A survey on composition operators on some function spaces

We investigate some types of composition operators, linear and not, and conditions for some spaces to be mapped into themselves and for the operators to satisfy some good properties.


Introduction
The following non-linear operators, C f , x → f • x and S h , x(·) → h(·, x(·)) called, respectively, the (autonomous) composition operator and the (nonautonomous) superposition operator, have been widely studied. They especially appear in the process of solving certain non-linear integral equations. For instance, in [4] and [5], the authors show that existence and uniqueness results for solutions of non-linear integral equations of Hammerstein-Volterra type

s)h(s, x(s))ds, (t ≥ 0)
and of Abel-Volterra type are closely related to existence and uniqueness results for solutions of operator equations involving C f and S h . Also, for example, in [29], it is proved, for the integral equation of Volterra type in the Henstock setting, that the existence of a continuous solution depends, among other conditions, on the property of mapping continuous functions into Henstock-integrable functions, satisfied by the involved non-autonomous superposition operator; in [15], the authors provide, in the Henstock-Kurzweil-Pettis setting, existence and closure results for integral problems driven by regulated functions, both in single-and set-valued cases ( [14]). Hence, in many fields of non-linear analysis and its applications (in particular to integral equations), the following problem becomes of interest: Given a class X of functions, find conditions on (and eventually characterise) the functions under which the generated operators map the space X into itself.
The case of the operator S h is called in the literature Superposition Operator Problem ([8,10,11]) or, sometimes, Composition Operator Problem ( [4,6]) since it is also considered for the autonomous case C f and, in this simpler form as well, it is sometimes unexpectedly difficult. In addition to the action spaces of the non-linear operators C f and S h , boundedness and continuity are properties which have also been the object of several studies: many results analysing such properties for composition operators on function spaces, among which Lip, Lip γ , BV , BV p , AC and W 1,p , appeared in the last decades (see, for instance, the papers cited throughout this note).
This note is intended to serve as a survey on the state of the art of some aspects and to describe some further properties of the non-linear operators S h and C f (left composition operator), and the linear operator T f : x → x•f (right composition operator), discuss them and give examples. Clearly, the theory is wide and far from being complete.
This note is organised into four sections, including the introduction. In Sect. 2, we briefly introduce the investigated function spaces, and we recall some main properties.
In Sect. 3, we analyse the non-linear operators C f and S h . First, we investigate them on Lipschitz spaces and some spaces of functions of bounded variation, providing the main results in the literature with examples. Then, we focus on spaces of Baire functions. In particular, we show that when the operator C f maps the space of Baire functions into itself, then it is automatically continuous. We also characterise the non-linear operator S h which transforms Baire one functions into maps of the same type, and we show how to construct a function h easily which is not even Baire one but such that the associated operator S h maps the space of Baire functions into itself.
Sect. 4 is devoted to the linear composition operator T f . We start by investigating Lipschitz spaces and some spaces of functions of bounded variation.
In particular, our study shows that, unlike the case of left compositors, not all the investigated spaces have the same type of right compositors. Then, we study the linear operator T f on the space of Baire one functions and we develop some parallel results on the space of Baire two functions. Unlike the case of left Baire compositors which are the same for Baire classes of any order, and in particular for Baire one functions and Baire two functions, we show that it is not the case when we consider right composition. Namely, we show that right Baire one compositors do not coincide with right Baire two compositors.

Preliminary definitions
In this section, we collect some basic notations, definitions and results, which will be needed in the sequel.
By Lip ([a, b]) and Lip γ ([a, b]), we denote, respectively, the space of all Lipschitz functions on [a, b], and the space of all γ-Lipschitz (or Hölder continuous) functions on [a, b], endowed with the usual norms with

p-variation, Jordan variation, Riesz variation
.,n is an arbitrary finite system of non-overlapping intervals with a i , b i ∈ H, i = 1, ..., n.
If H has a minimal as well as a maximal element, then V p (f, H) is the supremum of the sums where min(H) = t 0 < t 1 < · · · < t n = max(H) and t i ∈ H, i = 0, ..., n.
From now on, in this paragraph, we consider f as a function defined on a closed interval of the real line, that is f : [a, b] → R.   [4,5,7] In the case of continuous functions, we have the following definition.
Recall the following useful (strict) inclusions.
1. For 1 < q < p < +∞, 2. Let 0 < γ < 1. Then . Hence, the previous theorem does not hold for p = 1 as a function in BV ([a, b]) usually does not need to be continuous and therefore nor absolutely continuous.

Baire functions
Let X be a Polish space, that is a separable and completely metrizable space. Recall that an F σ set is a countable union of closed sets, a G δ set is a countable intersection of open sets, and a G δσ set is a countable union of G δ sets ( [12]).

In every metrizable space, any open set is an
, for every x ∈ X. These functions are so called since they were first defined and studied by Baire ([9]). Clearly, each continuous function is of Baire class one.
In general, a real valued function g : X → R is said to be of Baire class n, n ∈ N, if there exists a sequence {g k } k∈N of functions of Baire class n − 1, Denote by B 0 (X) the collection of real valued continuous functions on X, that is B 0 (X) = C(X), and by B n (X), n ≥ 1, the collection of real valued Baire n functions on X.
Then, the following (strict) inclusions hold: Several equivalent definitions of Baire class one functions have been obtained already: it is well-known that "g is Baire one if and only if for every open set A, g −1 (A) is an F σ set", and that "g is Baire two if and only if for every open set A, g −1 (A) is a G δσ set" (see, for instance, [21] and [30]). Given g : X → R, the following are equivalent: 3. for every closed set C in X, the restriction g |C has a point of continuity in C.
Clearly, if a function g : X → R has countably many discontinuity points then it is Baire one. In particular, if g : [a, b] → R is monotone, or of bounded variation, then g is Baire one. In general, functions of Baire class one play an important role in applications. For example, semi-continuous functions and derived functions, all belong to this class ( [12,22]). Some interesting, very recent results concerning fixed points of Baire functions and the so called equi-Baire property can be found in [1] and [2].
If g is Baire one, then the set of points of continuity of g is a residual subset of X. This last property is not a characterisation as the following example shows ( [28]: page 148, Example IV).
Let C be the Cantor ternary set. The set C has Lebesgue measure zero and is of first category since it is nowhere dense. Let C 0 be the collection of the points of P which are not endpoints of the complementary intervals. Let f = χ C and g = χ C0 . Then f and g are continuous at points of [0, 1] \ C and discontinuous at points of C. But f is Baire one as it is the characteristic function of a closed set but g is not Baire one as g |C is discontinuous at every point.
Another well-known example of non Baire one functions is the Dirichlet function.
As g n has finitely many discontinuity points, it is of Baire class one. The Dirichlet map is the pointwise limit of the sequence {g n } n∈N . So, it is Baire two but not Baire one.
The following is a beautiful, natural characterisation of a Baire one function.
. Then the following statements are equivalent.
1. For any > 0, there exists a positive function δ on X such that  [24].
In the sequel, sometimes, when understood, in the above mentioned spaces, we omit X (we write, for example, B 1 rather than B 1 (X)).

Two types of non-linear operators: (left) composition operators and superposition operators
Recall that an operator between two normed spaces is said to be bounded if it maps bounded sets into bounded sets. Clearly, unlike the case of linear operators, in the non-linear case, the two properties of being bounded and being continuous are not equivalent. They are not even, in general, related: a non-linear operator may be continuous without being bounded, or bounded without being continuous.
where g : J → R is an arbitrary function, is called the (autonomous) composition operator generated by the function f . It is usually denoted by C f . Hence, where g : J → R is an arbitrary function, is called the (non-autonomous) superposition operator generated by the function h. It is usually denoted by S h . Hence, for each g : ).

Composition operators
Composition operators on Lipschitz functions and some spaces of functions of bounded variation.   Note that the equivalence between conditions (a) and (d) of Theorem 3.1.2 is a particular case of the following more general result involving Sobolev spaces, which follows from Theorem 1 in [27]: Theorem 3.1.5. Let 1 ≤ q ≤ p < ∞, and let f : R → R be a Borel function. Then the composition operator C f maps the space if and only if f satisfies the local Lipschitz condition ( ). Moreover, the operator C f is bounded and the following inequality holds: Some spaces behave well with respect to the composition operator, as the following results show: Remark 3.1.7. Let 0 < γ ≤ 1. As example 5.25 in [4] shows, there exists a composition operator C f that maps Lip γ ([0, 1]) into itself but is not continuous. In order to have the continuity of C f , extra properties have to be satisfied by the generating function f . In [18] the authors prove that C f is continuous on Lip γ ([a, b]) if and only if f ∈ C 1 (R). In Theorem 5.26 of [4], the authors prove that the continuity of C f , defined from Lip γ ([a, b]) into itself, is equivalent to its uniform continuity on bounded subsets.
Given a space X of real functions defined on a real interval J, in accordance with the terminology used in [30] by Zhao in the particular case of Baire functions, we say that a function f : R → R is a left X compositor if f • g belongs to X whenever g is an element of X. Hence, we can re-write Theorem 3.1.2 as a characterisation of left compositors for some spaces.
compositors are all the same, namely they all coincide with the collection of all maps satisfying ( ).
From Theorem 3.1.8 and the fact that the composition, the sum and the product of two functions satisfying the local Lipschitz condition ( ) still satisfy the local Lipschitz condition ( ), we have the following proposition. Lip γ ([a, b]). Then, the following hold.
1. If f : R → R and g : R → R are left X compositors then so is the sum f + g.

2.
If f : R → R and g : R → R are left X compositors then so is the product fg. 3. If f : R → R and g : R → R are left X compositors then so is the composition f • g.

Composition operators on spaces of Baire functions
Let g : R → R and f : R → R. If g is a Baire one function and f is continuous, then the composition function f • g is Baire one but, as it is well-known, the composition of two Baire one functions is not necessarily Baire one. Here is a well-known example. Then f • g is the Dirichlet function, that is not Baire one.
Notice that, by taking f = χ (0,1] and g the same as above, we still have that f • g is the Dirichlet function. This also shows, as f has a finite number of discontinuity points (namely, exactly one: x = 0), that the last claim in [30] is not true.
Thus, as done above for other spaces, it is natural to ask which functions f have the property that their composition with any Baire one function is still of Baire class one, that is C f maps the space of Baire one functions in itself.
Remark 3.1.12. As already mentioned, Zhao, in [30], calls a function f : R → R for which C f (B 1 ) ⊆ B 1 a left Baire one compositor. That is, f is a left Baire one compositor if and only if f • g is Baire one whenever g is a Baire one function.
The following result follows from Theorem 3 of [17], in the case of Baire one functions and, more generally, from Theorem D of [20], for Baire functions of class n. We have that, in the case of Baire functions, the composition operator behaves well. Namely, the following holds. Theorem 3.1.14. Let n ∈ N. Let f : R → R. If C f maps the space B n into itself then it is automatically continuous with respect to pointwise convergence.
Proof. As C f maps the space B n into itself, by Theorem 3.1.13, f is continuous. Assume that {g k } k∈N is a sequence in B n converging pointwise to a function g of Baire class n. Then C f (g k ) = f • g k is a sequence of Baire functions of class n pointwise converging to the Baire class n function C f (g) = f • g.

Recall that a subset A of B n is said to be bounded if each element h in
A natural question arises: what about right compositors in all the previous spaces? Clearly, right composition g → g • f , defined with a suitable f and on suitable spaces, is linear. This question is investigated in Sect. 4.

Superposition operators on Lipschitz functions and some spaces of functions of bounded variation
As for the case of composition operators, we are interested in investigating which spaces are mapped by S h into themselves and, in general, in the properties of S h . Natural sufficient conditions are the following: In particular, the function h is then necessarily continuous on [a, b] × R.
In [26], the author presents necessary and sufficient conditions for the continuity of a non-autonomous superposition operator in the BV ([a, b]) case:  ([a, b]) be fixed. The following conditions are equivalent: (i) the superposition operator S h is continuous at x; is continuous at u = x(t) and for every > 0 there exists δ > 0 such that, for every k ∈ N, every partition a = t 0 < · · · < t k = b of the interval [a, b], and every finite sequence u 0 , u 1 , . . . , The following result is a special case of Theorem 3.8 in [13] (when X = BV ϕ with the Young function ϕ(t) = t p ).
In [13], the authors provide the following interesting example of an operator S h mapping BV into itself without being bounded or continuous. They take As far as we know, up to now, no characterisation of the functions h is known for the associated operator S h to map BV p ([a, b]) into itself, with p > 1.
Other interesting results concerning the operator S h on the spaces mentioned above and on other spaces like, for instance, Sobolev spaces and Besov spaces, also in higher dimensions, can be found, for example, in [8,10,11].  h(x, g(x)). More precisely, for any g ∈ B 1 ([0, 1]), the following are equivalent: 1. for every positive , there exists a positive Baire one function δ on [0, 1] such that |t − s| < min{δ(t), δ(s)} implies |h(t, g(t)) − h(s, g(s))| < ; 2. the function h(·, g(·)) ∈ B 1 ([0, 1]). Proof. This follows from the fact that the composition of a continuous function with a Baire one function is a Baire one function.

A type of linear operators: right composition operators
where g : I → R is an arbitrary function on I, is the right composition operator generated by f . We, hereby, denote it by T f . Hence, for each function g : As in the case of non-linear operators, we are interested in finding conditions on f in order for T f to map a space of functions X into itself.

Right composition operators on Lipschitz functions and some spaces of functions of bounded variation
In [19], right BV compositors are completely characterised.   All previous results are proved, in [5] and [19], on the unit interval [0, 1], but, of course, it is the same thing if we work on any interval [a, b]. In this general setting, we also prove the following results:  ([a, b]) into itself. Moreover, the operator T f is automatically bounded.
As is arbitrary, g • f is absolutely continuous, that is g • f ∈ AC ([a, b]). Next, we prove the continuity (boundedness). Let {g n } n∈N and g be functions in AC ([a, b]) with lim n→+∞ g n − g AC = 0. As Hence, the thesis. 1. If f and g are k−continuous functions, then so is the sum f + g.
2. If f and g are k−continuous functions, then so is the product fg.

f is a right Baire one compositor.
Another characterisation of right Baire one compositors, involving k−continuous functions, is given in [16]: Next, we give a simple example of a k-continuous function that is not continuous.   Proof. Let f be a continuous function. Then f is also k−continuous. Let g be a right Baire one compositor, then, by Theorem 4.0.13, g is a k−continuous function. Hence, from the properties in Proposition 4.0.11, it follows that f + g and fg are k−continuous functions. By Theorem 4.0.13, this is equivalent to saying that f + g and fg are right Baire one compositors.
Next, we give a characterisation of right Baire two compositors. 1. For any G δ set C, f −1 (C) is a G δσ . 2. For any G δσ set C, f −1 (C) is a G δσ . 3. The operator T f maps the space B 2 into itself (that is, the function f : R → R is a right Baire two compositor).
Proof. (1) and (2) are clearly equivalent as every G δ set is a G δσ set.
(3) ⇒ (1) Suppose f is a right Baire two compositor and C is a G δ set. Let g be the characteristic function of C. As C is both F σδ and G δσ , g is Baire two. Hence g • f is Baire two. Therefore f −1 (C) is a G δσ set because f −1 (C) = (g • f ) −1 (0, 3 2 ). The following example shows that there exist functions which are right Baire two compositors but not right Baire one compositors.
Example 4.0.19. It is well-known that Q is an F σ set, but not a G δ set. As every F σ set is a G δσ set, Q is a G δσ set. Moreover, R \ Q = R \ ∪ q∈Q {q} = ∩ q∈Q (R \ {q}) is a G δ set but not an F σ set. As every G δ set is a G δσ set, R \ Q is a G δσ set. Consider the function f = χ R\Q . Then f −1 ({1}) = R \ Q. As {1} is closed, from condition (1) of Theorem 4.0.12, it follows that f is not a right Baire one compositor. Now, let C be a G δ set. Then if 0, 1 / ∈ C and hence f −1 (C) is a G δσ set. From condition (1) of Theorem 4.0.18, it follows that f is a right Baire two compositor. AEM material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creativecommons.org/licenses/by/4.0/.
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