* * * * *

Let be an operator between vector spaces satisfying some linearity condition. We fix Banach subspaces and fo and and of . If the restrictions and are operators into and , respectively, satisfying some norm estimates, then an interpolation theorem tells us that we can often find “intermediate” Banach spaces between and such that , for each , is a bounded operator into with its norm depending canonically on the norm estimates at the endpoints.

Contents

• Lecture 1 – Strong Endpoint Estimates: The theorems of Riesz-Thorin and Stein. If a linear map is bounded as an operator from to and from to , respectively, the the Riesz-Thorin interpolation theorem establishes the boundedness of as an operator from to , where and depend on the exponents and the parameter . As an immediate application, we shall prove the Hausdorff-Young inequality for the Fourier transform and the generalized Young’s inequality. We also mention the variable-parameter version of Riesz-Throin, proven by E. Stein in his 1955 thesis.
  1. The Fourier transform
  2. The Riesz-Thorin interpolation theorem
  3. Applications: two inequalities
  4. Interpolation on analytic families of operators
• Lecture 2 – Weak Endpoint Estimates: The theorems of Marcinkiewicz and Fefferman-Stein. What if an operator just fails to be bounded at the endpoints? In the second part of the talk, we shall consider the weak-type operators, which are bounded as an operator from a Lebesgue space to a weak Lebesgue space. The prototypical examples of such operators are the Hardy-Littlewood maximal function in the theory of differentiation and the Hilbert transform in the theory of singular integrals. We also mention an extension of Stein’s interpolation theorem by C. Fefferman, which allows the endpoint-operator to be bounded from a Lebesgue space to a Hardy space.
  1. The Hardy-Littlewood maximal function
  2. The Marcinkiewicz interpolation theorem
  3. Application: the Hilbert transform
  4. Interpolation on Hardy Spaces: and

Lecture 1. Strong Endpoint Estimates: The theorems of Riesz-Throin and Stein

The first theorem on our list is the interpolation theorem of M. Riesz and Thorin, concerning operators on Lebesgue spaces. The theorem depends on complex-analytic methods for its proof, whence the operators must take complex scalars. The general theorems of its kind, pioneered by Calderón, are known as complex interpolation.

- 1. The Fourier transform -

To motivate the Riesz-Thorin interpolation theorem, we briefly review the and theory of the Fourier transform on Euclidean spaces. For each , we define the Fourier transform by setting

where is the standard dot product of and . We immediately see that

which shows that the Fourier transform is a bounded operator from to . Straightforward computation yields the following basic properties of the Fourier transform—here we let and :

1. If , then .
2. If , then .
3. If , then .

A natural setting for the study of the Fourier transform is the Schwartz space , which is the collection of smooth functions on that are rapidly decreasing. The formal definition is stated succinctly with the aid of multi-indices, which are vectors in with nonnegative integer components. Each multi-index comes with the following set of shorthand notations:

We are now ready to state the

Definition. The Schwartz space on is the collection of -functions such that

for every pair of multi-indices and .

We remark that contains , hence there are plenty of Schwartz functions. In fact, is dense in for . Furthermore, the Fourier transform of a Schwartz function is again a Schwartz function.

An important result in the theory of the Fourier transform is the Fourier inversion theorem. We shall only need a simple case of the theorem: see pp.5-17 of [SW71] for a discussion of the general version.

Theorem 1 (Fourier inversion theorem). If and , then

for almost every .

Proof. We first note that the Fourier transform of the Gaussican is again the Gaussian, whence Theorem 1 holds for . Setting

for each , we see that

whereby the theorem also holds for all .

We also observe that a simple application of the Fubini-Tonelli theorem yields the following multiplication formula for :

Lastly, we will need the following standard fact, whose proof can be found, for example, in page 243 of [Fol99] or page 63 of [Ste70].

Lemma 2 (Approximation to the identity).Let and set

for each . If for each , then

for every with a fixed .

We are now ready to prove Theorem 1. Consider the following modification of the inversion formula:

.

Since , the dominated convergence theorem implies that

.

We set and note that

by the multiplication formula. Since , we see that

where . Observing that

we conclude from Lemma 2 that

The theorem now follows.

Note that the inversion theorem can be reformulated as

If , then the inversion theorem implies that

Therefore,

whence by the multiplication formula we obtain

This implies that the Fourier transform restricted to is an isometry in the -norm. Since the Schwartz space is dense in , there exists a unique norm-preserving extension of the Fourier transform onto all of . We concede that

- 2. The Riesz-Thorin interpolation theorem -

We have thus seen that the Fourier transform is a bounded linear operator from to and from to , each with the operator norm at most 1. Since the natural “intermediate Banach spaces” of and are for , an interpolation theorem for our example should give us a norm estimate of the Fourier transform operator on .

Note, however, that the Fourier transform and the Fourier transform are, formally speaking, two different operators. We must find a way to describe an operator that takes, in a suitable sense, multiple domains. Taking a cue from how we defined the Fourier transform, we shall consider an operator defined on a collection of functions that is dense in each .

Definition. Let and be -finite measure spaces and fix a vector space of -measurable complex-valued functions on that

• contains characteristic functions of sets of finite measure, and
• is closed under truncation of , viz., a -measurable complex-valued  function defined by

  

for some nonnegative real numbers and . Given , a linear operator on into a vector space of -measurable complex-valued functions on is of type if there exists a constant such that

for all . The smallest such is the -norm of .

Since is dense in , each linear operator of type can be extended uniquely to the operator with the same operator norm. With this definition, we can now state the Riesz-Thorin interpolation theorem as follows:

Theorem 3 (Riesz-Thorin). If a linear operator is of type with -norm and of type with -norm , then is of type with -norm for each , where

We shall need the following lemma, which is the complex-analytic method we have advertised:

Lemma 4 (Hadamard’s three lines lemma). Let be a holomorphic function in the interior of the closed strip

and a bounded continuous function on . If on the line and if on the line , then, for each , we have the inequality

on the line .

The proof of the lemma is an instance of a widely-used maximum modulus principle argument, known as the Phragmén–Lindelöf principle.

Proof of Lemma. We define holomorphic functions

so that as . We wish to show that on .

We first note that on the lines and . Moreover, is bounded above on , and is bounded below on , hence we have the bound on . Noting the inequality

,

we see that converges uniformly to 0 as . We can therefore find such that for all and , whence by the maximum modulus principle we have on the rectangle

.

It follows that on for each , and letting yields the desired result.

We are now ready to present a proof of the Riesz-Thorin interpolation theorem. We write to denote the conjugate exponent of , so that

The following proof is taken from pp. 181-183 of [SW71]:

Proof of Theorem 3. Fix . For notational convenience, we let

With this notation, we set

so that

We shall first prove the theorem for simple functions, which evidently belong to . By the Riesz representation theorem, we have

for each simple function , where the supremum is taken over all simple functions of -norm at most one. Therefore, it suffices to show that

for each such . If , then there is nothing to prove, and so we can assume by renormalization of that . We therefore set out to establish for all simple functions with .

We now suppose that

are two simple functions satisfying the above conditions. We also assume without loss of generality that and , so that and . Write and and set

for each . Then

is an entire function such that

We note, in particular, that is holomorphic in the interior of and is continuous on . Since is linear, we see that

whence is bounded on .

We now furnish a bound for on the lines and . Note first that by Hölder’s inequality. Observing the identities and on the line , we see that

Since is of type , it thus follows that

on the line . A similar computation establishes the bound on the line , whence by Hadamard’s three lines theorem we have the inequality

on the line . Setting , we have

which is the desired inequality.

Having established the theorem for simple functions, we now prove the theorem for the general . To this end, we shall furnish a sequence of simple functions such that

for then Fatou’s lemma yields the inequality

Therefore, the task of proving the theorem reduces to finding such a sequence.

We assume without loss of generality that and . Let be the truncation of , defined as

and another truncation. contains all truncations of , and and are bounded by , hence and . We can find a monotonically increasing sequence converging to , which satisfies

by the monotone convergence theorem. If and are truncations of defined in the same way as and , then we have

Since is of types and , we have

We can then find a subsequence of converging almost everywhere to , whence we may as well assume that the full sequence converges almost everywhere to . Similarly, we can find a subsequence of such that converges almost everywhere to , whence the sequence defined by setting

is the desired sequence. This completes the proof of the Riesz-Thorin interpolation theorem.

- 3. Applications: two inequalities -

We now return to the task of establishing the -boundedness of the Fourier transform. The Fourier transform operator on is of type and of type with the norm at most one in each case. It then follows from the Riesz-Thorin interpolation theorem that the Fourier transform operator is of type with -norm at most one for all . We therefore have the following inequality:

Corollary 5 (Hausdorff-Young). If , then

for all .

We also extend the classical inequality of Young, which can be stated as follows:

Theorem 6 (Young). If , then

for all and .

This implies that the convolution operator is of type with -norm at most . We also note that Hölder’s inequality yields the estimate

whence is of type with -norm at most . It then follows that is of type with the operator norm at most for all . We therefore have the following inequality:

Corollary 7 (Young). If such that

then

for all and

- 4. Interpolation on analytic families of operators -

 We note that the proof of Riesz-Thorin was a study of the function

where was holomorphic in virtue of and depending holomorphically on the complex variable . The two endpoint norm estimates on the operator gave bounds on the lines and , whence the three lines lemma yielded the interpolated bound on . The corresponding bound for was then obtained by a “duality argument”, using the Riesz representation theorem.

What if we add the to the operator and study

instead? If the operator “varies holomorphically” over the strip

so that is holomorphic in , then we could argue as above to obtain an interpolation theorem that allows for varying operators. A collection of such operators is called an analytic family of operators, following the traditional practice of referring to complex analysis as the theory of analytic functions.

Formally, we suppose that , for each , is a linear operator on the space of simple functions in into measurable functions on such that for all simple functions in and all simple functions in . The family of operators is said to be admissible if the function

is holomorphic in the interior of , continuous on , and admits a constant such that

The interpolation theorem of Stein can then be stated as follows:

Theorem 8 (Stein). If is an admissible family of linear operators with the norm estimates

for all simple functions in such that and satisfies

for all and some , then, for , there exists a constant such that

for all such simple functions f, where

In fact, we can take

We will not discuss the formal proof here: see, for example, pp.206-209 of [SW71]. See also Terry Tao’s blog post, which offers a detailed discussion of the theorem.

Lecture 2. Weak Endpoint Estimates: The theorems of Marcinkiewicz and Fefferman-Stein

We now shift gears to study the classical real interpolation theorem of Marcinkiewicz. Unlike Stein’s interpolation theorem, the Marcinkiewicz interpolation theorem does not allow the operator to vary. The improvement, however, is that we may now interpolate with weaker endpoint conditions—the extent of which we shall make precise in what follows. Due to the real-variable nature of the proof, we may take the scalars to be either real or complex.

- 1. The Hardy-Littlewood maximal function -

Recall the differentiation theorem of Lebesgue, which states that we have the almost-everywhere pointwise convergence

for all . Crucial in establishing this theorem is the Hardy-Littlewood maximal function

and the corresponding weak-type inequality

where is a constant independent of and . This inequality cannot be improved for general -functions. Consider, for example, the action of the maximal operator on , a one-dimensional characteristic function. Since

setting yields the equality.

Surprisingly, the weak -bound, combined with the -bound

is sufficient to guarantee that the maximal operator can be extended to a bounded operator for all . The proof of this fact can be generalized considerably, and we are led to yet another interpolation theorem—the Marcinkiewicz interpolation theorem.

- 2. The Marcinkiewicz interpolation theorem -

Before we state the interpolation theorem, let us introduce a way to measure the size of a measurable function in a global manner.

Definition. The distribution function of a measurable function on a -finite measure space is

.

With this notation, we can write the weak-type inequality of the Hardy-Littlewood maximal function as

In general, we define weak-type operators as follows:

Definition. Let and be -finite measure spaces and fix a vector space of -measurable functions on that

1. contains characteristic functions of sets of finite measure, and
2. is closed under truncation of .

An operator on into a vector space of -measurable functions is subadditive if

for almost every , whenever . Given and , a subadditive operator is of weak type if there exists a constant such that

for all . If , the above inequality is replaced by the condition

The smallest such is the weak -norm of  .

Of course, is dense in , and so each subadditive operator of weak type can be extended uniquely to a “bounded” operator on with the same norm. What, then, is the target space of ? Evidently, we need a larger space than the classical Lebesgue spaces. Natural substitutes in this case are the weak Lebesgue spaces, which we define below:

Definition. For , the  weak Lebesgue space is the collection of all -measurable functions such that

is finite.

We remark that is not a norm. In fact, the weak Lebesgue spaces are merely quasi-normed vector spaces. A subadditive operator of weak type can nevertheless be extended uniquely to an operator , which is bounded in the sense that for all , where is the weak -norm of .

We also note that the definition of linear operators of type can be extended in an obvious manner to include subadditive operators. Every operator of type is then of weak type .  The interpolation theorem of Marcinkiewicz can now be stated as follows:

Theorem 9 (Marcinkiewicz). Fix such that

• ,
• ,
• , and
• .

If a subadditive operator is of weak type with weak $(p_0,q_0)$-norm and of weak type with weak -norm , then is of type for each , where

In the interest of time, we will not present a proof of the interpolation theorem. Instead, we merely point out a few key features of the theorem. We first note that, unlike Riesz-Thorin, there is no nice estimate for the interpolated bound. It is known, for example, that the -norm satisfies the inequality

provided that and . see, for example, pp. 31-34 of [Gra10] for a proof of this bound.

The classical proof of the theorem makes use of Hardy’s inequality and relies heavily on the notion of non-increasing rearrangement of a measurable function on that preserves the distribution function and the -norm of $h$. Such a rearrangement is given by

on . Appendix B of [Ste70] contains a proof of the interpolation theorem for .

The proper setting for the weak-type endpoint estimates is theLorentz space , which is a collection of -measurable functions on such that the Lorentz norm

is finite—here $0 < p,q \leq \infty$. The off-diagonal Marcinkiewicz interpolation theorem, due to Zygmund, provides an interpolation result over Lorentz spaces and can be found in Chapter 5, section 3 of [SW71]. An improvement of this result, due to Grakafos, can be found in of [Gra10].

- 3. Application: the Hilbert transform -

We now apply the interpolation theorem to the study of the Hilbert transform, which is, in a sense, the only singular integral operator in one dimension. First, a definition:

Definition. The Hilbert transform of is

If we set , then the Hilbert transform can be written as a convolution operator

.

In the sense of tempered distributions, the Fourier transform of is , and so

It thus follows from Plancherel’s theorem that

and so the Hilbert transform is of type (2,2).

It can also be shown that the Hilbert transform is of weak type (1,1). The proof, which can be found in pp. 187-188 of [SW71], requires a basic knowledge of Fourier analysis on tempered distributions and will not be reproduced here. We assume the weak bound here and apply the Marcinkiewicz interpolation theorem to obtain the interpolated bound

for .

Using the above result, we can obtain, “by duality”, the -bound for all . Recall that if such that

then and . We fix , , and such that . Since $latex  K \in latex L^2(\mathbb{R})$, the double integral

converges absolutely. Therefore, we may rewrite the above integral as the iterated integral

Using the bound for —with the kernel instead of but with the same constant —we can check that belongs to . Furthermore, its norm is bounded by , which is at most . By Hölder’s inequality, we have

whence taking the supremum over all yields

The duality argument presented above is quite general and can be applied to a large class of operators known as singular integrals. Indeed, we have the following theorem:

Theorem 10 (Calderón–Zygmund). Let . If for almost every and , then the operator

is of type for all . The -norm of only depends on p, B, and d; in particular, it does not depend on .

The idea of the proof is as follows. The weak-type (2,2) bound is established via a Plancherel argument, and the famous Calderón–Zygmund decomposition is used to establish the weak-type (1,1) bound. Marcinkiewicz establishes the interpolated bound between (1,1) and (2,2), and a duality argument gives a bound for the rest. For a detailed discussion, see Chapter 2 of [Ste70].

- 4. Interpolation on Hardy spaces: and -

The idea of allowing for weaker endpoint estimates can be applied in the context of complex interpolation as well. As a concluding remark, we mention here the Fefferman’s extension of Stein’s interpolation theorem, which allows for varying operators. The natural substitutes for the Lebesgue spaces in this setting are the (real-variable) Hardy spaces, which behave much nicer under the action of various operators used frequently in harmonic analysis. Formally, the Hardy space is a collection of tempered distributions such that, for some with , the maximal function

is in . It turns out that for .

Note, however, that . In particular, the Hilbert transform maps into . The interpolation theorem, as was introduced in [FS72], replaces the bound in Stein’s interpolation theorem with an bound. By a similar duality argument as the one given above, this interpolation theorem can be extended to for all . The key fact is that the dual of is the John-Nirenberg space of bounded mean oscillation (BMO)—this was one of the main theorems in [FS72].

We will not discuss the interpolation theory on Hardy spaces any further. See Chapters 3 and 4 of [Ste93] for a detailed exposition of real-variable Hardy-space theory: a Fefferman-Stein type interpolation theorem is proved in Chapter 4, section 5.

References

• [FS72] Charles Fefferman and Elias M. Stein, “ spaces of several variables”, Acta Math. 129 (1972), no. 3-4, 137-193.
• [Fol99] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second ed., John Wiley & Sons, 1999.
• [Gra10] Loukas Grafakos, Classical Fourier Analysis, Springer, 2010.
• [SS05] Elias M. Stein and Rami Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, 2005.
• [SS11] Elias M. Stein and Rami Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis, Princeton University Press, 2011.
• [Ste70] Elias M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
• [Ste93] Elias M. Stein, Harmonic Analysis, Princeton University Press, 1993.
• [SW71] Elias M. Stein and Guido Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971.
• [Tao11] Terence Tao, “Stein’s interpolation theorem“, What’s new (http://terrytao.wordpress.com/), May 3, 2011.
• [Wol03] Thomas H. Wolff, Lectures on Harmonic Analysis, American Mathematical Society, 2003.

* * * * *

Part 1. Triangular Fourier Series

What is a triangular Fourier series? Let us suppose that is a -periodic function on . If for some , we know that the classical Fourier series

converges to , both in the -sense (M. Riesz) and in the pointwise almost-everywhere sense (L. Carleson and R. Hunt). These results can be phrased in terms of boundedness of certain operators. For one, the -convergence happens if and only if the operator

is bounded on for all . Similarly, the almost-everywhere convergence happens if and only if the operator

is bounded on .

So then, we have a clear notion of the convergence

.

Why not take the square, and obtain:

.

We expand the left-hand side as follows:

.

The last term is the convolution .

We can now ask ourselves the following questions: (1) Does the following convergence happen?

(2) Similarly, does the following convergence happen?

where ?

As it turns out, we indeed have both convergence results, in the -sense (M. Lacey and C. Thiele), and in the almost-everywhere pointwise sense (C. Muscalu, T. Tao, C. Thiele). Equation (1) is, in fact, the simplest example of triangular Fourier series, whose name is perhaps best explained by the following graph of the range of summation:

, when .

To establish the -theorem, we need to prove that the operator

is bounded uniformly in .  Similarly, establishing the pointwise almost-everywhere convergence amounts to proving that the operator

is bounded uniformly in . More generally, we can consider the multilinear operators

.

Part 2. The Physical Reality

Why do we care about triangular Fourier series? Let us imagine that we have particles (bodies) on the plane, which move in circular orbits. For the sake of simplicity, we assume that the trajectories of the particles are concentric at 0. We can then write to denote the position of the jth particle on the complex plane; we note immediately that . Finally, let us also assume that each is distinct from one another.

Suppose that the particles influence each other in the following way, where for each :

.

In this case, we can write , where

We can now ask the following:

Question 1. Do we have for all ?

To answer this question, we shall embed the system into a more general system

,

where . We can then ask the similar question:

Question 2. Do we have for all and for almost every ?

These systems are called AKNS systems, which are connected to mathematical physics, KdV, NLS, and so on.

Let us now solved the problem posed above. An ansatz is that the solutions are of the form

for some . If this is the case, then we have

where is an n-by-n matrix such that

.

We consider first, as it is as difficult as the whole problem. In fact, if we can understand this case in full generality, then we will have obtained an understanding of the whole problem.

Since we hope to simplify the system further, we would like as many zeroes in our matrix as possible. In other words, we solve the following system:

.

This yields a 2-by-2 upper triangular system

,

whence we obtain

.

If we work in the simple subcase of , then we can solve this system by hand using linear algebra:

This yields, , which implies that

,

whence

.

The above identity yields the bound

.

We would like to have , so that the particles will not fly away. One way to obtain this bound is to have be an -function, whence our problem reduces to investigating the boundedness of

We can consider another continuous analogue of the Carleson operator introduced in the first part of the talk, known as Carleson’s operator:

By Carleson-Hunt, we know that is bounded for . Let us study this operator further.

If for some , and if is the conjugate exponent of , then Carleson-Hunt yields

,

whence we have

by the Hausdorff-Young inequality. Therefore, we have the following inequality:

.

We note, however, that the boundedness of

is given by Menshow-Zygmund (1930s) and so does not require Carleson-Hunt. (If , then we’d need Carleson, which requires much more effort.) So then, what does Carleson by itself give? First and foremost, we have the following inequality for :

.

If and for some , then once again Carleson gives the above estimate. Indeed, the Fourier transform of an function decays very, very slowly.

The conclusion is that the question posed above has an affirmative answer if we have a potential in for some . Furthermore, the question also has an affirmative answer if we have a potential which is the Fourier transform of an -function for some .

We might ask ourselves whether this is true in general. In other words, if, for some , the functions are either themselves or are Fourier transforms of -functions, could we say that no particles fly away? The first question has an affirmative answer, by a result of M. Christ and A. Kiselev from about ten years ago. The second question is a bit more complicated. For simplicity’s sake, we shall consider the problem in . The general case of our problem considers an n-by-n upper triangular matrix, which yields the operator

.

In general, the above operator is bounded. There is also the Carleson variation of this operator—this just means we put a hat on each .

* * * * *

Recall that we adopt the notation of Agnew and Morse [AM38] and write to denote the following:

1. is a real vector space;
2. is a sublinear functional on ;
3. is a linear functional defined on a subspace of ;
4. is dominated by , viz., for all ;
5. is a collection of endomorphisms on ;
6. is invariant under each , viz., for each ;
7. is invariant under each , viz., for each ;
8. is invariant under each , viz., for each .

Furthermore, if is satisfied, we write to denote the collection of extensions of on such that is still dominated by , and that

for all . In this notation, the classical Hahn-Banach theorem can be phrased as follows:

Theorem 1 (Hahn-Banach). Let be a real vector space, and , where is the identity mapping. If , then is nonempty.

Recall that a semigroup is a set with an associative binary operation. In the last talk, we have proved the following symmetry-invariant extension of Hahn-Banach:

Theorem 2 (Agnew-Morse). Let be a semigroup of commuting endomorphisms on a real vector space . If , then is nonempty.

For a proof, see pages 25-27 of Lax [Lax02].

We also recall the key example from the last talk. We write to denote the space of all real-valued sequences, and to denote the space of all bounded real-valued sequences. The classical Banach limit extend the usual limit in the following manner:

Theorem 3. There exists an operator , called a generalized limit or a Banach limit, such that

1. The generalized limit agrees with the usual limit if the usual limit exists;
2. ;
   
3. for any ;
   
4. .

We now see that the above theorem is an easy consequence of the Agnew-Morse theorem. Indeed, the Agnew-Morse theorem furnishes an extension of the regular limit operator that is invariant under the translation operator. A slight modification of the Agnew-Morse theorem allows us to extend the limit operator even further. First, some definitions:

Definition 4. is the space of all real-valued sequences. Let .

• is the translation operator

  .

• is the Hölder mean operator

  .

• For each integer , the map is the -fold iteration operator

  ,

  where each term is repeated times.

Definition 5. We define the following subspaces of $X^*$..

• is the space of all convergent sequences with limit 0.
• is the space of all convergent sequences.
• is the space of all bounded sequences.
• For each integer , is the space of all such that .
• is the set of all sequences in such that , that is, is dominated by asymptotically.
• .

The extended result is the following, which is Theorem 4 in [MWW69]:

Theorem 6. On , there exists a positive linear functional such that is the limit of the sequence whenever . Furthermore, on is invariant under all elements in the semigroup generated by and , and on is invariant under all elements in the semigroup generated by , , and , where .

We omit the proof of this result. Interested readers can find a proof in [MWW69]. We merely point out the variant of the Agnew-Morse theorem required to establish the above theorem. To state the variant, we need some definitions.

Definition 7. A set of endomorphisms on is said to act on a subset of commutatively to within left -factors if each and every furnish such that

.

Definition 8. Let be s set of endomorphisms on and a subset of . We say that is left-solvable over if there exists a sequence

of endomorphisms such that, for each , the set acts on commutatively to within left -factors, and if is commutative.

Here is the required variant, which is Theorem 2 in [MWW69]; interested readers can find a proof in [MWW69]:

Theorem 9. Let be a partially ordered linear space and a set of order-preserving linear transformations of into itself. Let

be a set of -invariant subspaces ( may be infinity) such that the union is . We assume that, for each , either

1. is left-solvable over , and for each , there is an such that ; or
2. for each , there is a such that ; and for each and such that and are in , there are such that

   .

Then every -invariant positive linear functional on can be extended to a -invariant positive linear functional on .

As another application of the Agnew-Morse theorem, we consider generalizations of the Lebesgue measure:

Theorem 10. There exists a nonnegative finitely additive set function that is invariant under rotation.

A proof was presented in the talk, which is taken directly from [Lax02]. Interested readers can find a proof in pages 33-34 of [Lax02]. Powered by Theorem 9, we can generalize the above theorem to the following:

Theorem 11. Let be a compact Hausdorff space and a semigroup of continuous endomorphisms on containing the identity such that is right-solvable over itself. Then there is a Baire measure on which is -invariant and ergodic.

This is Theorem 6 in [MWW69]. No proof of the above result was presented in the talk. The paper contains a partial proof.

References

• [AM38] A. G. Agnew and A. P. Morse, “Extensions of linear functionals, with applications to limits, integrals, measures, and densities,” Annals of Mathematics 39 (1938), no. 1, 20-30.
• [Lax02] Peter D. Lax, Functional Analysis, John Wiley & Sons. 2002.
• [MWW69] E. J. McShane, R. B. Warfield, and V. M. Warfield. “Invariant extensions of linear functionals, with applications to measures and stochastic processes,” Pacific Journal of Mathematics 28 (1969), no. 1, 121-142.

* * * * *

Definition 1. Let be a real vector space. A sublinear functional on is a real-valued function on satisfying

1. Positive homogeneity. for all and ;
2. Subadditivity. for all .

A linear functional is a sublinear functional that is additive, viz.,

for all .

Theorem 2 (Hahn, 1927; Banach, 1932). If a linear functional on a subspace of a real vector space is
dominated by a sublinear functional on , then there exists an extension of on still dominated by the same sublinear functional.

Proof. Extend the linear functional by one dimension and use transfinite induction.

We now discuss a well-known application of the Hahn-Banach theorem, known as Banach limits, which extends the notion of limit to a more general class of sequences.

Definition 3. is the normed linear space of all bounded real-valued sequences with the norm

.

Of course, not every bounded sequence converges. The classical Banach limit extends the notion of limit as follows:

Theorem 4. There exists an operator , called a generalized limit or a Banach limit, such that

1. The generalized limit agrees with the usual limit if the usual limit exists;
2. ;
   
3. for any ;
   
4. .

Proof. We define by setting

This is a sublinear functional on . If is the left-translation map

,

then for all . Let be the space of convergent real-valued sequences, which is a linear subspace of . We define a linear functional by

.

Then for all , and for all . We invoke the Hahn-Banach theorem to furnish an extension of such that is dominated by and that is invariant under left translation. In particular,

,

for each satisfies the inequality . It thus follows that is the desired map.

Note that one of the key properties of the limit operator was “invariance under the translation operator,” viz.,

.

Keeping this in mind, we introduce a new notation, which will bring out the key elements of the Hahn-Banach theorem. The following is a slightly modified version of the notation employed in Agnew and Morse [AM38].

We shall write to denote the following:

1. is a real vector space;
2. is a sublinear functional on ;
3. is a linear functional defined on a subspace of ;
4. is dominated by , viz., for all ;
5. is a collection of endomorphisms on ;
6. is invariant under each , viz., for each .
7. is invariant under each , viz., for each ;
8. is invariant under each , viz., for each .

If , then we write to denote the set of extensions of on such that is still dominated by , and that

for all . With the new shorthand, we can rephrase the classical Hahn-Banach theorem as follows:

Theorem 5 (Hahn-Banach, as stated in [AM38]).  Let be a real vector space, and , where is the identity mapping. If , then is nonempty.

The main theorem of the talk is the symmetry-invariant version of the Hahn-Banach theorem known as the Agnew-Morse theorem. The theorem was first proven by A. G. Agnew and A. P. Morse in 1938 [AM38], and was subsequently generalized by E. J. McShane, R. B. Warfield, and V. M. Warfield in 1969 [MWW69]. We state here the minor generalization of the McShane-Warfield-Warfield formulation of the theorem, as stated in [Lax02]:

Theorem 6 (Agnew-Morse, 1937; McShane-Warfield-Warfield, 1969; Lax, 2002). Let be a real vector space, and be a collection of endomorphisms on that commute, viz.,

for any pair of operators and in . If , then is nonempty.

A remark is in order. If is invariant under a pair of operators , then is also invariant under . Moreover, if and $\Lambda_s$ commute with every operator in , then so does . We may thus enlarge by adding the identity and all finite products of the elements therein; this turns into a semigroup, which we now define:

Definition 7. A semigroup is a set with an associative binary operation.

For example, with the usual addition is a semigroup. We note that the term “semigroup” in functional analysis usually refers to the following:

Definition 8. A one-parameter semigroup of operators over a Banach space $latexX$ is a family of bounded operators such that

1.   ;
2. for all $t,s \geq 0$.

Note that a one-parameter semigroup is essentially a “representation” of the semigroup on the Banach space : it is a homomorphism of into the collection of endomorphisms on .

Let us now return to the main theorem. We have shown that proving Theorem 6 amounts to proving the following variant:

Theorem 9 (Agnew-Morse, the semigroup version)
Let be a semigroup of commuting endomorphisms on a real vector space . If , then is nonempty.

This is the statement of the Agnew-Morse found in [MWW69]. The following proof is taken from [Lax02], pp.25-27:

Proof. Let denote a convex combination of operators in , viz.,

,

where is a nonnegative sequence whose sum is 1, and a sequence of operators in . We set

,

where the infimum is taken over all convex combinations . This is our bounding sublinear functional, as we shall show.

Note that the convex hull of , the collection of all convex combinations , closed under composition. For each convex combination , we have the following inequality:

. (Eq. 1)

Therefore, .

We claim that is a sublinear functional. Indeed,

,

and so is positive homogeneous. To see that is subadditive, we pick arbitrary elements and of . By the definition of , each furnishes a pair of maps and in the convex hull of such that

and . (Eq. 2)

Since and commute, we have

.

The subadditivity of implies that

,

and

by (Eq. 1). It thus follows from the estimate (Eq. 2) that

,

whence is subadditive as was claimed.

We now show that $\sigma$ dominates . By the invariance of and under each operator in , we have the following equality for each convex combination and each element of $laetx Y$:

;

that is, is invariant under each . Since is invariant under each as well, we have

for each . It thus follows that

for all , whence by the Hahn-Banach theorem that there is an extension of on still dominated by .

We claim that is invariant under all operators in . To show this, we use an averaging trick. For a fixed , we set

for each . Since is a semigroup, is in the convex hull of . Therefore, we have

for each . The last equality follows from

,

which is just the geometric series. The subadditivity of implies that

,

and we have

by the invariance of under all operators in . It thus follows that

.

Letting , we now see that

.

Since dominates , the above inequality implies that

,

whence by linearity

.

Replacing with , we also have

,

and so

.

It thus follows that , or

,

as was claimed.

It now suffices to observe that dominates , hence .

Next time, we shall consider a few applications of the Agnew-Morse theorem.

References

• [AM38] A. G. Agnew and A. P. Morse, “Extensions of linear functionals, with applications to limits, integrals, measures, and densities,” Annals of Mathematics 39 (1938), no. 1, 20-30.
• [Lax02] Peter D. Lax, Functional Analysis, John Wiley & Sons. 2002.
• [MWW69] E. J. McShane, R. B. Warfield, and V. M. Warfield. “Invariant extensions of linear functionals, with applications to measures and stochastic processes,” Pacific Journal of Mathematics 28 (1969), no. 1, 121-142.

Let us recall that a topological space is disconnected if there exists a disjoint pair of open subsets and of whose union is , and that is connected otherwise. A discrete-valued map is a continuous map from a topological space to a discrete space , which is a topological space in which every subset is open. An alternate characterization of connectedness is as follows:

Proposition 1 (Alternate characterization of connectedness). A topological space is connected if and only if every discrete-valued map on is constant.

The proof is quite simple. If is connected, then the preimage of an element in the image of is nonempty, open, and closed: therefore, must be the whole space. Conversely, if is not connected, then we can find a disjoint pair of open subsets and of whose union is , whence the map which is 0 on and 1 on is a nonconstant discrete-valued map on .

Why would anyone think of such an alternate definition? Glen Wilson offered the following perspective, which I very much like. First off, following the “categorical” way of thinking, we submit that describing a collection of object in terms of maps between them—as opposed to the objects themselves—is a good thing. But why constant maps? We can consider “labeling” each connected component in a topological space by quotienting out by its connected components. The resulting space is a discrete space, and the quotient map is a surjective continuous map that is constant if and only if is a singleton. Of course, this happens precisely when has one connected component, i.e., if is connected.

This alternate characterization leads us to swift proofs of the key properties of connected space. Let us first consider the following

Proposition 2. The continuous image of a connected space is connected.

Here is a one-liner for the proof: If is a connected space, a continuous map, and any discrete-valued map, then the composition must be constant, and so is connected. The following triumph of intuition also admits a devastatingly simple proof:

Proposition 3. If a collection of connected sets share a point, then the union is connected.

Here, any discrete-valued map must be constant on each connected set, and the value of must be the same because they all share a point: it follows that is constant on the union. Another illustrative example is as follows:

Proposition 4. If is a connected subset of a topological space , and if is a subset of such that , then is connected.

Again, the proof is very short. Any discrete-valued map on is a continuous, hence sequentially continuous, map that is constant on , whence we conclude that the value of on any limit point of is the same as the value takes on . It follows that is constant on , hence on .

Added 8/25/2011: I have purged the blog once again (I left one post—Connectedness and Discrete-Valued Maps—intact). Let’s see how long this lasts.