A New Method of Extension of Local Maps of Banach Spaces. Applications and Examples
Genrich Belitskii, Victoria Rayskin

TL;DR
This paper introduces blid maps as a new tool to extend local maps in Banach spaces lacking smooth bump functions, enabling advances in local analysis and applications like derivative reconstruction and solving cohomological equations.
Contribution
It proposes blid maps as an alternative to bump functions for extending local maps in non-smooth Banach spaces, expanding the scope of local analysis techniques.
Findings
Blid maps enable extension of smooth local maps in spaces without bump functions.
Application to reconstruct maps from derivatives at a point.
Facilitates finding global solutions to cohomological equations.
Abstract
A known classical method of extension of smooth local maps of Banach spaces uses smooth bump functions. However, such functions are absent in the majority of infinite-dimensional Banach spaces. This is an obstacle in the development of local analysis, in particular in the questions of extending local maps onto the whole space. We suggest an approach that substitutes bump functions with special maps, which we call blid maps. It allows us to extend smooth local maps from non-smooth spaces, such as . As an example of applications, we show how to reconstruct a map from its derivatives at a point, for spaces possessing blid maps. We also show how blid maps can assist in finding global solutions to cohomological equations having linear transformation of argument.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
A New Method of Extension of Local Maps of Banach Spaces. Applications and Examples
Genrich Belitskii
Department of Mathematics and Computer Science
Ben Gurion University of the Negev
P.O.B. 653
Beer Sheva, 84105
Israel
and
Victoria Rayskin
Department of Mathematics
Tufts University
Medford, MA 02155-5597
Abstract.
A known classical method of extension of smooth local maps of Banach spaces uses smooth bump functions. However, such functions are absent in the majority of infinite-dimensional Banach spaces. This is an obstacle in the development of local analysis, in particular in the questions of extending local maps onto the whole space. We suggest an approach that substitutes bump functions with special maps, which we call blid maps. It allows us to extend smooth local maps from non-smooth spaces, such as . As an example of applications, we show how to reconstruct a map from its derivatives at a point, for spaces possessing blid maps. We also show how blid maps can assist in finding global solutions to cohomological equations having linear transformation of argument.
2010 Mathematics Subject Classification:
Primary 26E15; Secondary 46B07, 58Bxx
The authors thank the referees for helpful suggestions.
1. Introduction
With the advancement of dynamical systems and analysis, the complexity of global analysis became evident. This stimulated the development of techniques for the study of local properties of a global problem. One of the methods of localization is based on the functions with bounded support. The history of applications of functions vanishing outside of a bounded set goes back to the works of Sobolev ([S]) on generalized functions.111Sobolev was a student of N.M. Guenter, and this work was probably influenced by Guenter. However, Guenter was accused in the development of ”abstract” science at the time when the USSR was desperate for an applied theory for the creation of atomic weapons. Guenter was forced to resign from his job. Later, functions with bounded support were used by Kurt Otto Friedrichs in his paper of 1944. His colleague, Donald Alexander Flanders, suggested the name mollifiers. Friedrichs himself acknowledged Sobolev’s work on mollifiers stating that: ”These mollifiers were introduced by Sobolev and the author”. A special type of mollifier, which is equal to 1 in the area of interest and smoothly vanishes outside of a bigger set, we call a bump function.
There are many examples, where bump functions are used for the study of local properties of dynamical systems in . For instance, see [N] and [St]. J. Palis in his work [P] considers bump functions in Banach spaces. He proves the existence of Lipschitz-continuous extensions of local maps with the help of Lipschitz-continuous bump functions. However, Z. Nitecki ([N]) points out that generally speaking, the smoothness of these extensions may not be higher than Lipschitz.
This is an obstacle in the local analysis of dynamical systems in infinite-dimensional spaces. The majority of infinite-dimensional Banach spaces do not have smooth bump functions. In the works [B], [B-R], [R] we discuss the conditions when two diffeomorphisms on some Banach spaces are locally -conjugate. To construct the conjugation, we use bounded smooth locally identical maps. We call them blid maps. Blid maps are the maps that substitute bump functions and allow localization of Banach spaces.
The main objectives of this paper are to present blid maps (Section 2) and to introduce the questions of existence of smooth blid maps on various infinite dimensional Banach spaces and their subsets (Sections 3, 5). One of the important questions that arises in this topic is the following: Which infinite dimensional spaces possess smooth blid maps?
As an application example, for spaces possessing blid maps, we prove an infinite-dimensional version of the Borel lemma on a reconstruction of map from the derivatives at a point (Section 4.1). We also show how blid maps assist in the proof of decomposition lemmas, frequently used in local analysis (Section 4.2). Finally, in Section 4.3 we apply blid maps to the investigation of cohomological equations with a linear transformation of an argument, which frequently arise in the normal forms theory.
We discuss a possibility of extension of a map , where is a neighborhood of a point in a space . More precisely, does there exist a mapping defined on the entire , which coincides with in some (smaller) neighborhood? Because we do not specify this neighborhood, there arises the following notion of a germ at a point.
Let be a real Banach space, be either a real or complex one, and be a point. Two maps and from neighborhoods and of the point into are called equivalent if there is a neighborhood of such that both of the maps coincide on . A germ at is an equivalence class. Therefore, every local map from a neighborhood of into defines a germ at . Sometimes in the literature it is denoted by , although in general we will use the same notation, , as for the map.
We consider Fréchet -maps with . All notions and notations of differential calculus in Banach spaces we borrow from [C].
For a given -germ at , we pose the following questions.
Question 1.1**.**
Does its global -representative (i.e., a -map defined on the whole ) exist?
Question 1.2**.**
Assume that has local representatives with bounded derivatives. Does there exist a global one with the same property?
It was shown in the works [DH] and [HJ] that there exist separable Banach spaces that do not allow -smooth extension of a local -representative222We thank reviewers of this paper for the reference to this result..
Below, without loss of generality we assume that .
Usually for extension of local maps described in Question 1.1 and Question 1.2 bump functions are used. The classical definition of a bump function ([S]) is a non-zero bounded -function from to having a bounded support. We use a similar modified definition which is more suitable for our aims. Namely, a bump function at [math] is a -map which is equal to in a neighborhood of [math] and vanishing outside of a lager neighborhood . If is a local representative of a -germ defined in a neighborhood () then
[TABLE]
is a global -representative of the germ , and it solves at least the Question 1.1. If, in addition, all derivatives of are bounded on the entire , then (1.1) solves both of the Questions. If these functions do exist for any , then every germ has a global representative.
A continuous bump function exists in any Banach space. It suffices to set , where is a continuous bump function at zero on the real line. Let be an even integer. Then
[TABLE]
is a -bump function at zero on . Here is a -bump function on the real line.
However, if is not an even integer, then space does not have -smooth () bump functions (see [M]). The Banach-Mazur theorem states that any real separable Banach space is isometrically isomorphic to a closed subspace of . Consequently, the space of does not have smooth bump functions at all (see [K] or [M]). Following V.Z. Meshkov ([M]), we will say that a space is -smooth, if it possesses a -bump function.333Usually, a space is called smooth if it satisfies a similar property related to the smoothness of a norm (see, for example, [F-M]). However, we will adopt the definition of [M].
Example 1.3**.**
The real function
[TABLE]
defines a (which is even analytic) germ at zero. In spite of the absence of smooth bump functions, the germ has a global representative. To show this, let be a real -function on the real line such that
[TABLE]
Then the -function
[TABLE]
coincides with in the ball and is a global representative of the germ with bounded derivatives of all orders.
2. Blid maps
Definition 2.1**.**
A -blid map for a Banach space is a global Bounded Local Identity at zero -map .
In other words, is a global representative of the germ at zero of identity map such that
[TABLE]
The existance of blid maps allows locally defined mappings to be extended to the whole space.
Theorem 2.2**.**
Let a space possesses a -blid map . Then for every Banach space and any -germ at zero from to there exists a global -representative. Moreover, if all derivatives of are bounded, and contains a local representative bounded together with all its derivatives, then it has a global one with the same property.
Proof.
Let for all , and for . Further, let be a representative of a germ defined on a neighborhood , and let a closed ball . The map
[TABLE]
is a -blid map also, and its image is contained in . Therefore the map
[TABLE]
is well-defined on the whole space , and it coinsides with the map in the neighborhood . The map is a global -representative of the germ . If both of the maps and are bounded together with all of their derivatives, then possesses the same property. This completes the proof. ∎
3. Examples
Let us present spaces having blid maps.
Let be -smooth, and let be a -bump function at zero. Then
[TABLE]
is a -blid map. If the bump function is bounded together with all its derivatives, then has the same property.
- 2.
Let be the space (a Banach algebra) of all continuous functions on a compact Hausdorff space with
[TABLE]
and let h be a -bump function on the real line. Then the map
[TABLE]
is -blid map with bounded derivatives of all orders.
Indeed, since is locally equal to 1, is a local identity. Let be a positive real number such that for all . We can always find such , because bump functions have bounded support.
Then,
[TABLE]
Also,
[TABLE]
Similarly, one can show boundedness of all higher order derivatives.
- 3.
More generally, let be a subspace such that
[TABLE]
for any bump function on the real line.
Then (3.1) defines a -blid map with bounded derivatives. For example, any ideal of the algebra satisfies (3.2).
- 4.
Let be the space (which is also a Banach algebra) of all -functions on a smooth compact manifold with or without boundary, and
[TABLE]
Then (3.1) gives a -blid map with bounded derivatives. The same holds for any closed subspace satisfying (3.2). As above, may be an ideal of the algebra.
- 5.
Let possess a -blid map , and a subspace444By definition, a subspace of a Banach space is always closed. of be -invariant. Then the restriction is a -blid map on .
- 6.
Let be a bounded projector and possess -blid map . Then the restriction is a - blid map on , while the restriction is a -blid map on . Consequently, if is a subspace, such that there exists another subspace of , so that these two form a complementary pair, then possesses a blid map.
Corollary 3.1**.**
Let a space be as in items 1-6. Then for any Banach space and any -germ at zero there is a global -representative. If the germ contains a local representative with bounded derivatives, then there is a global one with the same property.
4. Applications
4.1. The Borel lemma for Banach spaces
Let be a linear space over a field () and be a linear space over a field (). A map is called a polynomial homogeneous map of degree if there is a -linear map
[TABLE]
such that . For the given , the map is not unique, but there is a unique symmetric one. We will assume that is symmetric. Then, the first derivative of at a point , is a linear map , and can be calculated by the formula
[TABLE]
In general, the derivative of order , is a homogeneous polynomial map of degree and equals
[TABLE]
In particular,
[TABLE]
does not depend on . And lastly, for , .
It follows that all derivatives of at zero are zero, except for the order .
The latter equals .
Now, let and be Banach spaces. Recall that must be real, while can be real or complex. Let be a local map. Then
[TABLE]
for any is a polynomial map of degree , and it is continuous and even . Therefore, for some we have the estimate
[TABLE]
Question 4.1**.**
Given a sequence of continuous polynomial maps from to of degree , does there exist a -germ , which satisfies (4.1) for all ?
The classical Borel lemma states, that given a sequence of real numbers , there is a function on the real line such that . This means that the answer to the Question 4.1 is affirmative for . The same is true for finite-dimensional and .
Theorem 4.2** (The Borel lemma).**
Let a Banach space X possess a -blid map with bounded derivatives of all orders. Then for any Banach space Y and any sequence of continuous homogeneous polynomial maps from to there is a -map with bounded derivatives of all orders such that (4.1) is satisfied for all
Proof.
Let be a -blid map at zero with bounded derivatives on . Set . For a given the map is a -blid map also.
Then the map belongs to , and all its derivatives at [math] are zero, except for the order . The latter equals to . In addition, all derivatives of the map are bounded, and the derivative of order allows the following estimate
[TABLE]
with constants depending only on the maps , , and not depending on a choice of . Therefore, under an appropriate choice of the series
[TABLE]
converges in topology to a map from to . It is clear that
[TABLE]
This equality proves the statement. ∎
Corollary 4.3**.**
Let a space X be as in items 1-6 of Section 3. Then for any Banach space Y and any sequence of continuous homogeneous polynomial maps from to there is a -map with bounded derivatives of all orders such that (4.1) is satisfied for all
4.2. Decomposition lemmas
In this section we will state lemmas that are useful in local and global analysis. We will use them in Section 4.3, where we present solution of the cohomological equation.
Let possesses a -blid map , and can be decomposed into a sum of two subspaces,
[TABLE]
Then there is a bounded projector with , and the bounded projector onto . One can write
[TABLE]
Denote by and the corresponding -blid maps on and .
Let now be a Banach space. Recall that a -map is called flat on a subset if it vanishes on together with all its derivatives.
Lemma 4.4**.**
Let all derivatives of be bounded on . Let a map be bounded with all derivatives on every bounded subset, and be flat at zero. Then there is a decomposition and a neighborhood of zero such that () has the same boundedness property and is flat on the intersection ().
Proof.
Let and
[TABLE]
Then are flat on in a neighborhood
[TABLE]
and
[TABLE]
while
[TABLE]
Additionally, the maps satisfy an estimate
[TABLE]
Therefore, the series
[TABLE]
converges in the -topology under an appropriate choice of . Its sum is a -map from flat on . Equations (4.3) and (4.4) imply
[TABLE]
As a result, the map
[TABLE]
is flat on when . ∎
Lemma 4.5**.**
Let a -map vanish in a neighborhood of zero. Then there is a decomposition
[TABLE]
into a sum of maps vanishing on strips and respectively.
Proof.
Let as . One can choose and a -blid map to be such that
[TABLE]
Then the -map
[TABLE]
vanishes as , while the map
[TABLE]
vanishes as ∎
Note that in the Lemma 4.5 we do not assume boundedness of derivatives of map .
4.3. Cohomological equations in Banach spaces
Given a map , the equation
[TABLE]
will be called cohomological equation with respect to the unknown function .
Various versions of this equation are well-known and have been studied in multiple articles. Yu.I. Lyubich in the article [L] presents a very broad overview of this area and considers (4.5) in a non-smooth category. For a discussion of smooth cohomological equations we recommend the book [B-T].
In the present work, we consider a linear space over a field , and a linear map . These cohomological equations are often studied in the theory of normal forms.
Consider a homogeneous polynomial map of degree and linear map . Let us also look for a solution in a polynomial form, . Then we arrive at the equality
[TABLE]
where . If the operator is invertible, then (4.6) has a solution for every . Otherwise, additional restrictions on arise. If is finite-dimensional, then the invertability of is provided by the absence of the resonance relations
[TABLE]
Therefore, the absence of resonance relations for all ensures the solvability of (4.6) for any and any If , then the mentioned condition implies also that is hyperbolic, i.e., its spectrum does not intersect the unit circle in . Consider now (4.5) with real finite-dimensional space , and . If all equations (4.6) are solvable for , then (4.5) is called formally solvable at zero. It is known that if is hyperbolic, invertible authomorphism, then formal solvability implies solvability. Our aim is to prove a similar assertion in the infinite-dimensional case.
So, let be a continuous invertible hyperbolic linear operator. Then, there is a direct decomposition in a sum of -invariant subspaces such that spec lies inside of the circle, while spec lies outside. Moreover, there is an equivalent norm in such that
[TABLE]
with some If (4.5) has a solution, then , and we assume this condition to be fulfilled.
Proposition 4.6**.**
Let one of the subspaces , be trivial, then (4.5) has a global -solution for any -function , with bounded derivatives on every bounded subset .
Proof.
Assume . Then the series
[TABLE]
converges in the space of -functions, since
[TABLE]
for every bounded subset .
Similarly, if , then the series
[TABLE]
leads to a solution we need. ∎
If both of the spaces are non-trivial, then the existence of solutions requires additional assumptions and constructions. As above, is a continuous homogeneous polynomial of degree . Let be a solution of (4.5). Differentiating both of its sides, we arrive at equation
[TABLE]
where , and is a linear map in the Banach space of continuous homogeneous polynomials. It arises after -multiple differentiation of the function . If all equations (4.9n) have continuous solutions, then we say that (4.5) is formally solvable at zero. The latter a-priory takes place if identity does not belong to . The opposite condition can be considered as an infinite-dimensional version of resonance relations.
Theorem 4.7**.**
Let be a hyperbolic linear automorphism, and possesses a -blid map with bounded derivatives on . If all derivatives of f are bounded on every bounded subset, and (4.5) is formally solvable at zero (i.e. all equations (4.9n) have continuous solutions), then there exists a global -solution .
Proof.
First, we will show that a formal solvability implies a local one, i.e., implies existence of (a global) -function such that (4.5) holds in a neighborhood of zero.
Let be continuous solutions of (4.9n). We will build a -function , following the Borel lemma. The substitution reduces (4.5) to the equation
[TABLE]
where is flat at zero. Lemma 4.4 provides the decomposition
[TABLE]
together with a neighborhood . Let be a blid map. One can assume that . Consider the pair of equations:
[TABLE]
Then, the estimates (4.8) imply that the series
[TABLE]
converge since is flat on . The series present solutions to the first and second equations (4.11) correspondingly.
The (global) function satisfies (4.10) in a neighborhood of the origin, i.e. it is local solution for this equation. In turn, the function is a local solution of the initial equation (4.5).
Now we will prove that local solvability implies a global one. Let be a local solution. The substitution reduces (4.5) to the equation
[TABLE]
where vanishes in a neighborhood of zero. Let be a decomposition described by Lemma 4.5. Consider the series
[TABLE]
Since , for any bounded set there is a number such that for . Hence, if , then for all , and (4.13) represents a global smooth function. By the same arguments the series
[TABLE]
is a global smooth function. The sum is a smooth global solution of (4.12). This construction completes the proof. ∎
5. More examples and open questions
One of the important questions of extensions of local maps on Banach spaces is the following. Do Banach spaces without smooth blid maps exist? Recently, affirmative answer was presented in [DH] (also see [HJ]). The authors proved that there exist Banach spaces that do not allow -extension (and hence the -blid map).
Question 5.1**.**
For which spaces do smooth blid maps exist? Do they exist on , with non-even ?
Some other open questions related to the generalization of Theorem 2.2 we discuss below.
How can we extend germs of maps defined at a closed subset ? For this construction we need to define smooth blid maps at . More precisely, generalizing the definition of germs at a point, we will say that maps and from neighborhoods and of into are equivalent, if they coincide in a (smaller) neighborhood of . Every equivalence class is called a germ at . We pose the same question. Given a -germ at , does there exist a global representative? Assume there exist a -map whose image is contained in a neighborhood of and which is equal to the identity map in a smaller neighborhood. Such maps we call smooth blid maps at S. Then every local map defined in can be extended on the whole . It suffices to set .
In the next example, we construct the map for a segment (in particular, for a ball).
Example 5.2**.**
Let be a set of all functions whose graphs are contained in a closed , where is chosen in such a way that . Let be a -function, which is equals to in a neighborhood of and vanishes outside of a bigger set. Then, for an arbitrary
[TABLE]
is a -blid map for .
If for some , then can be thought of as a segment .
In particular, given and a constant , setting and , we obtain the ball .
Every -germ at contains a global representative.
Note, this example has an obvious generalization to segments and balls in .
The Question 5.1 and Example 5.2 bring us to the next question.
Question 5.3**.**
For which pairs do similar constructions exist? In particular, can a smooth blid map be constructed for any bounded subset of a space possessing a smooth blid map? For example, we do not know whether a smooth blid map can be constructed for a sphere .
Question 5.4**.**
The Borel lemma for finite-dimensional spaces is a particular case of the well-known Whitney extension theorem from a closed set . What is an infinite-dimensional version of the Whitney theorem?
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[B] G. Belitskii, The Sternberg theorem for a Banach space, Funct. Anal. Appl., 18 (1984), 238–239. MR 0757253 (86b:58097), doi:10.1007/BF 01086163 .
- 2[B-T] G. Belitskii, V. Tkachenko, One-dimensional functional equations, Oper. Th.: Adv. and Appl., 144, Birkhäauser Verlag, 2003.
- 3[B-R] G. Belitskii, V. Rayskin, Equivalence of families of diffeomorphisms on Banach spaces , Math. preprint archive, UT Austin, 07-71. https://www.ma.utexas.edu/mp arc-bin/mpa?yn=07-71 .
- 4[C] H. Cartan, Calcul Différentiel, Hermann, Paris, 1967 178 pp. MR 0223194 (36 #6243)
- 5[DH] S. D’Alessandro and P. Hajek, Polynomial algebras and smooth functions in Banach spaces , Journal of Functional Analysis 266 (2014), 1627-1646.
- 6[F-M] R. Fry, S. Mc Manus, Smooth bump functions and the geometry of banach spaces: A brief survey , Expositiones Mathematicae, 20(2) (2002) 143–183, https://doi.org/10.1016/S 0723-0869(02)80017-2 · doi ↗
- 7[HJ] P. Hajek and M. Johanis, Smooth Analysis in Banach Spaces, Walter de Gruyter, Gmb H, Berlin , 2014 497 pp.
- 8[M] V.Z. Meshkov, Smoothness properties in Banach spaces, Studia Mathematica, 63 (1978), 111–123. MR 0511298 (80b:46027), http://matwbn.icm.edu.pl/ksiazki/sm/sm 63/sm 6319.pdf .
