Throughout the paper, we set E a Banach space and . denotes the space of all strongly measurable E-valued functions that are defined on Ω with the norm
(bounded mean oscillation, see [25, 26]) is the space of all E-valued local integrable functions with the norm
where B ranges in the class of the balls in , is the Lebesgue measure of B and is the average .
For and , we set
where B ranges in the class of balls with radius ρ.
We will say that a function is in if . We will call the VMO modulus of f.
Note that if , where C is the set of complex numbers, then and coincide with John-Nirenberg class BMO and Sarason class VMO, respectively.
The Banach space E is called a UMD-space if the Hilbert operator
is bounded in , (see, e.g., [27]). UMD spaces include, e.g., , spaces and Lorentz spaces , .
Let
A linear operator A is said to be φ-positive (or positive) in a Banach space E with bound if is dense on E and
for , , I is an identity operator in E and is the space of bounded linear operators in E. Sometimes will be written as and denoted by . It is known [[28], §1.15.1] that there exist fractional powers of the positive operator A. Let denote the space with the graphical norm
Let and be two Banach spaces. A set is called R-bounded (see [14, 23]) if there is a positive constant C such that for all and , ,
where is a sequence of independent symmetric -valued random variables on .
Let denote the Schwartz class, i.e., the space of all E-valued rapidly decreasing smooth functions on . Let F denote the Fourier transformation. A function is called a Fourier multiplier from to if the map , is well defined and extends to a bounded linear operator
The set of all multipliers from to will be denoted by . For , it will be denoted by .
Let
Definition 1 A Banach space E is said to be a space satisfying a multiplier condition if for any , the R-boundedness of the set implies that Ψ is a Fourier multiplier in , i.e., for any .
Definition 2 The φ-positive operator A is said to be an R-positive in a Banach space E if there exists such that the set
is R-bounded.
A linear operator is said to be positive in E uniformly in x if is independent of x, is dense in E and
for all , .
Let denote the space of all compact operators from to . For , it is denoted by . Assume and E are two Banach spaces and is continuously and densely embedded into E. Let m be a natural number. (the so-called Sobolev-Lions type space) denotes a space of all functions possessing the generalized derivatives such that is endowed with the norm
For the space will be denoted by . It is clear to see that
Let s be a positive parameter. We define in the following parameterized norm:
Function satisfying equation (1) a.e. on is said to be a solution of the problem (1) on .
From [21] we have the following theorem.
Theorem A1 Suppose the following conditions are satisfied:
-
(1)
E is a Banach space satisfying the multiplier condition with respect to and A is an R-positive operator in E;
-
(2)
are n-tuples of nonnegative integer numbers such that
-
(3)
is a region such that there exists a bounded linear extension operator from to .
Then the embedding
is continuous and there exists a positive constant
such that
for all and .
Theorem A2 Suppose all conditions of Theorem A1 are satisfied. Assume Ω is a bounded region in and . Then, for , the embedding
is compact.
In a similar way as in [[2], Theorem 2.1], we have the following result.
Lemma A1 Let E be a Banach space and . The following conditions are equivalent:
-
(1)
;
-
(2)
f is in the BMO closure of the set of uniformly continuous functions which belong to VMO;
-
(3)
.
For , , , consider the commutator operator
Proof Indeed, we observe that if with VMO modulus η, there exists a constant C such that for so that the E-valued usual mollifiers converge to f in the BMO norm. More precisely, given with VMO modulus , we can find a sequence of E-valued functions converging to f in E-valued BMO spaces as with VMO moduli such that . In a similar way, other cases are derived. □
From [[26], Theorem 1] and [[29], Corollary 2.7], we have the following.
Theorem A3 Let E be a UMD space and . Then is a bounded operator in , .
From Theorem A3 and the property (2) of Lemma A1, we obtain, respectively:
Theorem A4 Assume all conditions of Theorem A3 are satisfied. Also, let and let η be the VMO modulus of a. Then, for any , there exists a positive number such that
Theorem A5 Let E be a UMD space, and uniformly R-positive in E. Moreover, let , . Then the following commutator operator is bounded in :
Note that singular integral operators in E-valued spaces were studied, e.g., in [30].
Theorem A6 Assume all conditions of Theorem A5 are satisfied and η is a VMO modulus of .
Then, for any , there exists a positive number such that
Consider the nonlocal BVP for parameter-dependent DOE with constant coefficients
where , a, , are complex numbers, , , s is a positive and λ is a complex parameter; and A is a linear operator in E. Let be roots of the equation , be a 2m-dimensional matrix and be a determinant of the matrix , where
It is known that (see, e.g., [[24], §1.15]) if the operator A is φ-positive in E, then operators , generate the following analytic semigroups:
Let
From [[19], Theorem 1] and [[22], Theorem 3.2], we obtain the following.
Theorem A7 Assume the following conditions are satisfied:
-
(1)
E is a Banach space satisfying the multiplier condition with respect to ;
-
(2)
A is an R-positive operator in E for and ;
-
(3)
, , , and .
Then
-
(1)
for , , and for sufficiently large , the problem (2) has a unique solution . Moreover, the following coercive uniform estimate holds:
-
(2)
For , the solution is represented as
(3)
where are uniformly bounded operators in E and
where
Consider the BVP for DOE with variable coefficients
where is a complex-valued function, , , are complex numbers, s is a positive and λ is a complex parameter, and is a linear operator in E.
Let , be roots of the equation , be a 2m-dimensional matrix and be a determinant of the function matrix , where
In the next theorem, we consider the case when principal coefficients are continuous. The well-posedness of this problem occurs in studying of equations with VMO coefficients. From [[19], Theorem 3] and [[22], Theorem 3.2], we get the following.
Theorem A8 Suppose the following conditions are satisfied:
-
(1)
E is a Banach space satisfying the multiplier condition with respect to ;
-
(2)
, , , and a.e. ;
-
(3)
, and for a.e. ;
-
(4)
is a uniformly R-positive operator in E and
Then, for , and for sufficiently large , there is a unique solution of the problem (4). Moreover, the following coercive uniform estimate holds: