Multipliers and embedding operators with application to abstract differential equat{\i}ons

TL;DR

This paper develops operator-valued multiplier theorems in weighted Lebesgue-Bochner spaces and applies them to establish embedding theorems and maximal regularity for abstract elliptic and parabolic equations.

Contribution

It introduces new multiplier theorems of Mikhlin and Marcinkiewicz--Lizorkin type for operator-valued functions in weighted spaces, leading to advances in regularity theory.

Findings

Established Mikhlin and Marcinkiewicz--Lizorkin type multiplier theorems in weighted Lebesgue-Bochner spaces.

Derived embedding theorems in Sobolev-Lions type spaces using these multiplier results.

Proved maximal regularity properties for abstract elliptic and parabolic equations.

Abstract

In this paper, Mikhlin and Marcinkiewicz--Lizorkin type operator-valued multiplier theorems in weighted Lebesgue-Bochner spaces are studied. By using this results embedding theorems in Sobolev-Lions type spaces is obtained. Moreover, maximal regularity properties of abstract elliptic and parabolic equations are derived

Equations

$${\Vert D^{\alpha }u\Vert }_{L_{p,\gamma }\left(\Omega ;E\left(A^{1-|\alpha :l|-\mu }\right)\right)}\le C_{\mu }\left[h^{\mu }{\Vert u\Vert }_{W_{p,\gamma }^l\left(\Omega ;E\left(A\right),E\right)}+h^{-\left(1-\mu \right)}{\Vert u\Vert }_{L_{p,\gamma }\left(\Omega ;E\right)}\right]$$

$$l=\left(l_1,l_2,...,l_n\right),\alpha =\left({\alpha }_1,{\alpha }_2,...,{\alpha }_n\right),\left|\alpha :l\right|=\sum _{k=1}^n\frac{{\alpha }_k}{l_k},$$

$$0<h\le h_0<\infty ,0<\mu <1-\left|\alpha :l\right|.$$

$$Lu=\sum _{|\alpha |=2l}{a}_{\alpha }{D}^{\alpha }u+Au+\sum _{|\alpha |<2l}{A}_{\alpha }\left(x\right)D^{\alpha }u+\lambda u=f,$$

$$\sum _{|\alpha |\le 2l}{\left|\lambda \right|}^{1-\frac{|\alpha |}{2l}}{\Vert D^{\alpha }u\Vert }_{L_{p,\gamma }\left(R^n;E\right)}+{\Vert Au\Vert }_{L_{p,\gamma }\left(R^n;E\right)}\le C{\Vert f\Vert }_{L_{p,\gamma }\left(R^n;E\right)}.$$

$${\partial }_{t}u+\sum _{|\alpha |=2l}{a}_{\alpha }{D}^{\alpha }u+Au=f\left(t,x\right),t\in \left(0,\infty \right)\text{, }x\in R^n,$$

$$u\left(0,x\right)=0,x\in R^n$$

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

$${\Vert f\Vert }_{L_{\infty ,\gamma }\left(\Omega ;E\right)}=\text{ess}\underset{x\in \Omega }{\sup }{\Vert f\left(x\right)\Vert }_E\gamma \left(x\right)\text{ for }p=\infty .$$

$$\underset{Q}{\sup }\left(\frac{1}{\left|Q\right|}\underset{Q}{\int }\gamma \left(x\right)dx\right){\left(\frac{1}{\left|Q\right|}\underset{Q}{\int }{\gamma }^{-\frac{1}{p-1}}\left(x\right)dx\right)}^{p-1}\le C$$

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

$${\Vert \sum _{i,j=1}^N{\alpha }_{ij}{\epsilon }_i{\epsilon }_j^{\mid }{x}_{ij}\Vert }_{L_2\left(\Omega \times {\Omega }^{\mid };E\right)}dy\le \alpha {\Vert \sum _{i,j=1}^N{\epsilon }_i{\epsilon }_j^{\mid }{x}_{ij}\Vert }_{L_2\left(\Omega \times {\Omega }^{\mid };E\right)}$$

$$S_{\phi }=\left\{\xi ;\xi \in \mathbb{C}\text{, }\left|\arg \xi \right|\le \phi \right\}\cup \left\{0\right\},0\le \phi <\pi .$$

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

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

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

$$F\left(u\right)\left(\phi \right)=u\left(F\left(\phi \right)\right)\text{, }\phi \in S\left(R^n\right).$$

$$\Psi \in C^{\left(l\right)}\left(R^n;B\left(E_1,E_2\right)\right)$$

$${\Vert F^{-1}\Psi \left(\xi \right)Fu\Vert }_{L_{q,\gamma }\left(R^n;E_2\right)}\le C{\Vert u\Vert }_{L_{p,\gamma }\left(R^n;E_1\right)}$$

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

$$R\left\{{\left|\xi \right|}^{|\alpha |}{D}_{\xi }^{\alpha }\Psi \left(\xi \right):\xi \in R^n\setminus \left\{0\right\},\right\}\le K_{\alpha }<\infty$$

$${\Vert u\Vert }_{W_{p,\gamma }^m\left(\Omega ;E_0,E\right)}={\Vert u\Vert }_{L_{p,\gamma }\left(\Omega ;E_0\right)}+\sum _{k=1}^n{\Vert \frac{{\partial }^{l_k}u}{\partial x_k^{l_k}}\Vert }_{L_{p,\gamma }\left(\Omega ;E\right)}<\infty .$$

$$X=L_{p,\gamma }\left(R^n;E_1\right)\text{ and }Y=L_{p,\gamma }\left(R^n;E_2\right).$$

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

$$R\left\{{\xi }^{\beta }{D}_{\xi }^{\beta }M\left(\xi \right):\xi \in R^n\setminus \left\{0\right\}\right\}\le C_{\beta }<\infty$$

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

$$R\left\{{\left|\xi \right|}^{|\beta |}{D}_{\xi }^{\beta }M\left(\xi \right):\xi \in R^n\setminus \left\{0\right\}\right\}\le C_{\beta }<\infty$$

$$Tu=\sum _{j=1}^{\infty }{T}_j{\Delta }_j^{E_1}u$$

$$\Vert T\Vert \le C_{E_1}{C}_{E_2}R\left(\left\{T_j:j\in {\mathbb{Z}}^n\right\}\right).$$

$$S:=\left\{\sum _{k=0}^{\infty }{T}_k{\Delta }_k:T_k\in Q\text{ such that }{T}_k{\Delta }_k={\Delta }_k{T}_k\text{ for all }k\in \mathbb{N}\right\}$$

Full text

Multipliers and **embedding operators with application to abstract differential equatıons **

Veli B. Shakhmurov

Okan University, Department of Mechanical engineering, Akfirat, Tuzla 34959 Istanbul, Turkey,

E-mail: veli.sahmurov@okan.edu.tr

ABSTRACT

In this paper, Mikhlin and Marcinkiewicz–Lizorkin type operator-valued multiplier theorems in weighted Lebesgue-Bochner spaces are studied. Using these results one derives embedding theorems in $E_{0}$-valued weighted Sobolev-Lions type spaces $W_{p,\gamma}^{l}\left(\Omega;E_{0},E\right)$, where $E_{0}$, $E$ are two Banach spaces, $E_{0}$ is continuously and densely embedded into $E.$ One proves that, there exists a smoothest interpolation space $E_{\alpha},$ between $E_{0}$ and $E$, such that the differential operator $D^{\alpha}$ acts as a bounded linear operator from $W_{p,\gamma}^{l}\left(\Omega;E_{0},E\right)$ to $L_{p,\gamma}\left(\Omega;E_{\alpha}\right)$. By using these results the $L_{p,\gamma}-$separability properties of elliptic operators and regularity properties of appropriate degenerate differential operators are studied. In particular, we prove that the associated differential operator is positive and also is a generator of an analytic semigroup. Moreover, the maximal $L_{p,\gamma}$-regularity properties of Cauchy problem for abstract parabolic equation and system of infinity many parabolic equations is obtained.

**AMS: **

**47Axx, 46E35, 47A50, 42B37, 42B15 **

**Key Words: **Banach space-valued functions; Operator-valued multipliers; embedding of Sobolev-Lions spaces; Differential-operator equations; Interpolation of Banach spaces;

**1. Introduction **

Fourier multipliers in vector-valued function spaces have been studied e.g. in $\left[4\right]$, $\left[18\right],$ $\left[31\right].$ Operator-valued Fourier multipliers have been investigated in $\left[5\right],$ $\left[8-11\right]$ and $\left[29\right].$ Mikhlin type Fourier multipliers in scalar weighted spaces have been studied e.g. in $\left[13\right]$, $\left[28\right]$. Moreover, operator-valued Fourier multipliers in weighted abstract $L_{p}$ spaces were investigated e.g. in $\left[2\right]$ and $\left[16\right].$ In $\left[\text{6, 12, 13}\right]$ singular integral operators with operator-valued kernel were studied in weighted $L_{p}$-spaces. Embedding theorems in vector-valued function spaces are studied e.g. in $\left[\text{14, 15}\right]$, $\left[\text{20-26}\right]$. Regularity properties of differential-operator equations (DOEs) have been studied e.g. in $\left[1\right]$, $\left[2\right],$ $\left[7,\text{ }8\right],$ $\left[22-25\right]$, $\left[29-30\right].$ A comprehensive introduction to DOEs and historical references may be found in $\left[1\right]$ and $\left[30\right].$

In this paper, operator-valued multiplier theorems in $E-$valued weighted Lebesque spaces are obtained. These multiplier theorems are used to show the boundedness of embedding operator in the anisotropic Sobolev-Lions space $W_{p,\gamma}^{l}\left(\Omega;E\left(A\right),E\right)$, i.e. under some conditions we prove that the differential operator $u\rightarrow D^{\alpha}u$ is bounded from $W_{p,\gamma}^{l}\left(\Omega;E\left(A\right),E\right)$ to $L_{p,\gamma}\left(\Omega;E\left(A^{1-\left|\alpha:l\right|}\right)\right)$ and the following Ehrling-Nirenberg-Gagilardo type sharp estimate holds

[TABLE]

for $u\in W_{p,\gamma}^{l}\left(\Omega;E\left(A\right),E\right)$, where $A$ is a positive operator in $E$ and

[TABLE]

[TABLE]

This fact generalizes and improves the results $\left[\text{3, \S\ 9, 27, \S\ 1.7}\right]$ for scalar Sobolev space, the result $\left[14\right]$ for one dimensional Sobolev-Lions spaces and the results $\left[15\right]$, $\left[22\right]$ for Hilbert-space valued class. Finally, we consider the differential-operator equation

[TABLE]

where $a_{\alpha}$ are complex numbers, $A$, $A_{\alpha}\left(x\right)$ are linear operators in a Banach space $E$ and $\lambda$ is a complex parameter.

We say that the problem $\left(1.1\right)$ is $L_{p,\gamma}\left(R^{n};E\right)$-separable if there exists a unique solution $u\in W_{p,\gamma}^{2l}\left(R^{n};E\left(A\right),E\right)$ of $\left(1.1\right)$ for all $f\in L_{p,\gamma}\left(R^{n};E\right)$ and there exists a positive constant $C$ depend only on $p$ and $\gamma$ such that the following coercive uniform estimate holds

[TABLE]

Estimate $\left(1.2\right)$ implies that if $f\in L_{p,\gamma}\left(R^{n};E\right)$ and $u$ is a solution of $\left(1.1\right)$ then all terms of equation $\left(1.1\right)$ belong to $L_{p,\gamma}\left(R^{n};E\right)$ (i.e. all terms are separable in $L_{p,\gamma}\left(R^{n};E\right)$). The above estimate implies that the inverse of the differential operator generated by $\left(1.1\right)$ is bounded from $L_{p,\gamma}\left(R^{n};E\right)$ to $W_{p,\gamma}^{2l}\left(R^{n};E\left(A\right),E\right).$

By using the separability properties of $\left(1.1\right)$ we show that the Cauchy problem for the parabolic equation

[TABLE]

[TABLE]

is well-posed in weighted spaces $L_{\mathbf{p},\gamma}\left(R^{n};E\right)$ with mixed norm, where $\mathbf{p=}\left(p,p_{1}\right)$.

The paper is organized as follows. In Section 2, the necessary tools from Banach space theory and some background materials are given. In sections 3, the multiplier theorems in vector-valued weighted Lebesque spaces are proved. In Section 4, by using these multiplier theorems, embedding theorems in $E$-valued weighted Sobolev type spaces are shown. Finally, in sections 5-8 the separability properties of $\left(1.1\right)$, $\left(1.3\right)$ and also regularity properties of appropriate degenerate differential operators are established.

**2. Notations and background **

Let $E$ be a Banach space and let $\gamma=\gamma\left(x\right),$ $x=\left(x_{1},x_{2},...,x_{n}\right)$ be a positive measurable function on the measurable subset $\Omega\subset R^{n}.$ Let $L_{p,\gamma}\left(\Omega;E\right)$ denote the weighted Lebesgue-Bochner space, i.e. the space of all strongly measurable $E-$valued functions that are defined on $\Omega$ with the norm

[TABLE]

[TABLE]

For $\gamma\left(x\right)\equiv 1,$ the space $L_{p,\gamma}\left(\Omega;E\right)$ will be denoted by $L_{p}=L_{p}\left(\Omega;E\right).$

The weight $\gamma$ is said to be satisfy an $A_{p}$ condition, i.e. $\ \gamma\in A_{p},$ $1<p<\infty$, if there is a positive constant $C$ such that

[TABLE]

for all for all cubes $Q$ $\subset R^{n}.$

The Banach space $E$ is called a UMD-space and written as $E\in$ UMD if only if the Hilbert operator

[TABLE]

is bounded in the space $L_{p}\left(R,E\right),$ $p\in\left(1,\infty\right)$ (see e.g. $\left[8\right]$). UMD spaces include $L_{p}$, $l_{p}$ spaces, Lorentz spaces $L_{pq},$ $p,$ $q\in\left(1,\infty\right)$ and Morrey spaces (see e.g. $\left[20\right]$).

A Banach space $E$ has a property ($\alpha$) (see e.g. $\left[\text{19}\right]$) if there exists a constant $\alpha$ such that

[TABLE]

for all $N\in\mathbb{N}\mathbf{,}$ $x_{i,j}\in E,$ $\alpha_{ij}\in\left\{0,1\right\},$ $i,j=1,2,...,N,$ and all choices of independent, symmetric, $\left\{-1,1\right\}-$ valued random variables $\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{N},$ $\varepsilon_{1}^{\prime},\varepsilon_{2}^{\prime},...,\varepsilon_{N}^{\prime}$  on probability spaces $\Omega,$ $\Omega^{\prime}.$ For example the spaces $L_{p}\left(\Omega\right),$ $1\leq p<\infty$  has the property ($\alpha$).

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

[TABLE]

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 $E_{2}.$ For $E_{1}=E_{2}=E$ it will be denote by $B\left(E\right).$

A linear operator $A$ is said to be positive in a Banach space $E$, with bound $M$, if $D\left(A\right)$ is dense in $E$ and

[TABLE]

with $\xi\in S_{\varphi},$ $\varphi\in\left[0,\right.\left.\pi\right),$ where $M$ is a positive constant and $I$ is an identity operator in $E.$ Sometimes instead of $A+\xi I$, we will write $A+\xi$ or $A_{\xi}.$ It is known $\left[\text{27, \S 1.15.1}\right]$ there exist fractional powers $A^{\theta}$ of the positive operator $A.$

Definition 2.1. A positive operator $A$ is said to be $R-$positive in the Banach space $E$ if there exists $\varphi\in\left[0,\pi\right.\left.{}\right)$ such that the set

[TABLE]

is $R$-bounded (see e.g. $\left[29\right]$).

Let $E\left(A^{\theta}\right)$ denote the space $D\left(A^{\theta}\right)$ with graphical norm defined as

[TABLE]

Let $\left(E_{1},E_{2}\right)_{\theta,p}$denote the interpolation space obtained from $\left\{E_{1},E_{2}\right\}$ by the $K-$method $\left[\text{27, \S 1.3.1}\right]$, where $\theta\in\left(0,1\right),$ $p\in\left[1,\infty\right.\left.{}\right)$.

We denote by $D\left(R^{n};E\right)$ the space of $E-$valued $C^{\infty}-$ function with compact support, equipped with the usual inductive limit topology and $S\left(E\right)=S\left(R^{n};E\right)$ denote the $E-$valued Schwartz space of rapidly decreasing smooth functions. For $E=\mathbb{C}$ we simply write $D\left(R^{n}\right)$ and $S=S\left(R^{n}\right)$, respectively. Let $D^{\prime}\left(R^{n};E\right)$ $=B\left(D\left(R^{n}\right),E\right)$ denote the space of $E-$valued distributions and let $S^{\prime}\left(E\right)=S^{\prime}\left(R^{n};E\right)$ denote a space of linear continued mapping from $S\left(R^{n}\right)$ into $E.$ The Fourier transform for $u\in S^{\prime}\left(R^{n};E\right)$ is defined by

[TABLE]

Let $\gamma$ be such that $S\left(R^{n};E_{1}\right)$ is dense in $L_{p,\gamma}\left(R^{n};E_{1}\right).$ A function

[TABLE]

is called a multiplier from $L_{p,\gamma}\left(R^{n};E_{1}\right)$ to $L_{q,\gamma}\left(R^{n};E_{2}\right)$ if there exists a positive constant $C$ such that

[TABLE]

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

We denote the set of all multipliers fom $L_{p,\gamma}\left(R^{n};E_{1}\right)$ to $L_{q,\gamma}\left(R^{n};E_{2}\right)$ by $M_{p,\gamma}^{q,\gamma}\left(E_{1},E_{2}\right).$ For $E_{1}=E_{2}=E$ we denote the $M_{p,\gamma}^{q,\gamma}\left(E_{1},E_{2}\right)$ by $M_{p,\gamma}^{q,\gamma}\left(E\right).$

A set $K\subset B(E_{1},E_{2})$ is called $R$-bounded (see e.g. $[$8, § 3.1$]$) if there is a constant $C>0$ such that for all $T_{1},T_{2},...,T_{m}\in K$ and $u_{1},u_{2},...,u_{m}\in E_{1},$ $m\in\mathbb{N}$

[TABLE]

where $\left\{r_{j}\right\}$ is a 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 the $R-$bound of $K$ and denoted by $R\left(K\right).$

**Definition 2.2. **The Banach space $E$ satisfies the multiplier condition with respect to $p\in\left(1,\infty\right)$ and to the weighted function $\gamma$ if for all $\Psi\in C^{\left(n\right)}\left(R^{n};B\left(E\right)\right)$ the inequality

[TABLE]

for $\alpha=\left(\alpha_{1},\alpha_{2},...,\alpha_{n}\right),$ $\alpha_{i}\in\left\{0,1\right\}$ implies that $\Psi\in M_{p,\gamma}^{p,\gamma}\left(E\right).$

Note that, if $E_{1}$ and $E_{2}$ are UMD spaces and $\gamma\left(x\right)\equiv 1,$ then by virtue of operator valued multiplier theorems (see e.g $\left[9-12\right],$ $\left[30\right]$) we obtain that $\Psi$ is a Fourier multiplier in $L_{p}\left(R^{n};E\right).$

Let$\ \Omega$ be a domain on $R^{n}$ and let $l=\left(l_{1},l_{2},...,l_{n}\right)\in\mathbb{N}^{n}$. Assume $E_{0}$ is continuously and densely belongs to $E.$ Here, $W_{p,\gamma}^{l}\left(\Omega;E_{0},E\right)$ denotes the anisotropic weighted Sobolev-Lions type space of functions $u\in L_{p,\gamma}\left(\Omega;E_{0}\right)$ which have generalized derivatives $\frac{\partial^{l_{k}}u}{\partial x_{k}^{l_{k}}}\in L_{p,\gamma}\left(\Omega;E\right)$ with norm

[TABLE]

For $l_{1}=l_{2}=...=l_{n}=m$ we denote $W_{p,\gamma}^{l}\left(\Omega;E_{0},E\right)$ by $W_{p,\gamma}^{m}\left(\Omega;E_{0},E\right)$ as a isotropic weighted Sobolev-Lions space.

3. Operator-valued multiplier results in weighted Lebesque spaces

Let $E_{1}$, $E_{2}$ be Banach spaces. We put

[TABLE]

By following Theorems 3. 6 and 3.7 of $\left[9\right]$ we will show the following multiplier theorems:

Theorem 3.1. Let $\gamma\in A_{p},$ $p\in\left(1,\infty\right)$. Assume $E_{1}$, $E_{2}$ are UMD spaces with property ($\alpha$) and let

[TABLE]

If

[TABLE]

for all $\beta=\left(\beta_{1},\beta_{2},...,\beta_{n}\right),$ $\beta_{i}\in\left\{0,1\right\},$ then $M$ is a multiplier from $X$ to $Y$ with $\left\|M\right\|_{B\left(X,Y\right)}\leq C\mathop{\displaystyle\sum}\limits_{\beta_{i}\in\left\{0,1\right\}}C_{\beta}.$

If $n=1$ the result remains true without $E_{1}$ having property ($\alpha$).

Theorem 3.2. Let $\gamma\in A_{p},$ $p\in\left(1,\infty\right)$. Let $E_{1}$, $E_{2}$ be UMD spaces and let

[TABLE]

If

[TABLE]

for all $\beta=\left(\beta_{1},\beta_{2},...,\beta_{n}\right),$ $\beta_{i}\in\left\{0,1\right\},$ then $M$ is a multiplier from $X$ to $Y$ with $\left\|M\right\|_{B\left(X,Y\right)}\leq C\mathop{\displaystyle\sum}\limits_{\beta_{i}\in\left\{0,1\right\}}C_{\beta}.$

To prove Theorem 3.1 we need the following result:

The following Propositions A1 and A2 are due to Clément, de Pagter, Sukochev and Witvliet, see $[5]$.

**Proposition A1. ** Let $\Delta_{j}^{E_{1}}$ and $\Delta_{j}^{E_{2}}$ be unconditional Schauder decompositions of the Banach spaces $E_{1}$ and $E_{2}$ respectively, with unconditional constants $C_{E_{1}}$ and $C_{E_{2}}$. Further let $\left\{T_{j}:j\in\mathbb{Z}^{n}\right\}$ be an $R-$bounded family in $B(E_{1},E_{2})$ with $T_{j}\Delta_{j}^{E_{1}}=\Delta_{j}^{E_{2}}T_{j}\Delta_{j}^{E_{1}}$ for all $j\in\mathbb{N}.$ Then the series

[TABLE]

converges for every $u\in E_{1}$ and defines a bounded operator $T:E_{1}\rightarrow E_{2}$ with

[TABLE]

**Proposition A2. **Assume $E$ is a Banach space that has property($\alpha$), $\Delta=\left\{\Delta_{k}\right\}_{k=1}^{\infty}$ is an unconditional Schauder decomposition and $Q\subset B\left(E\right)$ is an $R$-bounded collection of operators. Then the set

[TABLE]

is $R$-bounded in $E$.

Let $\Omega\subset R^{n}$. By using the same reasoning as used in $\left[\text{5, Lemma 3.17}\right]$ we have:

Lemma 3.1. Let $\gamma\in A_{p}$, $p\in\left(1,\infty\right)$. Assume $E$ is a Banach spaces. For $\phi\in L_{\infty}\left(\Omega\right)$ we denote by $M_{\phi}=M_{\phi}^{X}$ the associated multiplication operator in $X$ $=L_{p,\gamma}\left(\Omega;E\right)$. Then the collection

[TABLE]

is $R-$bounded in $X.$

From Lemma 3.1 we obtain

**Corollary 3.1. Let $\gamma\in A_{p},$ $p\in\left(1,\infty\right)$. **Assume $E_{1}$ and $E_{2}$ are Banach spaces. For $\phi\in L_{\infty}\left(\Omega\right)$ we denote by $M_{\phi}^{X}$ and $M_{\phi}^{Y}$ the associated multiplication operators in $X=L_{p,\gamma}\left(\Omega;E_{1}\right)$ and $Y=L_{p,\gamma}\left(\Omega;E_{2}\right)$ respectively. If the set $K\subset B(X,Y)$ is $R-$bounded, then the family

[TABLE]

is $R-$bounded as well.

For $k=nr+j$, $r\in\mathbb{Z}$, $j\in\left\{1,2,...,n\right\}$ let

[TABLE]

[TABLE]

For $\nu=\left(\nu_{1},\nu_{2},...,\nu_{n}\right)\in\mathbb{Z}^{n}$ let

[TABLE]

From $\left[\text{2, Proposition A}_{4}\right]$ we have:

Lemma 3.2. Let $\gamma\in A_{p}$, $p\in\left(1,\infty\right)$ and let $E$ be a UMD space (respectively, UMD space with property ($\alpha$)). Then for any choice of signs $\varepsilon_{k}$, $k\in\mathbb{Z}$ (respectively, $\varepsilon_{k}$, $k\in\mathbb{Z}^{n}$ ) the function $\psi:R^{n}\rightarrow\mathbb{C}$ with $\psi\left(\xi\right)=\varepsilon_{k}$ for $\xi\in\mathbb{D}_{k}$ (respectively, $\psi\left(\xi\right)=\varepsilon_{\nu}$ for $\xi\in\mathbb{D}_{\nu},$ $\nu\in\mathbf{\bigtriangleup}_{\nu}$) is a $M_{p,\gamma}^{p,\gamma}\left(E\right)$ multiplier.

Let $E$ be a Banach space. The ($n-$dimensional) Riesz projection operator $R$ is defined by

[TABLE]

where $\chi\left(\Omega\right)$ denotes the characteristic function of $\Omega$ $\subset R^{n}.$

Let

[TABLE]

where $F_{j}$ denote the one-dimensional Fourier transform with respect to variable $x_{j}$ and $\chi_{j}$ denotes the characteristic function of the halfspace

[TABLE]

Lemma 3.3. Assume $\gamma\in A_{p}$ for $p\in\left(1,\infty\right)$ and $E$ is a UMD space. Then $R$ defines a bounded operator in $L_{p,\gamma}\left(R^{n};E\right).$

**Proof. **Since $\gamma\in A_{p}$, then by $\left[\text{12, Corollary 2.10}\right]$ (or $\left[\text{6, Theorem 4}\right]$ ) the Hilbert operator is bounded in $L_{p,\gamma}\left(R;E\right).$ It is known that $R_{1}=\frac{1}{2\pi i}\left(i\pi I-H\right)$, where $I$ is the identity operator. By using this relation we obtain that Riesz projection operator $R_{1}$ is bounded in $L_{p,\gamma}\left(R;E\right)$. Hence, one-dimensional Riesz projection $R_{j}$ also are defined bounded operators in $L_{p,\gamma}\left(R;E\right)$. It is not hard to see that

[TABLE]

i.e. $R$ is bounded operator in $L_{p,\gamma}\left(R;E\right).$

For $j=\left(j_{1},j_{2},...j_{n}\right)\in\mathbb{Z}^{n}$ let $D_{j}$ be the dyadic interval associated with $j$, i.e.

[TABLE]

and

[TABLE]

where

[TABLE]

Consider the operator

[TABLE]

Lemma 3.4. Assume $\gamma\in A_{p}$ for $p\in\left(1,\infty\right)$ and $E$ is a UMD space. Then for each $a,b$ $\in R^{n}$ the operator $f\rightarrow\Phi_{a,b}f$ is bounded in $L_{p,\gamma}\left(R^{n};E\right)$. Moreover, the set $\left\{\Phi_{a,b}:a,b\in R^{n}\right\}$ is an $R-$bounded family in $B\left(L_{p,\gamma}\left(R^{n};E\right)\right)$.

**Proof. **We first look at characteristic functions of sets of the form

[TABLE]

We can $F^{-1}\chi_{C_{a}}Ff$ expressed as:

[TABLE]

for

[TABLE]

We see that the set $\left\{\Phi_{a}:a\in R^{n}\right\}$ is $R-$bounded in view of Proposition 3.1. Setting $C_{b}=\prod\limits_{k=1}^{n}\left[-\infty,\right.\left.b_{k}\right)$ we analogously get that the set $\left\{\Phi_{b}:b\in R^{n}\right\}$ is $R-$bounded as well, where

[TABLE]

Since $\Phi_{a,b}=\Phi_{a}\Phi_{b},$ the result follows because the pointwise product of $R-$bounded sets is again $R-$bounded.

Assume $E_{1}$ and $E_{2}$ are UMD spaces. We put

[TABLE]

Let $\left\{A_{j}:j\in\mathbb{Z}^{n}\right\}$ be a decomposition of $(0,\infty)^{n}$ in intervals such that for each compact $K\in(0,\infty)^{n}$ the set $\left\{A_{j}\cap K:j\in\mathbb{Z}^{n}\right\}$ is finite. Assume further that the families $\left\{\Delta_{j}^{X_{1}}:j\in\mathbb{Z}^{n}\right\}$ and $\left\{\Delta_{j}^{Y_{1}}:j\in\mathbb{Z}^{n}\right\}$ of the corresponding Fourier multipliers, i.e

[TABLE]

are unconditional Schauder decompositions of $X_{1}$, $Y_{1}$ respectively, where $F_{E}$ and $F_{E}^{-1}$ denote the Fourier and inverse Fourier transforms. For $k\in\mathbb{N}$ we now cut each interval $A_{j}$ in $2^{kn}$ smaller ones by decomposing it in each coordinate direction into $2^{k}$ pieces. These new smaller intervals are denoted by $A_{j,l}^{k},$ where $j\in\mathbb{Z}^{n}$ and $l\in\left\{0,1,...,2^{k}-1\right\}^{n}.$

Let $M$ be a function on $R^{n}$ with values in a Banach space $B(E_{1},E_{2})$. Assume that $M$ is constant operator on the intervals $A_{j,l}^{k}$, and denote by $M_{j,l}^{k}$, the corresponding value of $M$. Next we show that an operator-valued function which is constant on the $A_{j,l}^{k}$ ’s is a Fourier multiplier from $X$ to $Y$ if it satisfies a certain inequality involving $R-$bounds.

**Proposition 3.1. **Assume $\gamma\in A_{p}$ for $p\in\left(1,\infty\right)$ and $E_{1}$, $E_{2}$ are UMD spaces. Further let $M:$ $R^{n}$ $\rightarrow B(E_{1},E_{2})$ be a function which is constant on each $A_{j,l}^{k}$ and zero on $R^{n}\smallsetminus\left(0,\infty\right)^{n}.$ Assume that

[TABLE]

for every multiindex $\beta\in\left(0,1\right)^{n}$ and $k\in\mathbb{Z}.$ Then $M$ is a Fourier multiplier from $X$ into $Y$. The norm of $T=F_{E_{2}}^{-1}MF_{E_{1}}$ may be estimated by

[TABLE]

where $C_{X}$ and $C_{Y}$ are the unconditional constants and $C_{Q}$ is the $R$–bound found in Lemma 3.4.

**Proof. **By Lemma 3.4, each $\chi_{A_{j,l}^{k}}$ is a Fourier multiplier in $X$. We denote the operators $F_{E_{1}}\chi_{A_{j,l}^{k}}F_{E_{1}}^{-1}$ by $\Delta_{j,l}^{k}.$ For $f\in S\left(R^{n};E_{1}\right)$ we get

[TABLE]

Then by using the same reasoning as used in $\left[\text{9, Theorem 3.3}\right]$ we obtain

[TABLE]

where $T_{j}$ are operators defined by

[TABLE]

Since $M_{j,l}^{k}\Delta_{j,l}^{k}=\Delta_{j}^{Y}M_{j,l}^{k}$ and $\Delta_{j,l}^{k}\Delta_{j}^{X}=\Delta_{j}^{X}\Delta_{j,l}^{k},$ we have $\Delta_{j,l}^{k}\Delta_{j}^{X}=T_{j}\Delta_{j}^{X}=\Delta_{j}^{Y}T_{j}\Delta_{j}^{X}.$ Moreover, since $\left\{\Delta_{j}^{X}:j\in\mathbb{Z}^{n}\right\}$ and $\left\{\Delta_{j}^{Y}:j\in\mathbb{Z}^{n}\right\}$ are unconditional Schauder decompositions of the spaces $X$, $Y$ respectively and $S\left(R^{n};E_{1}\right)$ is dense in $X$, it remains to prove that the family $\left\{T_{j}:j\in\mathbb{Z}^{n}\right\}$ is $R-$bounded. This step is derived as in $\left[\text{9, Theorem 3.3}\right]$, i.e. we show that

[TABLE]

Then in view of Proposition A1 we have $T\in B\left(X;Y\right)$ with

[TABLE]

In a similar way as $\left[\text{9, Proposition 3.4}\right]$ it can be shown the following proposition. It will be used to prove the Mikhlin theorem by approximating the given function $\Psi:R^{n}\rightarrow B(X,Y)$ by piecewise constant multipliers and is a generalization of the same result from $\left[7\right]$ for unweighted spaces $L_{p}\left(R^{n};E\right)$.

**Proposition 3.2. **Assume $\gamma\in A_{p}$ for $p\in\left(1,\infty\right)$ and $E_{1}$, $E_{2}$ are Banach spaces. Let $M$, $M_{N}\in L_{1}^{loc}\left(R^{n},B\left(E_{1},E_{2}\right)\right)$ be Fourier multipliers from $X$ to $Y$ such that $M_{N}\rightarrow M$ in $L_{1}^{loc}\left(R^{n},B\left(E_{1},E_{2}\right)\right)$. If $E_{2}$ reflexive and the sequence

[TABLE]

is uniformly bounded in $B\left(X,Y\right),$ then the operator $T:=F_{E_{2}}^{-1}MF_{E_{1}}$ is a bounded operator from $X$ to $Y$ with

[TABLE]

The next lemma states that the family of dyadic intervals in $R^{n}$ can be used to build up an unconditional Schauder decomposition of $R\left(X\right)$ provided $E$ is a UMD space with property ($\alpha$).

Lemma 3.5. Assume $\gamma\in A_{p}$ for $p\in\left(1,\infty\right)$ and $E$ is a UMD space. For $j=\left(j_{1},j_{2},...j_{n}\right)\in\mathbb{Z}^{n}$ let $D_{j}$ be the dyadic interval defined by $\left(3.1\right)$ and $\Delta_{j}:=F^{-1}\chi_{D_{j}}F.$ Then:

(a) If $n=1$, then the family $\left\{\Delta_{j}:j\in\mathbb{Z}^{n}\right\}$ is an unconditional Schauder decomposition of $X_{1}=R\left(L_{p,\gamma}\left(R^{n};E\right)\right);$

(b) If $E$ has property ($\alpha$), then the assertion of part (a) is true for arbitrary $n$.

**Proof. **(a) It is clear that the $\Delta_{j}$’s are projections in $L_{p,\gamma}\left(R^{n};E\right)$ and that $\Delta_{j}\Delta_{j^{\prime}}$ $=$ $\delta_{jj^{\prime}}\Delta_{j}$. Let $1,2,...$ be any enumeration of $\mathbb{Z}$. We have to prove that

[TABLE]

This convergence is clear for $f\in$ $S(0,\infty;E)$. In view of a $3\varepsilon$–argument it remains to show that the set$\{T_{N}:N\in\mathbb{N}\}$ is uniformly bounded. To this aim we define the function $m_{N}:\mathbb{R}\rightarrow\mathbb{R}$ by

[TABLE]

By Proposition A4 of $[2]$ we get that each $m_{N}\left(x\right)$ is a Fourier multiplier in $L_{p,\gamma}\left(R^{n};E\right).$ Moreover, the proof the Proposition A4 in $[2]$ shows that the family $\left\{F^{-1}m_{N}F\right\}$ is uniformly bounded. Hence, we get

[TABLE]

[TABLE]

This gives the assertion (a). By Proposition A2 we get that the collection $\left\{\mathop{\displaystyle\sum}\limits_{k\in G}\Delta_{k}:G\subset\mathbb{Z}\right\}$ is $R-$bounded which in view of Proposition A1 yields that the product of two unconditional Schauder decompositions is again an unconditional Schauder decomposition. The general case now follows by induction.

**Proof of Theorem 3.1. **Without loss of generality we assume $M\left(\xi\right)=0$ for $\xi\notin\left(0,\infty\right)^{n}.$ To apply Propositions 3.1 and 3.2 we use the decomposition of Lemma 3.5 to approximate $M$. Now, we cut each $D_{j}$ into $2^{nk}$ pieces and define

[TABLE]

where

[TABLE]

In view of Proposition 3.1 we have to estimate the $R-$bounds

[TABLE]

for all $\beta\in\left(0,1\right)^{n}$ independently of $k$. For $\beta=$ $\left(0,0,...,0\right)$ this expression is trivially bounded by $R(\{M(\xi),$ $\xi\neq 0\})$. For $\beta\neq 0$ let $i$ be the smallest index with $\beta_{i}$ $=1$. Every $\nu$ with $\nu_{i}=0$ and $\nu\leq\beta$ has a term $\tilde{\nu}$ with $\tilde{\nu}_{m}=\nu_{m}$ for $m\neq i$ and $\tilde{\nu}_{i}=1$. Now, by using the same reasoning as used in the proof of Theorem 3.6 of $\left[9\right]$ by Corollary 3.1 we get the desired estimate

[TABLE]

[TABLE]

which completes the proof.

Remark 3.1. If $E_{1}$ does not have property ($\alpha$), we can use another decomposition of $R^{n}$ to get an unconditional Schauder decomposition of $L_{p,\gamma}\left(R^{n};E_{1}\right)$. But without property ($\alpha$) we have to impose stronger conditions on $M$ to get $L_{p,\gamma}$ boundedness of the corresponding multiplier operator.

**Proof of Theorem 3.2. **For $j\in\mathbb{Z}$, let $s\left(j\right)\in\mathbb{Z}$ and $t=t\left(j\right)\in\left\{1,2,...n\right\}$ be the unique numbers satisfying $j=ns+t$. Set

[TABLE]

and define $\Delta_{j}=F^{-1}\chi_{D_{j}}F.$ Let $j=ns+t$ be the unique representation of $j.$ For $k\in\mathbb{Z}$, $r\in\mathbb{Z}^{n}$ with $0\leq r_{i}\leq 2^{k}-1$ define the operator $M_{j,r}^{k}$ by

[TABLE]

where

[TABLE]

Then, by reasoning as the proof of Theorem 3.7 in $\left[9\right]$ we get the assertion.

4. Embeding theorems in Sobolev-Lions type spaces

The embedding of Sobolev-Lions spaces play important roll in the regularity theory of PDE with operator coefficients. In this section, we show continuity of embedding operators in anisotropic Sobolev-Lions spaces.

Let

[TABLE]

[TABLE]

[TABLE]

From $\left[\text{22, Lemma 3.1}\right]$ we have

Lemma 4.1. Assume $A$ is a $\varphi-$ positive linear operator on a Banach space $E$. Then for any $h>0$ and $0\leq\mu\leq 1-\varkappa$ the operator-function

[TABLE]

is bounded in $E$ uniformly with respect to $\xi\in R^{n}$ and$\ h>0$ i.e. there exists a constant $C_{\mu}$ such that

[TABLE]

for all $\xi\in R^{n}$ and $h>0.$

One of main result of this section is the following:

Theorem 4.1. Let $\gamma\in A_{p}$ for $p\in\left(1,\infty\right).$ Assume $E$ is an UMD space and $A$ is a $\varphi-$ positive operator in $E$. Then for $0\leq\mu\leq 1-\varkappa$ the embedding

[TABLE]

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

[TABLE]

for $u\in Y$ and $0<h\leq h_{0}<\infty.$

Proof. It is clear to see that

[TABLE]

[TABLE]

Hence, denoting  $Fu$ by $\hat{u},$ we get from $\left(4.3\right)$ the following estimate

[TABLE]

[TABLE]

where $C_{1}$, $C_{2}$ are positive constants depending only of $p$ and $\gamma$. Similarly, there exist positive constants $M_{1}$ and $M_{2}$ such that for $u\in Y$ we have

[TABLE]

Therefore, for proving the inequality $\left(4.2\right)$ it suffices to show

[TABLE]

[TABLE]

Therefore, the inequality $\left(4.2\right)$ will follow if we prove the following estimate

[TABLE]

for $u\in Y,$ where

[TABLE]

Due to positivity of $A,$ the operator function $G\left(\xi\right)$ has a bounded inverse in $E$ for all $\xi\in R^{n}$ and $h>0.$ So, we can set

[TABLE]

The inequality $\left(4.5\right)$ will follow immediately from $\left(4.6\right)$ if we can prove that the operator-function $\Psi_{h}=\xi^{\alpha}A^{1-\varkappa-\mu}G^{-1}\left(\xi\right)$ is a multiplier in $M_{p,\gamma}^{q,\gamma}\left(E\right)$ uniformly with respect to $h.$ So, by Theorem 3.1 it suffices to show that the set

[TABLE]

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

[TABLE]

By Lemma 4.1 there exists a constant  $C_{\mu}>0$ such that the following uniform estimate holds

[TABLE]

Let first, $\beta=\left(\beta_{1},...\beta_{n}\right)$ where  $\beta_{k}=1$ and $\beta=0$ for $j\neq k$. Then, by using the resolvent properties of $A$ we obtain

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Repeating the above process, we obtain that there exists a constant  $C_{\mu}>0$  depending only $\mu$ such that

[TABLE]

for $\beta=\left(\beta_{1},...\beta_{n}\right)$, $\beta_{k}\in\left\{0,1\right\}$ and for all $\xi\in R^{n}$, $\xi\neq 0$.  Due to $R$-positivity of $A$ and by $\left(4.9\right)$ we obtain that the set

[TABLE]

is $R$ bounded uniformly in $h$. Then, by virtue of Kahane’s contraction principle $\left[\text{8, Lemma 3.5}\right]$ and by $\left(4.9\right)$ we obtain that the set

[TABLE]

is uniformly $R$-bounded. Moreover, by using the inequalities of moment for positive operators and Young’s we get that

[TABLE]

where

[TABLE]

Then thanks to $R$-boundedness of $B_{i}\left(\xi,\lambda\right)$ we have

[TABLE]

for all $\xi_{1},\xi_{2},...,\xi_{m}\in R^{n}$, $\eta_{j}=\left(\xi_{j1,}\xi_{j2},...,\xi_{jn}\right)\in R^{n},$ $u_{1,}u_{2},...,u_{m}\in E$, $m\in\mathbb{N}$, where $\left\{r_{j}\right\}$ is a sequence of independent symmetric $\left\{-1,1\right\}$-valued random variables on $\left[0,1\right]$. Thus, in view of Kahane’s contraction principle, additional and product properties of $R$-bounded operators and $\left(4.10\right)$, $\left(4.11\right)$ we obtain

[TABLE]

[TABLE]

The estimate $\left(4.12\right)$ implies $R$-boundedness of the set $B\left(\xi,h\right)$, which implies the assertion.

It is possible to state Theorem 4.1 in a more general setting. For this aim, we use the concept of extension operator.

Condition 4.1. Let $\gamma\in A_{p}$ for $p\in\left(1,\infty\right)$. Let $A$ be a positive operator in UMD space $E.$ Assume a region  $\Omega\subset R^{n}$ such that there exists bounded linear extension operator$\ B$ from $W_{p,\gamma}^{l}\left(\Omega,E\left(A\right),E\right)$ to $Y$ for $1<p<\infty.$

Remark 4.1. If  $\Omega\subset R^{n}$ is a region satisfying the strong  $l-$horn condition  (see $\left[3\right]$, p.117 for $E=\mathbb{C},$ $A$ $=I$  and $\gamma\left(x\right)\equiv 1$) then for $1<p<\infty$ there exists a bounded linear extension operator from  $W_{p}^{l}\left(\Omega\right)=W_{p}^{l}\left(\Omega;\mathbb{C},\mathbb{C}\right)$ to $W_{p}^{l}\left(R^{n}\right)=W_{p}^{l}\left(R^{n};\mathbb{C},\mathbb{C}\right).$

Theorem 4.2. Assume conditions of Theorem 4.1 and Condition 4.1 are satisfied. Then for $0\leq\mu\leq 1-\varkappa$ the embedding

[TABLE]

is continuous and there exists a constant $C_{\mu}$ depending only of $\mu,$ $p$, $\gamma$ such that

[TABLE]

[TABLE]

for $u\in W_{p,\gamma}^{l}\left(\Omega;E\left(A\right),E\right)$ and $0<h\leq h_{0}<\infty.$

Proof. It is suffices to prove the estimate $\left(4.10\right).$ Let $B$ is a bounded linear extension operator from  $W_{p,\gamma}^{l}\left(\Omega;E\left(A\right),E\right)$ to $W_{p,\gamma}^{l}\left(R^{n};E\left(A\right),E\right)$ and let $B_{\Omega}$ be the restriction operator from  $R^{n}$ to $\Omega.$ Then for any $u\in W_{p,\gamma}^{l}\left(\Omega;E\left(A\right),E\right)$ we have

[TABLE]

[TABLE]

[TABLE]

Result 4.1. Assume the conditions of Theorem 4.2 are satisfied. Then for $u\in W_{p,\gamma}^{l}\left(\Omega;E\left(A\right),E\right)$ we have the following multiplicative estimate

[TABLE]

Indeed, setting

[TABLE]

in $\left(4.10\right)$ we obtain $\left(4.11\right).$

Theorem 4.3. Suppose conditions of Theorem 4.1 are hold. Then for $0<\mu<1-\varkappa$ the embedding

[TABLE]

is continuous and there exists a constant $C_{\mu}$ depending only of $\mu,$ $p$, $\gamma$ such that

[TABLE]

for $u\in Y$ and $0<h\leq h_{0}<\infty.$

**Proof. **It is sufficient to prove the estimate $\left(4.12\right)$ for $u\in Y.$ By definition of interpolation spaces $\left(E\left(A\right),E\right)_{\varkappa+\mu,p}$ (see $\left[\text{27, \S 1.14.5}\right]$) the estimate $\left(4.12\right)$ is equivalent to the inequality

[TABLE]

[TABLE]

By multiplier properties, the inequality $\left(4.13\right)$ will follow immediately if we will prove that the operator-function

[TABLE]

is a multiplier from $X$ to $L_{p,\gamma}\left(R^{n};L_{p}\left(R_{+};E\right)\right).$ This fact is proved by the same manner as Theorem 4.1. Therefore, we get the estimate $\left(4.12\right).$

In a similar way, as the Theorem 4.2 we obtain

Theorem 4.4. Suppose conditions of Theorem 4.2 are hold. Then for $0<\mu<1-\varkappa$ the embedding

[TABLE]

is continuous and there exists a constant $C_{\mu}$ depending only of $\mu,$ $p$, $\gamma$ such that

[TABLE]

for $u\in W_{p,\gamma}^{l}\left(\Omega;E\left(A\right),E\right)$ and $0<h\leq h_{0}<\infty.$

Result 4. 2. Suppose the conditions of Theorem 4.2 are hold. Then for $u\in W_{p,\gamma}^{l}\left(\Omega;E\left(A\right),E\right)$ we have the following multiplicative estimate

[TABLE]

Indeed setting $h=\left\|u\right\|_{L_{p,\gamma}\left(\Omega;E\right)}.\left\|u\right\|_{W_{p,\gamma}^{l}\left(\Omega;E\left(A\right),E\right)}^{-1}$ in $\left(4.14\right)$ we obtain $\left(4.15\right).$

From Theorem 4.2 we obtain

Result 4.3. Assume the conditions of Theorem 4.2 are satisfied for $l_{1}=l_{2}=\ldots=l_{n}=m.$ Then for $0\leq\mu\leq 1-\varkappa$ the embedding

[TABLE]

is continuous and there exists a constant $C_{\mu}$ depending only of $\mu,$ $p$, $\gamma$ such that

[TABLE]

for $u\in W_{p,\gamma}^{m}\left(\Omega;E\left(A\right),E\right)$ and $0<h\leq h_{0}<\infty$ where

[TABLE]

Result 4.3. If $E=H$, where $H$ is a Hilbert space and  $p_{k}=q_{k}=2,$ $\Omega=\left(0,T\right),$ $l_{1}=l_{2}=\ldots=l_{n}=m$ , $A=A^{\times}\geq c^{2}I,$ $\gamma\left(x\right)\equiv 1$ then we obtain the well known Lions-Peetre $\left[\text{14}\right]$ result.  Moreover, the result of Lions-Peetre is improving even in the one dimensional case and for non selfedjoint positive operators $A.$

From Theorems 4.2 we obtain

Result 4.4. Suppose the conditions of Theorem 4.2 are satisfied for $\gamma\left(x\right)\equiv 1.$ Then for $0\leq\mu\leq 1-\varkappa$ the embedding

[TABLE]

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

[TABLE]

for $u\in W_{p}^{l}\left(\Omega;E\left(A\right),E\right)$ and $0<h\leq h_{0}<\infty$.

Moreover, if $\Omega$ is a bounded domain in $R^{n}$ and $A^{-1}$ is a compact operator in $E,$ then for $0<\mu\leq 1-\varkappa$ the embedding

[TABLE]

is compact.

If $E=\mathbb{C},$ $A=I$, $\gamma\left(x\right)\equiv 1$ we get the embedding $D^{\alpha}W_{p}^{l}\left(\Omega\right)\subset L_{p}\left(\Omega\right)$ proved in $\left[3\right]$ for Sobolev spaces $W_{p}^{l}\left(\Omega\right).$

Let $s>0.$ Consider the following sequence space (see e.g. $\left[\text{27, \S\ 1.18}\right]$)

[TABLE]

with the norm

[TABLE]

Note that,  $l_{q}^{0}=l_{q}.$ Let $A$ be infinite matrix defined in $l_{\nu}$ such that  $D\left(A\right)=l_{q}^{s},$ A=\left[\delta_{ij}2^{si}\right],\where  $\delta_{ij}=0$, when  $i\neq j,$  $\delta_{ij}=1,$ when $i=j=1,2,...,\infty.$

It is clear to see that the operator $A$ is positive in $l_{q}$. From Theorem 4.2 we obtain the following results:

**Result 4.5. **Suppose the conditions of Theorem 4.2 are satisfied for $E=\mathbb{C}$. Then for $0\leq\mu\leq 1-\varkappa$, $1<p<\infty$ the embedding

[TABLE]

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

[TABLE]

for $u\in W_{p,\gamma}^{l}\left(\Omega,l_{q}^{s},l_{q}\right)$ and $0<h\leq h_{0}<\infty.$

**Result 4.6. **Suppose the conditions of Theorem 4.2 are hold for $E=\mathbb{C}$. Then for $0<\mu\leq 1-\varkappa,$ $1<p<\infty$ the embedding

[TABLE]

is compact.

**Result 4.7. **For $0\leq\mu\leq 1-\varkappa$, $1<p<\infty$ the embedding

[TABLE]

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

[TABLE]

for $u\in W_{p}^{l}\left(\Omega,l_{q}^{s},l_{q}\right)$ and $0<h\leq h_{0}<\infty.$

Note that, these results haven’t been obtained with classical method until now.

5. Separable differential operators in weighted Lebesque spaces

Firstly, consider the leading part of the equation $\left(1.1\right)$, i.e. consider the following equation

[TABLE]

where $a_{\alpha}$ are complex numbers, $l\in\mathbb{N},$ $A$ is a linear operator in a Banach space $E$ and $\lambda$ is a complex parameter.

Let

[TABLE]

**Condition 5.1. ** Let

[TABLE]

for $0\leq\varphi_{1}<\pi;$

(b) There exists the positive constat $M_{0}$ so that

[TABLE]

In this section we prove the following result

Theorem 5.1. Suppose the following conditions hold:

(1) Condition 5.1 is hold;

(2) $\gamma\in A_{p}$ for $p\in\left[1,\infty\right]$;

(3) $A$ is a $R-$positive operator in UMD space $E$ for $0\leq\varphi<\pi-\varphi_{1}$.

Then for all $\ f\in X$ and $\lambda\in S\left(\varphi_{1}\right)$ equation $\left(6.1\right)$ has an unique solution $u$ that belongs to space $Y$ and the coercive uniform estimate holds

[TABLE]

Proof. By applying the Fourier transform to the equation $\left(5.1\right)$ we get

[TABLE]

where

[TABLE]

Since $K\left(\xi\right)\in S\left(\varphi_{1}\right)$ for all $\xi\in R^{n},$ the operator $A+K\left(\xi\right)$ is invertible in $E$. So, we obtain that the solution of the equation $\left(5.3\right)$ can be represented in the form

[TABLE]

By using $\left(5.4\right)$ we have

[TABLE]

[TABLE]

Hence, it is suffices to show that the operator-functions

[TABLE]

[TABLE]

are multipliers in $X.$ To see this, it is suffices to show that the following collections

[TABLE]

are $R-$bounded in $E$ uniformly in $\lambda$, where

[TABLE]

Due to $R-$positivity of $A$, the set

[TABLE]

is $R$-bounded. Moreover, by using the same reasoning as used in the proof of Theorem 4.1 and in view of (3) condition we obtain that the set

[TABLE]

is $R$-bounded uniformly in $\lambda\in S\left(\varphi_{1}\right)$. Then by virtue of of Kahane’s contraction principle, by product properties of the collection of $R$ -bounded operators (see e.g. Lemma 3.5., Proposition 3.4. in $\left[8\right]$) and due to $R-$positivity of operator $A$ we obtain

[TABLE]

[TABLE]

The estimates $\left(5.4\right)$ by Theorem 3.1 imply that the operator functions $\sigma_{1,\lambda}\left(\xi\right)$ and $\sigma_{2,\lambda}\left(\xi\right)$ are $L_{p,\gamma}\left(R^{n};E\right)$ multipliers.

Let $L_{0}$ denote the differential operator in $X$ that generated by problem $\left(5.1\right)$ for $\lambda=0,$ that is

[TABLE]

The estimate $\left(5.2\right)$ implies that the operator $L_{0}$ has a bounded inverse from $X$ into $Y.$ We denote by $L$ differential operator in $X$ that generated by problem $\left(1.1\right)$, i.e.

[TABLE]

Theorem 5.2. Suppose all conditions of Theorem 5.1 are hold and

[TABLE]

Then for all $\ f\in X$ and $\lambda\in S\left(\varphi_{1}\right)$ with sufficiently large $\left|\lambda\right|$ equation $\left(1.1\right)$ has an unique solution $u$ that belongs to space $Y$ and the uniform coercive estimate holds

[TABLE]

Proof. In view of condition on $A_{\alpha}\left(x\right)$ and by virtue of Theorem 4.1 there is $h>0$ such that

[TABLE]

[TABLE]

for $u\in Y$. Then from estimates $\left(5.2\right)$ and $\left(5.6\right)$ for $u\in Y$ we have

[TABLE]

Since $\left\|u\right\|_{X}=\frac{1}{\lambda}\left\|\left(L_{0}+\lambda\right)u-L_{0}u\right\|_{X}$ for $u\in Y$ we get

[TABLE]

[TABLE]

From estimates $\left(5.7\right)$ and $\left(5.8\right)$ for $u\in Y$ we obtain

[TABLE]

Then choosing $h$ and $\lambda$ such that $Ch^{\mu}<1,$ $CMh^{-\left(1-\mu\right)}<\lambda,$ from $\left(5.9\right)$ for sufficiently large $\lambda$ we have

[TABLE]

Since we have the relation

[TABLE]

so by using the estimates $\left(5.5\right),$ $\left(5.10\right)$ and the perturbation theory of linear operators we obtain the assertion.

From Theorem 5.2 we obtain the following results:

**Result 5.1. **Assume the conditions of Theorem 5.2 are satisfied. Then there exists a constant $C_{1}$ and $C_{2}$ depending only on $p$, $\gamma$ such that

[TABLE]

for all $u\in Y$ and for sufficiently large $d>0.$

**Result 5.2. **Assume the conditions of Theorem 5.2 are satisfied. Then the resolvent operator $\left(L+\lambda\right)^{-1}$ satisfies the following coercive sharp estimate holds

[TABLE]

for $\lambda\in S\left(\varphi_{1}\right).$

The** Result 5.2 implies that operator $L$ **is positive operator in $X$. Then by virtue of $\left[\text{27, \S 1.14.5}\right]$ the operator $L$ is a generator of an analytic semigroup in $X$ for $\varphi\in\left(\frac{\pi}{2},\pi\right).$

**6. The Cauchy problem for abstract parabolic equation **

Consider now, the Cauchy problem $\left(1.3\right).$ In this section we obtaın the existence and uniqueness of the maximal regular solution of problem $\left(1.3\right)$. First all of we show

**Theorem 6.1. **Assume the conditions of Theorem 5.1 are satisfied. Then the operator $L_{0}$ is $R$-positive in $X.$

**Proof. **Theorem 5.1 implies that the operator $L_{0}$ is positive in $X$. We have to prove the $R$-boundedness of the set

[TABLE]

From Theorem 5.1 we have

[TABLE]

for $f\in X,$ where

[TABLE]

By definition of $R$-boundedness, it is sufficient to show that the operator function $\Phi\left(\xi,\lambda\right)$ (depended on variable $\lambda$ and parameters $\xi,$ $\varepsilon$ ) is uniformly bounded multiplier in $X.$ In a similar manner one can easily show that $\Phi\left(\xi,\lambda\right)$ is multiplier in $X.$ Then, by definition of $R$-boundedness we have

[TABLE]

[TABLE]

for all $\xi_{1},\xi_{2},...,\xi_{m}\in R^{n}$, $\lambda_{1},\lambda_{2},...,\lambda_{m}\in S_{\varphi},$ $f_{1,}f_{2},...,f_{m}\in X$, $m\in\mathbb{N}$, 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 $\sigma\left(\lambda\right)$ is $R$-bounded.

For $\mathbf{p=}\left(p,p_{1}\right)$, $R_{+}^{n+1}=R_{+}\times R^{n},$ $F=L_{\mathbf{p,}\gamma}\left(R_{+}^{n+1};E\right)$ will be denoted the space of all $E$-valued $\mathbf{p}$-summable weighted functions with mixed norm, i.e. the space of all measurable functions $f$ defined on $R_{+}^{n+1}$ for which

[TABLE]

Analogously, $F_{0}=W_{\mathbf{p,}\gamma}^{1,2l}\left(R_{+}^{n+1},E\left(A\right),E\right)$ denotes the Sobolev-Lions space with corresponding mixed norm, i.e.

[TABLE]

[TABLE]

The main result of this section is the following:

Theorem 6.2. Assume all conditions of Theorem 5.1 hold for $\varphi\in\left(\frac{\pi}{2},\pi\right)$ and $p_{1}\in\left(1,\infty\right)$. Then for $f\in F$ problem $\left(1.3\right)$ has a unique solution $u\in F_{0}$ satisfying

[TABLE]

Proof. So, the problem $\left(1.3\right)$ can be expressed as

[TABLE]

By the Result 5.2 the operator $L_{0}$ is positive in $X$. The Theorem 6.1 implies that $L_{0}$ is $R-$positivity in $X$ for $\varphi\in\left(\frac{\pi}{2},\pi\right).$ Then by virtue of $\left[\text{29, Th. 4.10}\right]$ we obtain that, for $f\in L_{p_{1}}\left(R_{+};X\right)$ the Cauchy problem $\left(6.2\right)$ has a unique solution $u\in F_{0}$ satisfying

[TABLE]

In view of Result 5.1 the operator $L_{0}$ is separable in $X,$ i.e, the estimate $\left(6.3\right)$ implies $\left(6.1\right)$.

**7. Degenerate abstract differential equations **

Let us consider the problem

[TABLE]

where $A$, $A_{\alpha}$ are linear operators in a Banach space $E$ and $\lambda$ is a complex parameter, where

[TABLE]

here $\gamma_{k}\left(x\right)$ are positive measurable functions on $R^{n}.$

Let

[TABLE]

[TABLE]

Here,

[TABLE]

Let

[TABLE]

Remark 7.1.

Under the substitution

[TABLE]

the spaces $X$ and $Y$ are mapped isomorphically onto the weighted spaces $L_{p,\tilde{\gamma}}\left(R^{n};E\right)$, $W_{p,\tilde{\gamma}}^{2l}\left(R^{n};E\left(A\right),E\right),$ where

[TABLE]

Moreover, under the transformation $\left(7.3\right)$ the problem $\left(7.1\right)$ is mapped to the undegenerate problem $\left(1.1\right)$ considered in the weighted space $L_{p,\tilde{\gamma}}\left(R^{n};E\right)$.

**Condition 7.1. **Assume $\left(7.1\right)$ holds and $\gamma_{k}\left(x_{k}\left(\tau_{k}\right)\right)\in A_{p}$ for $k=1,2,...,n$ and $p\in\left(1,\infty\right).$

From Theorem 5.2 and Remark 7.1 we obtain the following results:

**Result 7.1. **Assume the conditions of Theorem 5.2 are satisfied. Then for all $\ f\in X$ and $\lambda\in S\left(\varphi_{1}\right)$ with sufficiently large $\left|\lambda\right|$ equation $\left(1.1\right)$ has an unique solution $u$ that belongs to $Y$ and the uniform coercive estimate holds

[TABLE]

Let $G$ denote the operator in $X$ generated by the problem $\left(7.1\right).$

**Result 7.2. **Assume the conditions of Theorem 5.2 and the Condition 7.1 are satisfied. Then the resolvent operator $\left(L+\lambda\right)^{-1}$ satisfies the following sharp estimate

[TABLE]

for $\lambda\in S\left(\varphi_{1}\right).$

The** Result 5.2 implies that operator $G$ **is positive operator in $X$. Then by virtue of $\left[\text{27, \S 1.14.5}\right]$ the operator $G$ is a generator of an analytic semigroup in $X$ for $\varphi\in\left(\frac{\pi}{2},\pi\right).$

Consider the Cauchy problem for degenerate parabolic equation

[TABLE]

[TABLE]

where $a_{\alpha}$ are complex numbers and $A$ is a linear operator in a Banach space $E.$

For $\mathbf{p=}\left(p,p_{1}\right)$, let $\Phi=L_{\mathbf{p}}\left(R_{+}^{n+1};E\right)$ denotes $L_{\mathbf{p,}\gamma}\left(R_{+}^{n+1};E\right)$ for $\gamma\left(x\right)\equiv 1.$ Analogously, $\Phi_{0}=W_{\mathbf{p,}\gamma}^{1,\left[2l\right]}\left(R_{+}^{n+1},E\left(A\right),E\right)$ denotes the Sobolev-Lions space with corresponding mixed norm, i.e.

[TABLE]

[TABLE]

From Theorem 6.2 and Remark 7.1 we obtain the following results:

**Result 7.3.  **Assume all conditions of Theorem 5.1 and the Condition 7.1 are satisfied for $\varphi\in\left(\frac{\pi}{2},\pi\right)$ and $p_{1}\in\left(1,\infty\right)$. Then for all $f\in\Phi$ problem $\left(7.4\right)$ has a unique solution $u\in\Phi_{0}$ satisfying

[TABLE]

8. Maximal regularity properties of infinite many system of parabolic equations

Consider the Cauchy problem for infinite many system of parabolic equations

[TABLE]

[TABLE]

where $a_{\alpha}$ and $a_{ij}$ are complex numbers.

**Condition 8.1. **Let

[TABLE]

Let

[TABLE]

[TABLE]

[TABLE]

Here,

[TABLE]

Theorem 8.1. Assume the Conditions 5.1 and 8.1 are satisfied. Then for all $f\left(x\right)=\left\{f_{i}\left(x\right)\right\}_{1}^{\infty}\in X_{\mathbf{p},q}$ problem $\left(8.1\right)-\left(8.2\right)$ has a unique solution $u=\left\{u_{i}\left(x\right)\right\}_{1}^{\infty}$ that belongs to space $Y_{\mathbf{p},q}$ and the coercive sharp estimate holds

[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 $\left(8.4\right)$ and $\left(8.5\right)$ we get

[TABLE]

i.e., the operator $A$ is $R$-positive in $l_{q}.$ Hence, by Theorem 6.2 we obtain the assertion.

Remark 8.1. There are a lot of $R-$positive operators in different concrete Banach spaces. Therefore, putting concrete Banach spaces instead of $E,$ and concrete differential, pseudo differential operators, or finite, infinite matrices instead of $A,$ by virtue of Theorems 5.2  and 6.2 we can obtained the different class of maximal regular partial differential equations or system of equations.

Acknowledgements

The author would like to express a gratitude to Dr. Neil. Course for his useful advice in English in preparing of this paper

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