On Integral Operators with Operator Valued Kernels

Here Lq-Lp boundedness of integral operator with operator-valued kernels is studied and the main result is applied to convolution operators. Using these results Besov space regularity for Fourier multiplier operator is established.


Introduction
It is well-known that, solutions of inhomogeneous differential and integral equations are represented by integral operators. In order to investigate stability properties of these problems it is important to have boundedness of corresponding integral operators in the studied function spaces. For instance, the boundedness of Fourier multiplier operators play crucial role in the theory of linear PDE especially in the study of maximal regularity for elliptic and parabolic PDE. For exposition of the integral operators with scalar valued kernels see [3] and for the application of multiplier theorems see [2] .
Maria Girardi and Lutz Weis [4] recently proved that integral operator k(t, s)x Y dµ(t) ≤ C 1 x X for all x ∈ X, sup t∈T S k * (t, s)y * X * dν(s) ≤ C 2 y * Y * for all y * ∈ Y * are satisfied. Inspired from [4] we will show that (1) defines a bounded linear operator K : L q (S, X) → L p (T, Y ) if the kernel k : T × S → B(X, Y ) satisfies the following conditions . Here X and Y are Banach spaces over the field C and X * is the dual space of X. The space B(X, Y ) of bounded linear operators from X to Y is endowed with the usual uniform operator topology. Now let us state some important notations from [4] . A subspace Y of X * τ -norms X, where τ ≥ 1, provided It is clear that if Y τ -norms X then the canonical mapping Let (T, T , µ) and (S, S , ν) be positive measure spaces and ε(S, X) will denote the space of finitely-valued and finitely supported measurable functions from S into X i.e.

, Proposition 2.2] and [4, Lemma 2.3]) .
A vector valued function f : * given by Often [E(X)] * = E * (X * ), for example, provided X has the Radon-Nikodym property. Let us remind an important fact that if X is reflexive or if X is separable, then X has the Radon-Nikodym property.

L q → L p estimates for Integral Operators
In this section we identify conditions on operator-valued kernel k : T × S → B(X, Y ), extending theorems in [4] so that To prove our main result we shall use interpolation theorems of L p spaces. Therefore, we will study L 1 (S, X) → L θ (T, Y ) and L θ ′ (S, X) → L ∞ (T, Y ) boundedness of integral operator (1). The following two conditions are natural measurability assumptions on k : T × S → B(X, Y ).
Note that if k satisfies the above condition then for each f ∈ ε(S, X), there is Theorem 2.3. Suppose 1 ≤ θ < ∞ and the kernel k : T × S → B(X, Y ) satisfies Condition 2.1 and Condition 2.2. Then integral operator (1) acting on ε(S, X) extends to a bounded linear operator x i 1 Ai (s) ∈ ε(S, X) be fixed. Taking into account the fact that 1 ≤ θ, using the general Minkowski-Jessen inequality and assumptions of the theorem we obtain Theorem 2.5. Let Z be a subspace of Y * that τ -norms Y. Suppose 1 ≤ θ < ∞ and k : T × S → B(X, Y ) satisfies Condition 2.1 and Condition 2.4. Then integral operator (1) acting on ε(S, X) extends to a bounded linear operator Proof. Suppose f ∈ ε(S, X) and y * ∈ Z are fixed. Let T f , T y * ∈ full T be corresponding sets due to Condition 2.1 and Condition 2.4. If t ∈ T f ∩ T y * then by using Hölder's inequality and assumptions of the theorem we get .
In [4, Lemma 3.9] authors slightly improved interpolation theorem [1, Thm 5.1.2] . The next lemma is a more general form of [4,Lemma 3.9] . Lemma 2.6. Suppose a linear operator Then, for 1 q − 1 p = 1 − 1 θ and 1 ≤ q < θ θ−1 ≤ ∞ the mapping K extends to a bounded linear operator Proof. Let us first consider conditional expectation operator Hence, by Riesz-Thorin theorem ([1, Thm 5.1.2]) we have Now, taking into account (3) and using the same reasoning as in the proof of [4, Lemma 3.9] one can easily show assertion of this lemma.
Theorem 2.7. (Operator-valued Schur's test) Let Z be a subspace of Y * that τ -norms Y and 1 Proof. Combining Theorem 2.3, Theorem 2.5 and Lemma 2.6 we obtain assertion of the theorem.
Remark 2.8. Note that choosing θ = 1 we get original results in [4] . For L ∞ estimates (it is more delicate and based on ideas from geometry Banach spaces) and weak continuity and duality results see [4] . The next corollary plays important role in the Fourier Multiplier theorems. Corollary 2.9. Let Z be a subspace of Y * that τ -norms Y and 1 It is easy to see that k : R n → B(X, Y ) satisfies Condition 2.1, Condition 2.2 and Condition 2.4 with respect to Z. Thus, assertion of corollary follows from Theorem 2.7.

Fourier Multipliers of Besov spaces
In this section we shall indicate importance of Corollary 2.9 in the theory of Fourier multipliers (FM). Thus we give definition and some basic properties of operator valued FM and Besov spaces.
Let us define the partition of unity {ϕ k } k∈N0 of functions from S(R n , R). Suppose ψ ∈ S(R, R) is a nonnegative function with support in [2 −1 , 2], which satisfies Let 1 ≤ q ≤ r ≤ ∞ and s ∈ R. The Besov space is the set of all functions f ∈ S ′ (R n , X) for which is finite; here q and s are main and smoothness indexes respectively. The Besov space has significant interpolation and embedding properties: where m ∈ N and C s (X) denotes the Holder-Zygmund spaces.
Definition 3.1. Let X be a Banach space and 1 ≤ u ≤ 2. We say X has Fourier Definition 3.3. Let (E 1 (R n , X), E 2 (R n , Y )) be one of the following systems, where 1 ≤ q ≤ p ≤ ∞ (L q (X), L p (Y )) or (B s q,r (X), B s p,r (Y )). A bounded measurable function m : R n → B(X, Y ) is called a Fourier multiplier from E 1 (X) to E 2 (Y ) if there is a bounded linear operator (6) The uniquely determined operator T m is the FM operator induced by m. Note that if T m ∈ B(E 1 (X), E 2 (Y )) and T * m maps E * 2 (Y * ) into E * 1 (X * ) then T m satisfies the weak continuity condition (6).
For definition of Besov spaces and its basic properties we refer to [5].
Since (5) can be written in the convolution form Corollary 2.9 and Corollary 3.2 can be applied to obtain L q (R n , X) → L p (R n , Y ) regularity for (5).
Theorem 3.4. Let X and Y be Banach spaces having Fourier type u ∈ [1, 2] and p, q ∈ [1, ∞] so that 0 ≤ 1 q − 1 p ≤ 1 u . Then there is a constant C depending only on F u,n (X) and F u,n (Y ) so that if : a > 0 .
choosing appropriate a and using (4) we obtain S (B (Y * , X * )) and M u (m) = M u (m * ). Thus, in a similar manner as above, we get for some constant C 2 depending on F u,n (X * ). Hence by (8-9) and Corollary 2.9 for all p, q ∈ [1, ∞] so that 0 ≤ 1 q − 1 p ≤ 1 u . Now, taking into account the fact that S (B (X, Y )) is continuously embedded in B (B(X, Y )) and using the same reasoning as [5,Theorem 4.3] one can easily prove the general case m ∈ B and weak continuity of T m .
Theorem 3.5. Let X and Y be Banach spaces having Fourier type u ∈ [1, 2] and p, q ∈ [1, ∞] are so that 0 ≤ 1 q − 1 p ≤ 1 u . Then, there exist a constant C depending only on F u,n (X) and F u,n (Y ) so that if m : R n → B(X, Y ) satisfy then m is a FM from B s q,r (R n , X) to B s p,r (R n , Y ) and T m B s q,r →B s p,r ≤ CA for each s ∈ R and r ∈ [1, ∞].
Taking into consideration the Theorem 3.4 one can easily prove above theorem in a similar manner as [5,Theorem 4.3].
The following corollary provides a practical sufficient condition to check (10).
Using the fact that W l u (R n , B(X, Y )) ⊂ B n( 1 u + 1 p − 1 q ) u,1 (R n , B(X, Y )), the above lemma can be proven in similar fashion as [5, Lemma 4.10].
Choosing θ = ∞ in the Lemma 3.6 we get the following corollary: then m is a FM from B s q,r (R n , X) to B s p,r (R n , Y ).