**EMBEDDING THEOREMS IN BANACH-VALUED** *B-SPACES* **AND MAXIMAL** *B-REGULAR DIFFERENTIAL-OPERATOR* **EQUATIONS**

VELI B. SHAKHMUROV

*Received 28 September 2004; Revised 8 November 2005; Accepted 4 May 2006*

The embedding theorems in anisotropic Besov-Lions type spaces*B*^{l}* _{p,θ}*(R

*;*

^{n}*E*0,E) are stud- ied; here

*E*0 and

*E*are two Banach spaces. The most regular spaces

*E*

*α*are found such that the mixed diﬀerential operators

*D*

*are bounded from*

^{α}*B*

^{l}*(R*

_{p,θ}*;E0,*

^{n}*E) toB*

^{s}*(R*

_{q,θ}*;*

^{n}*E*

*α*), where

*E*

*α*are interpolation spaces between

*E*0and

*E*depending on

*α*

*=*(α1,α2,. . .,α

*n*) and

*l*

*=*(l1,l2,. . .,l

*n*). By using these results the separability of anisotropic diﬀerential-operator equations with dependent coeﬃcients in principal part and the maximal

*B-regularity*of parabolic Cauchy problem are obtained. In applications, the infinite systems of the quasielliptic partial diﬀerential equations and the parabolic Cauchy problems are stud- ied.

Copyright © 2006 Veli B. Shakhmurov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**1. Introduction**

Embedding theorems in function spaces have been studied in [8,35,37,38]. A com- prehensive introduction to the theory of embedding of function spaces and historical references may be also found in [37]. In abstract function spaces embedding theorems have been investigated in [4,5,10,17,21,27,34,40]. Lions and Peetre [21] showed that if

*u**∈**L*2

0,*T;H*0

, *u*^{(m)}*∈**L*2(0,T;H), (1.1)

then

*u*^{(i)}*∈**L*2

0,*T;*^{}*H,H*0

*i/m*

, *i**=*1, 2,. . .,m*−*1, (1.2)
where*H*0,*H* are Hilbert spaces,*H*0is continuously and densely embedded in*H, where*
[H0,H]*θ*are interpolation spaces between*H*0and*H* for 0*≤**θ**≤*1. The similar questions
for anisotropic Sobolev spaces*W*_{p}* ^{l}*(Ω;H0,

*H),*Ω

*⊂*R

*and for corresponding weighted*

^{n}Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 16192, Pages1–22 DOI10.1155/JIA/2006/16192

spaces have been investigated in [28–31] and [23,24], respectively. Embedding theorems
in Banach-valued Besov spaces have been studied in [4,5,27,32]. The solvability and
spectrum of boundary value problems for elliptic diﬀerential-operator equations (DOE’s)
have been refined in [3–7,13,28–33,39,40]. A comprehensive introduction to DOE’s and
historical references may be found in [15,18,40]. In these works, Hilbert-valued function
spaces essentially have been considered. The maximal*L** _{p}*regularity and Fredholmness of
partial elliptic equations in smooth regions have been studied, for example, in [1,2,20]

and for nonsmooth domains studied, for example, in [16,26]. For DOE’s the similar problems have been investigated in [13,28–32,36,39,40].

Let*E*0,*E*be Banach spaces such that*E*0is continuously and densely embedded in*E.*

In the present paper,*E-valued Besov spacesB*^{l+s}* _{p,θ}*(R

*;E0,E)*

^{n}*=*

*B*

^{s}*(R*

_{p,θ}*;E0)*

^{n}*∩*

*B*

^{l+s}*(R*

_{p,θ}*;E) are introduced and called Besov-Lions type spaces. The most regular interpolation class*

^{n}*E*

*α*

between*E*0 and *E*is found such that the appropriate mixed diﬀerential operators*D** ^{α}*
are bounded from

*B*

^{l+s}*(R*

_{p,q}*;*

^{n}*E*0,E) to

*B*

^{s}*(R*

_{p,q}*;E*

^{n}*). By applying these results the maximal regularity of certain class of anisotropic partial DOE with varying coeﬃcients in Banach- valued Besov spaces is derived.*

_{α}The paper is organized as follows.Section 2collects notations and definitions.Section 3presents the embedding theorems in Besov-Lions type spaces

*B*^{s+l}_{p,q}^{}*R** ^{n}*;E0,E

^{}

*.*(1.3)

Section 4contains applications of the underlying embedding theorem to vector-valued
function spaces. Section 5is devoted to the maximal regularity (in *B*^{s}* _{p,q}*(R

*;E)) of the certain class of anisotropic DOE with variable coeﬃcients in principal part. Then by us- ing these results the maximal*

^{n}*B-regularity of the parabolic Cauchy problem is shown. In*Section 6these DOE are applied to BVP’s and Cauchy problem for the finite and infinite systems of quasielliptic and parabolic PDEs, respectively.

**2. Notations and definitions**

Let*E*be a Banach space. Let*L**p*(Ω;*E) denote the space of all strongly measurableE-valued*
functions that are defined onΩ*⊂**R** ^{n}*with the norm

*f**L**p*(Ω;E)*=* *f*(x)^{}_{E}^{p}*dx*

1/ p

, 1*≤**p <**∞*,
*f**L**∞*(Ω;E)*=*ess sup

*x**∈*Ω

*f*(x)^{}_{E}^{}, *x**=*

*x*1,x2,. . .,x*n*

*.*

(2.1)

The Banach space*E*is said to be a*ζ-convex space (see [9,*11,12,19]) if there exists
on*E**×**E*a symmetric real-valued function*ζ(u,v) which is convex with respect to each of*
the variables, and satisfies the conditions

*ζ(0, 0)>*0, *ζ*(u,*v)**≤ **u*+*v*, for*u** ≤*1*≤ **v**.* (2.2)

A*ζ-convex spaceE*is often called a UMD-space and written as*E**∈*UMD. It is shown in
[9] that the Hilbert operator

(H f)(x)*=*lim

*ε**→*0

*|**x**−**y**|**>ε*

*f*(y)

*x**−**yd y* (2.3)

is bounded in*L**p*(R;E), *p**∈*(1,*∞*) for those and only those spaces*E, which possess the*
property of UMD spaces. The UMD spaces include, for example,*L** _{p}*,

*l*

*spaces and the Lorentz spaces*

_{p}*L*

*pq*,

*p,q*

*∈*(1,

*∞*).

**Let C be the set of complex numbers and let**
*S**ϕ**=*

*λ;λ**∈***C,***|*argλ*−**π**| ≤**π**−**ϕ*^{}*∪ {*0*}*, 0*< ϕ**≤**π.* (2.4)
A linear operator*A*is said to be a*ϕ-positive in a Banach spaceE, with boundM >*0 if
*D(A) is dense onE*and

(A*−**λI)*^{−}^{1}^{}_{L(E)}*≤**M*^{}1 +*|**λ**|**−*1

(2.5)
with*λ**∈**S**ϕ*,*ϕ**∈*(0,π],*I*is identity operator in*E, andL(E) is the space of all bounded*
linear operators in*E. SometimesA*+*λI*will be written as*A*+*λ*and denoted by*A**λ*. It is
known [37, Section 1.15.1] that there exist fractional powers*A** ^{θ}*of the positive operator

*A. LetE(A*

*) denote the space*

^{θ}*D(A*

*) with the graphical norm*

^{θ}*u**E(A** ^{θ}*)

*=*

*u** ^{p}*+

^{}

*A*

^{θ}*u*

^{}

^{p}^{}

^{1/ p}, 1

*≤*

*p <*

*∞*,

*−∞*

*< θ <*

*∞*

*.*(2.6) Let

*E*0and

*E*be two Banach spaces. By (E0,E)

*σ,p*, 0

*< σ <*1, 1

*≤*

*p*

*≤ ∞*we will denote the interpolation spaces obtained from

*{*

*E*0,E

*}*by the

*K*-method (see, e.g., [37, Section 1.3.1] or [10]).

Let*S(R** ^{n}*;E) denote a Schwartz class, that is, the space of all

*E-valued rapidly decreasing*smooth functions

*ϕ*on

*R*

*.*

^{n}*E*

*=*

**C will be denoted by**

*S(R*

*). Let*

^{n}*S*

^{}(R

*;E) denote the space of*

^{n}*E-valued tempered distributions, that is, the space of continuous linear operators from*

*S(R*

*) to*

^{n}*E.*

Let*α**=*(α1,α2,. . .,α*n*),*α**i*are integers. An*E-values generalized functionD*^{α}*f* is called
a generalized derivative in the sense of Schwartz distributions of the generalized function

*f* *∈**S*^{}(R* ^{n}*,E) if the equality

*D*^{α}*f*,ϕ^{}*=*(*−*1)^{|}^{α}^{|}^{}*f*,D^{α}*ϕ*^{} (2.7)
holds for all*ϕ**∈**S(R** ^{n}*).

By using (2.7) the following relations
*F*^{}*D*^{α}_{x}*f*^{}*=*

*iξ*1

*α*1

,. . .,^{}*iξ**n*

*α**n**f*, *D*^{α}_{ξ}^{}*F(f*)^{}*=**F*^{}*−**ix**n*

*α*1

,. . .,^{}*−**ix**n*

*α**n*

*f*^{} (2.8)
are obtained for all *f* *∈**S*^{}(R* ^{n}*;E).

Let*L*^{∗}* _{θ}*(E) denote the space of all

*E-valued function spaces such that*

*u*

*L*

^{∗}*(E)*

_{θ}*=*

_{∞}

0

*u(t)*^{}^{θ}_{E}*dt*
*t*

1/θ

*<**∞*, 1*≤**θ <**∞*, *u**L*^{∗}*∞*(E)*=* sup

0<t<*∞*

*u(t)*^{}_{E}*.* (2.9)

Let*s**=*(s1,s2,. . .,*s**n*) and*s**k**>*0. Let*F*denote the Fourier transform. Fourier-analytic rep-
resentation of*E-valued Besov space onR** ^{n}*is defined as

*B*^{s}_{p,θ}^{}*R** ^{n}*;E

^{}

*=*

*u**∈**S*^{}^{}*R** ^{n}*;E

^{},

*u*

*B*

^{s}*(R*

_{p,θ}*;E)*

^{n}*=*
*F*^{−}^{1}

*n*
*k**=*1

*t*^{κ}^{k}^{−}^{s}^{k}^{}1 +^{}*ξ**k*^{κ}^{k}

*e*^{−}^{t}^{|}^{ξ}^{|}^{2}*Fu*

*L*^{∗}* _{θ}*(L

*p*(R

*;E))*

^{n},

*p**∈*(1,*∞*), *θ**∈*[1,*∞*],κ*k**> s*_{k}

*.*

(2.10)

It should be noted that the norm of Besov space do not depend onκ*k*. Sometimes we
will write*u**B*^{s}* _{p,θ}*in place of

*u*

*B*

^{s}*(R*

_{p,θ}*;E).*

^{n}Let*l**=*(l1,l2,. . .,l*n*),*s**=*(s1,s2,. . .,s*n*), where*l**k*are integers and*s**k*are positive numbers.

Let*W*^{l}*B*^{s}* _{p,θ}*(R

*;E) denote an*

^{n}*E-valued Sobolev-Besov space of all functionsu*

*∈*

*B*

^{s}*(R*

_{p,θ}*;E) such that they have the generalized derivatives*

^{n}*D*

^{l}

_{k}

^{k}*u*

*=*

*∂*

^{l}

^{k}*u/∂x*

_{k}

^{l}

^{k}*∈*

*B*

^{s}*(R*

_{p,θ}*;E),*

^{n}*k*

*=*1, 2,. . .,n with the norm

*u**W*^{l}*B*^{s}* _{p,θ}*(R

*;E)*

^{n}*=*

*u*

*B*

^{s}*(R*

_{p,θ}*;E)+*

^{n}*n*

*k*

*=*1

*D*^{l}_{k}^{k}*u*^{}_{B}*s*

*p,θ*(R* ^{n}*;E)

*<*

*∞*

*.*(2.11) Let

*E*0is continuously and densely embedded into

*E.W*

^{l}*B*

^{s}*(R*

_{p,θ}*;E0,E) denotes a space of all functions*

^{n}*u*

*∈*

*B*

^{s}*(R*

_{p,θ}*;E0)*

^{n}*∩*

*W*

^{l}*B*

^{s}*(R*

_{p,θ}*;E) with the norm*

^{n}*u**W*^{l}*B*^{s}_{p,θ}*= **u**W*^{l}*B*^{s}* _{p,θ}*(R

*;E0,E)*

^{n}*=*

*u*

*B*

^{s}*(R*

_{p,θ}*;E0)+*

^{n}*n*

*k*

*=*1

*D*_{k}^{l}^{k}*u*^{}

*B*^{s}* _{p,θ}*(R

*;E)*

^{n}*<*

*∞*

*.*(2.12) Let

*l*

*=*(l1,l2,. . .,l

*),*

_{n}*s*

*=*(s1,s2,. . .,s

*), where*

_{n}*s*

*are real numbers and*

_{k}*l*

*are positive num- bers.*

_{k}*B*

^{l+s}*(R*

_{p,θ}*;*

^{n}*E*0,E) denotes a space of all functions

*u*

*∈*

*B*

^{s}*(R*

_{p,θ}*;E0)*

^{n}*∩*

*B*

^{l+s}*(R*

_{p,θ}*;E) with the norm*

^{n}*u*_{B}^{s+l}_{p,θ}_{(R}^{n}_{;E}_{0}_{,E)}*= **u**B*^{s}* _{p,θ}*(R

*;E0)+*

^{n}*u*

_{B}

^{l+s}

_{p,θ}_{(R}

^{n}_{;E)}

*.*(2.13) For

*E*0

*=*

*E*the space

*B*

^{l+s}*(R*

_{p,θ}*;E0,E) will be denoted by*

^{n}*B*

^{l+s}*(R*

_{p,θ}*;E).*

^{n}Let*m*be a positive integer.*C(*Ω;E) and*C** ^{m}*(Ω;E) will denote the spaces of all

*E-valued*bounded continuous and

*m-times continuously diﬀerentiable functions on*Ω, respec- tively. We set

*C** _{b}*(Ω;

*E)*

_{=}*u*_{∈}*C(Ω;E), lim*

*|**x**|→∞**u(x) exists*

*.* (2.14)

Let*E*1and*E*2be two Banach spaces. A functionΨ*∈**C** ^{m}*(R

*;*

^{n}*L(E*1,E2)) is called a multi- plier from

*B*

^{s}*(R*

_{p,θ}*;*

^{n}*E*1) to

*B*

^{s}*(R*

_{q,θ}*;*

^{n}*E*2) for

*p*

*∈*(1,

*∞*) and

*q*

*∈*[1,

*∞*] if the map

*u*

*→*

*Ku*

*=*

*F*

^{−}^{1}Ψ(ξ)Fu,

*u*

*∈*

*S(R*

*;*

^{n}*E*1), is well defined and extends to a bounded linear operator

*K*:*B*^{s}_{p,θ}^{}*R** ^{n}*;

*E*1

*−→**B*^{s}_{q,θ}^{}*R** ^{n}*;E2

*.* (2.15)

The set of all multipliers from*B*^{s}* _{p,θ}*(R

*;E1) to*

^{n}*B*

^{s}*(R*

_{q,θ}*;E2) will be denoted by*

^{n}*M*

^{q,θ}*(s,E1,*

_{p,θ}*E*2).

*E*1

*=*

*E*2

*=*

*E*will be denoted by

*M*

^{q,θ}*(s,E). The multipliers and operator-valued mul- tipliers in Banach-valued function spaces were studied, for example, by [25], [37, Section 2.2.2.], and [4,11,12,14,22], respectively.*

_{p,θ}Let

*H*_{k}*=*

Ψ*h**∈**M*^{q,θ}_{p,θ}^{}*s,E*1,E2

,*h**=*

*h*1*h*2,. . .,*h*_{n}^{}*∈**K*^{} (2.16)
be a collection of multipliers in*M*^{q,θ}* _{p,θ}*(s,E1,E2). We say that

*H*

*k*is a uniform collection of multipliers if there exists a constant

*M*0

*>*0, independent on

*h*

*∈*

*K, such that*

*F*^{−}^{1}Ψ*h**Fu*^{}_{B}*s*

*p,θ*(R* ^{n}*;E2)

*≤*

*M*0

*u*

*B*

_{q,θ}*(R*

^{s}*;E1) (2.17) for all*

^{n}*h*

*∈*

*K*and

*u*

*∈*

*S(R*

*;E1).*

^{n}Let*β**=*(β1,β2,. . .,β* _{n}*) be multiindexes. We also define

*V*

_{n}*=*

*ξ**=*

*ξ*1,*ξ*2,. . .,*ξ*_{n}^{}*∈**R** ^{n}*,

*ξ*

_{i}*=*0,

*i*

*=*1, 2,

*. . .,n*

^{},

*U*

*n*

*=*

*β*:*|**β**| ≤**n*^{}, *ξ*^{β}*=**ξ*1^{β}^{1}*ξ*2^{β}^{2},*. . .,ξ**n*^{β}* ^{n}*,

*ν*

*=*1

*p*

^{−}1

*q.* (2.18)

*Definition 2.1. A Banach spaceE*satisfies a*B-multiplier condition with respect to* *p,q,*
*θ, ands*(or with respect to *p,θ, ands* for the case of *p**=**q) when*Ψ*∈**C** ^{n}*(R

*;L(E)), 1*

^{n}*≤*

*p*

*≤*

*q*

*≤ ∞*,

*β*

*∈*

*U*

*, and*

_{n}*ξ*

*∈*

*V*

*if the estimate*

_{n}*ξ*1^{β}^{1}^{+}^{ν}*ξ*2^{β}^{2}^{+}* ^{ν}*,. . .,

^{}

*ξ*

*n*

^{β}

^{n}^{+}

^{ν}*D*

*Ψ(ξ)*

^{β}^{}

_{L(E)}*≤*

*C*(2.19)

impliesΨ*∈**M*^{q,θ}* _{p,θ}*(s,E).

*Remark 2.2.* Definition 2.1is a combined restriction to*E,p,q,θ, ands. This condition*
is suﬃcient for our main aim. Nevertheless, it is well known that there are Banach spaces
satisfying the*B-multiplier condition for isotropic case andp**=**q, for example, the UMD*
spaces (see [4,14]).

A Banach space*E*is said to have a local unconditional structure (l.u.st.) if there exists a
constant*C <**∞*such that for any finite-dimensional subspace*E*0of*E*there exists a finite-
dimensional space*F*with an unconditional basis such that the natural embedding*E*0*⊂**E*
factors as*AB*with*B*:*E*0*→**F,A*:*F**→**E, and**A**B** ≤**C. All Banach lattices (e.g.,L** _{p}*,

*L*

*p,q*, Orlicz spaces,

*C[0, 1]) have l.u.st.*

The expression*u**E*1∼*u**E*2means that there exist the positive constants*C*1and*C*2

such that

*C*1*u*_{}_{E}_{1}_{≤ }*u*_{}_{E}_{2}_{≤}*C*2*u*_{}_{E}_{1} (2.20)
for all*u**∈**E*1*∩**E*2.

Let*α*1,α2,. . .,α*n*be nonnegative and let*l*1,l2,*. . .,l**n*be positive integers and let
1*≤**p**≤**q**≤ ∞*, 1*≤**θ**≤ ∞*, *|**α:.l**| =*

*n*
*k**=*1

*α**k*

*l**k*, κ*=*
*n*
*k**=*1

*α**k*+ 1/ p*−*1/q
*l**k* ,
*D*^{α}*=**D** ^{α}*1

^{1}

*D*

*2*

^{α}^{2},. . .,D

_{n}

^{α}

^{n}*=*

*∂*

^{|}

^{α}

^{|}*∂x*1^{α}^{1}*∂x*2^{α}^{2},. . .,∂x^{α}*n** ^{n}*,

*|*

*α*

*| =*

*n*

*k*

*=*!

*α**k**.*

(2.21)

Consider in general, the anisotropic diﬀerential-operator equation
(L+*λ)u**=*

*|**α:.l**|=*1

*a**α*(x)D^{α}*u*+*A**λ*(x)u+ ^{}

*|**α:.l**|**<1*

*A**α*(x)D^{α}*u**=* *f* (2.22)
in*B*^{s}* _{p,θ}*(R

*;E), where*

^{n}*a*

*α*are complex-valued functions and

*A(x),A*

*α*(x) are possibly un- bounded operators in a Banach space

*E, here the domain definitionD(A)*

*=*

*D(A(x)) of*operator

*A(x) does not depend onx. Forl*1

*=*

*l*2

*=*,. . .,

*=*

*l*

*we obtain isotropic equations containing the elliptic class of DOE.*

_{n}The function belonging to space*B*^{s+l}* _{p,θ}*(R

*;E(A),*

^{n}*E) and satisfying (2.22) a.e. onR*

*is said to be a solution of (2.22) on*

^{n}*R*

*.*

^{n}*Definition 2.3. The problem (2.22) is said to be aB-separable (orB*^{s}* _{p,θ}*(R

*;E)-separable) if the problem (2.22) for all*

^{n}*f*

*∈*

*B*

^{s}*(R*

_{p,θ}*;E) has a unique solution*

^{n}*u*

*∈*

*B*

^{s+l}*(R*

_{p,θ}*;E(A),*

^{n}*E) and*

*Au**B*^{s}* _{p,θ}*(R

*;E)+*

^{n}^{}

*|**α:l**|=*1

*D*^{α}*u*^{}_{B}*s*
*p,θ*

*R** ^{n}*;E

*≤*

*C*

*f*

*B*

^{s}*(R*

_{p,θ}*;E)*

^{n}*.*(2.23)

Consider the following parabolic Cauchy problem

*∂u(y,x)*

*∂y* + (L+*λ)u(y,x)**=**f*(y,*x),* *u(0,x)**=*0, *y**∈**R*+,*x**∈**R** ^{n}*, (2.24)
where

*L*is a realization diﬀerential operator in

*B*

^{s}*(R*

_{p,θ}*;E) generated by problem (2.22), that is,*

^{n}*D(L)**=**B*^{s+l}_{p,θ}^{}*R** ^{n}*;

*E(A),E*

^{},

*Lu*

*=*

*|**α:.l**|=*1

*a**α*(x)D^{α}*u*+*A(x)u*+ ^{}

*|**α:.l**|**<1*

*A**α*(x)D^{α}*u.* (2.25)
We say that the parabolic Cauchy problem (2.24) is said to be a maximal*B-regular,*
if for all *f* *∈**B*^{s}* _{p,θ}*(R

^{n+1}_{+};E) there exists a unique solution

*u*satisfying (2.24) almost every- where on

*R*

^{n+1}_{+}and there exists a positive constant

*C*independent on

*f*, such that it has the estimate

*∂u(y,x)*

*∂y*

*B*^{s}* _{p,θ}*(R

*+ ;E)+*

^{n+1}*Lu*

*B*

^{s}*(R*

_{p,θ}*+ ;E)*

^{n+1}*≤*

*C*

*f*

*B*

^{s}*(R*

_{p,θ}*+ ;E)*

^{n+1}*.*(2.26)

**3. Embedding theorems**

In this section we prove the boundedness of the mixed diﬀerential operators*D** ^{α}* in the
Besov-Lions type spaces.

*Lemma 3.1. LetAbe a positive operator in a Banach spaceE, letbbe a positive number,*
*r**=*(r1,r2,. . .,*r**n**),α**=*(α1,*α*2,. . .,α*n**), andl**=*(l1,*l*2,. . .,l*n**), whereϕ**∈*(0,π],*r**k**∈*[0,b],*l**k*

*are positive andα*_{k}*,k**=*1, 2,. . .,n, are nonnegative integers such thatκ*= |*(α+*r) :l**| ≤**1.*

*For 0< h**≤**h*0*<**∞**and 0**≤**μ**≤*1*−*κ*the operator-function*

Ψ(ξ)*=*Ψ*h,μ*(ξ)*=**ξ*1^{r}^{1}*ξ*2^{r}^{2},*. . .,*^{}*ξ*_{n}^{}^{r}* ^{n}*(iξ)

^{α}*A*

^{1}

^{−}^{κ}

^{−}

^{μ}*h*

^{−}

^{μ}^{}

*A*+

*η(ξ)*

^{}

^{−}^{1}(3.1)

*is a bounded operator inEuniformly with respect toξandh, that is, there is a constantC*

_{μ}*such that*

Ψ*h,μ*(ξ)^{}_{L(E)}*≤**C** _{μ}* (3.2)

*for allξ**∈**R*^{n}*, where*

*η**=**η(ξ)**=*
*n*
*k**=*1

*ξ**k*^{l}* ^{k}*+

*h*

^{−}^{1}

*.*(3.3)

*Proof. Since**−**η(ξ*)*∈**S(ϕ), for allϕ**∈*(0,π] and*A*is a*ϕ-positive inE, then the operator*
*A*+*η(ξ) is invertiable inE. Let*

*u**=**h*^{−}^{μ}^{}*A*+*η(ξ)*^{}^{−}^{1}*f .* (3.4)

Then

Ψ(ξ)*f*^{}_{E}*=*(hA)^{1}^{−}^{κ}^{−}^{μ}*u*^{}_{E}*h*^{−}^{(1}^{−}^{μ)}^{}*h*^{1/l}^{1}*ξ*1^{α}^{1}^{+r}^{1},. . .,^{}*h*^{1/l}^{n}*ξ**n*^{α}^{n}^{+r}^{n}*.* (3.5)
Using the moment inequality for powers of positive operators, we get a constant*C**μ*de-
pending only on*μ*such that

Ψ(ξ)*f*^{}_{E}*≤**C**μ**h*^{−}^{(1}^{−}^{μ)}*hAu*^{1}^{−}^{κ}^{−}^{μ}*u*^{κ+μ}*h*^{1/l}^{1}*ξ*1^{α}^{1}^{+r}^{1},. . .,^{}*h*^{1/l}^{n}*ξ**n*^{α}^{n}^{+r}^{n}*.* (3.6)
Now, we apply the Young inequality, which states that*ab**≤**a*^{k}^{1}*/k*1+*b*^{k}^{2}*/k*2for any positive
real numbers*a,b*and*k*1,*k*2with 1/k1+ 1/k2*=*1 to the product

*hAu*^{1}^{−}^{κ}^{−}^{μ}

*u*^{κ+μ}*h*^{1/l}^{1}*ξ*1^{α}^{1}^{+r}^{1},*. . .,*^{}*h*^{1/l}^{n}*ξ**n*^{α}^{n}^{+r}^{n}

(3.7)
with*k*1*=*1/(1*−*κ*−**μ),k*2*=*1/(κ+*μ) to get*

Ψ(ξ)*f*^{}_{E}*≤**C*_{μ}*h*^{−}^{(1}^{−}^{μ)}^{}(1*−*κ*−**μ)*^{}*hAu*^{}

+(κ+μ)^{}*h*^{1/l}^{1}^{}*ξ*1^{(α}^{1}^{+r}^{1}^{)/(}^{κ}^{+μ)},. . .,^{}*h*^{1/l}^{n}^{}*ξ**n*^{(α}^{n}^{+r}^{n}^{)/(}^{κ}^{+μ)}*u*
*.*
(3.8)
Since

*n*
*i**=*1

*α**i*+*r**i*

(κ+*μ)** ^{=}*
1
κ+

*μ*

*n*
*i**=*1

*α**i*+*r**i*

*l*_{i}* ^{=}*
κ

κ+*μ** ^{≤}*1, (3.9)

there exists a constant*M*0independent on*ξ, such that*
*ξ*1^{(α}^{1}^{+r}^{1}^{)/(}^{κ}^{+μ)},*. . .,*^{}*ξ*_{n}^{}^{(α}^{n}^{+r}^{n}^{)/(}^{κ}^{+μ)}*≤**M*0

1 +

*n*
*k**=*1

*ξ*_{k}^{}^{l}^{k}

(3.10)
for all*ξ**∈**R** ^{n}*. Substituting this on the inequality (3.8) and absorbing the constant coeﬃ-
cients in

*C*

*, we obtain*

_{μ}*ψ(ξ*)*f*^{}*≤**C**μ*

*h*^{μ}

*Au*+

*n*
*k**=*1

*ξ**k*^{l}^{k}*u*

+*h*^{−}^{(1}^{−}^{μ)}*u*

*.* (3.11)

Substituting the value of*u*we get
*ψ(ξ)f*^{}*≤**C*_{μ}*h*^{μ}

*A*^{}*A*+*η(ξ)*^{}^{−}^{1}*f*^{}+
*n*
*k**=*1

*ξ*_{k}^{}^{l}^{k}^{}*A*+*η(ξ*)^{}^{−}^{1}*f*^{}

+*h*^{−}^{(1}^{−}^{μ)}^{}

*A*+*η(ξ)*^{}^{−}^{1}*f*^{}*.*

(3.12)

By using the properties of the positive operator*A*for all*f* *∈**E*we obtain from (3.12)

Ψ(ξ)*f*^{}_{E}*≤**C**μ**f**E**.* (3.13)

*Lemma 3.2. LetEbe a UMD space with l.u.st.,p**∈*(1,*∞**),θ**∈*[1,*∞**] and let for allk,j**∈*
(1,n)

*s*_{k}

*l**k*+*s**k*+ *s**j*

*l**j*+*s**j* *≤*1. (3.14)

*Then the spacesB*^{l+s}* _{p,θ}*(R

*;E) and*

^{n}*W*

^{l}*B*

^{s}*(R*

_{p,θ}*;*

^{n}*E) are coincided.*

*Proof. In the first step we show that the continuous embeddingW*^{l}*B*^{s}* _{p,θ}*(R

*;*

^{n}*E)*

*⊂*

*B*

^{l+s}*(R*

_{p,θ}*;*

^{n}*E) holds, that is, there is a positive constantC*such that

*u*_{B}^{l+s}_{p,θ}_{(R}^{n}_{;E)}*≤**C**u**W*^{l}*B*^{s}* _{p,θ}*(R

*;E) (3.15) for all*

^{n}*u*

*∈*

*W*

^{l}*B*

^{s}*(R*

_{p,θ}*;E). For this aim by using the Fourier-analytic definition of an*

^{n}*E-*valued Besov space and the space

*W*

^{l}*B*

^{s}*(R*

_{p,θ}*;E) it is suﬃcient to prove the following estimate:*

^{n}
*F*^{−}^{1}

*n*
*k**=*1

*t*^{κ}^{k}^{−}^{l}^{k}^{−}^{s}^{k}^{}1 +^{}*ξ**k*^{κ}^{k}

*e*^{−}^{t}^{|}^{ξ}^{|}^{2}*Fu*^{}

*L**θ**p*

*≤**C*^{}*F*^{−}^{1}
*n*
*k**=*1

*t*^{κ}^{k}^{−}^{s}^{k}^{}1 +^{}*ξ**k*^{κ}^{k}

*e*^{−}^{t}^{|}^{ξ}^{|}^{2}*Fυ*^{}

*L**θ**p*

, (3.16) where

*L**θ p**=**L*^{∗}_{θ}^{}*L**p*

*R** ^{n}*;

*E*

^{},

*υ*

*=*

*F*

^{−}^{1}

1 +
*n*
*k**=*1

*ξ*_{k}^{l}^{k}

*Fu.* (3.17)

To see this, it is suﬃcient to show that the function
*φ(ξ*)*=*

*n*
*k**=*1

1 +^{}*ξ**k*^{l}^{k}^{+s}^{k}^{+δ}^{n}

*k**=*1

1 +^{}*ξ**k*^{s}^{k}^{+δ}^{−}^{1}
1 +

*n*
*k**=*1

*ξ**k*^{l}^{k}*−*1

, *δ >*0 (3.18)
is Fourier multiplier in*L**p*(R* ^{n}*;E). It is clear to see that for

*β*

*∈*

*U*

*n*and

*ξ*

*∈*

*V*

*n*

*ξ*1^{β}^{1}*ξ*2^{β}^{2},. . .,^{}*ξ**n*^{β}^{n}*D*^{β}*φ(ξ)*^{}_{L(E)}*≤**C.* (3.19)
Then in view of [41, Proposition 3] we obtain that the function*φ*is Fourier multiplier in
*L** _{p}*(R

*;*

^{n}*E).*

In the second step we prove that the embedding*B*^{l+s}* _{p,θ}*(R

*;E)*

^{n}*⊂*

*W*

^{l}*B*

^{s}*(R*

_{p,θ}*;E) is contin- uous. In a similar way as in the first step we show that for*

^{n}*s*

_{k}*/(l*

*+*

_{k}*s*

*) +*

_{k}*s*

_{j}*/(l*

*+*

_{j}*s*

*)*

_{j}*≤*1 the function

*ψ(ξ)**=*
_{n}

*k**=*1

1 +^{}*ξ*_{k}^{}^{s}^{k}^{+δ}^{}1 +
*n*
*k**=*1

*ξ*_{k}^{}^{l}^{k}_{n}

*k**=*1

1 +^{}*ξ*_{k}^{}^{l}^{k}^{+s}^{k}^{+δ}^{}

*−*1

(3.20)
is Fourier multiplier in*L**p*(R* ^{n}*;E). So, we obtain for all

*u*

*∈*

*B*

^{l+s}*(R*

_{p,θ}*;E) the estimate*

^{n}
*F*^{−}^{1}

*n*
*k**=*1

*t*^{κ}^{k}^{−}^{s}^{k}^{}1 +^{}*ξ**k*^{κ}* ^{k}*
1 +

*n*
*k**=*1

*ξ*_{k}^{l}^{k}

*e*^{−}^{t}^{|}^{ξ}^{|}^{2}*Fu*
*L**θ**p*

*≤**C*^{}*F*^{−}^{1}
*n*
*k**=*1

*t*^{κ}^{k}^{−}^{l}^{k}^{−}^{s}^{k}^{}1 +^{}*ξ**k*^{κ}^{k}

*e*^{−}^{t}^{|}^{ξ}^{|}^{2}*Fu*^{}

*L**θ**p*

*.*

(3.21)

It implies the second embedding. This completes the prove ofLemma 3.2.

*Theorem 3.3. Suppose the following conditions hold:*

(1)*Eis a UMD space with l.u.st. satisfying theB-multiplier condition with respect to*
*p,q**∈*(1,*∞**),θ**∈*[1,*∞**], ands**=*(s1,s2,. . .,s*n**), wheres**k**are positive numbers;*

(2)*α**=*(α1,*α*2,. . .,α_{n}*),l**=*(l1,l2,. . .,l_{n}*), whereα*_{k}*are nonnegative,l*_{k}*are positive integers,*
*ands**k* *such thats**k**/(l**k*+*s**k*) +*s**j**/(l**j*+*s**j*)*≤**1 fork,j**=*1, 2,. . .,*nand 0**≤**μ**≤*1*−*κ*,*κ*=*

*|*(α+ 1/ p*−*1/q) :*l**|**;*

(3)*Ais aϕ-positive operator inE, whereϕ**∈*(0,π] and 0*< h**≤**h*0*<**∞**.*
*Then the following embedding*

*D*^{α}*B*^{l+s}_{p,θ}^{}*R** ^{n}*;E(A),

*E*

^{}

*⊂*

*B*

_{q,θ}

^{s}^{}

*R*

*;E*

^{n}^{}

*A*

^{1}

^{−}^{κ}

^{−}

^{μ}^{}(3.22)

*is continuous and there exists a positive constantC*

_{μ}*depending only onμ, such that*

*D*^{α}*u*^{}_{B}*s*

*q,θ*(R* ^{n}*;E(A

^{1}

^{−}^{κ}

^{−}*))*

^{μ}*≤*

*C*

_{μ}^{}

*h*

^{μ}*u*

_{B}

^{l+s}*p,θ*(R* ^{n}*;E(A),E)+

*h*

^{−}^{(1}

^{−}

^{μ)}*u*

*B*

^{s}*(R*

_{p,θ}*;E)*

^{n}(3.23)

*for allu**∈**B*^{l+s}* _{p,θ}*(R

*;E(A),E).*

^{n}