Stability of the inverses of interpolated operators with application to the Stokes system

We study the stability of isomorphisms between interpolation scales of Banach spaces, including scales generated by well-known interpolation methods. We develop a general framework for compatibility theorems, and our methods apply to general cases. As a by-product we prove that the interpolated isomorphisms satisfy uniqueness-of-inverses. We use the obtained results to prove the stability of lattice isomorphisms on interpolation scales of Banach function lattices and demonstrate their application to the Calderón product spaces as well as to the real method scales. We also apply our results to prove solvability of the Neumann problem for the Stokes system of linear hydrostatics on an arbitrary bounded Lipschitz domain with a connected boundary in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document}, n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 3$$\end{document}, with data in some Lorentz spaces Lp,q(∂Ω,Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p,q}(\partial \Omega , \mathbb {R}^n)$$\end{document} over the set ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document} equipped with a boundary surface measure.


Introduction
The study of a special class of operators by Fredholm [22], in his research on integral operators, initiated the theory of Fredholm operators and the theory of Riesz operators. These theories were developed in inseparable connection with several other aspects of the Banach space theory of operators. In particular, the study of the local spectral theory of operators, including the study of the decay of the eigenvalues of operators on Banach spaces. It is also important to note that the theory of Fredholm operators found applications in problems of partial differential equations. The study of stability properties of interpolated operators is a central task in abstract interpolation theory. This is motivated by applications in many areas of analysis, including PDE's. It is worth noting here that the Fredholmness property is not stable under interpolation in general, however, it is known that it is locally stable for the real method and the complex method of interpolation.
We point out that in our recent paper [8], it is shown that stability of Fredholm properties of interpolated operators can be proved in a non-trivial way via the stability of isomorphisms for some class of interpolation methods. This result motivates the aim of this paper, which is to develop a very general framework on stability and the local uniqueness-of-inverse properties of interpolated isomorphisms acting between interpolated Banach spaces. We will be mainly interested in the most general cases of abstract scales of interpolation spaces. The idea to prove general results is motivated by useful applications to the solvability of partial differential equations lifted from some particular cases, such as the Hilbert space L 2 or L p -spaces, to a more general class of spaces. In the last section we will give applications of our results to the solvability of the Neumann problem for the Stokes system of linear hydrostatics.
In order to discuss some results and ideas in more detail, we introduce some notations used in the paper. As usual, we let [ · ] θ stand for the complex method of interpolation. The real method is denoted by ( · ) θ,q with θ ∈ (0, 1) and q ∈ [1, ∞]. For basic notation of interpolation theory, we refer to [9] and [10].
If a Banach space X is intermediate with respect to a Banach couple X we denote by X 0 the closure of X 0 ∩ X 0 in X . We recall that a mapping F : B → B, from the category B of all couples of Banach spaces into the category B of all Banach spaces is said to be an interpolation functor (or method) if, for any couple X := (X 0 , X 1 ), the Banach space F(X 0 , X 1 ) is intermediate with respect to X (i.e., X 0 ∩ X 1 ⊂ F( X ) ⊂ X 0 + X 1 ), and T : F(X 0 , X 1 ) → F(Y 0 , Y 1 ) for all T : (X 0 , X 1 ) → (Y 0 , Y 1 ); here, as usual, the notation T : (X 0 , X 1 ) → (Y 0 , Y 1 ) means that T : X 0 + X 1 → Y 0 + Y 1 is a linear operator, such that the restrictions of T to the space X j is a bounded operator from X j to Y j , for both j = 0 and j = 1. An interpolation functor F is said to be regular on a Banach couple X whenever F( X ) 0 = F( X ) and is said to be regular if it is regular on any Banach couple.
An operator T : (X 0 , X 1 ) → (Y 0 , Y 1 ) between Banach couples is said to be invertible whenever the restriction T | X j : X j → Y j is invertible (i.e., T is an isomorphism of X j onto Y j ) for each j ∈ {0, 1}. We point out that in what follows we will often omit the domain of the restricted operator.
Notice that by the closed graph theorem, for any Banach couples X and Y , If C may be chosen independently of X and Y , then F is called a bounded (more precisely C-bounded) interpolation functor and it is called exact if C = 1. All interpolation functors considered in this paper will be bounded. The roots of stability properties of interpolated operators are related to the remarkable theorem of Shneiberg [39] which states that, if T : (X 0 , X 1 ) → (Y 0 , Y 1 ) is a bounded linear operator between compatible couples of complex Banach spaces, then the set c of all θ ∈ (0, 1), for which the operator T :

invertible between Calderón interpolation spaces is open, and so it is the union of countably many disjoint open intervals of invertibility of T
For the real method ( · ) θ,q of interpolation with θ ∈ (0, 1) and q ∈ [1, ∞], it is known (see [40] for q < ∞ and [30] for q = ∞) that, for a fixed q ∈ [1, ∞], the set q of all θ ∈ (0, 1) for which the operator T : (X 0 , X 1 ) θ,q → (Y 0 , Y 1 ) θ,q is invertible is open. Thus, q is the union of countably many disjoint open intervals of invertibility of T , q := j∈J (a j,q , b j,q ). Note also that in the article [5] it is shown that the set q does not depend on q, and therefore intervals (a j,q , b j,q ) do not depend on q.
It is known that, in general, the real and complex methods yield different spectra of interpolated operators, due to the fact that there exist a Banach couple (X 0 , X 1 ) and an operator T : (X 0 , X 1 ) → (X 0 , X 1 ), such that the operator T : (X 0 , X 1 ) θ,1 → (Y 0 , Y 1 ) θ,1 is invertible, but T : [X 0 , X 1 ] θ → [X 0 , X 1 ] θ is not invertible (see [1,Example 12]). From the main result of this paper it follows that, if the operator T : [X 0 , X 1 ] θ → [X 0 , X 1 ] θ is invertible for some θ ∈ (0, 1), then T : (X 0 , X 1 ) θ,q → (Y 0 , Y 1 ) θ,q is invertible for all q ∈ [1, ∞]. This implies that for any interval of invertibility (a i , b i ) ⊂ c of an operator T , acting between complex interpolation spaces, there exists an interval (a j,q , b j,q ) ⊂ q of T acting between the real interpolation spaces, such that (a i , b i ) ⊂ (a j,q , b j,q ).
The main purpose of this work is to provide a unified general approach to abstract compatibility theorems of stronger type than the Albrecht-Müller result for operators between Banach spaces generated by abstract interpolation methods. We introduce a key notion, that of a locally stable family of interpolation functors {F θ } θ∈(0,1) (for an exact definition we refer to Sect. 4), and we prove that a certain class of interpolation methods introduced by Cwikel-Kalton-Milman-Rochberg in [16] are locally stable. In particular, the Calderón complex family {[ · ] θ } θ∈(0,1) as well as the Lions-Peetre real family of interpolation functors {( · ) θ,q } θ∈(0,1) for all q ∈ [1, ∞] are locally stable. Moreover, this unified general approach allows us to find conditions under which the subtle compatibility result holds. Under these conditions (see Theorem 4.7) we prove that if U ⊂ (0, 1) is an open interval of invertibility of T (i.e., such that T θ is invertible for all θ ∈ U ), then for any θ , θ ∈ U the inverse operators T −1 θ and T −1 We also show that the complex method, {[ · ] θ } θ∈(0, 1) , and the real method, {( · ) θ,q } θ∈(0,1) , for any 1 ≤ q ≤ ∞ satisfy the conditions of Theorem 4.7. In particular, from this it follows that the statement of Theorem 8.1 in the mentioned above paper [27] is correct.
It is worth pointing out that among several motivations for studying compatibility problems, there are important applications to PDE's. The roots for these problems are in Calderón's paper [13] in which it is proved that, if ( , A, μ) is a σ -finite measure space and T : L p (μ) → L p (μ) is a bounded operator for 1 < p < ∞, which is invertible for p = 2, then T is also invertible when 2 − ε < p < 2 + ε, for some small ε > 0. In fact, careful analysis of Calderón's proofs gives the compatibility of inverses, that is, there exists some small ε > 0 such that for all p, q ∈ (2 − ε, 2 + ε), the inverse T −1 considered on the space L p (μ) is compatible with T −1 considered on L q (μ), when both operators are restricted to L p (μ) ∩ L q (μ). In [36] a very useful application was given, to solvability of the Dirichlet problem with data in L p (∂ ), for the equation u = 0 in , u = f and ∂u/∂n = g on ∂ , in a bounded Lipschitz domain ⊂ R n . In the work [27], by Kalton-Mayaboroda-Mitrea, compatibility results were used to variants of the Dirichlet problem as well as the Neumann problem for the Laplacian in L p (∂ )-spaces. We provide applications of our results. At first in Sect. 5, we study the stability of lattice isomorphisms on interpolation scales of Banach function lattices. We prove under mild assumptions a surprising result that for Banach function lattices generated by the Calderón products isomorphism on one space of the scale implies that it is also a lattice isomorphism on both all Claderón product spaces of the interior of scale and the scale of the real interpolation spaces.
In the last Sect. 6 of the paper, we combine results from Fabes, Kenig and Verchota [21], with our results on stability of Fredholm property of interpolated operators, to show the solvability of the Neumann problem for the Stokes system of linear hydrostatics on Lipschitz domain in R n , with boundary values in some vector-valued Lorentz space L p,q (∂ , R n ) over the boundary ∂ of , equipped with the boundary surface measure σ .
Throughout the paper we will use standard notation. As usual, for a given Banach space X we denote by L(X ) the Banach space of all bounded linear operators on X equipped with the uniform norm. If X and Y are Banach spaces such that X ⊂ Y and the inclusion map id : X → Y is bounded, then we write X → Y . We write X ∼ = Y whenever X = Y , with equality of norms.

Notation and preliminary results
We introduce the basic notations and definitions to be used throughout this work. We will use complex methods of interpolation introduced by Calderón in his fundamental paper [12].
Let S := {z ∈ C; 0 < Rez < 1} be an open strip in the plane. For a given θ ∈ (0, 1) and any couple X = (X 0 , X 1 ) we denote by F( X ) the Banach space of all bounded continuous functions f :S → X 0 + X 1 on the closureS that are analytic on S, and R t → f ( j + it) ∈ X j is a bounded continuous function, for each j ∈ {0, 1}, and endowed with the norm The lower complex interpolation space is defined by [ X ] θ := { f (θ ); f ∈ F( X )} and is endowed with the quotient norm. This definition is slightly different from those in [9,12], however it gives the same interpolation spaces (see, e.g., [12]). We recall that in the original definition it is required in addition that f ∈ F( X ) satisfies We also recall the basic constructions and results of [16] which we will use here, and we refer to this paper for more details. Let Ban be the class of all Banach spaces over the complex field. A mapping X : Ban → Ban is called a pseudolattice, or a pseudo-Z-lattice, if (i) For every B ∈ Ban the space X (B) consists of B-valued sequences {b n } := {b n } n∈Z modelled on Z ; (ii) Whenever A is a closed subspace of B it follows that X (A) is a closed subspace of X (B) ; (iii) There exists a positive constant C = C(X ) such that, for all A, B ∈ Ban and all bounded linear operators T : A → B and every sequence {a n } ∈ X (A), the sequence {T a n } ∈ X (B) and satisfies the estimate for each m ∈ Z and all {b n } ∈ X (B).
Important examples of pseudolattices are the Fourier spaces FC and FL 1 ; the space UC of unconditionally convergent series; the space WUC of weakly unconditionally convergent sequences. For more information on these spaces and their applications, we refer to Janson's article [25].
For every Banach couple B = (B 0 , B 1 ) and every Banach couple of pseudolattices Following [16], for every s in the annulus A := {z ∈ C; 1 < |z| < e}, we define the Banach space B X ,s to consist of all elements of the form b = n∈Z s n b n (convergence It is easy to check that the map B → B X ,s is an interpolation functor. We will consider mainly couples X = (X 0 , X 1 ) of Banach pseudolattices which are translation invariant, i.e., such that for any Banach space B we have for all {b n } ∈ X j (B), each k ∈ Z and j ∈ {0, 1}. Here and in what follows S denotes the left-shift operator on two-sided (vector-valued) sequences defined by S{b n } = {b n+1 }.
Following [16], X = (X 0 , X 1 ) is said to be a rotation-invariant Banach couple of pseudolattices whenever the rotation map {b n } → {e inτ b n } is an isometry of X j (B) onto itself for every real τ and every Banach space B.
Let X = (X 0 , X 1 ) be a couple of pseudolattices and let B = (B 0 , B 1 ) be a Banach couple. For our purposes it will be convenient to express a natural correspondence between elements in the space J ( X , B) and certain analytic functions defined on A with values on B 0 + B 1 . To see this we define the space F X ( B) to consist of all vector valued analytic functions f b : A → B 0 + B 1 , which has the Laurent series expansion given by B) is a Banach space, the uniqueness theorem for analytic functions The following useful lemma is obvious, but we include a proof. Lemma 2.1 Let X = (X 0 , X 1 ) be a Banach couple of rotation-invariant pseudolattices. Then, for every Banach couple B = (B 0 , B 1 ) and all s ∈ A, we have Our hypothesis yields Since f (s) = f (|s|) ∈ B X ,|s| , f (s) ∈ B X ,|s| and this proves (i).
Define f by f (z) = f (ze −iϕ ) for all z ∈ A. Our hypothesis gives that f ∈ F X ( B). Combining the above facts yields f (s) = f (|s|) = x and this proves (ii).
(iii). It is enough to observe that the proofs of (i) and (ii) yields We note that the above lemma shows if X = (X 0 , X 1 ) is a Banach couple of rotationinvariant pseudolattices, then for any s = e θ+iϕ with θ ∈ (0, 1) and ϕ ∈ [0, 2π), we have that B X ,s ∼ = B X ,e θ for any Banach couple B.
We point out that, concerning interpolation methods, the idea of [16] was to show that a large family of interpolation methods have a suitable complex analytic structure that could be used for methods that a priori do not seem to have one. This essential fact is used deeply in our paper. Note that with the right choices of pseudolattice couples (X 0 , X 1 ), we recover the classical methods of interpolation (see [16] for more details). In particular let s = e θ with 0 < θ < 1. If X 0 = X 1 = p with 1 ≤ p ≤ ∞, the space B X ,s coincides with the Lions-Peetre real J -method space B θ, p;J (see, e.g., [33, p. 41] where this space is denoted by s( p, θ, B 0 ; p, θ − 1, B 1 ).
It is well known that (B 0 , B 1 ) θ, p;J = (B 0 , B 1 ) θ, p up to equivalence of norms (see [9,Chap. 3] Here, as usual, for any Banach couple X = (X 0 , X 1 ) the Peetre K -functional is defined by Let X be a Banach space intermediate with respect to a Banach couple X = (X 0 , X 1 ).
The Gagliardo completion or relative completion of X with respect to X is the Banach space X c of all limits in X 0 + X 1 of sequences that are bounded in X and endowed with the norm x X c = inf{sup k≥1 x k X }, where the infimum is taken over all bounded sequences {x k } in X whose limit in X 0 + X 1 equals x. We will use the well-known fact (see [10,Lemma 2.2.21]) that for any Banach couple (X 0 , X 1 ) we have If X = (FC, FC), then B X ,s coincides, to within equivalence of norms, with the [15]). If X = (UC, UC), then B X ,s is the ± method space B θ ∼ = B 0 , B 1 θ (see [35, p. 176]). If we replace UC by WUC, we obtain the Gustavsson-Peetre variant of B 0 , B 1 θ which is denoted by B; θ (see [23, p. 45], [25]).

The uniqueness of inverses on the intersection of a couple
Throughout the paper, for an operator T : X → Y between Banach couples and every ω ∈ A, we often denote by T ω the restriction T | X X ,ω : X X ,ω → Y X ,ω . For simplicity of notation, we write T θ instead of T e θ for any θ ∈ (0, 1).
In the further presentation, δ denotes the function given in the annulus A by the formula We now state the main results of this section for operators between spaces generated by interpolation constructions described in the previous section. Theorem 3.1 Let X = (X 0 , X 1 ) be a Banach couple of translation-invariant pseudolattices and let T : X → Y be an operator between complex Banach couples. Assume that T : Moreover, the following upper estimate for the norm of T −1 ω holds: In the case when X = (X 0 , X 1 ) is a couple of translation-and rotation-invariant pseudolattices we obtain the following variant of the Albrecht-Müller result. Theorem 3.2 Let X = (X 0 , X 1 ) be a couple of translation-and rotation-invariant pseudolattices and let T : X → Y be an operator between complex Banach couples. Assume that T θ * : X X ,e θ * → Y X ,e θ * is invertible for some θ * ∈ (0, 1). Then where η(θ * ) = δ(e θ * ). Moreover, T −1 θ agrees with T −1 θ * on Y 0 ∩ Y 1 and for any θ ∈ I .
To prove this theorem we will need some preliminary results. We start with a more precise cancellation principle from [16] stated in the lemma below. Careful analysis of the proof of lemma 3.1 in [16] gives the required estimate with a constant depending on the parameter s ∈ A, but not on the couple of translation-invariant pseudolattices. We omit the proof here and for a detailed proof we refer to [7].
Let X = (X 0 , X 1 ) be a couple of pseudolattices and B = (B 0 , B 1 ) be a complex Banach couple. Now, we introduce special maps and spaces which will play an essential role. Given s ∈ A, the continuous map δ s : F X ( B) → B 0 + B 1 is given by The kernel of δ s is denoted by N s ( B). Clearly, the map δ s : In what follows we will apply a result from [30]. For the reader's convenience, we state this result. To do this we need to recall some fundamental definitions from the theory of distances between closed subspaces of Banach spaces.
Let U be a Banach space. For two given closed subspaces U 0 , Let U , V be Banach spaces and let U 0 , U 1 and V 0 , V 1 be closed subspaces of U and V , respectively. Let H be a linear bounded operator from U to V which maps U j to In what follows the next theorem is the crucial tool. The proof is a straightforward minor modification of the proof of Theorem 9 in [30].

Theorem 3.4 Suppose that H : U → V maps U j to V j for each j ∈ {0, 1}, and the quotient operator H
Let X be a Banach couple of pseudolattices, B a Banach couple, let "dist" be the distance defined on closed subspaces of the space F X ( B), and let s, ω ∈ A. Then we define The following variant of a result from [30] is relevant to our purposes.

Theorem 3.5 For all s, ω ∈ A one has
where the supremum is taken over all complex Banach couples B.
Proof Given a complex Banach couple B, we have In particular we have Now observe that and so Combining the above facts with the triangle inequality yields that, for all ω ∈ A, Since ε is arbitrary, we get and this completes the proof.
We are ready for the proof of Theorem 3.1.

Proof of Theorem 3.1
For ω ∈ A define the operator by the formula where T : Now we fix s ∈ A. Then, from Theorem 3.5, we conclude that for Combining the above with Theorem 3.4 applied to the Banach spaces To prove the estimate for the norm of T −1 ω for all ω ∈ W , we first observe that following the above notation, it follows from the equation (2) that H U →V = T X → Y and .
To finish the proof, we apply Theorem 3.4 to get the required norm estimate of T −1 ω for all ω ∈ W .
We isolate the following lemma for further reference.

Lemma 3.6
Let X = (X 0 , X 1 ) be a couple of pseudolattices and let Y be a Banach couple. Then, for every ω ∈ A, the operator V ω : is injective and has closed range with R( Proof We first remark that our hypothesis on X yields that a function Applying the uniqueness theorem for an analytic function in a domain gives f = 0 in A. It is obvious that the range satisfies Thus, we get that g = V ω f and so the desired equality We prove a lemma which will play a key role in the proof of the main result, Theorem 3.2. In the proof we will use some methods from [1,Theorem 4]. We recall that if S : X → Y is a bounded linear operator between Banach spaces, then, the so-called lower bound of S is defined by It is obvious that γ (S) > 0 if and only if S is injective and the range R(S) of S is a closed subspace in Y .

Proposition 3.7 Let X be a couple of pseudolattices and let X
Proof From Lemma 3.6, it follows that the injective operator is an analytic function.
We will adopt notation from Theorem 3.1. Thus, we will consider operators We note that Let c 1 and c be positive constants such that where T −1

It follows from Theorem 3.1 that there exists an open neighbourhood
We claim that an open neighborhood U ⊂ A of s given by satisfies the required statements, i.e., there exist analytic functions g : To see this fix k ∈ F X ( Y ) and observe that, if g(ω) = ∞ n=0 g n (ω − s) n and h(ω) = ∞ n=0 h n (ω − s) n are the Taylor expansions of g and h about s, then the solution of the required equation with g and h in the form given above reduces to the solution of the following recurrence equations generated by the sequences {g n } ⊂ F X ( X ) and {h n } ⊂ F X ( Y ) of Taylor's coefficients of g and h, respectively such that the series g(ω) = ∞ n=0 g n (ω − s) n and h(ω) = ∞ n=0 h n (ω − s) n converge in U .
Our hypothesis on the invertibility of T s : X X ,s → Y X ,s implies that Hence, for Clearly this yields (by T f 0 (s) = 0 and g 0 + N s ( and We claim that there exists To see this observe that, for all h ∈ F X ( Y ), we have According to Lemma 3.6, we can find (by Then by estimate (3) and (4), one has As a consequence, we deduce that the claim holds for h 0 .
Continuing the process, we construct sequences {g n } ⊂ F X ( X ) and {h n } ⊂ F X ( Y ) such that, for each n ∈ N we have This implies that the functions g : are analytic in U and satisfy the desired statement. This completes the proof. Now we are ready to prove Theorem 3.2.
Proof For a fixed y ∈ Y 0 ∩ Y 1 , let k be a constant function given by k(z) = y for all z ∈ A. Since k ∈ F X ( Y ), it follows from Proposition 3.7 that there exist an open neighborhood U ⊂ A of s and analytic functions g : such that, for all ω ∈ U and all z ∈ A, we have Define a function g : U → X 0 + X 1 by Then g is analytic in U and T ( g(ω)) = y by the above formula. Further, In particular, this implies that the analytic function g is constant on an open arc of the circle with the center at 0 and radius |s| which is contained in U . Thus g is constant in U by the uniqueness theorem. Hence T −1 ω y is independent of ω ∈ U . Combining the obvious inequality, with norm estimates of inverse operators given in Theorem 3.1 gives the desired conclusion about the invertibility of T θ :

The uniqueness of inverses on the intersection of interpolated Banach spaces
The main result of Sect. 3, Theorem 3.2, motivates a natural question related to uniqueness of inverses between interpolated spaces in an abstract setting. Before we formulate the question we introduce a key definition. A family {F θ } θ∈(0,1) of interpolation functors is said to be locally stable if for any Banach couples A = (A 0 , A 1 ) and B = (B 0 , B 1 ) and for every operator S : . An immediate consequence of Theorem 3.2 is the following:

is a Banach couple of translation-and rotationinvariant pseudolattices, then the following family of interpolation functors
Let {F θ } θ∈(0,1) be a locally stable family of interpolation functors and T : union of open disjoint intervals. These intervals we will call intervals of invertibility of T with respect to the family {F θ } θ∈(0,1) .
Let I ⊂ (0, 1) be any interval of invertibility of T . In this section we are interested in the following question: is it true that for any θ , θ ∈ I the inverses T −1 We point out that this problem is very important for PDEs (see, for example, discussions in [27]).
We will often use the following simple proposition.
be Banach couples and let T : A → B be an invertible operator. Then the following conditions are equivalent: The same arguments show that (ii) ⇒ (i). Since G( A) = A 0 + A 1 is an interpolation functor, the implication (iii) ⇒ (ii) follows. Now we are ready to state and prove the following result.
Our hypothesis that the family of functors {F θ } θ∈(0,1) is locally stable implies that T −1 θ 0 y n = T −1 θ 1 y n for each n ∈ N. Letting x n := T −1 θ 0 y n one has x n → T −1 θ 0 y in F θ 0 ( X ). We also have that x n → T −1 θ 1 y in F θ 1 ( X ). In consequence, the sequence {x n } converges to elements T −1 θ 0 y and T −1 θ 1 y in X 0 + X 1 . Thus T −1 θ 0 y = T −1 θ 1 y as required. 1) is a family of regular K -functors, then from Remark 3.6.5 in [10] it easily follows that this condition is fulfilled. In particular, it is true for families of functors given by

Remark 4.4 The condition that
In the next proposition we show that under an approximation hypothesis on (Y 0 , Y 1 ) the density condition required in Theorem 4.3 holds. Recall that the functor F θ is said to be of type θ if for any Banach couple A = (A 0 , A 1 ), we have the continuous inclusions Then, for any pair of regular interpolation functors F θ 1 and F θ 2 of type θ 1 and θ 2 , respectively, we have that Proof At first we note that as functors F θ 1 and F θ 2 are of type θ 1 and θ 2 , respectively, then there exists a constant C > 0 such that, for each j ∈ {0, 1}, we have Hence, we get that for all y ∈ Y 0 ∩ Y 1 and each j ∈ {0, 1}, Moreover, by the interpolation property, it follows that sup Since the functors are regular, for any y Hence, for i = 0 and i = 1 It then follows that, for every y We note that Lions [32] showed that a very wide class of Banach couples satisfy the approximation condition used in the above proposition. We will say that a family of interpolation functors {F θ } θ∈(0,1) satisfies the global ( )-condition if for any Banach couple A = (A 0 , A 1 ) and for any θ 0 , θ 1 with 0 < θ 0 < θ 1 < 1, we have continuous inclusions where the Gagliardo completion Here θ 0 <θ<θ 1 F θ ( A) denotes a Banach space consisting of all elements a such that, for every θ ∈ (θ 0 , θ 1 ), we have a ∈ F θ ( A) and sup In what follows we will use the next obvious observation.
Indeed, from invertibility of the operator T on the whole interval I , local stability of the family F θ and compactness of the interval [θ 0 , θ 1 ], we get that Hence from the estimate (9) above, we obtain Thus, using the right-hand continuous inclusion in the definition of the global ( )condition, we conclude thatx To finish the proof, we decompose the element x as Invertibility of the operator T on F θ j ( X ) implies injectivity of T on F θ j ( X ) c for each j ∈ {0, 1}. This implies that both x 0 −x and x 1 +x are equal to zero. Consequently x = 0 and so the operator T : To show applications to complex and real interpolation methods of the above results we need a lemma.  θ∈(0,1) . At first we note that it is shown in [24] that for any Banach couple (A 0 , A 1 ) we have where [A 0 , A 1 ] λ θ is the "periodic" interpolation space with λ = 2π . It follows immediately from the definition of the periodic interpolation space that with norm of the inclusion map less or equal than 1. Analysis of the Cwikel paper [15, p. 1008] shows that with norm of the inclusion map less or equal than C(θ ). Standard calculus shows that there exists a positive constant K independent of θ such that Taking all this together yields that the family {F θ } := {( · ) (FC,FC),e θ } θ∈(0,1) satisfies where the constants of equivalence of norms are bounded on any compact subinterval of (0, 1). To finish it is enough to apply Corollary 4.1 and Proposition 4.6. Now we consider the case {G θ } := {( · ) θ,q } θ∈(0,1) for any fixed 1 ≤ q ≤ ∞. Put {F θ } := {( · ) ( q , q ),e θ } θ∈(0,1) . It was shown in [16] that F θ ( A) = G θ ( A) up to equivalence of norms. Standard calculus show that there exist positive constants C 1 > 0 and C 2 > 0, independent on θ , such that Again applying Corollary 4.1 and Proposition 4.6 we are done.
From Theorem 4.7 we also obtain the compatibility theorem for the family {[ · ] θ } θ∈(0,1) of complex interpolation methods. Theorem 4.10 Let T : (X 0 , X 1 ) → (Y 0 , Y 1 ) be an operator between couples of complex Banach spaces and let I ⊂ (0, 1) be an interval of invertibility of T with respect to the family {[ · ] θ } θ∈(0,1) of complex interpolation methods. Then, for any θ 0 , θ 1 ∈ I the inverse operators T −1 θ 0 and T −1 Proof As in the proof of Theorem 4.9 it is enough to prove the global ( )-condition for the family {[ · ] θ } θ∈(0,1) for arbitrary Banach couple (A 0 , A 1 ): holds with equality of norms for any θ 0 , θ 1 , λ ∈ (0, 1) (see [9] and [15]). Hence, for This proves that We now show the right-hand continuous inclusion shown above. First note that the proof of Theorem 4.7.1 in [9] shows that for any x ∈ [A 0 , Applying the formula (1) from Sect. 2 for the Gagliardo completions, we have and hence sup θ 0 <θ<θ 1 Thus we conclude that the second required continuous inclusion holds. This completes the proof.

Theorem 4.11
Let {F θ } θ∈(0,1) be a family of locally stable interpolation functors of type θ that satisfies the global ( )-condition and the reiteration condition. Let T : Proof Since the family {F θ } θ∈(0,1) is locally stable and the operator T : Then from Theorem 4.7 it follows that the inverse operators T −1 θ 0 and T −1

Hence Proposition 4.2 (iii) implies the invertibility of the operator
To complete the proof it remains to note that as {F θ } θ∈(0,1) is a family of functors of type θ , then the reiteration theorem for the real method yields (see [9,Theorem 3.5

.3])
From this theorem we immediately obtain the next important result.

Theorem 4.12
Let T : (X 0 , X 1 ) → (Y 0 , Y 1 ) be an operator between couples of complex Banach spaces. If T : We conclude with the following result about the connections between the spectra σ of interpolated operators. The result is an immediate consequence of Theorem 4.11. Theorem 4.13 Let X = (X 0 , X 1 ) be a Banach couple of translation-and rotationinvariant pseudolattices and let the family {F θ } := {( · ) X ,e θ } θ∈(0,1) be such that the reiteration condition holds for a complex Banach couple (X 0 , X 1 ). If {F θ } satisfies the global ( )-condition for (X 0 , X 1 ) then for any operator T : X → X and all q ∈ [1, ∞] we have As a consequence, we obtain the following corollary. Corollary 4.14 Let (X 0 , X 1 ) be a couple of complex Banach spaces. Then for any operator T : (X 0 , X 1 ) → (X 0 , X 1 ) and for all q ∈ [1, ∞] we have We conclude with the following remark. In [1, Example 12], Albrecht and Müller gave an example of a Banach couple X and an operator T : X → X for which σ (T , X θ,1 ) = σ (T , [ X ] θ ).

Interpolation of lattice isomorphisms
In this section we apply our results from Sect. 4 to prove the stability of invertibility of positive operators on interpolation scales of Banach function lattices. In particular, we obtain that if a positive operator at one point of a scale of Calderón product spaces has a positive inverse then it has a positive inverse for all interior points of this scale.
We start with some required definitions. Let ( , A, μ) be a complete σ -finite measure space and let Y be a Banach space. Throughout the rest of the paper, L 0 (μ, Y ) denotes the space of equivalence classes of strongly measurable Y -valued functions on , equipped with the topology of convergence in measure (on sets of the finite μ-measure). In the case Y = R we write L 0 (μ) instead of L 0 (μ, R). By a Banach lattice over ( , A, μ) (or in L 0 (μ)), we will mean a Banach space X ⊂ L 0 (μ) which is an ideal in L 0 (μ), that is, if | f | ≤ |g| a.e., where g ∈ X and f ∈ L 0 (μ), then f ∈ X and f X ≤ g X . Below a Banach lattice X is called a Banach function lattice if X contains an element h such that h > 0 μ-almost everywhere.
In what follows we will use the well-known fact that every Banach function lattice X over a measure space ( , A, μ) is order dense in L 0 (μ), that is, for every f ∈ L 0 (μ)\{0} with 0 ≤ f there exists g ∈ X \{0} satisfying 0 ≤ g ≤ f (see [28,Lemma 1,p

. 95]).
A Banach lattice X is said to have the Fatou property if for any sequence { f n } of nonnegative elements from X such that f n ↑ f for f ∈ L 0 ( ) and sup n≥1 f n X < ∞, one has f ∈ X and f n X ↑ f X .
Let X and Y be Banach lattices. A linear mapping T : X → Y is said to be positive (resp., a lattice homomorphism where f ∨g := sup{ f , g}). Clearly, a lattice homomorphism belongs to the class of positive operators. We also note that there are many equivalent characterizations of lattice homomorphism. Throughout this section we will apply a useful easily verified characterization which states that T : X → Y is a lattice homomorphism if and only if |T f | = T | f | holds for all f ∈ X (see [3,Theorem 7.2] for details).
A lattice homomorphism T : X → Y which is also a bijection is called a lattice isomorphism. We will use the following characterization: a linear bijection T : X → Y is a lattice isomorphism if and only if T and T −1 are both positive (see [3,Theorem 7.3]).
An operator T : X → Y between two Banach lattices is said to be a lattice embedding if T is a lattice homomorphism and it is also an embedding, i.e., there exist two positive constants A and B satisfying Following [26], we say that a positive operator T : In what follows an operator T : (X 0 , X 1 ) → (Y 0 , Y 1 ) between couples of Banach lattices is said to be positive if T : X j → Y j is positive for j ∈ {0, 1}.
Recall that a linear subspace F ⊂ L 0 (μ) is a sublattice if f , g ∈ F implies f ∨g ∈ F. It is important to note that the range of lattice homomorphism is a sublattice but, in general, it is not an ideal. It should be pointed out that a key point in our study of lattice isomorphisms is demonstrating that under some conditions the range of a lattice embedding between Banach function spaces generated by interpolation functor is an ideal.
In this section we consider interpolation functors F such that F( X ) is a Banach lattice for any couple X of Banach lattices. Clearly, this holds for every exact interpolation functor. If F is not exact, we can introduce an equivalent norm · * on F( X ) under which F( X ) becomes a Banach lattice. This norm is given by the formula We start with the following useful technical lemma.
Lemma 5.1 Let F be an interpolation functor and let X = (X 0 , X 1 ) and Y = (Y 0 , Y 1 ) be couples of Banach function lattices. Suppose that T : X → Y is a positive linear operator such that T : F( X ) → F( Y ) is an injective lattice homomorphism. Then T : In addition, if f ∈ X 0 0 + X 0 1 and T f ≥ 0, then f ≥ 0, that is, the formal inverse of T defined on T (X 0 0 + X 0 1 ) is positive.
We claim that the above equality holds for all f ∈ X 0 0 + X 0 1 . Indeed, from the regularity of the couple (X 0 0 , X 0 1 ) it follows that for any f ∈ X 0 In view of |T f n | = T | f n | for f n ∈ X 0 ∩ X 1 , we get |T f | = T | f | as required and so T is a lattice homomorphism on X 0 0 + X 0 1 . To establish that T is injective on X 0 0 + X 0 1 , we assume that T f = 0 for some Since X 0 ∩ X 1 is a Banach function lattice, it follows by order density of X 0 ∩ X 1 in L 0 (μ) that there exists g ∈ X 0 ∩ X 1 ⊂ F( X ) such that 0 ≤ g ≤ | f | and g = 0. Consequently, 0 ≤ T g ≤ T | f | and so T g = 0, which contradicts the injectivity of T on F( X ). Now suppose that for a given f ∈ X 0 0 + X 0 1 one has T f ≥ 0. Using the proven fact This completes the proof.
Below we will call a locally stable family {F θ } θ∈(0,1) of interpolation functors of type θ stable if it satisfies the global ( )-condition and the reiteration condition (see (6) and (8) in Sect. 4). Using Lemma 5.1 and Theorem 4.7 we obtain the following result.
Theorem 5.2 Let X = (X 0 , X 1 ) and Y = (Y 0 , Y 1 ) be couples of Banach function lattices, let T : (X 0 , X 1 ) → (Y 0 , Y 1 ) be a positive operator and let I ⊂ (0, 1) be an interval of invertibility of T with respect to the stable family of functors is a lattice isomorphism for some θ * ∈ I , then for all θ ∈ I the operator is a lattice isomorphism. Moreover, for any θ 0 , θ 1 ∈ I , the inverse operators T −1 θ 0 and T −1 θ 1 are positive and agree on Proof Note that F θ ( X ), F θ ( Y ) are Banach function lattices for all θ ∈ (0, 1). Since is a lattice isomorphism and F θ * ( X ) ⊂ X 0 + X 0 1 , then from Lemma 5.1 it follows that T : is invertible for all θ ∈ I , therefore it is a lattice isomorphism. Thus the inverse operator T −1 θ : F θ ( Y ) → F θ ( X ) is positive. The second statement of the theorem follows from Theorem 4.7.
We will need the following lemma.

Lemma 5.3
Let X = (X 0 , X 1 ) and Y = (Y 0 , Y 1 ) be couples of Banach function lattices and let T : X → Y be a positive operator. Suppose that F and G are interpolation functors such that F( X ), G( X ) ⊂ X 0 0 + X 0 1 and G is regular on Y . If

is a lattice isomorphism and T : G( X ) → G( Y ) is a cone embedding, then T (G( X )) is a closed ideal of G( Y ).
Proof Since T : G( X ) → G( Y ) is a cone embedding, therefore there exists a positive constant C such that Hence from Lemma 5.1 we get that T (G( X )) is a closed subspace of G( Y ). We claim that T (G( X )) is an ideal. Clearly, it is enough to show that if g is a measurable function such that 0 ≤ g ≤ |T f | for some f ∈ G( X ), then there exists h ∈ G( X ) is a lattice isomorphism, and so T has a positive bounded inverse. Thus for each n ∈ N there exists a nonnegative f n ∈ F( X ) ⊂ X 0 0 + X 0 1 such that T f n = g n . We have 0 ≤ T f n = g n ≤ g ≤ |T f | = T (| f |), and so Lemma 5.1 yields 0 ≤ f n ≤ | f | for each n ∈ N. This shows that the sequence { f n } belongs to G( X ) and from (10) and the construction of {g n } it follows that { f n } is the Cauchy sequence in G( X ) which converges to some h ∈ G( X ). Clearly, T h = g and so this completes the proof. Now we are ready to prove the main result of this section on the stability of lattice isomorphisms between interpolation scales of Banach function lattices.
Let us now show that for any θ ∈ (0, 1) the operator (12) is also a lattice isomorphism. Since T : F θ * ( X ) → F θ * ( Y ) is a lattice isomorphism and the couples X , Y are regular, it follows from Lemma 5.1 that T : X 0 + X 1 → Y 0 + Y 1 is an injective lattice homomorphism. Therefore, the operator T : F θ ( X ) → F θ ( Y ) is also an injective lattice homomorphism. Thus it remains to show that T is surjective. Fix g ∈ F θ ( Y ). Then by the density of Y 0 ∩ Y 1 in F θ ( Y ) we can find a sequence {g n } in Y 0 ∩ Y 1 such that g n → g in F θ ( Y ) as n → ∞. Clearly, g n ∈ Y θ,1 for each n ∈ N, and so from the invertibility of the operator (11) with q = 1 we conclude that there exists a sequence { f n } in X θ,1 such that T f n = g n . Since then T f n = g n ∈ T (F θ ( X )) for each n ∈ N. Hence Lemma 5.3 yields that g ∈ T (F θ ( X )), that is, T (F θ ( X )) = F θ ( Y ). This completes the proof.
We will show applications of Theorem 5.4 to Claderón product spaces. Recall that the Calderón product space X (θ ) It is well known (see [12]) that X (θ ) := X 1−θ 0 X θ 1 is a Banach lattice endowed with the norm We are now in a position to state an application of the Theorem 5.4. Theorem 5.5 Let X = (X 0 , X 1 ) and Y = (Y 0 , Y 1 ) be regular couples of Banach function lattices with the Fatou property. Let T : X → Y be a linear positive operator such that for some parameter θ * ∈ (0, 1) the operator T : X (θ * ) → Y (θ * ) is a lattice isomorphism. Then for any θ ∈ (0, 1) and q ∈ [1, ∞] the operators To prove the above Theorem 5.5 we will use the following result that states that, under some mild conditions for Banach function lattices, a cone embedding at one point of the scale of Calderón product spaces is also a cone embedding at all points in the interior of the scale. We point out that this is an unpublished result proved by M. Milman (we refer to a private communication). For the reader's convenience we include the proof of this result. Proposition 5.6 Let T : (X 0 , X 1 ) → (Y 0 , Y 1 ) be a positive operator between couples of Banach function lattices with the Fatou property. Assume that an operator is a cone embedding for some θ * ∈ (0, 1). Then T : Y θ 1 is a cone embedding for all θ ∈ (0, 1). Proof Notice that for any couple of Banach lattices with the Fatou property (E 0 , E 1 ) and for every θ ∈ (0, 1), E(θ ) = E 1−θ 0 E θ 1 is a Banach function lattice with the Fatou property (see [34]). If (E 0 , E 1 ) is a couple of Banach function lattices with the Fatou property then the extrapolation formula of Cwikel-Nilsson [17,Theorem 3.5] gives Since the Calderón construction is an interpolation method for positive operators, then is a bounded operator for all θ ∈ (0, 1). It is given that T : is a cone embedding, that is, there is δ > 0 such that for all positive f ∈ X (θ * ), we have We will need the following easily verified reiteration formula that is valid for all couples of Banach lattices and for all α, θ 0 and θ 1 in (0, 1): where β = (1 − α)θ 0 + αθ 1 . We will also use the property of any positive operator P : X → Y between Banach lattices which says that if 0 ≤ x, y ∈ X and θ ∈ (0, 1), then (see, e.g., [31, p. 55]) We may assume without loss of generality that T X j →Y j ≤ 1 for j ∈ {0, 1}. Let 0 < θ * < θ < 1. Thus, we can find α ∈ (0, 1) such that θ * = αθ. Suppose f ∈ X (θ ) is nonnegative. Combining the Cwikel-Nilsson formula shown above with (17), we obtain In consequence, (15) and the mentioned extrapolation formula yield the required estimate The case 0 < θ < θ * < 1 can be proved similarly if we consider the couples (X 1 , X 0 ), . This completes the proof. Now we are ready to prove Theorem 5.5. In the proof, by E(C) we denote a standard complexification of a real Banach lattice E.
Proof Firstly, we recall the well-known result due to Shestakov [38] on Calderón's complex interpolation spaces which states that for any regular coupe (X 0 , X 1 ) of real Banach function lattices one has From this result follows an obvious observation: if T : is an operator between regular couples of real Banach lattices, then T :

The Neumann problem for the Stokes system
Considerable work has been done over the past decade to solve the Dirichlet and Neumann problem for Laplace's equation in a Lipschitz domain in R n (n ≥ 3) with data in L p (∂ ), or with one derivative in L p (∂ ). The result due to Coifman, McIntosh and Meyer [14] on the boundedness of the Cauchy integral on L p (1 < p < ∞) on any Lipschitz domain was a key for using the layer potential method in the study of these problems. Next, Calderón [13] introduced a new technique based on the study of invertibility of the classical layer potentials. These techniques were developed in the study the boundary value problems in Lipschitz domains for some systems of secondorder linear systems of partial differential equations. We refer to papers by Fabes, Kenig and Verchota [21], Pipher and Verchota [36] where, in particular, a history of achievements concerning the above problems is described. The aim of this section is to study the solvability of the Neumann problem for the Stokes system of linear hydrostatics on a Lipschitz domain in R n with data in Lorentz spaces. We combine results from the Fabes, Kenig and Verchota paper [21] with results on stability of the Fredholm property of interpolated operators from the papers [6,8]. Let us introduce the required notation.
We assume that is a bounded Lipschitz domain in R n , n ≥ 3; i.e., it is locally given by the domain above the graph of a Lipschitz function (see [21,[771][772] for more details). Denote by N the outward unit normal to , which is well defined with respect to the surface measure σ at a.e. point on ∂ .
For the convenience of the reader we recall some important definitions and notations. For simplicity of presentation, we consider a domain of the form := (x 1 , . . . , x n−1 , y); y > ϕ(x 1 , . . . , x n−1 ) , where ϕ : R n−1 → R is a Lipschitz function with Lipschitz constant L. Let us fix L > L. Then for every x ∈ ∂ , we denote by + (x) a vertical cone completely contained in =: + defined by and by − (x), we denote the reflection of + (x) contained in − := R n \ .
Next, given u : → R, the nontangential maximal function of u evaluated at boundary points of is defined by We say that u converges nontangentially σ -a.e. to a function f on ∂ if for σ -a.e.
x ∈ ∂ , The function f is called the nontangential boundary value of u and is denoted by u + . If u is defined in − and converges nontangentially at x ∈ ∂ − , then the respective limit is denoted by u − (x). Given functions u j : → R for 1 ≤ j ≤ n, the vector-valued function u : → R n is given by u(x) := (u 1 (x) . . . , u n (x)) for all x ∈ . We consider R n with the Euclidean norm. The above definitions of u and u ± have a natural counterparts to vector-valued functions ( u) and u ± . For the corresponding u, u will denote the ordinary Laplacian acting on each component and ∇ · u := ∇, u will denote the divergence of the vector u, where , denotes inner product in R n . For a function p : R n → R, as usual, ∇p will denote its gradient.
We recall that for a given Banach lattice E := E( ) over a measure space ( , μ) := ( , A, μ) and Banach space X , we denote by E(X ) := E( , X ) the Köthe-Bochner space of all strongly A-measurable functions x ∈ L 0 (μ, X ) such that x(·) X ∈ E. This is a Banach space under pointwise operations and the natural norm x E(X ) := x(·) X E . In the particular case when E is a Banach lattice over (∂ , σ ), then the Köthe-Bochner space E(R n ) contains all measurable functions f = ( f 1 , . . . , f n ) ∈ L 0 (∂ , R n ) equipped with the norm In the case E = L p over (∂ , σ ) with 1 ≤ p ≤ ∞, we write f p for short.
In what follows we say that the Neumann problem (N) in either + or − is uniquely solvable for g, g ∈ E(∂ , R n ), if there exists a unique function u ∈ C 2 ( ) n and a unique (up to a constant) function p ∈ C 1 ( ), satisfying the Stokes system (N). Based on the method of lower potentials, Fabes, Kenig and Verchota [21] proved that the Neumann problem (N) is uniquely solvable in + (resp., − ) for any g ∈ L 2 0 (∂ , R n ) (resp., g ∈ L 2 N (∂ , R n )). Thus it appears a natural question of whether there is a variant of this result for other function spaces than L 2 (∂ , R n ). It is important to note here that it is mentioned in [21, p. 771] that well-known arguments (see [13,18] and [19]) show that their result extends to the case of L p (∂ , R n )-spaces with p ∈ [2 − ε, 2 + ε] for some ε = ε( ) > 0.
We will combine these statements with results on stability of the Fredholm property of interpolated operators from recent papers [6] and [8] to get an answer to the question of the unique solvability of the Neumann problem (N) in the setting of some Lorentz spaces over (∂ , σ ). We continue to review background material by recalling the definitions and some basic properties of the layer potential for the Stokes system in an arbitrary bounded Lipschitz domain ⊂ R n , n ≥ 3.
Let (x) := jk (x) n j,k=1 be the matrix of fundamental solutions of the Stokes system and let q(x) := (q j (x)) n j=1 be the pressure vector, where Here, δ jk are the Kronecker-Delta functions and ω n denotes the surface area of the unit sphere in R n . We also define a corresponding potential for the pressure q by Fix f ∈ L p (∂ , R n ) with p = 2 and let S f denote the single layer potential with density f defined by From the works [21, p. 773] and [20], it is known that where C = C( , n). The layer potential S satisfies the following key trace formulas on ∂ : where K is a singular integral operator which is bounded on L q (∂ , R n ) for any 1 < q < ∞ (see [14] and [20] for more details). In addition u = S f and q = Q f satisfies In the proof of the main result of this section the following Theorem 6.1 on stability of the Fredholm property of operators acting on the interpolation spaces of real and complex methods will play a crucial role. It is a consequence of results from previous sections combined with results from the recent papers (see [6,8] and Shneiberg's well-known paper [39]). For the sake of completeness we recall that if X and Y are Banach spaces, then a bounded operator T : X → Y is said to be Fredholm if it has finite-dimensional kernel and its image has finite codimension. So from the above definition, it follows that its image is closed. The index of T is defined by ind (T ) := dim (ker T ) − codim T . Theorem 6.1 Let (X 0 , X 1 ) and (Y 0 , Y 1 ) be Banach couples and let θ * ∈ (0, 1). Then the following statements about an operator T : (ii) If for some p ∈ [1, ∞) the operator T : (X 0 , X 1 ) θ * , p → (Y 0 , Y 1 ) θ * , p is Fredholm, then for all q ∈ [1, ∞) the operator T : (X 0 , X 1 ) θ * ,q → (Y 0 , Y 1 ) θ * ,q is Fredholm and ker T | (X 0 ,X 1 ) θ * , p = ker T | (X 0 ,X 1 ) θ * ,q .
Proof The statement (i) follows from Theorem 1.1 in [8], the proof of which is based on Theorem 4.12 of the current paper. We point out that Theorem 4.12 was published before in arXiv (see [7]).
The statement (iii) is proved in [39].
We will use the interpolation theorem on vector-valued sublinear operators. Recall that if X and Y are Banach spaces, then a mapping S : X → L 0 ( , A, Y ) is said to be a sublinear operator if, for all x, y ∈ X and any scalars λ, we have S(λx) Y = |λ| Sx Y and S(x + y) Y ≤ Sx Y + Sy Y μ-a.e..
Applying the most general version of the Hahn-Banach extension theorem (see [3, Theorem 2.1]) it is possible to obtain the following result (see [11]). Theorem 6.2 Let (E 0 , E 1 ) be a couple of Banach function lattices over a measure space ( , A, μ) and let X and Y be Banach spaces. Then, for any exact interpolation functor F, the following statements are true: is a sublinear operator such that S f E j (Y ) ≤ C j f E j (X ) for some C j > 0 and all f ∈ E j (X ), j = 0, 1. Then S : (ii) F(E 0 (X ), E 1 (X )) ∼ = F(E 0 , E 1 )(X ).
Before proceeding, we recall that if ( , A, μ) is a σ -finite and complete measure space, p, q ∈ [1, ∞), then the Lorentz space L p,q ( ) is defined to be the space of all f ∈ L 0 ( ) such that We are now in a position to state the main result of this section.

Theorem 6.3
Let be a bounded Lipschitz domain in R n , n ≥ 3, with connected boundary. There exists ε 0 > 0, depending on the Lipschitz character of , such that, given g ∈ (− 1 2 I + K)(L p,q (∂ , R n )) with 2 − ε 0 < p < 2 + ε 0 and 1 ≤ q < ∞, the Neumann problem for the Stokes system (N) has a solution in − , and the solution satisfies the estimate where E := L p,q (∂ , R n ) and C depends only on the Lipschitz character of ∂ . In addition there exist unique u and p satisfying the condition (21).
To establish uniqueness assume that u is the solution of the Neumann problem (N) with ∂ u − ∂ν = g = 0. Then applying the formula (see [21, (1.4) we conclude that u is a constant and so by the condition (a), we get that p is constant. This completes the proof.