THE STRUCTURE OF INTERMEDIATE SPACES

AND REAL INTERPOLATION

Evgeniy Pustylnik

This is the text of several lectures delivered in University of Illinois at Chicago in March 1994 by invitation of prof. Yoram Sagher.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . 1

1. Basic notions . . . . . . . . . . . . . . . . . . . 2

2. Sum and intersection of Banach spaces . . . . . . . . . . 5

3. Special functionals connected with a couple . . . . . . . . . 7

4. Imbedding functions of an intermediate space . . . . . . . . 9

5. Lorentz and Marcinkiewicz spaces . . . . . . . . . . . . 13

6. K-spaces and K-functors . . . . . . . . . . . . . . . 15

7. J-functors . . . . . . . . . . . . . . . . . . . . 18

8. Weak-type interpolation . . . . . . . . . . . . . . . 22 Recommended literature . . . . . . . . . . . . . . . . 24

Introduction

The real interpolation methods are a most elaborated, say, finished part of the interpolation theory of linear operators. This theory is a very powerful instrument of the modern functional analysis having entailed many brilliant results. The real methods are due to Lions and Peetre (1964) which introduced a variant with numerical parameters. The next valuable contribution was due to V.Dmitriev (1972) who studied these methods in general form. And the last full stop was put in 1980 by Brudnyi and Krugliak who solved all the problems still having stayed in the real interpolation. To my pity, I cannot mention many other mathematicians who worked in this area and obtained various partial but important results.

Of cause, the theory lasts to develop, new problems are arising, but they are around, not in the core, and what is made, is finished and is classical. There are different approaches to explaining the real interpolation. I will propose you one which is due to myself. I have come to the real interpolation in the beginning of 70-th by studying general families of Banach spaces and the global structure of the collection of all intermediate spaces for a given Banach couple. Only lack of one property, so-called K-divisibility, prevented me from proving the main results in the real interpolation. This property is very complicated to prove and I regard it as a main achievement of Brudnyi and Krugliak. I’ll give you

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this property without proof and, using it, I’ll construct the most of the real interpolation theory. Now let us start.

Many problems of the linear functional analysis are reduced to the question about boundedness of some linear operator T : A → B. For instance, a linear differential or integral equation is written in the form T x = y, it is solved in the form x = T^{−1}y, and the question about existence of solutions and about their properties is reduced to the question about operator T^{−1} being bounded from a space containing y to a space containing x. Another example: any Fourier series x(t) = ∑ a_{i}e_{i}(t) defines a linear operator T x = {a_{i}} which alows us to connect properties of x(t) and of its Fourier coefficients. Applying to the harmonic analysis we have a basic definition of the spectrum as a complement to the set of points s in which the resolvent R(s) = (T − sI)^{−1} is a bounded operator.

In the modern analysis we almost do not meet a situation when an operator T is studied on the pair of spaces strongly fixed beforehand. As a rule, it is considered simultaneously on a family of spaces and we should extract those of spaces on which T is bounded. Inasmuch as the spaces in such families are very different with respect to difficulties of studying them, it would be useful to be able to transpose the properties of operator from space to space. To prove the boundedness of T on simple spaces and to obtain the same on other spaces with the help of some universal law. How it can be realized? We order all spaces with respect to some criterion and search a possibility to interpolate (from the endpoints to inside) or to extrapolate (from the inside to outside) the required assertions. The extrapolation began to develop only resently, mainly in the works of M.Milman and his colleagues, and we shan’t speak about it. And the interpolation will be our subject to discuss.

Basic notions

As a principal, historically first example we take the interpolation in L_{p}-spaces with the norm

∥x∥_{L_{p}} = (∫_{D} |x(t)|^{p}dμ)^{1/p} (1 ≤ p < ∞), ∥x∥_{L_{∞}} = ess sup_{D} ∥x(t)∥.

This spaces have a natural order with respect to the growth of p. The first result here was obtained by M.Riesz (1926) (only for the similar sequence spaces l_{p}) . In 1938 G.Thorin improved this result using a new idea which served later as a startpoint for the abstract interpolation theory on the first stage. He used the famous Hadamar’s theorem on the concavity of analitical function modulus in the annulus or in the strip. The first abstract interpolation methods also were based on this theorem and owing to this are called the complex methods (1960-64). And when Lions and Peetre in 1964 proposed for the interpolation another idea, not connected with the complex function properties, they called it, on the contrast, the real interpolation.

The interpolation in the L_{p}-spaces has a very nice geometrical interpretation. Let us put on the axes X and Y the values 1/p. Each point (x, y) of the square [0, 1] × [0, 1] will mean the pair of spaces L_{p_{0}}, L_{q_{0}} such that x = 1/p_{0}, y = 1/q_{0}. This point is said to be admissible for an operator T if T : L_{p_{0}} → L_{q_{0}}. Let us have another point (1/p_{1}, 1/q_{1}) to be admissible for the same operator. Then the interpolation theorem of Riesz-Thorin means that every point of the segment thus obtained is admissible.

I’ll give two classical examples of utilizating such interpolation.

A) The Fourier transform.

Ff(ξ) = ∫_{IR^{n}} f(x) exp(−i⟨x, ξ⟩)dx, ⟨x, ξ⟩ = x_{1}ξ_{1} + ⋯ + x_{n}ξ_{n}.

Then

|Ff(ξ)| ≤ ∫ |f(x)|dx, ∫ |Ff(ξ)|^{2}dξ = (2π)^{n} ∫ |f(x)|^{2}dx

(the second is the Parseval equation). So, the points (1,0) and (1/2, 1/2) are admissible for the operator F. Connecting them by a segment we obtain all points (1/p, 1/q) to be admissible if 1 ≤ p ≤ 2 and 1/p + 1/q = 1. If we write this as an inequality for norms, we obtain the famous Haussdorf-Young inequality: ∥Ff∥_{L_{q}} ≤ C∥f∥_{L_{p}}. Note that in the Riesz-Thorin theorem one also has an estimate for the norms in intermediate points, which I didn’t indicate and which enables to obtain C = (2π)^{n/q}.

B) The convolution operator.

T f(x) = ∫ k(x − y)f(y)dy = k ∗ f(x).

If k ∈ L_{σ} then due to Minkowski inequality we have ∥T f∥_{L_{σ}} ≤ ∥k∥_{L_{σ}}∥f∥_{L_{1}} and due to Hölder inequality we have ∥T f∥_{L_{∞}} ≤ ∥k∥_{L_{σ}}∥f∥_{L_{τ}} where 1/σ + 1/τ = 1. In other words, the convolution operator T : L_{1} → L_{σ} and T : L_{τ} → L_{∞}. Again we may use the interpolation theorem connecting the points (1, 1/σ) and (1/τ, 0) and taking the intermediate points of the segment. Thus we obtain that T : L_{p} → L_{q} if 1/q = 1/p − 1/τ and the norm ∥T ∥ ≤ ∥k∥_{L_{σ}}. So, we get another famous Young inequality ∥k ∗ f∥_{L_{q}} ≤ ∥k∥_{L_{σ}}∥f∥_{L_{p}}.

Both Minkowski and Hölder inequalities, which have been used above, hold not only for the L_{p} but for some wider classes of Banach spaces, and the interpolation theorem also are valid there. This allows to generalize the last result to other spaces, but we always obtain the same kind of spaces which was given in the conditions on the functions k and f. However, there exists a theorem with the same result under essentially weaker condition on the function k — for example, for the case k(z) = |z|^{−n/σ} in IR^{n}, when evidently k ∉ L_{σ}. Now such results also can be obtained by the so-called weak-type interpolation, which is a combination of the real interpolation methods with some subtle imbedding theorems. There are many unsolved problems in this area and this is the matter of my interest now in the scientific research.

As I have already said, the concept of Banach interpolation spaces is based on some ordering them. As it turns out, the most fruitful idea is to order spaces with respect to their imbedding one into another. By this we must consider not only the algebraic imbedding as in the set theory, but keep the safety of linear and topological structure. This means that equalities x + y = z and y = x should be independent of a taken space and that any sequence {x_{n}} converging in several spaces should have everywhere the same limit. All this may be conditioned by special axioms, we may consider injections instead of coincidence of elements, but this makes all discussions and proves very complicated and sometimes leads to paradoxes.

There is another, simplified way: to take as a basis some common Haussdorf linear topological space H and to consider only those Banach spaces which are subspaces of H such that their own topology is stronger than that of H. For example, when studying the Banach functional spaces we may take as H the metrical space of all measurable functions. In fact, we need not use and even mention the space H, we need only the above mentioned properties of Banach spaces. As a result, any algebraical imbedding A ⊂ B implies inequality for norms ∥x∥_{B} ≤ C∥x∥_{A}.

If now three spaces A, B, E are connected with a relation A ⊂ E ⊂ B, it is natural to call E intermediate between A and B. For example, the space L_{p}(D) is intermediate between L_{p_{0}} and L_{p_{1}} if p_{0} ≤ p ≤ p_{1} and mes D < ∞. If D = (0, ∞), the spaces L_{p} cannot be imbedded one into another, although the Riesz-Thorin interpolation theorem is true even in this case. That is why we should generalize the concept of being intermediate. We say two spaces A_{0}, A_{1} ⊂ H form a Banach couple A⃗ if they have non-zero intersection Δ(A⃗) = A_{0} ∩ A_{1}. This intersection becomes to be a Banach space under the norming ∥x∥_{Δ(A⃗)} = max(∥x∥_{A_{0}}, ∥x∥_{A_{1}}). We also define the sum Σ(A⃗) = A_{0} + A_{1} = {x| x = x_{0} + x_{1}; x_{0} ∈ A_{0}, x_{1} ∈ A_{1}} with the norm ∥x∥_{A_{0}+A_{1}} = inf(∥x_{0}∥_{A_{0}} + ∥x_{1}∥_{A_{1}}) where infimum is taken over all representations of x as above. A Banach space A is said to be intermediate for the couple A⃗ if Δ(A⃗) ⊂ A ⊂ Σ(A⃗). In such a sense the space L_{p} is always intermediate for the couple (L_{p_{0}}, L_{p_{1}}) if p_{0} ≤ p ≤ p_{1}.

The collection of all intermediate spaces for a couple A⃗ we shall denote by π(A⃗). It is easy to see that

π(A⃗) = π(B⃗) ⟺ Δ(A⃗) = Δ(B⃗) & Σ(A⃗) = Σ(B⃗).

A couple A⃗ is called ordered if A_{0} ⊂ A_{1}. In this case Δ(A⃗) = A_{0}, Σ(A⃗) = A_{1}, and we have the previous definition of being intermediate.

Among all intermediate spaces we extract two most important classes.

Definition 1. An intermediate space A is said to be interpolation for the couple A⃗ if for each linear operator T: Σ(A⃗) → Σ(A⃗) such that

∥T ∥_{A_{i}} = sup_{x∈A_{i}} ∥T x∥_{A_{i}}/∥x∥_{A_{i}} < ∞, i = 0, 1,

we have

∥T ∥_{A} ≤ M max_{i=0,1} ∥T ∥_{A_{i}}.

The class of all such spaces we denote by Int(A⃗).

Definition 2. If in Definition 1 we consider only the one-dimensional operators T, i.e.

T x = f(x)y, y ∈ Δ(A⃗), f ∈ (Σ(A⃗))^{∗}

then the space A is called partly interpolation. The class of all such spaces we denote by Part(A⃗). Of cause, Int(A⃗) ⊂ Part(A⃗).

These definitions deal only with one couple. More general concept relates to two different couples. The combination (A_{0}, A_{1}, A) = (A⃗, A) will be called a triple, if A ∈ π(A⃗). We say that a triple (A⃗, A) is interpolation for a triple (B⃗, B), if each linear operator T: Σ(A⃗) → Σ(B⃗), such that^{1}

∥T ∥_{A_{i}B_{i}} = sup_{x∈A_{i}} ∥T x∥_{B_{i}}/∥x∥_{A_{i}} < ∞, i = 0, 1,

is bounded from A to B and

∥T ∥_{AB} ≤ M max_{i=0,1} ∥T ∥_{A_{i}B_{i}}.

By the same manner we can generalize the definition of being partly interpolation.

In all definitions we have the defining inequality with some constant C, M etc. If such a constant may be taken equal 1, we’ll speak about exact imbedding or exact interpolation. The exactness can be always obtained by a suitable renorming of considered spaces.

Sum and intersection of Banach spaces

The sum and intersection operations are the most important for studying intermediate spaces. It is easy to check that the unit ball of the space A ∩ B is intersection of the unit balls of A and B. The unit ball of the space A + B is a closed convex hull of the unit balls of A and B. These operations are dual one to another. It means that (A + B)^{∗} = A^{∗} ∩ B^{∗} and (A∩B)^{∗} = A^{∗} +B^{∗} provided that A∩B is dense in A+B. The requirement of density is very important, since otherwise the conjugate spaces A^{∗} and B^{∗} together can be not a couple. One may take more general dual spaces, but this leads to various difficulties and is out of our lectures.

A couple A⃗ will be called regular if the space Δ(A⃗) is dense in the space Σ(A⃗). We need this property only when being compelled to use the dual spaces. Whenever it is possible we’ll try not to do this. Sometimes we need another property of the couple A⃗. It will be called mutually closed if the spaces A_{0} and A_{1} are complete with respect to their sum Σ(A⃗). This means that the unit balls of A_{0} and A_{1} are closed in the space Σ(A⃗). For instance, the unit ball of the space C of all continuous functions on some set D is not closed in any L_{p}(D) with finite p; its closure is the unit ball of the space L_{∞}(D).

The operations of sum and of intersection may be generalized to the case of arbitrary many spaces. The intersection Δ(A_{ω}), ω ∈ Ω is the set of elements x belonging to any A_{ω} with sup_{ω} ∥x∥_{A_{ω}} < ∞ and this supremum is taken as the norm in Δ(A_{ω}). The sum Σ(A_{ω}) is defined as an intersection of all Banach spaces in wich any A_{ω} is imbedded exactly, provided that at least one of such spaces really exists. Thus we have that

∥x∥_{Σ(A_{ω})} ≤ inf_{ω} ∥x∥_{A_{ω}}

for any x common for all A_{ω}. For a given set of spaces {A_{ω}} to be summable it is sufficient that they are intermediate in some couple A⃗ and their imbedding constants into the space Σ(A⃗) are bounded on the whole (we say they are uniformly imbedded into Σ(A⃗)). I leave to prove by yourself that in the case of two spaces this definition of the sum coincides with the former. It is also easy to check the assertions on the unit balls.

1For such an operator we’ll use the notation T : A⃗ → B⃗.

If all A_{ω} ∈ π(A⃗) then both Δ(A_{ω}) and Σ(A_{ω}) also are intermediate for the couple A⃗. We have even a more general assertion.

Theorem 1. Let each triple (A⃗, A_{ω}), ω ∈ Ω is exact interpolation for the corresponding triple (B⃗, B_{ω}). Then the triple (A⃗, Δ(A_{ω})) is exact interpolation for the triple (B⃗, Δ(B_{ω})) and the triple (A⃗, Σ(A_{ω})) is exact interpolation for the triple (B⃗, Σ(B_{ω})).

Proof. If we write the inequality ∥T x∥_{B_{ω}} ≤ ∥x∥_{A_{ω}} for all ω ∈ Ω and take supremum of both sides, we immediately obtain the first assertion of the theorem. To prove the second assertion, we define the space D ∈ π(A⃗) with the norm ∥x∥_{D} = \frac{1}{2}(∥x∥_{Σ(A_{ω})} + ∥T x∥_{Σ(B_{ω})}) and see that each summand in the brackets is no greater than inf ∥x∥_{A_{ω}}. Thus any A_{ω} is exactly imbedded in D and so the norm in D must be no greater than the norm in the sum Σ(A_{ω}). This implies the required inequality ∥T x∥_{Σ(B_{ω})} ≤ ∥x∥_{Σ(A_{ω})}. □

The next assertion gives us a partial case of space summing when the parameter is a continuous real variable.

Theorem 2. Let the spaces A_{t}, a < t < b, be uniformly imbedded in Σ(A⃗) and let the space Δ(A⃗) be dense in any A_{t}. Let ∥x∥_{A_{t}} be a continuous function of t for each x ∈ Δ(A⃗). Then the space Σ(A_{t}) consists of all x ∈ Σ(A⃗) representable in the form

(1)  x = ∫_{a}^{b} u(t)dt

where u(t) is a function with values in Δ(A⃗) such that the function ∥u(t)∥_{A_{t}} is summable with respect to t on (a, b). The norm in Σ(A⃗) is given by the equality

(2)  ∥x∥_{Σ(A_{t})} = inf ∫_{a}^{b} ∥u(t)∥_{A_{t}}dt

where infimum is taken over all representations of x as above.

Note that a may be equal to zero and b may be equal to infinity.

Proof. Let us denote the space of all x representable in the form (1) with the norm (2) by A. Fix some t = t_{0} and consider the functions

u_{ε}(t) = \{ \begin{matrix}x/2ε & if |t − t_{0}| ≤ ε \\ 0 & if |t − t_{0}| > ε \end{matrix}

where ε < min(|t_{0} − a|, |t_{0} − b|). Then (1) is fulfilled for u = u_{ε} hence

∥x∥_{A} ≤ lim_{ε→0} ∫_{a}^{b} ∥u_{ε}(t)∥_{A_{t}}dt = ∥x∥_{A_{t_{0}}}

in virtue of ∥x∥_{A_{t}} being continuous. So any A_{t} is exactly imbedded in A.

Let now B be some other Banach space such that all A_{t} are exactly imbedded in B. Then, for each x representable in the form (1),

∫_{a}^{b} ∥u(t)∥_{A_{t}}dt ≥ ∫_{a}^{b} ∥u(t)∥_{B}dt,

thus the integral (2) converges in B and x ∈ B. Further

∥x∥_{A} = inf ∫_{a}^{b} ∥u(t)∥_{A_{t}}dt ≥ inf ∥ ∫_{a}^{b} u(t)dt∥_{B} = ∥x∥_{B},

hence A is exactly imbedded in B. Since B is arbitrary, this finishes the proof. □

Special functionals connected with a couple

Now we should estimate the position of an intermediate space in the couple so that we’ll be able further to compare different spaces one with another. The fact of being intermediate can give us only the imbedding constants of a space A into the couple:

D∥x∥_{Σ(A⃗)} ≤ ∥x∥_{A} ≤ C∥x∥_{Δ(A⃗)}

but this is undoubtedly not enough for some serious estimation. And now we introduce some simple but very fruitful idea wich in the essence is due to Peetre. Recall that by αA one means the same space A with the norm multiplied by α. For any real α, β > 0 we consider the couple (αA_{0}, βA_{1}) and note that every A ∈ π(A⃗) is also intermediate for all of these couples. If we again consider the imbedding constants of A into such couples, they become functions of α and β and can give us much more information about the space A. In fact, Peetre used only couples (A_{0}, tA_{1}); two parameters are due to Ovchinnikov (1975).

First of all we study the spaces αA_{0} ∩ βA_{1} and αA_{0} + βA_{1}. The norms in these spaces are called J and K functionals:

J(α, β, x, A⃗) = ∥x∥_{αA_{0}∩βA_{1}}, K(α, β, x, A⃗) = ∥x∥_{αA_{0}+βA_{1}}.

As functions of α, β they are continuous, monotonically increasing and homogenious of degree 1. Further on we shall often meet such type of functions and we denote the class of them by 𝔐. The K-functional is much more important since it is defined for all x appearing in the couple and can characterize the couple on the whole. If for any Φ(α, β) ∈ 𝔐 there exists an element x ∈ Σ(A⃗) such that K(α, β, x, A⃗) ∼ Φ(α, β), we say the couple A⃗ is abundant or has the full store of K-functionals. By equivalence of two functions K, Φ we mean existence of constants C_{1}, C_{2} such that C_{1}K ≤ Φ ≤ C_{2}K.

A function Φ(α, β) ∈ 𝔐 is said to be quasi-power if there exists ε > 0 such that both the functions α^{−ε}Φ(α, β) and β^{−ε}Φ(α, β) are monotonically increasing. Recently N.Krugliak has proved that the existence of an element x ∈ Σ(A⃗) with a quasi-power Kfunctional is necessary and sufficient for the abundance of a couple A⃗. Now I show you some examples of calculating K-functional. For the simplicity I’ll consider only α = 1, β = t since always K(α, β, x, A⃗) = αK(1, β/α, x, A⃗). It is not difficult to check that K(1, t, x, A⃗) is an increasing concave function of t.

Example 1. The spaces L_{1}(0, ∞) and L_{∞}(0, ∞) form the couple which we denote by L⃗. We have already seen, that any L_{p} is intermediate and even interpolation space for this couple. Mityagin (1965) and Calderon (1966) have found out all interpolation spaces of this couple. They are so-called rearrangement invariant space (r.i.s.), i.e. spaces with some kind of the symmetry. A space E is r.i. if for any x(t) ∈ E and y(t) ∈ L_{1} + L_{∞} the inequality

mes{t, |y(t)| > λ} ≤ mes{t, |x(t)| > λ}, ∀λ > 0

implies that y ∈ E and ∥y∥_{E} ≤ ∥x∥_{E}. If the last inequality turns to equality for all λ > 0, the functions x and y are said to be equimeasurable and their norms must be equal. For instance, the norm of any characteristic function χ_{G} depends only on mes G. The function φ_{E}(t) = ∥χ_{G}∥_{E} with mes G = t is called a fundamental function of E. Any L_{p} is r.i.s. with φ_{L_{p}} = t^{1/p}; more general example is given by the Orlicz spaces. Among all nonnegative functions equimeasurable with x, there exists a nondecreasing one which is denoted by x^{∗}. By the help of x^{∗} we can express the K-functional:

K(1, t, x, L⃗) = ∫_{0}^{t} x^{∗}(s)ds.

In such form we can represent any increasing concave function φ(t) with φ(0) = 0 taking x(s) = x^{∗}(s) = φ′(s), hence the couple L is abundant.

Example 2. The r.i.spaces form only a part of the spaces intermediate for the couple L⃗. A wider class consists of so-called ideal Banach spaces or else Banach lattices. These names are given for the spaces where the norm of function is monotone with respect to its absolute value. This means that an inequality |x(t)| ≤ |y(t)| implies the norm inequality ∥x∥ ≤ ∥y∥. As an example we consider a weight space E^{w} with the norm ∥x∥_{E^{w}} = ∥xw∥_{E}, where E is a r.i.s. and w(t) is some nonnegative measurable function called a weight. For a couple of weight spaces E⃗ = (E^{v}, E^{w}) the K-functional is

K(1, t, x, E⃗) ∼ ∥ min(v, tw) x∥_{E}

with equivalence constant 2.

In the second example we can take an arbitrary measure set (D, dμ) instead of (0, ∞) with Lebesgue measure. For instance, it will be convenient for us in some cases to take (0, ∞) but with the measure dt/t. Mainly we consider the couples L⃗_{1} = (L_{1}, L_{1}^{1/t}) and L⃗_{∞} = (L_{∞}, L_{∞}^{1/t}) with the indicated measure. The K-functional for them can be written in a more explicit form:

K(1, t, x, L⃗_{1}) = ∫_{0}^{∞} min(1, t/s)x(s)ds/s
K(1, t, x, L⃗_{∞}) = x̂(t) ∼ sup_{s} x(s) min(1, t/s),

where x̂(t) is the least concave majorant of x(t). In both of these cases we also have the couples to be abundant.

Example 3. We continue example 2 taking D = IN with atomic measure equal 1 in each point. Thus we obtain the sequence space E with the sequence weight w = {w_{n}}. Let E = l_{∞}. If x = {x_{n}} then K(1, t, x, l_{∞}, l_{∞}^{w}) is the least concave envelope of the points (1/w_{n}, x_{n}) on the plain.The abundance of such couple may be not in the case if the sequence {w_{n}} tends to 0 very quickly. The necessary and sufficient condition here is sup(w_{n}/w_{n+1}) < ∞.

Imbedding functions of an intermediate space

Now we consider the imbedding constants of an intermediate space A into all couples (αA_{0}, βA_{1}), α, β > 0. For many purposes it will be more convenient to take the couples in the form (\frac{1}{α}A_{0}, \frac{1}{β} A_{1}) and define the imbedding constants in the form

C(α, β) = sup_{x∈Δ(A⃗)} \frac{∥x∥_{A}}{∥x∥_{\frac{1}{α} A_{0}∩ \frac{1}{β} A_{1}}} , D(α, β) = inf_{x∈A} \frac{∥x∥_{A}}{∥x∥_{\frac{1}{α} A_{0}+ \frac{1}{β} A_{1}}} .

By such a choice we have obtained two functions from 𝔐. At first sight they seems to be not having any connection with each other. As we’ll see further, this is not true even in general case. And in special cases the connection between them can be very tight. Let us see these functions in the same examples as above.

Example 1. Let E be r.i.s. with the fundamental function φ(t). Then after not difficult but tedious computation we obtain that

C(α, β) = D(α, β) = βφ(α/β).

Example 2. Take a couple of weight spaces (E^{v}, E^{w}) with continuous weights v(t), w(t) on (0, ∞). If for some constant k

(1/k) min(v(t), w(t)) ≤ z(t) ≤ k max(v(t), w(t)), ∀t > 0,

then the space E^{z} is intermediate for this couple and

C(α, β) ∼ sup_{t} z(t) min(α/v(t), β/w(t)),

D(α, β) ∼ inf_{t} z(t) max(α/v(t), β/w(t))

(equivalence with constant 2). If z = Φ(v, w) with some Φ ∈ 𝔐, then

C(α, β) ∼ D(α, β) ∼ Φ(α, β).

Sometimes it is enough only to estimate C and D functions from one side. If we can find out that for some Φ ∈ 𝔐 and for any x ∈ Δ(A⃗)

∥x∥_{A} ≤ Φ(∥x∥_{A_{0}}, ∥x∥_{A_{1}}),

then C(α, β) ≤ Φ(α, β). For estimating D-function one may use the duality between sums and intersections. If the couple A⃗ is regular and Δ(A⃗) is dense in A, we can pass to the conjugate spaces. If now for some Ψ ∈ 𝔐 and for any f ∈ Δ(A⃗^{∗}) = (Σ(A⃗))^{∗} we have an inequality

∥f∥_{A^{∗}} ≤ Ψ(∥f∥_{A_{0}^{∗}}, ∥f∥_{A_{1}^{∗}}),

then D(α, β) ≥ 1/Ψ(1/α, 1/β).

To prove this assertion we fix x ∈ Σ(A⃗) and take f ∈ (Σ(A⃗))^{∗} such that

f(x) = ∥x∥_{\frac{1}{α} A_{0}+ \frac{1}{β} A_{1}}, ∥f∥_{αA_{0}^{∗}∩βA_{1}^{∗}} = 1.

This is possible because (\frac{1}{α}A_{0} + \frac{1}{β} A_{1})^{∗} = αA_{0}^{∗} ∩ βA_{1}^{∗}. Then ∥f∥_{A_{0}^{∗}} ≤ 1/α, ∥f∥_{A_{1}^{∗}} ≤ 1/β. So, for any x ∈ A we obtain

\begin{aligned}\frac{∥x∥_{A}}{∥x∥_{\frac{1}{α} A_{0}+ \frac{1}{β} A_{1}}} = \frac{∥x∥_{A}}{|f(x)|} & ≥ inf_{x∈A} \frac{∥x∥_{A}}{|f(x)|} = \frac{1}{∥f∥_{A^{∗}}} \\ & ≥ \frac{1}{Ψ(∥f∥_{A_{0}^{∗}}, ∥f∥_{A_{1}^{∗}})} ≥ \frac{1}{Ψ(1/α, 1/β)} \end{aligned}

that proves the assertion after infimizing over all x ∈ A.

Example. Let T be an unbounded selfadjoint operator in a Hilbert space H such that (T x, x) > 0 for all x ∈ H. Consider couple A_{0} = H, A_{1} = D(T) with the norm ∥x∥_{A_{1}} = ∥T x∥. For any increasing concave function φ(t) we can define an operator function φ(T) with the domain D(φ(T)) which we regard as a Banach space A_{φ} with the norm ∥x∥_{A_{φ}} = ∥φ(T)x∥. Using the integral representation of φ(T) and Jensen inequality we obtain the so-called “moment inequality”

∥φ(T)x∥ ≤ ∥x∥ φ(∥T x∥/∥x∥)

which means that

∥x∥_{A_{φ}} ≤ ∥x∥_{A_{0}}φ(∥x∥_{A_{1}}/∥x∥_{A_{0}}).

This gives us immediately that C(α, β) ≤ αφ(β/α).

What is the dual couple in our case? We know that any Hilbert space is dual for itself but for the couple this approach is unacceptable since we must use the common inner product for all spaces under consideration. Taking the space H as a basis we obtain that A_{0}^{∗} = H and A_{1}^{∗} = D(T^{−1}) or it is the completion of H with respect to the norm ∥T^{−1}x∥ if T^{−1} is a bounded operator. By the analogous way we define the space A_{φ}^{∗} and obtain that^{2}

∥φ^{−1}(T)x∥ ≤ \frac{∥x∥}{φ(∥T^{−1}x∥/∥x∥)}

or in other words

∥x∥_{A_{φ}^{∗}} ≤ \frac{∥x∥_{A_{0}^{∗}}}{φ(∥x∥_{A_{0}^{∗}}/∥x∥_{A_{1}^{∗}})}.

This gives us that D(α, β) ≥ αφ(β/α) hence C(α, β) ≤ D(α, β).

The last inequality is not occasional. It is usual for any interpolation space and even for a greater class of spaces that will be seen from the next theorem.

2To check this we should put φ^{−1}(t) = ψ(t^{−1}).

Theorem 3. In order that C(α, β) ≤ D(α, β) for all α, β > 0 it is necessary and sufficient that A be an exact partly interpolation space for the couple A⃗ (i.e. interpolation with respect to one-dimensional operators acting in this couple).

Proof. a)sufficiency. We have to prove that, for any two elements x ∈ A and y ∈ A_{o} ∩ A_{1}, we always have

(3)  \frac{∥y∥_{A}}{∥y∥_{\frac{1}{α} A_{0}∩ \frac{1}{β} A_{1}}} ≤ \frac{∥x∥_{A}}{∥x∥_{\frac{1}{α} A_{0}+ \frac{1}{β} A_{1}}} .

Without loss of generality we may take ∥y∥_{\frac{1}{α} A_{0}∩ \frac{1}{β} A_{1}} = 1. Now let us take a functional f(x) with norm 1 in the conjugate space (\frac{1}{α}A_{0} + \frac{1}{β} A_{1})^{∗} and such that f(x) = ∥x∥_{\frac{1}{α} A_{0}+ \frac{1}{β} A_{1}}. Consider the one-dimensional operator T z = f(z)y and show that the restrictions of this operator on the spaces A_{0}, A_{1} have the norms not greater than 1.

Indeed,

∥T z∥_{A_{0}} = |f(z)| ∥y∥_{A_{0}} ≤ α∥z∥_{\frac{1}{α} A_{0}+ \frac{1}{β} A_{1}} ≤ ∥z∥_{A_{0}}

and so for A_{1}. Thus ∥T x∥_{A} ≤ ∥x∥_{A} and from this

∥y∥_{A} ≤ \frac{∥x∥_{A}}{|f(x)|} = \frac{∥x∥_{A}}{∥x∥_{\frac{1}{α} A_{0}+ \frac{1}{β} A_{1}}}

which proves (3).

b)necessity. Let (3) is true for any x ∈ A, y ∈ A_{0} ∩ A_{1}. Take some one-dimensional operator T x = f(x)y and define

M_{i} = sup_{x∈A_{i}} \frac{|f(x)|}{∥x∥_{A_{i}}} , i = 0, 1.

Then the norm of T on the space A_{i} is equal to M_{i}∥y∥_{A_{i}}. We have to show that

∥T x∥_{A} = max_{i=0,1} M_{i}∥y∥_{A_{i}}∥x∥_{A}.

It follows from (3) that

∥T x∥_{A} = |f(x)| ∥y∥_{A} ≤ \frac{|f(x)|}{∥x∥_{\frac{1}{α} A_{0}+ \frac{1}{β} A_{1}}} ∥y∥_{\frac{1}{α} A_{0}∩ \frac{1}{β} A_{1}}∥x∥_{A}.

Now we show that

\frac{|f(x)|}{∥x∥_{\frac{1}{α} A_{0}+ \frac{1}{β} A_{1}}} ≤ max(αM_{0}, βM_{1}).

Indeed, take arbitrary ε > 0 and let x_{0} ∈ A_{0}, x ∈ A_{1} be such that x = x_{0} + x_{1} and

∥x∥_{\frac{1}{α} A_{0}+ \frac{1}{β} A_{1}} ≥ (1 − ε) (∥x_{0}∥_{\frac{1}{α} A_{0}} + ∥x_{1}∥_{\frac{1}{β} A_{1}}) .

Then

\frac{|f(x)|}{∥x∥_{\frac{1}{α} A_{0}+ \frac{1}{β} A_{1}}} ≤ \frac{|f(x_{0})| + |f(x_{1})|}{(1 − ε)(\frac{1}{α}∥x_{0}∥_{A_{0}} + \frac{1}{β} ∥x_{1}∥_{A_{1}})}

≤ \frac{1}{1 − ε} max [\frac{α|f(x_{0})|}{∥x_{0}∥_{A_{0}}}, \frac{β|f(x_{1})|}{∥x_{1}∥_{A_{1}}}] ≤ \frac{1}{1 − ε} max(αM_{0}, βM_{1})

and it remains only to pass to limit as ε → 0.

So, we have

∥T x∥_{A} ≤ max(αM_{0}, βM_{1}) max(\frac{1}{α}∥y∥_{A_{0}}, \frac{1}{β} ∥y∥_{A_{1}})∥x∥_{A}.

Now we put α = M_{1}, β = M_{0}. Then

∥T x∥_{A} ≤ M_{0}M_{1} max [\frac{∥y∥_{A_{0}}}{M_{1}} , \frac{∥y∥_{A_{1}}}{M_{0}} ] ∥x∥_{A} = max(M_{0}∥y∥_{A_{0}}, M_{1}∥y∥_{A_{1}})∥x∥_{A}.

q.e.d. □

If the interpolation constant of the space A is greater than 1, the inequality for C and D function is fulfilled up to equivalence. The inverse inequality happens more seldom; it is caused not by properties of single space but by properties of the couple on the whole. We give only one result with sufficient condition.

Theorem 4. If a couple A satisfies inequality

inf_{α,β} sup_{x∈Δ(A⃗)} \frac{∥x∥_{αA_{0}+βA_{1}}}{∥x∥_{αA_{0}∩βA_{1}}} ≥ k > 0,

then for any intermediate space D(α, β) ≤ \frac{1}{k} C(α, β).

Proof. Let α, β > 0 be fixed. For abitrary ε > 0 we can find x ∈ Δ(A⃗) such that ∥x∥_{αA_{0}+βA_{1}} ≥ (k − ε)∥x∥_{αA_{0}∩βA_{1}}. It follows then immediately from the definitions of C and D functions that

\frac{D(1/α, 1/β)}{C(1/α, 1/β)} ≤ \frac{∥x∥_{αA_{0}∩βA_{1}}}{∥x∥_{αA_{0}+βA_{1}}} ≤ \frac{1}{k − ε},

and this proves our theorem if ε tends to 0. □

If in Theorem 4 we have k = 1, then for any exact interpolation (and even partly interpolation space) we obtain an equality D(α, β) = C(α, β). The couple (L_{1}, L_{∞}) from Example 1 is quite so. As another example we can take the couple (C, C^{1}) on [0,1].

Is there any connection between C, D-functions of different intermediate spaces in a

given couple? It’s evident that A ⊂^{1} B implies C_{A} ≥ C_{B} and D_{A} ≥ D_{B}. The inverse implication is not true and even the way for obtaining inverse relations is not clear. Nevertheless such relations do exist and the following theorem is valid.

Theorem 5. Let A, B ∈ π(A⃗) and

∫_{0}^{∞} \frac{C_{A}(α, β)}{D_{B}(α, β)} \frac{dβ}{β} = ∫_{0}^{∞} \frac{C_{A}(α, β)}{D_{B}(α, β)} \frac{dα}{α} < ∞.

Then B ⊂ A. In the case of ordered couple A_{0} ⊂ A_{1} the limits of the first integral are from 0 to α, of the second from β to ∞. If A_{1} ⊂ A_{0} then the limits of the first integral are from α to ∞, of the second from 0 to β.

Note that the equality of integrals is not an additional condition. It always follows from the fact that all C and D functions are homogenious of degree one.

The proof of this theorem is rather complicated. We’ll obtain it some later as a corollary from general assertions of the real interpolation theory. Now I only would like to note that in any abundant couple (and even under some weaker conditions) the imbedding of B into A is proper i.e. B ≠ A. This means that in such a couple there is no space A for which the ratio C(1, t)/D(1, t) is integrable with respect to the measure dt/t.

Lorentz and Marcinkiewicz spaces

Now we introduce two important types of intermediate spaces which play a role of marks or of a skeleton for π(A⃗). Let Φ, Ψ ∈ 𝔐.

Definition 3. The intersection of all spaces from π(A⃗) with C(α, β) ≤ Φ(α, β) is called a Lorentz space and is denoted by Λ_{Φ}(A⃗).

Definition 4. The sum of all spaces from π(A⃗) with D(α, β) ≥ Ψ(α, β) is called a Marcinkiewicz space and is denoted by M_{Ψ}(A⃗).

It follows from Definition 3 that Λ_{Φ}(A⃗) ⊃^{1} Φ(α, β)(\frac{1}{α}A_{0} ∩ \frac{1}{β} A_{1}) for any α, β > 0, hence

Λ_{Φ}(A⃗) ⊃ ∑ Φ(α, β)(\frac{1}{α}A_{0} ∩ \frac{1}{β} A_{1})

with summing over all α, β > 0. Really we have here an equation. Indeed, denoting the space on the right by G we obtain that

∥x∥_{G} ≤ inf_{α,β} Φ(α, β)∥x∥_{\frac{1}{α} A_{0}∩ \frac{1}{β} A_{1}}

i.e. C_{G}(α, β) ≤ Φ(α, β) and so G ⊃ Λ_{Φ}(A⃗). This means that

Λ_{Φ}(A⃗) = ∑_{α,β} Φ(α, β)(\frac{1}{α}A_{0} ∩ \frac{1}{β} A_{1}) = ∑_{t} Φ(1, t)(A_{0} ∩ \frac{1}{t}A_{1}).

Analogously, it follows from Definition 4 that

M_{Ψ}(A⃗) = Δ_{α,β}Ψ(α, β)(\frac{1}{α}A_{0} + \frac{1}{β} A_{1}) = Δ_{t} Ψ(1, t)(A_{0} + \frac{1}{t}A_{1}).

Now I note some simple properties of these spaces. The space Δ(A⃗) is dense in any Λ_{Φ}(A⃗) and any M_{Ψ}(A⃗) is complete with respect to Σ(A⃗). All of these spaces are exact interpolation in the couple A⃗. Moreover, the triple (A⃗, Λ_{Φ}(A⃗)) is exact interpolation for the triple (B⃗, Λ_{Φ}(B⃗)) and the triple (A⃗, M_{Ψ}(A⃗)) is exact interpolation for the triple (B⃗, M_{Ψ}(B⃗)) for any two couples A⃗ and B⃗ and any functions Φ, Ψ ∈ 𝔐. The norm in these spaces is monotonically increasing with respect to index hence any intermediate space A satisfies the imbeddings Λ_{Φ}(A⃗) ⊂ A ⊂ M_{Ψ}(A⃗) if C(α, β) ≤ Φ(α, β) and D(α, β) ≥ Ψ(α, β). From this we obtain the following main interpolation theorem.

Theorem 6. Let A ∈ π(A⃗) and C(α, β) ≤ Φ(α, β), D(α, β) ≥ Ψ(α, β). Then for any couple B⃗ the triple (B⃗, Λ_{Φ}(B⃗)) is interpolation for the triple (A⃗, A) and this last is interpolation for the triple (B⃗, M_{Ψ}(B⃗)). □

Immediately from Definition 4 we obtain the expression of the norm in Marcinkiewicz space:

∥x∥_{M_{Ψ}(A⃗)} = sup_{t} Ψ(1, t) K(1, \frac{1}{t}, x, A⃗).

The expression of the norm in Lorentz space can be also obtained from Definition 3 by the help of Theorem 2:

∥x∥_{Λ_{Φ}(A⃗)} = inf ∫_{0}^{∞} Φ(1, t) J(1, \frac{1}{t}, u(t), A⃗) dt

where infimum is taken over all representation of x in the form

x = ∫_{0}^{∞} u(t)dt, u(t) ∈ Δ(A⃗), ∀t > 0.

In the case of the couple (L_{1}, L_{∞}), these spaces coincide with the ordinary Lorentz and Marcinkiewicz rearrangement invariant spaces; from this they have got their names in general case. They are well-known also in some other cases. For the couple (C, C^{1}), the Marcinkiewicz spaces coincide with the Hölder spaces. I will also say some words about the couples L⃗_{1} and L⃗_{∞} which have been introduced in the examples above. It is not difficult to show even directly that

Λ_{Φ}(L⃗_{1}) = L_{1}^{φ}, M_{Φ}(L⃗_{∞}) = L_{∞}^{φ}, φ(t) = Φ(1, \frac{1}{t}).

I hope to obtain this also as a corollary from the main assertions of the real interpolation. The operations of sum and intersection possess the associative property hence

Σ(Λ_{Φ_{ω}}) = Λ_{min Φ_{ω}}, Δ(M_{Ψ_{ω}}) = M_{max Ψ_{ω}}.

In fact, we have rearranged two operations of sum or two operations of intersection that always is possible. But two operations of different kind cannot be rearranged so simply, and the problem of Δ(Λ_{Φ_{ω}}) and Σ(M_{Ψ_{ω}}) is not easy. Only resently, after Brudnyi and Krugliak have proved the property of K-divisibility, it became clear that these operations also give spaces of the same kind if the number of components is finite. Namely,

Δ(Λ_{Φ_{ω}}) = Λ_{max Φ_{ω}}, Σ(M_{Ψ_{ω}}) = M_{min Ψ_{ω}}

with not equal but only equivalent norms.

I think now it is just a moment to tell you about the K-divisibility which will soon play a decisive role in all the theory. Let it be known for some x ∈ Σ(A⃗) that K(1, t, x, A⃗) ≤ ∑ φ_{i}(t), i = 1, 2, ..., with some increasing concave functions φ_{i}. Then there exist elements x_{i} ∈ Σ(A⃗) such that x = ∑ x_{i} with convergence in Σ(A⃗) and for any x_{i}

K(1, t, x_{i}, A⃗) ≤ γφ_{i}(t), ∀t > 0.

Here γ is some universal constant essentially greater than 1. I advise you as an exercise to apply this property to the sum of Marcinkiewicz spaces by comparing the unit ball of the space M_{min Ψ_{ω}} with the convex hull of unit balls of the spaces M_{Ψ_{ω}}. When studying the Lorentz spaces we always may assume that the cuple A⃗ is regular since these spaces are defined only through elements from Δ(A⃗). Under this assumption any (Λ_{Φ}(A⃗))^{∗} = M_{Ψ}(A⃗^{∗}) with Ψ(α, β) = 1/Φ(1/α, 1/β). You should also take into consideration that sums and intersections are dual one to another in the finite case.

K-spaces and K-functors

If the number of components is infinite, the sums of Marcinkiewicz spaces and the intersections of Lorentz spaces give us some new spaces which we start to study now. We shall need one more definition.

Definition 5. An intermediate space A is called K-space if it has the K-monotone norm, that is, for any pair of elements x ∈ Σ(A⃗), y ∈ A, an inequality K(1, t, x, A⃗) ≤ K(1, t, y, A⃗) implies x ∈ A and ∥x∥_{A} ≤ λ∥y∥_{A} with some constant λ independent of x and y. If λ = 1 we add the word “exact ”.

Lemma 1. The sum and the intersection of any number of exact K-spaces also are Kspaces.

Proof. The assertion about intersection is obvious while the assertion about sum may be proved by the use of K-divisibility. Namely, let all A_{ω} be K-spaces, y ∈ Σ(A_{ω}), x ∈ Σ(A⃗) and K(1, t, x, A⃗) ≤ K(1, t, y, A⃗). By consideration of the unit ball of Σ(A_{ω}) and its connection with the unit balls of A_{ω} one can show that for arbitrary ε > 0 there exist y_{ω} ∈ A_{ω} such that y = ∑ y_{ω} and

∥y∥_{Σ(A_{ω})} ≥ (1 − ε) ∑ ∥y_{ω}∥_{A_{ω}}.

There is a subtlety here. The set of all ω may be not countable but the number of non-zero components in any representation of y as a sum always turns to be finite or countable. Since for any t > 0 the K-functional is a norm in Σ(A⃗), we obtain that

K(1, t, x, A⃗) ≤ K(1, t, y, A⃗) = K(1, t, ∑ y_{ω}, A⃗) ≤ ∑ K(1, t, y_{ω}, A⃗).

The property of K-divisibility gives us the existence of x_{ω} such that x = ∑ x_{ω} and K(1, t, x_{ω}, A⃗) ≤ γK(1, t, y_{ω}, A⃗) for any ω. Every A_{ω} is an exact K-space, so ∥x_{ω}∥_{A_{ω}} ≤ ∥γy_{ω}∥_{A_{ω}} and

∥x∥_{Σ(A_{ω})} ≤ ∑ ∥x_{ω}∥_{A_{ω}} ≤ γ ∑ ∥y_{ω}∥_{A_{ω}} ≤ \frac{γ}{1 − ε}∥y∥_{Σ(A_{ω})}.

q.e.d. □

Instead of exact K-property of spaces A_{ω} we could require the existence of a common constant λ in Definition 5 for all these spaces.

Theorem 7. Any Marcinkiewicz space is an exact K-space. If the couple A⃗ is mutually closed, then any Lorentz space also is a K-space.

Proof. The first part follows from Lemma 1 and from the representation of the norm in M_{Ψ}. The second part will be proved if we show that any A_{0} ∩ tA_{1} has K-monotone norm and this follows from the formula

∥x∥_{A_{0}∩tA_{1}} = sup_{s} max(1, t/s) K(1, s, x, A⃗)

which may be shown by direct calculation. Since for any s > 0 the K-functional is a norm on Σ(A⃗), the right term of this formula always defines a space being complete with respect to Σ(A⃗). Just this fact causes the requirement on the couple A⃗ to be mutually closed. □

Theorem 8. For a space A ∈ π(A⃗) to be K-space it is necessary and sufficient that A = Σ(M_{Ψ_{ω}}) for some function set {Ψ_{ω}} ⊂ 𝔐.

Proof. The sufficiency follows from Lemma 1. To prove the necessity define Ψ_{ω}(α, β) = 1/K(1/α, 1/β, y_{ω}, A⃗) for each y_{ω} from the closed unit ball of A. Then for any x from the unit ball of M_{Ψ_{ω}} we have

K(1, t, x, A⃗) ≤ 1/Ψ_{ω}(1, 1/t) = K(1, t, y_{ω}, A⃗)

thus ∥x∥_{A} ≤ ∥y_{ω}∥_{A} ≤ 1. This means that the unit balls of all M_{Ψ_{ω}} are contained in the unit ball of A which turns to be a union of them. All the more it may be regarded as a convex hull and the proof is finished. □

As a consequence we get that any K-space is interpolation. Now we’ll show that it can be obtained by means of a general construction which will be an object of our study.

Definition 6. A rule F which to any Banach couple A⃗ assigns some intermediate space F(A⃗) is called a functor. This functor is said to be interpolation if for any two couples A⃗ and B⃗ the corresponding triples (A⃗, F(A⃗)) and (B⃗, F(B⃗)) are interpolation one for another.

The expressions αA_{0} ∩ βA_{1} and αA_{0} + βA_{1} give us the simplest interpolation functors. The definions of Lorentz and Marcinkiewicz spaces may be also regarded as interpolation functors. If we take some ideal Banach space E and for any couple A⃗ define a space A⃗_{E}^{K} with the norm

∥x∥A⃗_{E}^{K} = ∥K(1, t, x, A⃗)∥_{E},

we again obtain an interpolation functor which we’ll call a K-functor. Any space it yields is a K-space. It turns out that some contrary assertion is also true.

Theorem 9. Any K-space A ∈ π(A⃗) may be represented in the form A = A⃗_{E}^{K}.

Proof. In Theorem 8 we have shown that the unit ball S_{A} of a given K-space A is a union of the unit balles S_{ω} of the spaces M_{Ψ_{ω}} for some set of indices {Ψ_{ω}}. We also have established yet that each space M_{Ψ_{ω}} has a form A⃗_{E_{ω}}^{K} with E_{ω} = L_{∞}^{ψ_{ω}} where ψ_{ω}(t) = Ψ_{ω}(1, 1/t). Let us take E = Σ(E_{ω}) and show that A = A⃗_{E}^{K}. (I leave you as an exercise to prove that the sum of ideal Banach spaces is also an ideal space.) If ∥x∥_{A} = 1 then x ∈ S_{ω} for some ω i.e. K(1, t, x, A⃗) ∈ S_{E_{ω}} ⊂ S_{E}. Thus x ∈ A⃗_{E}^{K} with norm not greater than 1.

In other direction, let x ∈ A⃗_{E}^{K} with unit norm, i.e. K(1, t, x, A⃗) ∈ S_{E}. Since S_{E} is a closed convex hull of S_{E_{ω}}, there exist functions f_{ω}(t) and numbers μ_{ω} such that K(1, t, x, A⃗) = ∑ μ_{ω}f_{ω}, f_{ω} ∈ S_{E_{ω}} and ∑ μ_{ω} = 1. Any f_{ω} satisfies inequality |f_{ω}(t)| ≤ 1/ψ_{ω}(t) and the right-hand function are quasiconcave that enables to use the K-divisibility. We obtain the expansion x = ∑ x_{ω} with K(1, t, x_{ω}, A⃗) ≤ 2γμ_{ω}/ψ_{ω}(t) ∈ 2γμ_{ω}S_{E_{ω}} hence ∥x∥_{A} ≤ ∑ ∥x_{ω}∥_{M_{Ψω}} ≤ 2γ. □

This theorem permits the ideal Banach space E to depend on the space A. Moreover, even for the same space A one can find K-representations with different spaces E. A question may be posed: under which conditions a given interpolation functor F is really a K-functor representable with a common ideal Banach space E? It turns out that the following property is necessary and suffitient: for any two couples A⃗ and B⃗ the inequality K(1, t, x, A⃗) ≤ K(1, t, y, B⃗) implies the norm inequality ∥x∥_{F(A⃗)} ≤ ∥x∥_{F(B⃗)}. In practice we get such a situation when the functor is described only in terms of K-functionals without additional attributes.

Another question: if we have a functor of such kind, how to find a defining space E? A way to do this also is due to Brudnyi and Krugliak.

Theorem 10. Any K-functor F may be represented as A⃗_{E}^{K} with E = F(L⃗_{∞}). Moreover, this representation is optimal in the meaning that for any other representation F(A⃗) = A⃗_{F}^{K} always E ⊂ F.

Proof. Recall that K(1, t, x, L⃗_{∞}) is the least concave majorant of |x|. Since K-functional of any element is concave, the norm of it in F and in (L⃗_{∞})_{F}^{K} is the same for each ideal Banach space F whereas the norms of other elements in F are not greater than those in (L⃗_{∞})_{F}^{K} = F(L⃗_{∞}). In what follows we’ll call E = F(L⃗_{∞}) the optimal parameter of a functor F. □

It is evident that E ⊂ F implies A⃗_{E}^{K} ⊂ A⃗_{F}^{K}. It follows from this that any relation between K-functors such as imbeddings or equalities should be checked only on the couple L⃗_{∞}. Let us demonstrate this principle on the following assertion.

Theorem 11(theorem of reiteration). If E_{0}, E_{1} are optimal parameters of some K-functors then

(4)  (A⃗_{E_{0}}^{K}, A⃗_{E_{1}}^{K})_{E}^{K} = A⃗_{(E_{0},E_{1})_{E}^{K}}^{K} .

Proof. As said above, we check this formula only on the couple L⃗_{∞}. Since E_{0}, E_{1} are optimal parameters, the left-hand side of the equality (4) in the case A⃗ = L⃗_{∞} takes a form

F = ((L⃗_{∞})_{E_{0}}^{K}, (L⃗_{∞})_{E_{1}}^{K})_{E}^{K} = (E_{0}, E_{1})_{E}^{K}.

Moreover, the space F is interpolation for the couple L⃗_{∞} since it was obtained as a superposition of several interpolation functors. Peetre has proved in 1971 that the couple L⃗_{∞} is a Calderon couple, i.e. all its interpolation spaces can be obtained by means of K-functors. Thus F = (L⃗_{∞})_{G}^{K} for some space G and by Theorem 11 the right-hand side of the equality (4) is equal to (L⃗_{∞})_{F}^{K} = F. □

As an example of this theorem application we consider the iteration of Marcinkiewicz functor. Let Ψ_{0}(α, β), Ψ_{1}(α, β), Θ(α, β) ∈ 𝔐. Then

M_{Θ}(M_{Ψ_{0}}(A⃗), M_{Ψ_{1}}(A⃗)) = M_{Ψ}(A⃗), Ψ(α, β) = Θ(Ψ_{0}(α, β), Ψ_{1}(α, β)).

J-functors

Like the K-functors, Lions and Peetre have proposed another real interpolation functor which is similar to the formula of norm in Lorentz spaces.

Definition 7. For a given ideal Banach space E the space A⃗_{E}^{J} is defined as a collection of all x ∈ Σ(A⃗) with finite norm

∥x∥A⃗_{E}^{J} = inf ∥J(1, t, u(t), A⃗)∥_{E}

where infimum is taken over all representations of x in the form

x = ∫_{0}^{∞}u(t)dt/t.

A Lorentz space Λ_{Φ} is a typical representative of J-functors. It has a form Λ_{Φ}(A⃗) = A⃗_{L_{1}^{φ}}^{J} where L_{1}^{φ} is taken with the measure dt/t and the weight φ(t) = Φ(1, 1/t). As well as for the Lorentz spaces, one can show with the help of K-divisibility that in the case of a mutually closed couple A⃗ any A⃗_{E}^{J} is a K-space hence any J-functor coincides with some K-functor. That is why the J-functors almost have not being used in their own form except the Lorentz spaces. In accordance with the main principle we can write that

A⃗_{E}^{J} = A⃗_{F}^{K}, F = (L⃗_{∞})_{E}^{J}.

This formula is valid even for the couple not mutually closed if the space E is not contained in L_{1} or in L_{1}^{1/t} (with the measure dt/t). Otherwise this relation should be slightly changed (Brudnyi-Krugliak, 1980):

A⃗_{E}^{J} = A⃗_{F}^{K} ∩ A_{0} if E ⊂ L_{1},  A⃗_{E}^{J} = A⃗_{F}^{K} ∩ A_{1} if E ⊂ L_{1}^{1/t}.

The connection between J and K functors enables to establish certain properties of J-functors, for instance, the reiteration theorem, completely analogous to Theorem 11. In the case of Lorentz spaces it takes a form

Λ_{Θ}(Λ_{Φ_{0}}(A⃗), Λ_{Φ_{1}}(A⃗)) = Λ_{Φ}(A⃗), Φ(α, β) = Θ(Φ_{0}(α, β), Φ_{1}(α, β)).

I recall that the creators of real interpolation methods Lions and Peetre have studied only a numerical version of them. From the modern point of view this means that they considered as parameters only the L_{p}-spaces with power weights

L_{p,θ} = {u(t) | ∥u∥ = (∫_{0}^{∞}(t^{−θ}u(t))^{p}dt/t)^{1/p} < ∞}, 0 < θ < 1.

Probably, the main achievement of Peetre in this theory was proof of the fact that for such parameters both J and K functors are coincident. His initial proof was very complicated since he hasn’t got any tool so powerful as K-divisibility. Using the couple L⃗_{∞} as a sufficient test we can now study the equivalence of J and K functors in more general form.

Lemma 2. For any ideal Banach space E we have an imbedding A⃗_{E}^{K} ⊂ A⃗_{E}^{J}.

Proof. The J and K functionals in the couple L⃗_{∞} can be easily written explicitly:

K(1, t, x, L⃗_{∞}) ∼^{2} sup_{s} |x(s)| min(1, t/s),

J(1, t, x, L⃗_{∞}) = sup_{s} |x(s)| max(1, t/s).

Taking u(t, s) = \frac{1}{2}x(s) min(s/t, t/s) we obtain

\begin{aligned}∫_{0}^{∞} u(t, s) dt/t = ∫_{0}^{s} \frac{1}{2}x(s)\frac{t}{s}dt/t & + ∫_{s}^{∞} \frac{1}{2}x(s)\frac{s}{t}dt/t \\ & = \frac{1}{2}x(s) + \frac{1}{2}x(s) = x(s), \end{aligned} a)

sup_{s} |u(t, s)| max(1, t/s) = \frac{1}{2} sup_{s} |x(s)| min(s/t, t/s) max(1, t/s)

= \frac{1}{2} sup_{s} |x(s)| min(1, t/s).

b)

Thus for any ideal Banach space E

∥J(1, t, u(t), L⃗_{∞})∥_{E} ≤ ∥K(1, t, x, L⃗_{∞})∥_{E}.

Our u gives us only one of the possible integral representations of x while the norm for J-functor is defined as minimum over all such representations. Hence it is still less and the required imbedding is proved. □

By the help of this lemma we are able now to prove Theorem 5 on imbeddings of intermediate spaces one into another that I have put off up to now. Taking E = L_{1}^{φ} with φ(t) = Φ(1, 1/t), we obtain an inequality for the norm in Lorentz spaces:

∥x∥_{Λ_{Φ}} ≤ ∫_{0}^{∞} Φ(1, 1/t) K(1, t, x, A⃗) dt/t.

If x ∈ M_{Ψ} then K(1, t, x, A⃗) ≤ ∥x∥_{M_{Ψ}}/Ψ(1, 1/t) and so

∥x∥_{Λ_{Φ}} ≤ ∫_{0}^{∞} \frac{Φ(1, 1/t)}{Ψ(1, 1/t)} dt/t ∥x∥_{M_{Ψ}}.

Taking Φ(α, β) = C_{A}(α, β), Ψ(α, β) = D_{B}(α, β), t = α/β, we obtain the desired result.

An imbedding inverse to that in Lemma 2 is not true in general and we shall prove it for a partial case.

Theorem 12. Let (L⃗_{1})_{F}^{K} = (L⃗_{∞})_{F}^{K} = E for some ideal Banach space F. Then the functors A⃗_{E}^{K} and A⃗_{E}^{J} are equivalent.

Proof. For an arbitrary u(ξ, s) such that

x(s) = ∫_{0}^{∞} u(ξ, s) dξ/ξ

we put

v(ξ) = J(1, ξ, u, L⃗_{∞}) = sup_{s} |u(ξ, s)| max(1, ξ/s)

and find

K(1, t, v, L⃗_{1}) = ∫_{0}^{∞} min(1, t/ξ) sup_{s} |u(ξ, s)| max(1, t/s) dξ/ξ

≥ ∫_{0}^{∞} sup_{s} |u(ξ, s)| min(1, t/s) dξ/ξ ≥ sup_{s}(∫_{0}^{∞} |u(ξ, s)| dξ/ξ min(1, t/s))

≥ sup_{s} |x(s)| min(1, t/s) = K(1, t, x, L⃗_{∞}).

This implies that

∥v∥_{E} ≥ ∥x∥_{E} = ∥x∥_{(L⃗_{∞})_{E}^{K}} .

Taking infimum over all possible u(ξ, s), we obtain the imbedding (L⃗_{∞})_{E}^{J} ⊂ (L⃗_{∞})_{E}^{K} that, together with Lemma 2, gives the required equivalence. □

As shown in 1981 by V.Dmitriev, the interpolation from the couple L⃗_{1} to the couple L⃗_{∞} may be described by some K-functor. This means that the representation of E which is required in Theorem 12, is equivalent to the property of a triple (L⃗_{1}, E) to be interpolation for the triple (L⃗_{∞}, E). The last fact is very simply to check since any linear operator from the couple L⃗_{1} to the couple L⃗_{∞} is an integral operator with bounded kernel. Among all such operators which norms do not exceed 1, there exists a maximal one

(5)  T x(t) = ∫_{0}^{∞} min(1, t/s) x(s) ds/s

which alone has to be examined on the space E.

Now I will recall two notions which we have slightly touched upon. The first is a rearrangement invariant space that has equal norms for all equimeasurable functions. As examples we have mentioned L_{p} and Orlicz spaces. The second is a quasi-power function. If there exists a number ε > 0 such that t^{−ε}θ(t) is an increasing function and t^{ε−1}θ(t) is a decreasing function, then the function θ(t) is said to be quasi-power. It is clear that any actual power t^{α}, 0 < α < 1, possess this property. An ideal Banach space E will be said to be quasi-rearrangement invariant if it has a norm ∥x(t)∥_{E} = ∥x(t)θ(1/t)∥_{F} for some rearrangement invariant space F and some quasi-power function θ.

Theorem 13. Every quasi-rearrangement invariant space E admits the representation required in Theorem 12.

Proof. We show that the maximal operator (5) is bounded on such spaces. Indeed, for any t > 0

T x(t) = ∫_{0}^{∞} min(1, 1/s) x(ts) ds/s

and

∥T x(t)∥_{E} ≤ (∫_{0}^{∞} min(1, 1/s) ∥σ_{s}∥_{E} ds/s) ∥x∥_{E}

where σ_{s} is a dilatation operator: σ_{s}x(t) = x(ts). Any rearrangement invariant space F is exact interpolation for the couple (L_{1}, L_{∞}), thus the space E is exact interpolation for the couple (L_{1}^{θ}, L_{∞}^{θ}). Hence we have to examine the operator σ_{s} only on the spaces L_{1}^{θ}, L_{∞}^{θ} where its norm is computed immediately and does not exceed max(s^{ε}, s^{1−ε}) with ε from the definition of θ as a quasi-power function. Using the interpolation property we obtain an inequality

∥σ_{s}∥_{E} ≤ max (s^{ε}, s^{1−ε})

for any quasi-rearrangement invariant space E (with its own ε). It remains to remark that the integral

∫_{0}^{∞} min(1, 1/s) max(s^{ε}, s^{1−ε}) ds/s

is finite for any ε > 0. □

The above mentioned result of Peetre follows from this theorem because any L_{p} is a rearrangement invariant space.

Weak-type interpolation

The most of results we have obtained are due to the simplicity of studied couples such as L⃗, L⃗_{1}, L⃗_{∞}. If we form new couples from intermediate and even interpolation spaces of them, we collide with great difficulties sometimes overwhelming. For instance, any Orlicz space can be obtained from the couple L⃗ by means of K-functors, but the interpolation in couples consisting of Orlicz spaces has been studied only recently and cannot be described by the real interpolation methods. That is why the reiteration theorem are so important for the interpolation theory. They show the properties inherited by the above mentioned new couples. We’ll be especially interested in the so-called weak-type interpolation.

Let us consider some couple A⃗. An operator T will be said to be of weak type (Φ, Ψ) if it is bounded from Λ_{Φ}(A⃗) to M_{Ψ}(A⃗). The weak-type interpolation relates to operators being simultaneously of two weak types (Φ_{0}, Ψ_{0}) and (Φ_{1}, Ψ_{1}). Using general interpolation properties of Lorentz and Marcinkiewicz spaces, we can obtain from this immediately that any such operator is also of weak type (Φ, Ψ) where

(6)  Φ(α, β) = Θ(Φ_{0}(α, β), Φ_{1}(α, β)), Ψ(α, β) = Θ(Ψ_{0}(α, β), Ψ_{1}(α, β))

with arbitrary function Θ ∈ 𝔐. However, in some cases we can establish stronger results due to special kinds of reiteration.

Theorem 14. Let the functions θ(t) = Θ(1, t) and κ(t) = Φ_{1}(1, t)/Φ_{0}(1, t) be quasi-power and let Φ be defined by (6). Then

Λ_{Θ}(M_{Φ_{0}}(A⃗), M_{Φ_{1}}(A⃗)) = Λ_{Φ}(A⃗), M_{Θ}(Λ_{Φ_{0}}(A⃗), Λ_{Φ_{1}}(A⃗)) = M_{Φ}(A⃗).

Proof. Inasmuch as we deal with Lorentz spaces we may consider only regular couples and use the duality between Lorentz and Marcinkiewicz spaces proving only the first relation. Since M_{Φ} ⊃ Λ_{Φ} for any Φ, we always have an imbedding Λ_{Θ}(M_{Φ_{0}}(A⃗), M_{Φ_{1}}(A⃗)) ⊃ Λ_{Φ}(A⃗). The inverse imbedding will be proved if we show that

∥x∥_{Λ_{Φ}} ≤ C Θ(∥x∥_{M_{Φ_{0}}} , ∥x∥_{M_{Φ_{1}}} ).

But we know that

∥x∥_{Λ_{Φ}} ≤ ∫_{0}^{∞} Φ(1, 1/t) K(1, t, x, A⃗) dt/t

and

K(1, t, x, A⃗) ≤ ∥x∥_{M_{Φ_{i}}} / Φ_{i}(1, 1/t), i = 0, 1.

Thus for any N > 0

∥x∥_{Λ_{Φ}} ≤ ∥x∥_{M_{Φ_{0}}} ∫_{0}^{N} Θ(1, κ(t)) dt/t + ∥x∥_{M_{Φ_{1}}} ∫_{N}^{∞} Θ(1/κ(t), 1) dt/t

≤ ∥x∥_{M_{Φ_{0}}} Θ(1, κ(N)) ∫_{0}^{1} \frac{Θ(1, κ(Nt))}{Θ(1, κ(N))}dt/t

+ ∥x∥_{M_{Φ_{1}}} Θ(1/κ(N), 1) ∫_{1}^{∞} \frac{Θ(1/κ(Nt), 1)}{Θ(1/κ(N), 1)}dt/t.

As in Theorem 13 we can show that both of the integrals here are finite and independent of N because of functions Θ, κ being quasi-power. So

∥x∥_{Λ_{Φ}} ≤ C (∥x∥_{M_{Φ_{0}}} Θ(1, κ(N)) + ∥x∥_{M_{Φ_{1}}} Θ(1/κ(N), 1))

and taking N such that κ(N) = ∥x∥_{M_{Φ_{1}}} / ∥x∥_{M_{Φ_{0}}} we prove the theorem. □

Together with the ordinary reiteration of Lorentz and Marcinkiewicz functors, Theorem 14 leads us to the following assertion.

Corollary. Let Λ_{Φ_{i}}(A⃗) ⊂ B_{i} ⊂ M_{Φ_{i}}(A⃗), i = 0, 1, and the functions Θ, κ be as in Theorem 14. Then Λ_{Θ}(B⃗) = Λ_{Φ}(A⃗), M_{Θ}(B⃗) = M_{Φ}(A⃗), where B⃗ = (B_{0}, B_{1}) and Φ is defined by (6). □

The imbeddings of B_{i} as in this corollary are ensured if both of the spaces B_{i} are partly interpolation for the couple A⃗. We have established this in Theorem 3. As functions Φ_{i} we may take then C_{B_{i}}(α, β) or D_{B_{i}}(α, β).

What functors else have an analogous property? It turns out that those are all Kfunctors with quasi-rearrangement invariant parameters.

Theorem 15. Let E be a rearrangement invariant space and E_{θ} = E^{θ(1/t)} for any quasipower θ. If all conditions of the last corollary are fulfilled, then B⃗_{E_{θ}}^{K} = A⃗_{E_{φ}}^{K} where θ(t) = Θ(1, t), φ(t) = Φ(1, t).

Proof. First of all, in reference to Calderon (1966), I will note that any rearrangement invariant space can be obtained applying some K-functor to the couple L⃗ = (L_{1}, L_{∞}). Thus E = L⃗_{G}^{K} with some ideal Banach space G. Because θ(t) is a quasi-power function, we may write that

Λ_{Θ}(B⃗) = B⃗_{L_{1}^{θ}}^{J} = B⃗_{L_{1}^{θ}}^{K}

and thus

(Λ_{Θ}(B⃗), M_{Θ}(B⃗))_{G}^{K} = (B⃗_{L_{1}^{θ}}^{K}, B⃗_{L_{∞}^{θ}}^{K})_{G}^{K} = B⃗_{(L_{1}^{θ},L_{∞}^{θ})_{G}^{K}}^{K} = B⃗_{E_{θ}}^{K}.

Just so may be shown that

(Λ_{Φ}(A⃗), M_{Φ}(A⃗))_{G}^{K} = A⃗_{E_{φ}}^{K},

but from the corollary we know yet that both of the left-hand sides of these relations are equal. Hence the right-hand sides of them also must be the same. □

Now we are able to prove a rather general theorem on weak-type interpolation.

Theorem 16. Under conditions of Theorem 15, let a linear operator T be of weak types (Φ_{0}, Ψ_{0}) and (Φ_{1}, Ψ_{1}) such that both the ratios Φ_{1}/Φ_{0} and Ψ_{1}/Ψ_{0} are quasi-power. Then T : A → B for any A ⊂ A⃗_{E_{φ}}^{K}, B ⊃ A⃗_{E_{ψ}}^{K} where φ(t) = Φ(1, t), ψ(t) = Ψ(1, t) with Φ, Ψ defined by (6).

Proof. This assertion is almost obvious now. Indeed, from the interpolation property of any K-functor, we have immediately that T : (Λ_{Φ_{0}}, Λ_{Φ_{1}})_{E_{θ}}^{K} → (M_{Ψ_{0}}, M_{Ψ_{1}})_{E_{θ}}^{K} and, in virtue of Theorem 15, the first of these spaces is equal to A⃗_{E_{φ}}^{K} while the second is equal to A⃗_{E_{ψ}}^{K}. □

This theorem becomes more powerful if to combine it with other interpolation functors like sum and intersection. Let Ω ⊂ 𝔐 is a set of quasi-power functions Θ(α, β) giving corresponding sets of functions Φ, Ψ representable in the form (6). Under conditions of Theorem 16, we obtain the continuity of T as an operator from A⃗_{E_{φ}}^{K} to A⃗_{E_{ψ}}^{K} for any pair of corresponding indices φ, ψ but the interpolation constants may depend on function Θ. Assume that these constants are uniformly bounded over all set Ω and let E = Σ(E_{φ}), F = Σ(E_{ψ}) with summing over above mentioned sets of indices. Then T : A⃗_{E}^{K} → A⃗_{F}^{K}. The same result is true when E = Δ(E_{φ}), F = Δ(E_{ψ}).

Consider now a partial case when ψ_{0}/φ_{0} = ψ_{1}/φ_{1} = σ(t). Then for any Θ ∈ Ω we have ψ = σφ and thus K(1, t, x, A⃗) ∈ F if and only if σ(t)K(1, t, x, A⃗) ∈ E. As a result, we need not know explicitly the set Ω — only a fact that E has a representation Σ(E_{φ}) or Δ(E_{φ}) over some admissible set of indices. This is enough to assirt that T : A⃗_{E}^{K} → B where B is the collection of all x ∈ Σ(A⃗) for which

∥x∥_{B} = ∥σ(t)K(1, t, x, A⃗)∥_{E} < ∞.

Recommended literature

1. Ju.A.Brudnyi, N.Ja.Krugljak, Interpolation Functors and Interpolation Spaces, North Holland Publish., 1991, p.720.

2. M.Cwikel, Monotonicity properties of interpolation spaces, Ark. Mat. 414 (1976), no. 1, 213–236; 419 (1981), no. 1, 123–136.

3. S.Janson, Minimal and maximal methods of interpolation, J.Funct. Anal. 444 (1981), no. 1, 50–73.

4. S.G.Krein, Ju.I.Petunin, E.M.Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978, p.400 (Russian); Engl. transl.: Amer. Math. Soc., Providence, 1982, p.375.

5. V.I.Ovchinnikov, The Method of Orbits in Interpolation Theory, Math. Reports, Vol. 1, Part 2, Harwood Acad. Publish., 1984, 349–516.

6. E.I.Pustylnik, Minimal and maximal intermediate Banach spaces, Ukrain. Mat. Zhurnal 429 (1977), no. 1, 129–137; Engl. transl.: Ukrainian Math. J. 429(1977), no. 1, 102–107.

7. E.I.Pustylnik, Estimation of the position of intermediate spaces for a Banach couple, Studia Math. 107 (1993), 137–155.