Nova S´erie

ON SOME GENERAL NOTIONS OF SUPERIOR LIMIT, INFERIOR LIMIT AND VALUE OF A DISTRIBUTION AT A POINT

J. Campos Ferreira Presented by J.P. Dias

Abstract: For distributions defined on open sets of R^{n}, we define and study
notions of superior limit, inferior limit, and a consequent concept of value at a point,
that is more general than that introduced by S. Lojasiewicz and considered in various
works of Mikusinski, Sebasti˜ao e Silva and the present author.

1 – Introduction

In this work we give definitions of the notions of superior limit and inferior limit of a real distribution of n variables at a point of its domain and study some properties of these notions, showing that they are well connected with the fundamental algebraic operations of distribution theory. For distributions of one variable these questions were studied in [4], the aim of the present work being essentially to extend to the case of several variables some results of that paper.

The concepts of superior and inferior limit generate naturally a notion of value of
a (real or complex) distribution at a point. This notion, keeping all the essential
properties of the homonymous concept considered in [3], [11], [12] and [14], is
much more general than this one (as can be seen in [4] and [8]). The extension
to the case of distributions defined on open sets ofR^{n} of other subjects treated
in [4] — namely applications to the integration of distributions — and also of
some questions studied in other papers ([3], [5] to [9] and [13]) will be the object
of future works.

Received: October 4, 1999.

AMS Subject Classification: 46F10.

Keywords: Distributions; limsup; liminf; value (at a point).

2 – Superior and inferior limits and value for strictly bounded distri- butions

Let n be a positive integer, I_{1}, I_{2}, ..., In open (non empty) intervals of R,
I =I_{1}× · · · ×In,a= (a1, ..., a_{n}) a point inI andI_{a}= (I1\{a1})× · · · ×(In\{an}).

We shall denote byBC(Ia) the (real) vector space of all bounded continuous real
functions defined on I_{a}. For F ∈ BC(Ia), F(a) and F(a) will be, respectively,
the upper and the lower limit of the functionF at the pointa; so, ifk · k is (for
instance) the euclidean norm onR^{n}, we shall have:

F(a) = lim

²→0^{+}sup^{n}F(x) : kx−ak< ², x∈Ia

o , F(a) = lim

²→0^{+}inf^{n}F(x) : kx−ak< ², x∈I_{a}^{o}.

For each j ∈ {1, ..., n} let ρ_{j} be the operator such that, forF ∈ BC(Ia) and
x∈I_{a},

(ρjF)(x) = 1
x_{j}−a_{j}

Z xj

aj

F(x_{1}, ..., x_{j−1}, ξ, x_{j+1}, ..., xn) dξ .
Then we have:

Proposition 2.1. ρj is a linear injection of BC(Ia) into itself; for each
F ∈BC(Ia), puttingG=ρ_{j}F, we have: F(a)≤G(a)≤G(a)≤F(a).

Proof: It is clear that G: I_{a} → Ris a continuous function and that (since
each value G(x) is a certain kind of mean value of the function F) it is also
bounded on I_{a}. To see that G(a) ≤ F(a) notice that, if λ is a real number
greater thanF(a), there exists ² >0 such that, for x∈ Ia and kx−ak< ², we
haveF(x)< λ; then, for the same values of x,

1
x_{j}−a_{j}

Z xj

aj

F(x_{1}, ..., x_{j−1}, ξ, x_{j+1}, ..., x_{n}) dξ < λ

and soG(a)< λ; so we haveG(a)≤F(a). The relationF(a)≤G(a) is obtained
in a similar way. Finally, since we have, for eachx∈I_{a}

F(x) = ∂

∂x_{j}

h(xj−a_{j})G(x)^{i},

we see thatρ_{j} is injective.

It is clear that each functionF ∈BC(Ia) (being defined almost everywhere on
Iand locally integrable on this interval) can be identified with a (real) distribution
defined onI; it is also clear that the distributions that correspond to two distinct
functions of the spaceBC(Ia) are always distinct distributions. So, denoting by
D_{R}^{0} (I) the space of the real distributions defined on I, we can write BC(Ia) ⊂
D_{R}^{0} (I). Now, let us denote by∂_{j} (forj∈ {1, ..., n}) the operator defined onD^{0}_{R}(I)
and such that

∂_{j}(g) =f iff f =D_{j}^{h}(xb_{j}−a_{j})g^{i} ,

where the symbolDj denotes derivation in order toxjin the sense of distributions and the accent over a variable means that it is a dummy variable.

It is obvious that, for eachF ∈BC(Ia) and eachj,∂_{j}(ρjF) =Fand also that
two operators,∂_{i}and∂_{j}, are always interchangeable. We shall put∂=∂_{1}∂_{2}· · ·∂_{n}
and, for eachp∈N,∂^{p}=∂_{1}^{p}∂_{2}^{p}· · ·∂_{n}^{p}. Now we can introduce the following defini-
tion: The distributionf is said to be strictly bounded at the pointa— and we
can writef ∈ B_{a}^{∗}(I) — if there existsF ∈BC(Ia) andp∈Nsuch thatf =∂^{p}F.
It is clear that, if p < q (p, q ∈N) and f = ∂^{p}F with F ∈BC(Ia), there exists
F_{1} ∈ BC(Ia) such that f = ∂^{q}F_{1}: it is sufficient to take F_{1} = ρ^{q−p}F, where
ρ^{q−p} =ρ^{q−p}_{1} · · ·ρ^{q−p}_{n} . From this it follows easily thatB_{a}^{∗}(I) is a subspace of the
real vector spaceD^{0}_{R}(I).

Obviously BC(Ia)⊂ B^{∗}_{a}(I). But a distribution that, on the setIa, is identical
to a function that does not belong to BC(Ia) may well be an element of the
space B^{∗}_{a}(I). For instance — with n = 1, I = R and a = 0 — the distribution
sin_{x}^{1} −D(x sin_{x}^{1}), that coincides with the function ^{1}_{x} cos_{x}^{1} in the set R\{0}, is
strictly bounded at the origin.

Now, let us introduce a result that is essential in the sequel:

Theorem 2.2. For each j ∈ {1, ..., n}, the operator ∂_{j}, restricted to the
spaceB^{∗}_{a}(I), is an automorphism of this vector space.

Before the proof of this theorem (and even before introducing the lemma that will precede that proof) it is convenient to remember the definition of the notion of pseudo-polynomial.

Let J = J_{1}×J2× · · · ×Jn be a (non degenerate) interval of R^{n} and r =
(r_{1}, r_{2}, ..., rn)∈N^{n}. A pseudo-polynomial defined onJ and of degree less thanr
is any functionP that can be put in the form:

P = P_{1}+P_{2}+· · ·+Pn ,

where, for each j ∈ {1, ..., n}, P_{j} is a “polynomial” in x_{j} of degree less than r_{j}
and with “coefficients” that are (real or complex) continuous functions defined

onJ and independent ofxj, that is
P_{j}(x) = P_{j}(x1, ..., x_{n}) =

rXj−1

k=0

a_{jk}(x1, ..., x_{j−1}, x_{j+1}, ..., x_{n})x^{k}_{j} ,
where the functionsa_{jk} are continuous on J.

It is well known that, ifF is a continuous function defined on J, the equality
D^{r}F = D_{1}^{r}^{1}D_{2}^{r}^{2}· · ·D_{n}^{r}^{n}F = 0

(whereD_{j} denotes, as usual, the operator of derivation in order tox_{j} in the sense
of distributions) is verified iff the functionF is a pseudopolynomial of degree less
thanr defined onJ.

Now we can prove the following lemma:

Lemma 2.3. Let J_{1}, J_{2}, ..., J_{n} be n intervals of R unbounded on the left
and J = J_{1}× · · · ×Jn. If the pseudo-polynomial P(x_{1}, ..., x_{n}) defined on J, has
the limit zero when each one of the variablesxj tends to −∞ (the other n−1
variables being fixed in arbitrary points of their domains) then P(x_{1}, ..., x_{n}) is
identically zero.

Proof (of the lemma): We shall use induction on the number of variables,n;

since the result is obvious forn= 1, we shall accept its truth in the case where
the number of variables is n−1 and we shall prove it for a pseudopolynomial
P(x_{1}, ..., xn) in the conditions of the hypothesis of the lemma. We can suppose
that the “polynomials” inx_{1}, x_{2}, ..., x_{n}, of whichP(x1, ..., x_{n}) is the sum, are all
of the same “degree”p−1, since it would always be possible to reduce ourselves
to that case adding, if necessary, some terms with a null coefficient.

So, let us suppose that we have
(1) P(x_{1}, ..., x_{n}) =

Xn

j=1 p−1X

k=0

a_{jk}(x_{1}, ..., x_{j−1}, x_{j+1}, ..., x_{n})x^{k}_{j}

and, chosen p distinct points ξ_{0}, ξ_{1}, ..., ξ_{p−1} in the interval J_{n}, let us consider
the system (with p equations in the unknowns a_{n,0}(x_{1}, ..., x_{n−1}), ...,
a_{n,p−1}(x_{1}, ..., x_{n−1})):

n−1X

j=1 p−1X

k=0

a_{jk}(x1, ..., x_{j−1}, x_{j+1}, ..., x_{n−1}, ξ_{l})x^{k}_{j} +

p−1X

k=0

a_{nk}(x1, ..., x_{n−1})ξ_{l}^{k} =

= P(x_{1}, ..., x_{n−1}, ξ_{l}) (l= 0, ..., p−1),

whose determinant is the Vandermonde determinant of (ξ_{0}, ..., ξ_{p−1}). We can
easily see that the solution of that system can be put in the form

ank(x_{1}, ..., x_{n−1}) =

=

p−1X

l=0

β_{kl}P(x_{1}, ..., x_{n−1}, ξ_{l}) +

n−1X

j=1 p−1X

l=0

b_{jkl}(x_{1}, ..., x_{j−1}, x_{j+1}, ..., x_{n−1})x^{l}_{j}
(k= 0, ..., p−1),
where the β_{kl} are constants and the functionsb_{jkl} are independent of x_{j} and of
x_{n}.

If we substitute these values in the equality (1) we obtain a new equality of the form:

n−1X

j=1 p−1X

l=0

cjl(x_{1}, ..., x_{j−1}, x_{j+1}, ..., xn)x^{l}_{j} =

= P(x_{1}, ..., x_{n}) −

p−1X

k=0 p−1X

l=0

β_{kl}P(x_{1}, ..., x_{n−1}, ξ_{l})x^{k}_{n}.
According to the hypothesis of the lemma, if the variable x_{j} tends to−∞, all
the other variables being fixed, the second member will tend to zero (for every
value ofj). So, for any value of x_{n} in the intervalJ_{n}, the first member (that is
a pseudo-polynomial inx_{1}, ..., x_{n−1} if xn is fixed) tends to zero whenever any of
its variables tends to−∞. Then it follows from the induction hypothesis that we
must have

P(x_{1}, ..., xn) =

p−1X

k=0 p−1X

l=0

β_{kl}P(x_{1}, ..., x_{n−1}, ξ_{l})x^{k}_{n} .

Now, taking into account that the first member — and then also the second, that
is a “polynomial” in xn — has the limit zero whenxn→ −∞, we can conclude
thatP(x_{1}, ..., x_{n}) is identically zero.

Proof (of Theorem 2.2): It is easily seen that the restriction of ∂j to the
spaceB_{a}^{∗}(I) (restriction that will be denoted by the same symbol, ∂_{j}) is a linear
operator from this space onto itself; it is also clear that, to prove that∂_{j} is one-
to-one, it will be enough to show that, from one equality of the form ∂^{p}F = 0
(withF ∈BC(Ia) and p∈N), it follows necessarily F = 0.

Suppose then that we have ∂^{p}F = 0 and denote by K_{1} the set of all points
x= (x_{1}, ..., xn) of the interval I such that xj > aj for every value of j; it is clear
thatK_{1} is one of the connected components of the setI_{a}. Denote also by F_{1} the
restriction of the functionF to the setK_{1}.

Now, starting from the equality ∂^{p}F_{1} = 0 (that follows immediately from
the hypothesis ∂^{p}F = 0), let us change the variables x_{1}, ..., x_{n} to new variables
u_{1}, ..., un, by means of the formulas:

u_{j} = log(xj −a_{j}) (j∈ {1, ..., n}) .

We shall obtain (on one intervalJ_{1}, that is the cartesian product ofnintervals
unbounded on the left) an equality of the form

(2) D^{p}G_{1}(u_{1}, ..., u_{n}) = 0
where

G_{1}(u_{1}, ..., un) = e^{u}^{1}^{+···+u}^{n}F_{1}(a_{1}+e^{u}^{1}, ..., an+e^{u}^{n}) .

Taking into account the equality (2) and the fact that G_{1} is a continuous
function on J_{1}, we see that G_{1} is a pseudo-polynomial defined on this interval;

since the function F is bounded we see also that G_{1} tends to zero when each
one of the variablesuj tends to −∞, the other variables staying fixed. Now the
lemma allows us to conclude thatG_{1}= 0 and so thatF = 0 onK_{1}.

We should prove analogously that F is equal to zero in each one of the other
connected components ofI_{a}(the change of variables to consider would be defined
in each case by the systemuj = log|xj −aj|,j∈ {1, ..., n}). So we can conclude
thatF is the null function on I_{a}.

For each j, let∂_{j}^{−1} be the inverse (of the restriction toB_{a}^{∗}(I)) of the operator

∂_{j}; for each p∈N, put∂^{−p} =∂_{1}^{−p}· · ·∂_{n}^{−p}. Moreover, iff is a distribution of the
space B_{a}^{∗}(I), put fp =∂^{−p}f. Then it is clear that we have fp ∈ B_{a}^{∗}(I) for every
value ofp and also f_{p}∈BC(Ia) if the value of p is sufficiently large.

For f ∈ B_{a}^{∗}(I) we shall call degree off at the point a — deg_{a}f — the least
value ofp∈Nsuch thatf_{p} ∈BC(I_{a}). Then , for eachf ∈ B_{a}^{∗}(I), we can consider
two sequences: {fp(a)} and {fp(a)} (with p∈N,p≥deg_{a}f), where f_{p}(a) [resp.

f_{p}(a)] denotes as before the upper limit [resp. lower limit] of the functionf_{p} at
the pointa.

From Proposition 2.1 it follows that, for p≥deg_{a}f, we have:

fp(a) ≤ f_{p+1}(a) ≤ f_{p+1}(a) ≤ fp(a) .
So we can introduce the following definitions:

Let f ∈ B_{a}^{∗}(I) and, for each p≥deg_{a}f, let fp =∂^{−p}f; then, the limit of the
sequence f_{p}(a) [resp. f_{p}(a)] will be called superior limit [resp. inferior limit] of
the distributionf at the pointaand will be denoted by lim sup

a f [resp. lim inf

a f].

For any distribution f ∈ B^{∗}_{a}(I) we have clearly: lim inf

a f ≤lim sup

a f. If the equality is verified, we say that the distribution f is strictly continuous at the point a and the common value of the superior and inferior limits, denoted by f(a), is called the value off at the pointa.

We shall use the symbol V_{a}^{∗}(I) to denote the set of all distributions that are
strictly continuous ata.

It is convenient to observe that, when f is a function in the space BC(Ia), the equalities

f(a) = lim sup

a f , f(a) = lim inf

a f

are not generally satisfied. For instance, with n= 1, I =R, a= 0 and h(x) =
sin_{x}^{1}, we have, as it is easily verified,

lim sup

0

h(x) = lim inf

0 h(x) = 0 andh(0) = 1, h(0) =−1.

In any case it is clear that, for f ∈BC(Ia), we always have:

(3) f(a) ≤ lim inf

a f ≤ lim sup

a f ≤ f(a) .

So, in a sense, the notions of superior and inferior limit of a distribution at a point do not precisely generalize the usual homonymous notions for functions, although, as we shall see in the next chapter, they share with them a lot of significant properties.

On the other hand, it follows immediately from the inequalities (3) that, if
the function f has a limit at the point a in the usual sense, then it belongs to
the spaceV_{a}^{∗}(I) and the valuef(a) coincides with that limit.

3 – Superior and inferior limits and value: the general case

Let A be an open set of R^{n},a a point in A and f a real distribution defined
inA.

We shall say that f isbounded at the point a, and we shall write f ∈ Ba(A),
iff there exists an open intervalI of R^{n} such that

i) a∈I ⊂A, and
ii) f_{|I} ∈ B^{∗}_{a}(I),

(wheref_{|I} denotes the restriction of f to the intervalI).

Now, we easily recognize the coherence of the following definitions: If f ∈ Ba(A) and ifI is an interval such that conditions i) and ii) are satisfied, the real number lim sup

a f_{|I} [resp. lim inf

a f_{|I}] will be called superior limit [resp. inferior
limit] of f at the point a, and will be denoted by lim sup

a f [resp. lim inf

a f]. If lim sup

a f = lim inf

a f, we shall say that f is continuous at the point a, with the value f(a) = lim sup

a f = lim inf

a f. The set of distributions defined in A and continuous at the pointawill be denoted by Va(A).

Suppose now that the open set Acoincides with an interval I containing the
point a: then we have obviously B^{∗}_{a}(I) ⊂ Ba(I) and V_{a}^{∗}(I) ⊂ Va(I), and these
inclusions are strict. For instance, if I is an unbounded open interval of R^{n}, it
is easy to see that each one of the coordinate functions f_{j}(x) = x_{j} defined on
I, although obviously continuous at each point of this interval, is not strictly
bounded at any one of these points. It is also clear that the superior and inferior
limits of a distribution that is strictly bounded at the pointaare the same if the
distribution is considered as an element ofB^{∗}_{a}(I) or as an element of Ba(I) (and
analogously for the value at the pointaof an element of the space V_{a}^{∗}(I)).

We shall state now some general properties of the concepts just defined. In many cases their proofs are so easy that we have decided to omit them.

Proposition 3.1. Let a ∈ R^{n} and A, A^{0} be two open sets of R^{n} such
that a ∈ A^{0} ⊂ A; let also f ∈ D_{R}^{0} (A). Then f ∈ Ba(A) [resp. f ∈ Va(A)] iff
f_{|A}^{0} ∈ B_{a}(A^{0}) [resp. f_{|A}^{0} ∈ V_{a}(A^{0})] and in that case

lim sup

a f = lim sup

a f_{|A}^{0}, lim inf

a f = lim inf

a f_{|A}^{0} [resp. f(a) =f_{|A}^{0}(a)].
In all the following propositions up to Corollary 3.11Awill continue to be an
open set ofR^{n}, andaa point in A.

Proposition 3.2. Let f ∈ Ba(A) [resp. f ∈ Va(A)], g ∈ D^{0}_{R}(A) and j ∈
{1,2, ..., n}; if there is an interval I such that a ∈ I ⊂ A and g_{|I} = ∂_{j}f_{|I}, then
g∈ Ba(A) [resp.g∈ Va(A)] and

lim sup

a g= lim sup

a f, lim inf

a g= lim inf

a f [resp. g(a) =f(a)] . Proposition 3.3. Let f, g ∈ Ba(A), h = f +g and denote by α and β, respectively, the smallest and the largest of the two numbers:

lim inf

a f + lim sup

a g and lim inf

a g+ lim sup

a f .

Thenh∈ Ba(A)and

lim inf

a f + lim inf

a g ≤ lim inf

a h ≤ α , β ≤ lim sup

a h ≤ lim sup

a f+ lim sup

a g . Corollary 3.4. If f ∈ Va(A),g∈ Ba(A) andh=f+g, then:

lim inf

a h=f(a) + lim inf

a g , lim sup

a h=f(a) + lim sup

a g .

Corollary 3.5. If f, g ∈ V_{a}(A) and h = f +g, then h ∈ V_{a}(A) and
h(a) =f(a) +g(a).

Proposition 3.6. Letf ∈ Ba(A),λ∈Rand g=λ f; then g∈ Ba(A) and lim sup

a g=λlim sup

a f, lim inf

a g=λlim inf

a f, if λ≥0 , lim sup

a

g=λlim inf

a f, lim inf

a g=λlim sup

a

f, if λ <0.

Corollary 3.7. If f ∈ Va(A), λ ∈ R and g = λ f, then g ∈ Va(A) and g(a) =λ f(a).

Corollary 3.8. Ba(A) is a vector subspace of the space D^{0}_{R}(A) of all real
distributions defined on A and V_{a}(A) is a vector subspace of B_{a}(A). The map
f 7→f(a), of the spaceVa(A)onto R, is linear.

Now we shall prove the following result:

Theorem 3.9. Let(^{1}) f ∈ Ba(A), ϕ∈ C_{R}^{∞}(A) and suppose that ϕ(a) = 0;

thenϕ f ∈ Va(A) and (ϕ f)(a) = 0.

Proof: Denote by K the set of all distributions h ∈ Ba(A) such that, for
every ϕ ∈ C_{R}^{∞}(A) with ϕ(a) = 0, we have ϕ h ∈ Va(A) and (ϕ h)(a) = 0. We
must prove thatK=Ba(A).

First note that, if g ∈ D^{0}_{R}(A) and if there exists an interval I such that
a∈I ⊂ A and g_{|I} ∈BC(Ia), then clearly g ∈ K. So, to conclude the proof, it
will be sufficient to show that, for anyf, g∈ D^{0}_{R}(A) and everyj∈ {1, ..., n}, if we

(^{1}) We denote by C_{R}^{∞}(A) [resp.C^{∞}(A)] the space of all real [resp. complex] infinitely dif-
ferentiable functions defined onA.

haveg∈K and if there is an intervalI (witha∈I ⊂A) such that f_{|I} =∂j(g_{|I}),
thenf ∈K. Suppose then thatg,I and j satisfy the conditions just stated and
that we havef_{|I} =∂j(g_{|I}); if ϕ∈C_{R}^{∞}(A) and ϕ(a) = 0 we shall have:

(4) ϕ f_{|I} = ϕ Dj

h(x_{b}j−aj)g_{|I}^{i} = ∂j(ϕ g_{|I})−(x_{b}j−aj) ∂ϕ

∂x_{j} g_{|I} .

As g∈K, it follows from the definition of this set that we have ϕ g ∈ Va(A)
and (ϕ g)(a) = 0. So, Proposition 3.1 shows the distribution ϕ g_{|I} is an element
of the spaceVa(I) with value zero at the pointa; then by Proposition 3.2 we have
also∂_{j}(ϕ g_{|I})∈ Va(I), and ∂_{j}(ϕ g_{|I})(a) = 0. On the other hand, as (xb_{j} −a_{j})_{∂x}^{∂ϕ}

j

belongs to the space C_{R}^{∞}(A) with value 0 at the point a, we shall have in a
similar way (x_{b}j−aj)_{∂x}^{∂ϕ}

jg_{|I} ∈ Va(I), with value zero at the same point. Then,
by means of the equality (4), Corollaries 3.5 and 3.7 and Proposition 3.1, we can
conclude that ϕ f ∈ V_{a}(A) and (ϕ f)(a) = 0, which means that f ∈K, finishing
the proof.

Proposition 3.10. Let f ∈ Ba(A),ϕ∈C_{R}^{∞}(A). Thenϕ f ∈ Ba(A)and
lim sup

a (ϕ f) =ϕ(a) lim sup

a

f, lim inf

a (ϕ f) =ϕ(a) lim inf

a f if ϕ(a)≥0 , lim sup

a (ϕ f) =ϕ(a) lim inf

a f, lim inf

a (ϕ f) =ϕ(a) lim sup

a

f if ϕ(a)<0 .

To verify this result it is sufficient to consider the equality ϕ f = ϕ(a)f + (ϕ−ϕ(a))f ,

and take into account Proposition 3.6, Theorem 3.9 and Corollary 3.4. From Proposition 3.10 it follows immediately:

Corollary 3.11. If f ∈ Va(A) and ϕ ∈ C_{R}^{∞}(A), then ϕ f ∈ Va(A) and
(ϕ f)(a) =ϕ(a)f(a).

The following properties concerning the tensor product, are also very natural:

Theorem 3.12. Let m and n be two positive integers, A [resp. B] be an
open set inR^{m} [resp. R^{n}], a∈A, b∈ B, f ∈ Ba(A), g ∈ Bb(B) and h =f ⊗g.

Thenh∈ B_{(a,b)}(A×B) and, putting
α_{∗} = lim inf

a f , α^{∗} = lim sup

a f , β_{∗} = lim inf

b g , β^{∗} = lim sup

b

g ,

we have:

lim inf

(a,b) h = min^{n}α_{∗}β_{∗}, α_{∗}β^{∗}, α^{∗}β_{∗}, α^{∗}β^{∗}^{o} ,
lim sup

(a,b)

h = max^{n}α_{∗}β_{∗}, α_{∗}β^{∗}, α^{∗}β_{∗}, α^{∗}β^{∗}^{o}.

Proof: From f ∈ Ba(A) it follows the existence of an open interval I such
that a ∈ I ⊂ A and f_{|I} ∈ B^{∗}_{a}(I); then, for p ≥ deg_{a}f_{|I}, we shall have f_{p} =

∂_{x}^{−p}f_{|I} ∈BC(Ia) (where, for every h∈ B^{∗}_{a}(I), we put

∂xh = Dx1· · ·Dxn

h(x_{b}_{1}−a_{1})· · ·(x_{b}n−an)h^{i} ,

∂_{x} being an automorphism ofB_{a}^{∗}(I) according to Theorem 2.2). Analogously, as
g∈ Bb(B), there exists one open interval J such that b∈J ⊂B and, supposing
p >deg_{b}g_{|J}, g_{p} =∂^{−p}_{y} g_{|J} ∈BC(Jb) (where ∂_{y} has the obvious meaning).

So, forp≥max{deg_{a}f_{|I},deg_{b}g_{|J}}we shall have also, puttingK=I×J (and
then K_{(a,b)} = I_{a}×J_{b}), h_{|K} =f_{|I} ⊗g_{|J} ∈ B_{(a,b)}^{∗} (K) since f_{p} ⊗g_{p} ∈ BC(K_{(a,b)})
and, with an obvious notation,

∂_{(x,y)}^{p} (fp⊗g_{p}) = ∂_{x}^{p}(fp)⊗∂_{y}^{p}(gp) = h_{|K} .
So we see thath∈B_{(a,b)}(A×B) (and also that deg_{(a,b)}h_{|K} ≤p).

Putting hp =fp⊗gp, we deduce easily that:

h_{p}(a, b) = min^{n}f_{p}(a)g_{p}(b), f_{p}(a)g_{p}(b), f_{p}(a)g_{p}(b), f_{p}(a)g_{p}(b)^{o} ,
h_{p}(a, b) = max^{n}f_{p}(a)g_{p}(b), f_{p}(a)g_{p}(b), f_{p}(a)g_{p}(b), f_{p}(a)g_{p}(b)^{o} .
Now, to complete the proof it is sufficient to letp→+∞.

As immediate consequences we have the following two corollaries:

Corollary 3.13. With the same notation of Theorem 3.12, if f ∈ Va(A) then:

lim inf

(a,b) h=f(a) lim inf

b g, lim sup

(a,b)

h=f(a) lim sup

b

g, if f(a)≥0 , lim inf

(a,b) h=f(a) lim sup

b

g, lim sup

(a,b)

h=f(a) lim inf

b g, if f(a)<0 .
Corollary 3.14. With the same notation of Theorem 3.12, iff ∈ Va(A)and
g∈ Vb(B),h∈ V_{(a,b)}(A×B) and h(a, b) =f(a)g(b).

Now, we are going to analyse some relations between the chief concepts that we are studying and the operation of composition. As we shall see, the changes of variables that are “well related” to those concepts possess some particular properties, which are convenient to consider immediately.

So, let A and B be two open sets inR^{n} and µa map from A to B; for each
x= (x1, ..., x_{n}) ∈A let µ(x) = y = (y1, ..., y_{n}) and suppose that the map µ can
be expressed by means of the system

y_{j} =p_{j}(x) =p_{j}(x_{1}, ..., x_{n}) (j∈ {1, ..., n}) ,

wherep_{j}∈C_{R}^{∞}(A). Let alsoa= (a_{1}, ..., a_{n}) be a fixed point in A,b= (b_{1}, ..., b_{n}) =
µ(a) and suppose that the jacobianJµ= ^{∂(p}_{∂(x}^{1}^{,...,p}^{n}^{)}

1,...,xn) does not vanish at the pointa.

Finally, suppose that there exists one open intervalI (witha∈I ⊂A) satisfying the conditions:

i) the restriction ofµ toI,µ_{|I}, is a diffeomorphism from I to the setµ(I);

ii) the jacobianJ_{µ}is different from 0 at each point of I;

iii) for eachj∈ {1, ..., n} and eachx= (x_{1}, ..., x_{n})∈I the conditions
pj(x_{1}, ..., xn) =bj and xj =aj

are equivalent(^{2}).

From this we deduce easily that, for eachj, there exists a functionϕj ∈C_{R}^{∞}(I),
taking onI values that are all strictly positive or all strictly negative, and such
that, in each pointx∈I we have

y_{j}−b_{j} = p_{j}(x)−b_{j} = (x_{j}−a_{j})ϕ_{j}(x) .
In order to get this result it is sufficient to observe that

y_{j}−b_{j} = p_{j}(x1, ..., x_{j−1}, x_{j}, x_{j+1}, ..., x_{n})− p_{j}(x1, ..., x_{j−1}, a_{j}, x_{j+1}, ..., x_{n})

= Z xj

aj

∂pj

∂x_{j} (x_{1}, ..., x_{j−1}, uj, x_{j+1}, ..., xn) duj ,
or, puttingu_{j}−a_{j} = (xj−a_{j})u^{∗}_{j},

y_{j}−b_{j} = (x_{j}−a_{j})
Z _{1}

0

∂p_{j}

∂x_{j}

³x_{1}, ..., x_{j−1}, a_{j}+ (x_{j}−a_{j})u^{∗}_{j}, x_{j+1}, ..., x_{n}^{´}du^{∗}_{j} ,

(^{2}) It is easy to see that, to assure the existence of an intervalI satisfying i), ii) and iii) it
is sufficient to suppose that, in some neighbourhood ofa(and for eachj∈ {1, ..., n}) we have
pj(x1, ..., xn) =bjifxj=aj.

where the function defined by the integral, that we shall denote byϕj, is clearly
of classC^{∞}.

Besides, it is obvious thatϕj cannot be zero at any pointx∈I wherepj(x)6=

b_{j}; butϕ_{j} cannot also be zero at any pointxwhere we havep_{j}(x) =b_{j} — and so,
by iii),xj =aj — because then the jacobianJµwould be zero at the same point,
in contradiction with ii). Being different from zero at each point of I,ϕ_{j} must
have a fixed sign on this interval. So we see that each one of the 2^{n} connected
components of the set I_{a} = {x ∈ I : (x1 −a_{1}) (x2 −a_{2})· · ·(xn−a_{n}) 6= 0} is
mapped by µ into a connected component of µ(Ia). For commodity, we shall
suppose in the sequel that the functions ϕ_{j} are all positive; without any loss of
generality we shall also suppose that, at every pointx∈I, the inequalities

1

2ϕj(a) < ϕj(x) < 2ϕj(a) (j∈ {1,2, ..., n}) are satisfied.

Before obtaining the chief result relating the superior and inferior limits with the change of variables, we shall state and prove four lemmas.

Lemma 3.15. Letxbe a point in the setI_{a}such that the open intervalJ_{y}^{0},
determined(^{3}) by the points b and y =µ(x) is contained in µ(I); let also Jx be
the interval determined by the points aand x, and let A_{x} =µ^{−1}(J_{y}^{0}). Then for
everyλ∈]0,1[there exists² >0such that, putting

J_{x}^{1−λ}= λ a+ (1−λ)J_{x} and J_{x}^{1+λ} =−λ a+ (1 +λ)J_{x} ,
we have J_{x}^{1−λ} ⊂A_{x} ⊂J_{x}^{1+λ} if kx−ak< ².

Proof: First observe that, without loss of generality, we can suppose that
the point a is the center of the interval I (since this interval could always be
substituted by a subinterval centered at that point); observe also that, to prove
the lemma, it is sufficient to consider the case where, for everyj, we havex_{j} > a_{j}
(if some of the values xj −aj were negative, we could reduce ourselves to the
first case by means of the change of variables (x1, ..., x_{n}) 7→ (x^{0}_{1}, ..., x^{0}_{n}) with
x^{0}_{j}−aj =|xj −aj|, for everyj). In these conditions we shall have clearly:

J_{x} = ^{n}u= (u1, ..., u_{n}) : ∀j 0< u_{j}−a_{j} < x_{j}−a_{j}^{o}
and

A_{x} = ^{n}u= (u_{1}, ..., u_{n}) : ∀j 0<(u_{j} −a_{j})ϕ_{j}(u)<(x_{j}−a_{j})ϕ_{j}(x)^{o} .

(^{3}) We say that the intervalK ⊂R^{n} is determined by the pointsu= (u1, u2, ..., un) and
v= (v1, v2, ..., vn) iffK={w= (w1, ..., wn) : ∀j,min{uj, vj}< wj<max{uj, vj}}.

Finally observe that if u∈Jx we haveku−ak<kx−ak (where k · k is still
the euclidean norm inR^{n}) and that, if u∈A_{x} thenku−ak<4kx−ak (as we
easily see taking into account that, for everyx∈I and everyj, we have assumed

1

2ϕ_{j}(a)< ϕ_{j}(x)<2ϕ_{j}(a)).

Now, given λ ∈ ]0,1[ we can determine γ in such a way that 0 < γ <

min{ϕ_{1}(a), ..., ϕn(a)} and also, for every j,
ϕ_{j}(a) +γ

ϕ_{j}(a)−γ < 1
1−λ

and then²^{0} >0 such that, for kx−ak< ²^{0} and j∈ {1, ..., n},
ϕj(a)−γ < ϕj(x) < ϕj(a) +γ .

Then, if u∈J_{x}^{1−λ} andkx−ak< ²^{0} we have also, for everyj,
ϕ_{j}(a)−γ < ϕ_{j}(u) < ϕ_{j}(a) +γ

(sinceJ_{x}^{1−λ} ⊂J_{x} and ku−ak<kx−ak foru∈J_{x}) and therefore:

u_{j}−a_{j}
xj−aj

ϕ_{j}(u)

ϕj(x) < u_{j}−a_{j}
xj−aj

ϕ_{j}(a) +γ

ϕj(a)−γ < 1 1−λ

u_{j}−a_{j}
xj−aj

< 1 .
From this it follows immediately that, forkx−ak< ²^{0}, we have J_{x}^{1−λ} ⊂Ax.

To obtain the other inclusion referred in the lemma, let us suppose again that
a numberλ∈]0,1[ was given and use it to determineγ^{0} >0 such that, for every
j,

ϕ_{j}(a)−γ^{0}

ϕ_{j}(a) +γ^{0} > 1
1 +λ .

Next, determine ²^{00}>0 such that, for kx−ak< ²^{00}, we have
ϕ_{j}(a)−γ^{0} < ϕ_{j}(x) < ϕ_{j}(a) +γ^{0} (for j∈ {1, ..., n}) .

Now, if u ∈ Ax and kx−ak < ^{²}_{4}^{00}, we shall have (by one of our previous
observations)ku−ak< ²^{00} and so, for every j:

ϕ_{j}(a)−γ^{0} < ϕ_{j}(u) < ϕ_{j}(a) +γ^{0} .

From this it follows 1 1 +λ

u_{j} −a_{j}
xj −aj

< u_{j} −a_{j}
xj −aj

ϕ_{j}(a)−γ^{0}

ϕj(a) +γ^{0} < u_{j}−a_{j}
xj−aj

ϕ_{j}(u)
ϕj(x) < 1

and then u ∈ J_{x}^{1+λ}. So, given λ ∈ ]0,1[, we shall have J_{x}^{1−λ} ⊂ A_{x} ⊂ J_{x}^{1+λ} for
everyxsuch that kx−ak< ²= min{²^{0},^{²}_{4}^{00}}.

Lemma 3.16. Let x, J_{x} and A_{x} be like in the preceding lemma and let
F ∈BC(Ia) with sup_{x∈I}_{a}F(x) =M and inf_{x∈I}_{a}F(x) =m >0. Then

x→alim Z

Ax

F(u)du Z

Jx

F(u)du

= 1.

Proof: Givenδ >0, determine λ∈]0,1[ such that M

m

h1−(1−λ)^{n}^{i}< δ and M
m

h(1+λ)^{n}−1^{i}< δ

and then ² > 0 such that, for kx−ak < ², we have (with the notation used in
Lemma 3.15)J_{x}^{1−λ}⊂A_{x}⊂J_{x}^{1+λ}.

Then, if kx−ak< ², Z

Jx^{1−λ}

F(u)du ≤ Z

Ax

F(u)du ≤ Z

Jx^{1+λ}

F(u)du and so, since we have (denoting byν, for instance, Jordan measure):

Z

Jx\Jx^{1−λ}

F(u)du Z

Jx

F(u)du

≤ M ν(Jx\J_{x}^{1−λ})

m ν(Jx) = M ν(Jx) [1−(1−λ)^{n}]
m ν(Jx) < δ
and analogously

Z

J_{x}^{1+λ}\Jx

F(u)du Z

Jx

F(u)du

≤ M^{h}(1 +λ)^{n}−1^{i}

m < δ

we can conclude that, forkx−ak< ²,

1−δ < 1− Z

Jx\Jx^{1−λ}

F(u)du Z

Jx

F(u)du

≤ Z

Ax

F(u)du Z

Jx

F(u)du

≤ 1 + Z

Jx^{1+λ}\Jx

F(u)du Z

Jx

F(u)du

< 1 +δ .

Now, let us recall and complete some of the notation that will be used in the
following results. A and B will still be two open sets of R^{n},µa map from A to
B,a∈A,b=µ(a),J will be an interval such that b∈J ⊂B,I an interval and
ϕ_{j}: I→R(j∈ {1, ..., n})nfunctions satisfying the conditions referred to before
Lemma 3.15. Without loss of generality we can suppose thatI ⊂µ^{−1}(J).

We shall also put I_{a} = {x ∈ I: (x1−a_{1})· · ·(xn−a_{n}) 6= 0}, J_{b} = {y ∈ J:
(y_{1}−b_{1})· · ·(yn−bn) 6= 0} and, for each Φ ∈ BC(Ia) [resp. Ψ ∈ BC(Jb)] and
eachx∈I_{a} [resp. y∈J_{b}]:

(ρa,xΦ)(x) = 1

(x_{1}−a_{1})· · ·(xn−a_{n})
Z x1

a1

· · · Z xn

an

Φ(u1, ..., u_{n}) du_{1}· · ·du_{n}

= 1

|x_{1}−a_{1}| · · · |xn−an|
Z

Jx

Φ(u)du

"

resp. (ρb,yΨ)(y) = 1

(y_{1}−b_{1})· · ·(yn−bn)
Z _{y}_{1}

b1

· · ·
Z _{y}_{n}

bn

Ψ(v_{1}, ..., vn) dv_{1}· · ·dvn

= 1

|y1−b_{1}| · · · |yn−b_{n}|
Z

J_{y}^{0}

Ψ(v)dv

# . Now we can state:

Lemma 3.17. Denoting by µ_{|I}_{a} the restriction ofµ to the set I_{a}, let G be
a function in the space BC(J_{b}) with positive infimum and F =G◦µ_{|I}_{a}. Then
F ∈BC(Ia) and putting

Fp =ρ^{p}_{a,x}F , Gp =ρ^{p}_{b,y}G and G^{∗}_{p}=Gp◦µ_{|I}_{a} ,

for each p there exists a continuous function λ_{p}: I_{a} → R such that G^{∗}_{p} =λ_{p}F_{p}
andlim_{x→a}λ_{p}(x) = 1.

Proof: It is obviousF ∈BC(I_{a}) (and thatF(a) =G(a),F(a) =G(a)>0).

To prove the lemma we shall use induction onp. As the case p = 0 (where we
can take λ_{0} = 1) is trivial, let us suppose that there exists a function λp in the
required conditions and consider the equality (where we denote as before byJ_{y}^{0}
the interval determined by the pointsb andy ∈J_{b}):

G_{p+1}(y) = 1

|y1−b_{1}| · · · |yn−b_{n}|
Z

J_{y}^{0}

G_{p}(v)dv .
Changing variables by means ofµ_{|I}_{a} we obtain

G^{∗}_{p+1}(x) = 1

|x_{1}−a_{1}| · · · |xn−a_{n}|ϕ_{1}(x)· · ·ϕ_{n}(x)
Z

Ax

G^{∗}_{p}(u)J_{µ}(u)du ,

whereAx =µ^{−1}(J_{y}^{0}). Now, by the induction hypothesis

G^{∗}_{p+1}(x) = 1

|x_{1}−a_{1}| · · · |xn−an|ϕ_{1}(x)· · ·ϕn(x)
Z

Ax

λp(u)Fp(u)Jµ(u) du
or, sinceA_{x} is connected andF_{p} positive inA_{x},

G^{∗}_{p+1}(x) = 1

|x_{1}−a_{1}| · · · |xn−an|ϕ_{1}(x)· · ·ϕn(x)λ_{p}(x)J_{µ}(x)
Z

Ax

F_{p}(u)du ,
wherex is a point inAx that tends toawhen x does. So, if we put

τp(x) = Z

Ax

Fp(u)du Z

Jx

F_{p}(u)du

and λ_{p+1}(x) = 1

ϕ_{1}(x)· · ·ϕn(x)λp(x)Jµ(x)τp(x) ,
we get finallyG^{∗}_{p+1} =λ_{p+1}F_{p+1}, where we easily see, taking into account Lemma
3.16, that lim_{x→a}λ_{p+1}(x) = 1.

Lemma 3.18. IfG∈BC(Jb) and F =G◦µ_{|I}_{a}, then
lim inf

a F = lim inf

b G , lim sup

a

F = lim sup

b

G .

Proof: If the infimum ofGis strictly positive, Lemma 3.17 implies immedi- ately that, for eachp,Gp(b) =Fp(a) and Gp(b) =Fp(a). Then:

lim sup

b

G = lim

p→∞Gp(b) = lim

p→∞Fp(a) = lim sup

a F

and the same for the inferior limits. Ifm= infy∈JbG(y)≤0, let us take a constant
csuch thatc >|m|and put Ψ(y) =G(y) +c(fory∈Jb) and Φ = Ψ◦µ_{|I}_{a}. Then
Ψ will be a function in the spaceBC(Jb) with infimum strictly greater than zero
and so, from what we have just seen, we shall have Φ∈BC(Ia) and

lim inf

a Φ = lim inf

b Ψ, lim sup

a Φ = lim sup

b

Ψ.

Since Φ(x) =F(x)+cfor eachx∈I_{a}, it follows, taking into account Corollary 3.4,
lim sup

a F = lim sup

a Φ−c = lim sup

b

Ψ−c = lim sup

b

G and analogously for the inferior limits.

Theorem 3.19. Let A, B, µ, a and b be like in the preceding lemmas, g∈ Bb(B) and f =g◦µ. Then f ∈ Ba(A)and

lim inf

a f = lim inf

b g , lim sup

a f = lim sup

b

g .

Proof: Denote by L the set of all distributions l∈ Bb(B) such that l◦µ ∈ Ba(A) and

lim sup

a (l◦µ) = lim sup

b

l , lim inf

a (l◦µ) = lim inf

b l .
We have to prove thatL=B_{b}(B).

Letg∈ Bb(B). If, for some intervalJ, withb∈J ⊂B, we haveg_{|J} ∈BC(Jb)
then, denoting by I an interval such that a∈ I ⊂A and I ⊂µ^{−1}(J), we shall
clearly havef_{|I} =g_{|J} ◦µ_{|I} ∈BC(Ia) and, by Lemma 3.18,

lim sup

a f_{|I} = lim sup

b

g_{|J}, lim inf

a f_{|I} = lim inf

b g_{|J} ;
from this it follows immediatelyf ∈ Ba(A) and

lim sup

a f = lim sup

b

g , lim inf

a f = lim inf

b g

that is,g∈L.

So, to conclude the proof, it will be sufficient to show that, if a distribution h
belongs to Land if k∈ D^{0}_{R}(B) is such that, for somej ∈ {1, ..., n} and on some
interval J (with b ∈ J ⊂ B) we have k_{|J} = ∂_{y}_{j}(h_{|J}) = D_{y}_{j}[(y_{b}_{j} −b_{j})h_{|J}], then
k∈L.

Suppose then thath,k,j andJ satisfy the conditions just stated and letI be
an interval such that a∈I ⊂ A and I ⊂µ^{−1}(J); suppose also, as usually, that
the restriction ofµtoI,µ_{|I}, can be expressed by means of the system

yi−bi = (xi−ai)ϕi(x) ,

with the ϕ_{i} strictly positive and of class C^{∞}. Then, putting h^{∗} = h ◦µ and
k^{∗}=k◦µwe shall have:

k_{|I}^{∗} = k◦µ_{|I} = ^{n}D_{y}_{j}^{h}(yb_{j} −b_{j})h_{|J}^{io}◦µ_{|I} =
Ã _{n}

X

i=1

∂_{x}_{i}

∂_{y}_{j} D_{x}_{i}

!h

(xb_{j}−a_{j})ϕ_{j}h^{∗}_{|I}^{i} .
But we easily see that, ifi6=j, we have inI:

∂_{x}_{i}

∂_{y}_{j} = (xb_{i}−a_{i})ω_{ij} ,

where the functionsωij: I →Rare of class C^{∞}; and also that,

(5) ∂x_{j}

∂y_{j}(a) = 1
ϕ_{j}(a) .
So, we have:

(6) k_{|I}^{∗} =
Xn

i=1i6=j

(xb_{j}−a_{j})ω(ij)(xb_{i}−a_{i})D_{x}_{i}(ϕjh^{∗}_{|I}) + ∂x_{j}

∂yj

D_{x}_{j}^{h}(xb_{j} −a_{j})ϕ_{j}h^{∗}_{|I}^{i} .

Now, as we have by hypothesish^{∗} ∈ Ba(A), we see (by Propositions 3.1, 3.10, 3.2
and Corollary 3.4) that, fori6=j, the distribution

(x_{b}i−ai)Dxi(ϕjh^{∗}_{|I}) = ∂xi(ϕjh^{∗}_{|I})−ϕjh^{∗}_{|I}

belongs to the spaceBa(I) and then (by Theorem 3.9 and Corollary 3.5) we can
conclude that the first term of the second member of (6) is continuous and has
value zero at the point a. On the other hand, taking into account equality (5)
and Propositions 3.2 and 3.10, we see that the distribution ^{∂x}_{∂y}^{j}

j ∂_{x}_{j}(ϕjh^{∗}_{|I}) has at
the point a the same superior and the same inferior limits as h^{∗}. From this it
follows easily thatk^{∗}∈ Ba(A) and that

lim sup

a k^{∗} = lim sup

a k_{|I}^{∗} = lim sup

a h^{∗}= lim sup

b

h = lim sup

b

k

and analogously for the inferior limits. This means that k ∈ L, concluding the proof.

As an immediate consequence we have:

Corollary 3.20. With the same notation of Theorem 3.19 suppose now that
g∈ V_{b}(B). Then f =g◦µ∈ Va(A) and f(a) =g(b).

As we saw, all the preceding definitions and results stated in this work concern
only real distributions; but it is quite clear that some of them are immediately
extensible to (complex) distributions. To prepare the obvious definitions we recall
that, as it is well known, iff is a (complex) distribution defined in an open set
Aof R^{n}, there exist two real distributions f_{1}, f_{2} ∈ D^{0}_{R}(A), uniquely determined,
such thatf =f_{1}+i f_{2}. Then we shall say that f is bounded [resp. continuous]

at the pointa∈ A — and we shall write f ∈ Ba(A) [resp.f ∈ Va(A)] — iff we