^{n};E\right).$ Hence, it suffices to show that operator-functions

[TABLE]

[TABLE]

are collections of multipliers in X\uniformly with respect to $t_{k}\in\left(0,\left.t_{0}\right]\right.$ and $\lambda\in S_{\varphi_{2}}.$ By virtue of $\left[\text{5, Lemma 2.3}\right],$ for $\lambda\in S_{\varphi_{1}}$ and $\nu\in S_{\varphi_{2}}$ with $\varphi_{1}+\varphi_{2}<\pi$ there is a positive constant $C$ such that

[TABLE]

By using the positivity properties of operator $A$ we get that

[TABLE]

is bounded for all $\xi\in R^{n}$, $\lambda\in S_{\varphi_{1}},$ $t_{k}\in\left(0\right.,\left.t_{0}\right]$ and

[TABLE]

By using Condition 2.1 and estimate $\left(2.6\right)$ we obtain that

[TABLE]

[TABLE]

Then by using the resolvent properties of positive operators and uniform estimate $\left(2.7\right)$ we obtain

[TABLE]

[TABLE]

where $I$ is an identity operator in $E.$ Moreover, by using the well known inequality

[TABLE]

for $\left|\beta\right|\leq m,$ $y_{k}>0$ and $\beta=\left(\beta_{1},\beta_{2},...,\beta_{n}\right)$ for all $u\in E$ we have

[TABLE]

[TABLE]

In view of estimate $\left(2.7\right)$ and by Condition 2.1 we get from the above inequality

[TABLE]

So, we obtain that the operator functions $\sigma\left(t,\lambda,\xi\right)$ and $\sigma_{\alpha}\left(t,\lambda,\xi\right)$ are uniformly bounded, i.e.,

[TABLE]

Due to $R$-positivity of $A,$ by $\left(2.8\right)$ and by Kahane’s contraction principle $\left[\text{6, Lemma 3.5}\right]$ we obtain that the set

[TABLE]

is uniformly $R$-bounded, i.e.,

[TABLE]

In a similar way we obtain

[TABLE]

for

[TABLE]

Consider the following sets

[TABLE]

[TABLE]

[TABLE]

In view of the $R-$positivity properties of operator $A$ and due to Kahane’s contraction, addition and product properties of the collection of $R$-bounded operators (see e.g. $\left[\text{7, 23}\right]$) and by $\left(2.9\right)$ for all $\left\{\xi^{\left(j\right)}\right\}\in R^{n}$, $\left\{\sigma_{\alpha}^{\beta}\left(t,\lambda,\xi^{\left(j\right)}\right)\right\}$, $j=1,2,...,\mu,$ $u_{1,}u_{2},...,u_{\mu}\in E$ and independent symmetric $\left\{-1,1\right\}-$valued random variables $r_{j}\left(y\right)$, $\mu\in\mathbb{N}$ we obtain the following uniform estimate

[TABLE]

[TABLE]

i.e.,

[TABLE]

Hence, we infer that the operator-valued functions $\sigma\left(t,\lambda,\xi\right)$ and $\sigma_{\alpha}\left(t,\lambda,\xi\right)$ are uniform $R-$bounded multipliers and it’s $R-$bounds are independent of $t$ and $\lambda$. By virtue of Preposition A0, the operator-valued functions \sigma\left(t,\lambda,\xi\right)\and $\sigma_{\alpha}\left(t,\lambda,\xi\right)$ are uniform collections of Fourier multipliers in $L_{p}\left(R^{n};E\right).$ So, we obtain that for all  $f\in L_{p}\left(R^{n};E\right)$ there is a unique solution of equation $\left(2.1\right)$ and estimate $\left(2.3\right)$ holds.

Let $O_{t}$ denote the operator in $X=L_{p}\left(R^{n};E\right)$ generated by problem $\left(2.1\right)$ for $\lambda=0$, i.e.,

[TABLE]

Theorem 2.1 and the definition of the space $H_{p}^{m}\left(R^{n};E\left(A\right),E\right)$ imply the following result:

**Result 2.1. **Assume all conditions of Theorem 2.1 are satisfied. Then there are positive constants $C_{1}$ and $C_{2}$ so that

[TABLE]

for $u\in Y.$ Indeed, if we put $\lambda=1$ in $\left(2.3\right),$ by Theorem 2.1 we get

[TABLE]

for $u\in Y$. Due to the closedness of $A$ and by the differential properties of the Fourier transform we have

[TABLE]

So, in view of estimate $\left(2.10\right)$ and by definition of $Y$ we obtain

[TABLE]

The first inequality is equivalent to the following estimate

[TABLE]

[TABLE]

So, it suffices to show that the operator functions

[TABLE]

are uniform Fourier multipliers in $X.$ This fact is proved in a similar way as in the proof of Theorem 2.1.

From Theorem 2.1 we have:

**Result 2.2. **Assume all conditions of Theorem 2.1 hold. Then, for all $\lambda\in S_{\varphi}$ the resolvent of operator $O_{t}$ exists and the following sharp uniform estimate holds

[TABLE]

Indeed, we infer from Theorem 2.1 that the operator $O_{t}+\lambda$ has a bounded inverse from $X$ to $Y.$ So, the solution $u$ of equation $\left(2.1\right)$ can be expressed as $u\left(x\right)=\left(O_{t}+\lambda\right)^{-1}f$ for all $f\in X.$ Then estimate $\left(2.3\right)$ implies the estimate $\left(2.11\right).$

**Result 2.3. **Theorem 2.1 particularly implies that the operator $O_{t}$ is positive in $X.$ Then the operators $O_{t}^{\sigma}$ are generators of analytic semigroups in $X$ for $\sigma\leq\frac{1}{2}$ (see e.g. $\left[\text{20, \S 1.14.5}\right]$).

Now consider the problem $\left(1.1\right).$ By using Theorem 2.1 and the perturbation theory of linear operators we have the following

Theorem 2.2. Assume all conditions of Theorem 2.1 are satisfied. Suppose $A_{\alpha}\left(x\right)A^{-\left(1-\frac{\left|\alpha\right|}{m}-\mu\right)}\in L_{\infty}\left(R^{n};B\left(E\right)\right)$ for $\mu\in\left(0,1-\frac{\left|\alpha\right|}{m}\right)$. Then for $f\in L_{p}\left(R^{n};E\right)$, $\lambda\in S_{\varphi_{2}},$ $0\leq\varphi_{2}<\pi-\varphi_{1}$, $\varphi_{1}+\varphi_{2}\leq\varphi$ and for sufficiently large $\left|\lambda\right|$ there is a unique solution $u$ of the equation $\left(1.1\right)$ belonging to $Y$ and the following coercive uniform estimate holds

[TABLE]

**Proof. **It is clear that $Q_{t}=O_{t}+L_{t},$ where $O_{t}$ is the operator in $L_{p}\left(R^{n};E\right)$ generated by problem $\left(2.1\right)$ for $\lambda=0$ and

[TABLE]

In view of the condition on $A_{\alpha}\left(x\right)$ and by the Theorem A1 for$\ u\in Y$ we have

[TABLE]

[TABLE]

[TABLE]

Then from estimates $\left(2.12\right),$ $\left(2.13\right)$ and for $u\in Y$ we obtain

[TABLE]

Since  $\left\|u\right\|_{X}=\frac{1}{\lambda}\left\|\left(O_{t}+\lambda\right)u+L_{t}u\right\|_{X}$  for $\lambda\in S_{\varphi_{2}}.$ Hence, for $u\in Y$ we get

[TABLE]

[TABLE]

From estimates $\left(2.13\right)-\left(2.15\right)$ for $u\in Y$ we obtain

[TABLE]

Then choosing $h$ and $\lambda$ such that $Ch^{\mu}<1,$  $C_{1}\left|\lambda\right|^{-1}h^{-\left(1-\mu\right)}<1$ from $\left(2.16\right)$ we obtain that

[TABLE]

From Theorem 2.1 and $\left(2.17\right)$ we get that the operator $\left(Q_{t}+\lambda\right)$ has a bounded inverse in $X.$ Moreover, it is clear that

[TABLE]

where $I$ is an identity operator in $X.$ Using relation $\left(2.18\right)$, estimates $\left(2.3\right)$, $\left(2.17\right)$ and perturbation theory of linear operators, we obtain that the operator $Q_{t}+\lambda$ has a bounded inverse from $X$ into $Y$ and the estimate $\left(2.12\right)$ holds.

3. The Cauchy problem for parabolic PsDOE with parameter

In this section, we shall consider the following Cauchy problem for the parabolic PsDO equation

[TABLE]

where $P_{t}\left(D\right)$ is the pseudo-differential operator defined by $\left(2.2\right)$ and $A$ is a linear operator in $E$, $t=\left(t_{1},t_{2},...,t_{n}\right)$, $t_{k}$ are positive parameters.

In this section, by applying Theorem 2.1 we establish the maximal regularity of the problem $\left(3.1\right)$ in $E-$valued mixed $L_{\mathbf{p}}$ spaces, where $\mathbf{p=}\left(p_{1},p\right).$

Let $O_{t}$ denote the operator generated by problem $\left(2.1\right).$ For this aim we need the following result:

**Theorem 3.1. **Suppose Condition 2. 1 hold, $E$ is an UMD space and the operator $A$ is $R$-positive in $E$ with respect to $\varphi$ with $0\leq\varphi<\pi-\varphi_{1}$. Then operator $O_{t}$ is uniformly $R$-positive in $L_{p}\left(R^{n};E\right).$

**Proof. **From Result 2.3 we obtain that the operator $O_{t}$ is positive in $X=L_{p}\left(R^{n};E\right)$. We have to prove the $R$-boundedness of the set

[TABLE]

From the proof of Theorem 2.1 we have

[TABLE]

where

[TABLE]

By reasoning as in the proof of Theorem 2.1, we obtain the following uniform estimate

[TABLE]

By definition of $R$-boundedness, it suffices to show that the operator function $\Phi\left(t,\xi,\lambda\right)$ ( which depends on variable $\lambda$ and parameters $\xi,$ $t$ ) is a multiplier in $L_{p}\left(R^{n};E\right)$ uniformly with respect to $\zeta$ and $t.$ Indeed, by reasoning as in Theorem 2.1 we can easily show that $\Phi\left(t,\xi,\lambda\right)$ is a uniform multiplier in $L_{p}\left(R;E\right).$ Then, by the definition of a $R$-bounded set we have

[TABLE]

[TABLE]

for all $\xi\in R$, $\lambda_{1},\lambda_{2}...\lambda_{m}\in S_{\varphi},$ $f_{1,}f_{2},...,f_{m}\in X$, $m\in N$, where $\left\{r_{j}\right\}$ is a sequence of independent symmetric $\left\{-1,1\right\}-$valued random variables on $\left[0,1\right]$. Hence, the set $\Phi\left(t,\xi,\lambda\right)$ is uniformly $R$-bounded.

Let $E$ be a Banach space. For $\mathbf{p=}\left(p\text{, }p_{1}\right),$ $Z=L_{\mathbf{p}}\left(R_{+}^{n+1};E\right)$ will denote the space of all $\mathbf{p}$-summable $E$-valued functions with mixed norm (see e.g. $\left[\text{2, \S\ 4}\right]$ for the complex-valued case), i.e., the space of all measurable $E$-valued functions $f$ defined on $R_{+}^{n+1}$, for which

[TABLE]

Let $E$ be a Banach space and $A$ be a positive operator in $E.$ Suppose, $l$ is a positive integer number. $W_{p}^{l}\left(R_{+};E\left(A\right),E\right)$ denotes the space of all functions $u\in L_{p}\left(R_{+};E\left(A\right)\right)$ possessing the generalized derivatives $u^{\left(l\right)}\in L_{p}\left(R_{+};E\right)$ with the norm

[TABLE]

Let $m$ be a positive number. $W_{\mathbf{p}}^{1,m}\left(R_{+}^{n+1};E\left(A\right),E\right)$ denotes the space of all functions $u\in L_{\mathbf{p}}\left(R_{+}^{n+1};E\left(A\right)\right)$ possessing the generalized derivative $D_{y}u=\frac{\partial u}{\partial y}\in Z$ with respect to $y$ and fractional derivatives $D_{x}^{\alpha}u\in Z$ with respect to $x$ for $\left|\alpha\right|\leq m$ with the norm

[TABLE]

where $u=u\left(y,x\right).$

Now, we are ready to state the main result of this section.

Theorem 3.2. Assume the conditions of Theorem 2.1 hold for $\varphi\in\left(\frac{\pi}{2},\pi\right)$. Then for $f\in Z$ problem $\left(3.1\right)$ has a unique solution

[TABLE]

satisfying the following unform coercive estimate

[TABLE]

Proof. By definition of $X=L_{p}\left(R^{n};E\right)$ and mixed space $L_{\mathbf{p}}\left(R_{+}^{n+1};E\right),$ $\mathbf{p=}\left(p\text{, }p_{1}\right)$, we have

[TABLE]

[TABLE]

Moreover, by definition of the space $W_{p}^{m}\left(R_{+};E\left(A\right),E\right)$ and by Result 2.1 we obtain

[TABLE]

[TABLE]

Hence, we deduced from the above equalities that,

[TABLE]

[TABLE]

Therefore, the problem $\left(3.1\right)$ can be expressed as the following Cauchy problem for the abstract parabolic equation

[TABLE]

By virtue of $\left[\text{1, Theorem 4.5.2}\right]$, the condition $E\in UMD$ implies $X\in UMD$ for $p\in\left(1,\infty\right)$. Then due to the $R-$positivity of $O_{t},$ by virtue of [23, Theorem 4.2] we obtain that for $f\in L_{p_{1}}\left(R_{+};X\right)$ equation $\left(3.3\right)$ has a unique solution $u\in W_{p_{1}}^{1}\left(R_{+};D\left(O_{t}\right),X\right)$ satisfying the following estimate

[TABLE]

From the Theorem 2.1, relation $\left(3.2\right)$ and from the above estimate we get the assertion.

**4. BVP for Anisotropic PsDE **

In this section, the maximal regularity properties of the anisotropic PsDE are studied.

Let $\tilde{\Omega}=\Omega\times R^{n}$, where $\Omega\subset R^{\mu}$ is an open connected set with compact $C^{2l}-$boundary $\partial\Omega$. Consider the BVP for the pseudo-differential equation

[TABLE]

[TABLE]

[TABLE]

where $u=u\left(x,y\right),\ P_{t}\left(D\right)$ is the pseudo differential operator defined by $\left(2.1\right)$ with respect to $x$ and

[TABLE]

where $\alpha=\left(\alpha_{1},\alpha_{2},...,\alpha_{\mu}\right)$, $\beta=\left(\beta_{1},\beta_{2},...,\beta_{\mu}\right)$ are nonnegative integer numbers, $t=\left(t_{1},t_{2},...,t_{n}\right)$ and $t_{k}$ are positive parameters.

If  $\tilde{\Omega}=R^{n}\times\Omega$, $\mathbf{p=}\left(p_{1},p\right)$, $L_{\mathbf{p}}\left(\tilde{\Omega}\right)$ will denote the space of all $\mathbf{p}$-summable scalar-valued functions with mixed norm ( see e.g.$\left[\text{2, \S\ 4}\right]$ ), i.e., the space of all measurable functions $f$ defined on $\tilde{\Omega}$, for which

[TABLE]

Analogously, $W_{\mathbf{p}}^{m,2l}\left(\tilde{\Omega}\right)$ denotes the anisotropic fractional Sobolev space with corresponding mixed norm, i.e., $W_{\mathbf{p}}^{m,2l}\left(\tilde{\Omega}\right)$ denotes the space of all functions $u\in L_{\mathbf{p}}\left(\tilde{\Omega}\right)$ possessing the fractional derivatives $D_{x}^{\alpha}u\in L_{\mathbf{p}}\left(\tilde{\Omega}\right)$ with respect to $x$ for $\left|\alpha\right|\leq m$ and generalized derivative $\frac{\partial^{2l}u}{\partial y_{k}^{2l}}\in L_{\mathbf{p}}\left(\tilde{\Omega}\right)$ with respect to $y$ with the norm

[TABLE]

Let $Q$ denote the operator generated by problem $\left(4.1\right)-\left(4.2\right)$, i.e.,

[TABLE]

[TABLE]

Let $\xi^{\prime}=\left(\xi_{1},\xi_{2},...,\xi_{\mu-1}\right)\in R^{\mu-1},$ $\alpha^{\prime}=\left(\alpha_{1},\alpha_{2},...,\alpha_{\mu-1}\right)\in Z^{\mu}$ and

[TABLE]

[TABLE]

**Condition 4.1. **Let the following conditions be satisfied;

(1) $b_{\alpha}\in C\left(\bar{\Omega}\right)$ for each $\left|\alpha\right|=2l$ and $b_{\alpha}\in L_{\infty}\left(\Omega\right)+L_{r_{k}}\left(\Omega\right)$ for each $\left|\alpha\right|=k<2l$ with $r_{k}\geq p_{1}$, $p_{1}\in\left(1,\infty\right)$ and $2l-k>\frac{l}{r_{k}};$

(2) $b_{j\beta}\in C^{2l-l_{j}}\left(\partial\Omega\right)$ for each $j$, $\beta$, $l_{j}<2l$, $p\in\left(1,\infty\right),$ $\lambda\in S_{\varphi},$ $\varphi\in[0,\pi);$

(3) for $y\in\bar{\Omega}$, $\xi\in R^{\mu}$, $\sigma\in S_{\varphi_{0}}$, $\varphi_{0}\in\left(0,\frac{\pi}{2}\right)$, $\left|\xi\right|+\left|\sigma\right|\neq 0$ let $\sigma+\sum\limits_{\left|\alpha\right|=2l}b_{\alpha}\left(y\right)\xi^{\alpha}\neq 0;$

(4) for each $y_{0}\in\partial\Omega$ local BVP in local coordinates corresponding to $y_{0}$

[TABLE]

[TABLE]

has a unique solution $\vartheta\in C_{0}\left(\mathbb{R}_{+}\right)$ for all $h=\left(h_{1},h_{2},...,h_{l}\right)\in\mathbb{C}^{l}$ and for $\xi^{{}^{\prime}}\in R^{n-1}$.

Suppose $\nu=\left(\nu_{1},\nu_{2},...,\nu_{n}\right)$ are nonnegative real numbers. In this section, we present the following result:

Theorem 4.1. Assume** **Condition 2.1 and Condition 4.1 are satisfied. Then for $\ f\in L_{\mathbf{p}}\left(\tilde{\Omega}\right)$, $\lambda\in S_{\varphi},$ $\varphi\in\left(0\right.,\left.\pi\right]$ problem $\left(4.1\right)-\left(4.2\right)$ has a unique solution $u\in W_{p}^{m,2l}\left(\tilde{\Omega}\right)$ and the following coercive uniform estimate holds

[TABLE]

Proof. Let $E=L_{p_{1}}\left(\Omega\right)$. It is known $\left[4\right]$ that $L_{p_{1}}\left(\Omega\right)$ is an $UMD$ space for $p_{1}\in\left(1,\infty\right).$ Consider the operator $A$ defined by

[TABLE]

Therefore, the problem $\left(4.1\right)-\left(4.2\right)$ can be rewritten in the form of $\left(2.1\right)$, where $u\left(x\right)=u\left(x,.\right),$ $f\left(x\right)=f\left(x,.\right)$ are functions with values in $E=L_{p_{1}}\left(\Omega\right).$ From $\left[\text{6, Theorem 8.2}\right]$ we get that the following problem

[TABLE]

[TABLE]

has a unique solution for $f\in L_{p_{1}}\left(\Omega\right)$ and arg $\eta\in S\left(\varphi_{1}\right),$ $\left|\eta\right|\rightarrow\infty.$ Moreover, the operator $A$ generated by $\left(4.3\right)$ is $R$-positive in $L_{p_{1}}.$ Then from Theorem 2.1 we obtain the assertion.

**5. The system of PsDE of infinite order **

Consider the following system of PsDEs of infinite order

[TABLE]

[TABLE]

where $P_{t}\left(D\right)$ is the pseudo-differential operator defined by $\left(2.2\right),$ $t=\left(t_{1},t_{2},...,t_{n}\right)$ and $t_{k}$ are positive parameters. Let $a_{ij}$ be real numbers and

[TABLE]

[TABLE]

[TABLE]

**Condition 5.1. **Let

[TABLE]

Let

[TABLE]

**Theorem 5.1. Assume Condition 2.1 and Condition 5.1 are satisfied. **Then, for $f\left(x\right)\in L_{p}\left(R^{n};l_{q}\right),$ $\left|\arg\lambda\right|\leq\varphi$, $\varphi\in\left(0\right.,\left.\pi\right]$ and for sufficiently large $\left|\lambda\right|,$ problem $\left(5.1\right)$ has a unique solution $u$ that belongs to the space $H_{p}^{m}\left(R^{n},l_{q}\left(A\right),l_{q}\right)$ and the following uniform coercive estimate holds

[TABLE]

[TABLE]

Proof. Let $E=l_{q},$ $A$ be a matrix such that $A=\left[a_{ij}\right],$ $i$, $j=1,2,...N.$ It is easy to see that

[TABLE]

where $D\left(\lambda\right)=\det\left(A-\lambda I\right)$, $A_{ji}\left(\lambda\right)$ are entries of the corresponding adjoint matrix of $A-\lambda I.$ Since the matrix $A$ is symmetric and positive definite, it generates a positive operator in $l_{q}$ for $q\in\left(1,\infty\right).$ For all $u_{1,}u_{2},...,u_{\mu}\in l_{q}$, $\lambda_{1},\lambda_{2},...,\lambda_{\mu}\in\mathbb{C}$ and independent symmetric $\left\{-1,1\right\}$-valued random variables $r_{k}\left(y\right)$, $k=1,2,...,\mu,$ $\mu\in\mathbb{N}$ we have

[TABLE]

[TABLE]

[TABLE]

Since $A$ is symmetric and positive definite, we have

[TABLE]

From $\left(5.2\right)$ and $\left(5.3\right)$ we get

[TABLE]

i.e., the operator $A$ is $R$-positive in $l_{q}.$ From Theorem 2.1 we obtain that problem $\left(5.1\right)$ has a unique solution $u\in$ $H_{p}^{m}\left(R^{n};l_{q}\left(A\right),l_{q}\right)$ for $f\in L_{p}\left(R^{n};l_{q}\right)$ and the following estimate holds

[TABLE]

From the above estimate we obtain the assertion.

Acknowledgements

The author would like to express a gratitude to Dr. Erchan Aptoula for his useful advices in English in preparing of this paper.

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