A Note on Operator Equations Describing the Integral
We study operator equations generalizing the chain rule and the substitution rule for the integral and the derivative of the type f ○ g + c = I (Tf ○ g ∙ Tg), f, g є C¹(R), (1) where T : C¹ (R) → C(R) and where I is defined on C(R). We consider suitable conditions on I and T such that (1) is w...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1067362016-10-04T03:02:34Z A Note on Operator Equations Describing the Integral König, H. Milman, V. We study operator equations generalizing the chain rule and the substitution rule for the integral and the derivative of the type f ○ g + c = I (Tf ○ g ∙ Tg), f, g є C¹(R), (1) where T : C¹ (R) → C(R) and where I is defined on C(R). We consider suitable conditions on I and T such that (1) is well-defined and, after reformulating (1) as V (f ○ g) = Tf ○ g ∙ Tg, f, g є C¹(R) (2) with V : C¹ (R) → C(R), give the general form of T, V and I. Simple initial conditions then guarantee that the derivative and the integral are the only solutions for T and I. We also consider an analogue of the Leibniz rule and study surjectivity properties there. Изучаем операторные уравнения, соответствующие цепному правилу и замене переменных f ○ g + c = I (Tf ○ g ∙ Tg), f, g є C¹(R), (1) где T : C¹(R) → C(R) и где I определен на C(R). Рассматриваем соответствующие условия на I и T такие, что (1) корректно определено и, после перенормировки (1) в форме V (f ○ g) = Tf ○ g ∙ Tg, f, g є C¹1(R) (2) с оператором V : C¹(R) → C(R), мы приводим общую форму T, V и I. Простые начальные условия гарантируют, что производная и интеграл являются единственными решениями для T и I. Также рассматриваем операторной аналог для правила Лейбница и изучаем его сюръективность. 2013 Article A Note on Operator Equations Describing the Integral / H. König, V. Milman // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 1. — С. 51-58. — Бібліогр.: 4 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106736 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We study operator equations generalizing the chain rule and the substitution rule for the integral and the derivative of the type f ○ g + c = I (Tf ○ g ∙ Tg), f, g є C¹(R), (1) where T : C¹ (R) → C(R) and where I is defined on C(R). We consider suitable conditions on I and T such that (1) is well-defined and, after reformulating (1) as V (f ○ g) = Tf ○ g ∙ Tg, f, g є C¹(R) (2) with V : C¹ (R) → C(R), give the general form of T, V and I. Simple initial conditions then guarantee that the derivative and the integral are the only solutions for T and I. We also consider an analogue of the Leibniz rule and study surjectivity properties there. |
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König, H. Milman, V. |
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König, H. Milman, V. A Note on Operator Equations Describing the Integral Журнал математической физики, анализа, геометрии |
| author_facet |
König, H. Milman, V. |
| author_sort |
König, H. |
| title |
A Note on Operator Equations Describing the Integral |
| title_short |
A Note on Operator Equations Describing the Integral |
| title_full |
A Note on Operator Equations Describing the Integral |
| title_fullStr |
A Note on Operator Equations Describing the Integral |
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A Note on Operator Equations Describing the Integral |
| title_sort |
note on operator equations describing the integral |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/106736 |
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A Note on Operator Equations Describing the Integral / H. König, V. Milman // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 1. — С. 51-58. — Бібліогр.: 4 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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2025-07-07T18:55:25Z |
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Journal of Mathematical Physics, Analysis, Geometry
2013, vol. 9, No. 1, pp. 51–58
A Note on Operator Equations Describing the Integral
H. König
Mathematisches Seminar Universität Kiel
24098 Kiel, Germany
E-mail: hkoenig@math.uni-kiel.de
V. Milman
School of Mathematical Sciences Tel Aviv University
Ramat Aviv, Tel Aviv 69978, Israel
E-mail: milman@post.tau.ac.il
Received July 23, 2012
We study operator equations generalizing the chain rule and the substi-
tution rule for the integral and the derivative of the type
f ◦ g + c = I (Tf ◦ g · Tg), f, g ∈ C1(R), (1)
where T : C1(R) → C(R) and where I is defined on C(R). We consider
suitable conditions on I and T such that (1) is well-defined and, after refor-
mulating (1) as
V (f ◦ g) = Tf ◦ g · Tg, f, g ∈ C1(R) (2)
with V : C1(R) → C(R), give the general form of T , V and I. Simple initial
conditions then guarantee that the derivative and the integral are the only
solutions for T and I. We also consider an analogue of the Leibniz rule and
study surjectivity properties there.
Key words: operator equation, chain rule, Leibniz rule, integral.
Mathematics Subject Classification 2010: 39B52(primary), 25A42, 34K30
(secondary).
This note is dedicated to the memory of the famous expert in Geometric
Functional Analysis M.I. Kadets. The second named author has blessed
memories of his personal contacts with the great personality of Mishail
Iosifovich Kadets, and considers himself fortunate that he has had this op-
portunity.
Partially supported by the Minkowski Center at the University of Tel Aviv, by the Alexander
von Humboldt foundation, by ISF grant 387/09 and BSF grant 2006079.
c© H. König and V. Milman, 2013
H. König and V. Milman
1. Introduction and Preliminary Discussion
Generalizing the chain rule D(f ◦ g) = Df ◦ g · Dg for f, g ∈ C1(R), we
studied in [AKM] the operator equation T (f ◦g) = Tf ◦g ·Tg for non-degenerate
operators T : C1(R) → C(R). These operators turned out to be of the form
Tf(x) =
H ◦ f(x)
H(x)
|f ′(x)|p{sgn f ′(x)}
for a suitable function H ∈ C(R), H > 0, a number p ≥ 0 and where the term
{sgn f ′(x)} may be present or not, cf. Theorem 1 of [AKM]. The more general
equation V (f ◦ g) = T1f ◦ g ·T2g ; f, g ∈ C1(R) for operators V, T1, T2 : C1(R) →
C(R) has, up to multiplication by continuous functions, very similar solutions,
cf. Theorem 3 of [KM2].
Looking at the indefinite integral J and the derivative D, the chain rule takes
the form
f ◦ g + c = J (Df ◦ g ·Dg), f, g ∈ C1(R),
with c being a constant. Motivated by this equation, we look for operators
T : C1(R) → C(R) and I defined on C(R) such that
f ◦ g ∼ I (Tf ◦ g · Tg) (1)
holds for all f, g ∈ C1(R). The equivalence ∼ has to be understood in a way so
that (1) yields a well-defined operator I: Assume, e.g., that ∼ means equality.
Then, choosing g = c to be a constant function yields for all f ∈ C1(R)
f(c) = I((Tf)(c) · Tc). (3)
Assuming that T is non-degenerate in the sense that for any c ∈ R there are
functions f1, f2 ∈ C1(R) with f1(c) = f2(c) and Tf1(c) 6= Tf2(c), we find that
I(Tf1(c) · Tc) = f1(c) = f2(c) = I(Tf2(c) · Tc),
so that either Tc = 0 holds for all constant functions or I will not be injective.
If Tc = 0, using (1) for f = c and general g and (3) for general f ∈ C1(R), we
arrive at the conclusion
c = I((Tc) ◦ g · Tg) = I(0) = I((Tf)(c) · Tc) = f(c).
Therefore, if Tc = 0, as in the case of the derivative and the indefinite integral,
the image of I should consist of classes of functions modulo the constants. Let
C ⊂ C1(R) denote the constant functions. To make (1) a meaningful equation
(and also motivated by the indefinite integral) we may require that there are
maps
52 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1
A Note on Operator Equations Describing the Integral
(a) I : C(R) → C1(R)/C and T : C1(R)/C → C(R) satisfying (1) with I being
injective.
For f ∈ C1(R) , denote [f ] := f + C ∈ C1(R)/C. Equation (1) then might
be interpreted as
[f ◦ g] = I (T [f ] ◦ g · T [g]) ; f, g ∈ C1(R). (1’)
Note here that [f ] ◦ g = [f ◦ g].
Alternatively, motivated by the definite integral, we may ask that there are op-
erators
(b) I : C(R) → C1(R) and T : C1(R) → C(R) and a fixed number c ∈ R such
that I is injective and
f ◦ g − (f ◦ g)(c) = I(Tf ◦ g · Tg); f, g ∈ C1(R) (1)
holds. In the next section we give precise statements describing the solutions
of the operator equations (1’) and (1).
2. Results for the Chain Rule
To state the results, we need the following notion of non-degeneracy of T .
Definition 1. A map T : C1(R) → C(R) is called non-degenerate provided
that there is y ∈ R such that for any x ∈ R there is f ∈ C1
b (R) with f(x) = y
and (Tf)(x) 6= 0. Here C1
b (R) denotes the half-bounded C1-functions on R, i.e.,
bounded from above or below (or both). We use a corresponding definition if T
acts as T : C1(R)/C → C(R).
In case (b) we have the following result.
Theorem 1. Assume that I : C(R) → C1(R) and T : C1(R) → C(R) are
operators such that for some fixed c ∈ R
f ◦ g − (f ◦ g)(c) = I(Tf ◦ g · Tg), f, g ∈ C1(R) (1)
holds. Suppose further that T is non-degenerate and that I is injective. Then
there are constants p > 0, d 6= 0 such that
Tf(x) = d |f ′(x)|p (sgn f ′(x)), f ∈ C1(R),
Ih(x) = d−2/p
x∫
c
|h(s)|1/p sgnh(s) ds, h ∈ C(R). (4)
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1 53
H. König and V. Milman
If T satisfies the initial conditions T (2Id) = 2 and T (3Id) = 3 (the constant
functions 2 and 3), we have that p = 1 and d = 1,
Tf(x) = f ′(x), Ih(x) =
x∫
c
h(s)ds.
Hence T is a generalized derivative and I a generalized definite integral. The
two initial conditions may be replaced by T (bId) = b for two different constants
b ∈ R different from 0 and 1. Case (a) leads to an analogue of the indefinite
integral.
Theorem 2. Assume that I : C(R) → C1(R)/C and T : C1(R)/C → C(R)
are operators such that
[f ◦ g] = I (T [f ] ◦ g · T [g]), f, g ∈ C1(R) (1’)
holds. Suppose further that T is non-degenerate and that there is W : C1(R) →
C(R) such that WI : C(R) → C(R) is injective. Then there are constants p > 0,
d 6= 0 and such that
T [f ](x) = d |f ′(x)|p (sgn f ′(x)), f ∈ C1(R),
Ih(x) = d−2/p
x∫
|h(s)|1/p sgn h(s) ds + C, h ∈ C(R) .
P r o o f of Theorem 1. Put d := T (Id), d ∈ C(R). Choose g = Id in (1)
to find that f − f(c) = I(d · Tf), where f(c) denotes the constant function with
value f(c). Since this holds for all f ∈ C1(R), d cannot be identically zero and
I is surjective onto the space C1
c (R) := {h ∈ C1(R)|h(c) = 0} of C1-functions
which are zero in c. Since I is injective by assumption, I is bijective as a map
I : C(R) → C1
c (R). Denote its inverse by Ṽ , Ṽ : C1
c (R) → C(R) and define
V : C1(R) → C(R) by V f := Ṽ (f − f(c)). Applying I−1 to (1) yields
V (f ◦ g) = Ṽ (f ◦ g − (f ◦ g)(c)) = Tf ◦ g · Tg (2)
for all f, g ∈ C1(R). Choosing g = Id and f = Id, respectively, we find that
V f = d Tf and V g = (d ◦ g) Tg, i.e. for any f ∈ C1(R),
V f = d ◦ f · Tf = d · Tf.
Since T is assumed to be non-degenerate, there is y ∈ R such that for any x ∈ R
there is f ∈ C1(R) with f(x) = y and Tf(x) 6= 0. By the preceeding equality,
d(y)(Tf)(x) = d(x)(Tf)(x), i.e., d is a constant function with constant d 6= 0
54 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1
A Note on Operator Equations Describing the Integral
since d was not identically zero. Define Sf := Tf/d. Then V f = d2Sf , Tf = dSf
and by (2)
S(f ◦ g) = Sf ◦ g · Sg, f, g ∈ C1(R)
holds. Since T is non-degenerate and d 6= 0, also S is non-degenerate. Hence by
Theorem 1 of [AKM] there is H ∈ C(R), H > 0 and p ≥ 0 such that either
Sf(x) =
H ◦ f
H
|f ′|p, p ≥ 0, f ∈ C1(R),
or
Sf(x) =
H ◦ f
H
|f ′|p sgn f ′, p > 0, f ∈ C1(R).
We indicate by brackets { } that the term sgn f ′ may be present or not in
the solution formulas. Then the operators V, T satisfying (2) with T being non-
degenerate are of the form
V f = d2 H ◦ f
H
|f ′|p{sgn f ′}, T f = d
H ◦ f
H
|f ′|p{sgn f ′} (5)
with H > 0, d 6= 0 and p ≥ 0.
Let b ∈ R. Applying (1) to g = Id and f as well as f + b yields
f − f(c) = I(d · T (f + b)) = I(d · Tf).
The injectivity of I together with d 6= 0 implies that T (f + b) = Tf , i.e., T does
not depend on shifts by b. Therefore (5) yields for f = Id that H(x + b) = H(x)
for all x ∈ R which means that H is constant. Therefore Tf = d |f ′|p{sgn f ′},
and choosing g = Id in (1) we have
f − f(c) = I(d Tf) = I(d2 |f ′|p {sgn f ′}) =: Ih.
Since I : C(R) → C1
c (R) is bijective and defined also on all negative functions,
the sgn f ′-term has to be present in the right side and p > 0 is required. To find a
formula for I, we have to solve h = d2 |f ′|p sgn (f ′), i.e., f ′ = d−2/p |h|1/p sgn (h).
Since Ih(c) = 0 is required, this gives that
Ih(x) = f(x)− f(c) = d−2/p
∫ x
c
|h(s)|1/p sgnh(s) ds.
Clearly these operators satisfy Eq. (1). In the case that additionally T (2Id) = 2
and T (3Id) = 3, we have p = 1, d = 1.
P r o o f of Theorem 2. Choosing g = Id in (1’) shows that I is surjective
onto C1(R)/C. Let V := I−1 : C1(R)/C → C(R). Then
V ([f ◦ g]) = T [f ] ◦ g · T [g]; f, g ∈ C1(R) (2’)
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1 55
H. König and V. Milman
holds. This is similar as in (2), however, here T and V are defined on function
classes only. Equation (2’) has similar solutions as (2) in terms of H, p and f ′,
cf. (5). The requirement that T [f ] depends only on the class [f ] = f + C again
implies that H is constant, being invariant under shifts by constants b. Then
with d, p as before
V [f ] = d2 |f ′(x)|p {sgn f ′}, T [f ] = d |f ′(x)|p {sgn f ′},
V [f ] = I−1[f ], [f ] = Ih. Again we solve
h = V [f ] = d2 |f ′|p {sgn f ′} (6)
also for non-positive functions h requires the term sgn f ′ to be present in V and
T . We have
f ′ = d−2/p |h|1/p sgn h
and hence
[f ](x) = d−2/p
x∫
|h(s)|1/p sgn h(s) ds + C, h ∈ C(R)
yields a solution [f ] = Ih of (6) and (1’).
3. Leibniz Rule
We now turn to the Leibniz rule operator equation
T (f · g) = Tf · g + f · Tg, f, g ∈ C1(R) (7)
where T : C1(R) → C(R) . It is known [KM1] that any operator T satisfying (7)
has the form
Tf = b f ′ + a f ln |f |, f ∈ C1(R), (8)
where b, a ∈ C(R) and 0 ln |0| := 0. The results for the chain rule operator
equation actually imply that the map T there is surjective. We will now study
surjectivity conditions for T satisfying (7): Let g ∈ C(R). We want to find
f ∈ C1(R) with Tf = g. Then Ig := f is a “generalized” integral in the Leibniz
rule sense. We prove:
Proposition 3. Assume T : C1(R) → C(R) satisfies the Leibniz rule
T (f1 · f2) = Tf1 · f2 + f1 · Tf2; f1, f2 ∈ C1(R) . (7)
Suppose that for all x ∈ R there are g1, g2 ∈ C1(R) with g1(x) = g2(x) and
(Tg1)(x) 6= (Tg2)(x) . Then T is surjective, i.e. Tf = g has a solution f ∈ C1(R)
for any g ∈ C(R) .
56 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1
A Note on Operator Equations Describing the Integral
P r o o f. Choose any g ∈ C(R). To find f ∈ C1(R) with Tf = g, we have
to solve the differential equation
Tf = b f ′ + a f ln |f | = g
in R, using (8). The assumption on T implies that b(x) 6= 0 for all x ∈ R . Let
A := a/b, G := g/b. Then A,G ∈ C(R) and
f ′ + A f ln |f | = G (9)
has to be solved for a suitable f ∈ C1(R). Locally solutions of (9) exist; we
only have to show that no singularity occurs on finite intervals. Assume that
on some bounded open interval J we have that f |J ≥ 2 holds. Since A and G
are continuous and bounded on J , we conclude from (9) that there is a constant
Mj > 0 such that f ′ ≤ MJ f ln |f |. The differential equation F ′ = MJ F ln |F | ,
however, has a bounded solution in J since for x0 ∈ J and initial value F (x0) =
f(x0) ≥ 2
MJ(x− x0) =
x∫
x0
dF (t)
F (t) ln |F (t)| =
F (x)∫
F (x0)
ds
s ln |s| = ln
ln F (x)
ln F (x0)
,
F (x) ≤ F (x0)exp(MJ (x−x0)).
By the generalized Gronwall inequality, cf. [H], Ch. III, Cor. 4.3, |f | ≤ |F | on J .
Therefore (9) admits a locally bounded solution f ∈ C1(R) . A similar argument
applies when f |J ≤ −2 holds.
We now claim that T is uniquely determined by its values Tf1 and Tf2 for
two functions f1, f2 ∈ C1(R) for which there is no open interval in R such that
either for some c1, c2 ∈ R
R | |f1(x)|c1 = |f2(x)|c2} or {x ∈ R | f1(x) ∈ {0, 1}}
or {x ∈ R | f2(x) ∈ {0, 1}}.
In this case
det
(
f ′1 f1 ln |f1|
f ′2 f2 ln |f2|
)
= f ′1f2 ln |f2| − f ′2f1 ln |f1|
= f1f2[(ln |f1|)′(ln |f2|)− (ln |f2|)′(ln |f1|)]
= (f1 ln |f1|)(f2 ln |f2|) · [(ln ln |f1|)′ − (ln ln |f2|)′] .
If (ln ln |f1| − ln ln |f2|)′ = 0 would hold on some open interval I ⊂ R , we
would get ln ln |f1| = ln ln |f2|+ ln c for some constant c > 0 and hence ln |f1| =
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1 57
H. König and V. Milman
c ln |f2| = ln |f2|c, so that |f1| = |f2|c would be true. Hence the above determinant
is non-zero in suitable points in arbitrarily small open intervals. If g1 = Tf1
and g2 = Tf2 are given, the continuous functions b and a in (10) are uniquely
determined by the linear equations for b(x) and a(x),
b(x)f ′j(x) + a(x) fj(x) ln |fj(x)| = gj(x)
in points x where the above determinant is non-zero, and outside these points by
a limiting argument using the continuity of b and a.
Acknowledgement. We would like to thank S. Kuksin for remarks concer-
ning the proof of Proposition 3.
References
[AKM] S. Artstein-Avidan, H. König, and V. Milman, The Chain Rule as a Functional
Equation. — J. Funct. Anal. 259 (2010), 2999–3024.
[H] Ph. Hartman, Ordinary Differential Equations. 2nd ed. Birkhäuser, 1982.
[KM1] H. König and V. Milman, Characterizing the Derivative and the Entropy Func-
tion by the Leibniz Rule, with an Appendix by D. Faifman. — J. Funct. Anal.
261 (2011), 1325–1344.
[KM2] H. König and V. Milman, Rigidity and stability of the Leibniz and the chain
rule. To appear in: Proc. Steklov Inst. Math. 280 (2013).
58 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1
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