# A Proof of Łojasiewicz’s Theorem

Authors: Namhoon Kim
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We give a necessary and sufficient condition for a primitive of a distribution to have the value at a point in the sense of Łojasiewicz. A formula defining the indefinite integral of a distribution with a basepoint is introduced, and further structural results are discussed.


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## Body

## 1. Introduction

LetD=D(R) be the topological C-vector space of complex valued compactly supported test functions on R, and let D′=D′(R) be the space of complex valued distributions on R. In the following discussion, a distribution f∈D′ is also denoted by f(x), and the dual pairing between f∈D′ and a test function ϕ∈D is denoted by either 〈f,ϕ〉 or 〈f(x),ϕ(x)〉. On the other hand, the letter x0 will always denote a point.According to Łojasiewicz [1], a distribution f∈D′ has the value c∈C at x0 if
(1)f(ax+x0)⟶c
in D′ as a→0. If such a value c exists at x0, we will say that f isevaluableat x0 and write f(x0)=c. For f to be evaluable at x0, it suffices for lim⁡a→0f(ax+x0) to exist in D′, as the limit can only be a constant. We can equivalently require that there exists c∈C such that lim⁡a→0+f(ax+x0)=c, as this entails lim⁡a→0-f(ax+x0)=c. Simply requiring the existence of lim⁡a→0+f(ax+x0) does not suffice, as the limit may in general be of the form c1+c2H(x-x0), where H is the Heaviside step function.One interesting consequence of this definition is the following.Theorem 1 (Łojasiewicz).
If a distributionf is evaluable at x0, then any primitive F of f is also evaluable at x0.This result is useful in various circumstances. For instance, if a distributionf is evaluable at a and b, then so is any primitive F of f, and we may define a definite integral of f as
(2)∫ab‍f=F(b)-F(a).
These ideas are connected with an interesting construction of distributional integral in the work of Estrada and Vindas [2].In view of the simplicity and naturality of Theorem1, the known proof is somewhat indirect. The argument follows as a corollary of a more difficult result of Łojasiewicz, which is stated in Theorem 5. The first purpose of this paper is to give a short and direct proof. We then arrive at a formula of the indefinite integral of a distribution with a basepoint. In fact, we can reverse the usual direction of reasoning and use the arguments developed along these lines to give a different proof of Theorem 5.Theorem5 is an example of a structure theorem, which is interesting in its own right and has a generalization involving the notion of the quasiasymptotic behavior [3]. In the last section, we study how variations of the definition of the value at a point lead to some other nice analogous structural results.


## 2. A Proof of Theorem1

In order to fix our notation, we briefly recall the following elementary notions [4]. Suppose we have a continuous family of distributions {fu}u∈I depending on a parameter u in an interval I, meaning that 〈fu,ϕ〉 is continuous in u for each ϕ∈D. If 〈fu,ϕ〉 is differentiable at u0∈I for each ϕ∈D, we say that {fu}u∈I is differentiable with respect to u at u0 and define ∂ufu|u=u0 by
(3)〈∂ufu|u=u0,ϕ〉=∂u〈fu,ϕ〉|u=u0.
Evidently ∂ufu|u=u0 is a distribution as it is the limit of distributions given by the difference quotients. Similarly, for a,b∈I, we define ∫ab‍fudu by
(4)〈∫ab‍fudu,ϕ〉=∫ab‍〈fu,ϕ〉du,
which is again a distribution, being the limit of distributions given by the Riemann sums. By pairing with test functions, it follows from the fundamental theorem of calculus that if {fu}u∈I and {Fu}u∈I are continuous families of distributions with ∂uFu|u=u0=fu0 for all u0∈I, then, for any a,b∈I,
(5)∫ab‍fudu=Fb-Fa.Let us note that, for any distributionf(x)∈D′, both {f(ax)}a∈(-∞,0) and {f(ax)}a∈(0,∞) are continuous families of distributions. If f is evaluable at x0=0, namely, if f(ax)→c as a→0, then {f(ax)}a∈R becomes a continuous family of distributions if we define f(0)=c. Our argument uses this simple observation.Proof of Theorem1.
Letf=∂F in D′ and suppose f is evaluable at x0=0. As seen above, {f(ax)}a∈R is a continuous family of distributions and so is the family {xf(ax)}a∈R. It is trivial to verify that the family {F(ax)}a∈(0,∞) is differentiable with respect to a∈(0,∞) with ∂aF(ax)|a=a0=xf(a0x). By (5), for u,u0∈(0,∞),
(6)∫uu0‍xf(ax)da=F(u0x)-F(ux).
The left-hand side is well defined for u∈R and gives a continuous family as u ranges over the real line, and thus, taking the limit u→0+ on both sides, we see that F(ux)→L(x) as u→0+ for some L(x)∈D′. Applying ∂x gives uf(ux)→L′(x), but clearly uf(ux)→0. We conclude that L(x) is a constant.It also follows that ifF is a primitive of a distribution f such that f(x0)=c, the family {F(ax+x0)}a∈R is differentiable with respect to a and we have ∂aF(ax+x0)|a=a0=xf(a0x+x0). In particular, ∂aF(ax+x0)|a=0=cx.


## 3. Distributions Integrable from a Basepoint

In the preceding proof, it is clear that the assumption thatf is evaluable at x0 was not entirely necessary. Let us say that f∈D′ isintegrable from  x0 if the following two conditions hold.(i)
Foru0>0,∫uu0‍xf(ax+x0)da converges in D′ as u→0+.(ii)
af(ax+x0)→0 in D′ as a→0+.By the same argument, this definition gives a necessary and sufficient condition for a primitiveF of f to be evaluable at x0. Indeed, if we set x0=0, then (i) is equivalent to the existence of L(x)≔lim⁡u→0+F(ux). In this case, since uf(ux)→L′(x) as u→0+, (ii) is equivalent to L(x) being a constant. We summarize this as follows.Proposition 2.
LetF be a distribution and let f=∂F. Then F is evaluable at x0 if and only if f is integrable from x0.We denote byDx0′ the space of all distributions integrable from x0. For f∈Dx0′, we define a distribution ∫x0x+x0‍f by the formula
(7)∫x0x+x0‍f≔lim⁡u→0+∫u1‍xf(ax+x0)da.
Let F be a primitive of f. For any u>0,
(8)∫u1‍xf(ax+x0)da=F(x+x0)-F(ux+x0),
and if f is integrable from x0, taking the limit u→0+, we have
(9)∫x0x+x0‍f=F(x+x0)-F(x0),
as F(x0) exists by Proposition 2. Replacing x with x-x0, we define the indefinite integral of f∈Dx0′ with basepoint x0 by
(10)∫x0x‍f≔lim⁡u→0+∫u1‍(x-x0)f(ax-ax0+x0)da.
It follows that we have ∫x0x‍f=F(x)-F(x0) and ∂x∫x0x‍f=f(x). We also note that ∫x0x‍f is evaluable with value 0 at x0.It is easy to see that iffn is a sequence in Dx0′, then fn→f in D′ for some f∈Dx0′ does not imply ∫x0x‍fn→∫x0x‍f in general. In order to remedy this, we introduce the following notions.Supposefn is a sequence in Dx0′. We say fn isbounded atx0 if, for each ϕ∈D,〈fn(ax+x0),ϕ(x)〉 is bounded independently of n as well as of a∈(0,1]. Let us say fn convergesboundedly to f∈Dx0′ if fn→f in D′ and fn-f is eventually bounded at x0. Finally, we say fn convergesuniformly to f∈Dx0′ if, for each ϕ∈D,〈fn(ax+x0),ϕ(x)〉 converges to 〈f(ax+x0),ϕ(x)〉 uniformly in a∈(0,1]. Clearly, uniform convergence implies bounded convergence.Lemma 3.
If a sequencefn in Dx0′ converges boundedly (resp., uniformly) to f∈Dx0′, then the sequence ∫x0x‍fn converges boundedly (resp., uniformly) to ∫x0x‍f.Proof.
Supposefn in Dx0′ converges to 0 with fn bounded at x0, and let Fn(x)=∫x0x‍fn. We have
(11)〈Fn(ax+x0),ϕ(x)〉=lim⁡u→0+∫u1‍〈axfn(abx+x0),ϕ(x)〉db=lim⁡u→0+∫aua‍〈fn(bx+x0),xϕ(x)〉db,
which shows 〈Fn(ax+x0),ϕ(x)〉 is bounded independently of n and of a∈(0,1], and by taking the limit in n under the integral sign, we see that Fn converges boundedly to 0. If fn in fact converges uniformly to 0, the uniform convergence of Fn is also apparent from the same expression.Let us writeFn⇉F on Ω to mean that Fn and F are continuous functions on Ω such that Fn converges to F uniformly on Ω. Let us also denote by ∫x0x‍:Dx0′→Dx0′ the map that sends f(x) to ∫x0x‍f.Lemma 4.
Letfn,f be distributions in Dx0′. If fn converges boundedly to f, then, for every bounded open neighborhood Ω of x0, there exists an integer k≥0 such that (∫x0x‍)kfn⇉(∫x0x‍)kf on Ω.Proof.
LetI be a compact interval containing Ω. We can find k≥0 and a sequence of continuous functions Fn,F on I such that fn=∂kFn,f=∂kF with Fn⇉F on I (see [5]). Thus, (∫x0x‍)kfn and Fn (resp., (∫x0x‍)kf and F) differ by polynomials Pn (resp., P) of degree <k on I. By Lemma 3, fn→f boundedly implies (∫x0x‍)kfn→(∫x0x‍)kf in D′, and since Fn→F in D′(Ω), we have Pn→P in D′(Ω), which is the case only when Pn⇉P on Ω. Hence (∫x0x‍)kfn⇉(∫x0x‍)kf on Ω.


## 4. Structure Theorem of Łojasiewicz

These ideas lead to a proof of another result of Łojasiewicz that we have already mentioned (cf. [1, 5, 6]). The proof given below seems illustrative in the sense that the implication in one direction is obtained by applying ∂x several times, and the converse is obtained by applying ∫x0x‍ several times.Theorem 5 (Łojasiewicz).
Letf∈D′. Then f(ax+x0)→c as a→0 if and only if f=∂kF for some k≥0, where F is a continuous function near x0 such that lim⁡x→x0(F(x)/(x-x0)k)=c/k!.Proof.
Letx0=0. If f=∂kF and F is continuous near 0 with lim⁡x→0F(x)/xk=c/k!, then F(ax)/ak→cxk/k! in D′ as a→0, and applying ∂xk we obtain f(ax)→c in D′. Conversely, suppose f(ax)→c in D′ as a→0. Letting fa(x)=f(ax), it is easily observed that fa(x) converges boundedly (in fact, uniformly) to c as a→0. By Lemma 4, there exist a neighborhood Ω of 0 and k≥0 such that (∫0x‍)kfa⇉(∫0x‍)kc=cxk/k! on Ω. As (∫0x‍)kfa=F(ax)/ak if F≔(∫0x‍)kf, we have F(ax)/ak⇉cxk/k! as a→0. For any fixed x≠0 in Ω, we have F(ax)/(ax)k→c/k! as a→0; namely, lim⁡a→0F(a)/ak=c/k!.


## 5. Further Structure Theorems

There are various notions of the value of a distribution at a point, some defined under stricter conditions with stronger properties while others applicable for more general distributions [7–11]. When a situation or an application demands some specific features from the evaluable distributions, one would like to know how the values that we obtain are associated with some structural qualities of the distributions. We now discuss some results of this type similar to Theorem 5.The works of Shiraishi and Itano give a notion of evaluation at a point with stricter properties than that of Łojasiewicz [7–9]. Let us call a sequence (fn) in D(Rd) a δ-sequence if there is a sequence of positive real numbers (an)→0 such that, ∀n∈N,(i)
fn(x)=0 for |x|≥an,(ii)
∫‍fn=1,(iii)
∫‍|fn| is bounded independently of n∈N.We say that a distributionT∈D′(Rd) has δ-valuec∈Catx0∈Rd if
(12)〈T,τx0fn〉⟶c
as n→∞ for all δ-sequences (fn), where (τx0fn)(x)=fn(x-x0). In fact, we can restrict this condition to real nonnegative δ-sequences (which are called δ-sequences in, e.g., [5, 7]) without affecting the definition. By the result in [7] (see also [12] for a proof based on ideas from nonstandard analysis), T∈D′(Rd) has δ-value c at x0 if and only if it can be represented as an L∞-function near x0 which is continuous at x0 with value c. Thus, we have T=c+Ψ, with
(13)esssup⁡|x-x0|<a|Ψ(x)|⟶0
as a→0+. As this condition is quite strong, we can regard this as the most conservative notion of the value of a distribution at a point.We can compare this with the previously discussed Łojasiewicz definition, as it is immediate that the Łojasiewicz value has the following sequential representation. Aδ-sequence (fn) of the form
(14)fn(x)=an-df(xan),
where f∈D(Rd) with ∫‍f=1 and an>0 with (an)→0, is called a model sequence. One sees that a distribution T has Łojasiewicz’s value c at x0 if and only if 〈T,τx0fn〉→c for all model sequences. A structural result given by Theorem 5 tells us that the condition imposed on T is much weaker.In this section we find a continuous family of classes of distributionsDp,x0 for 1≤p<∞ such that, for any 1≤q≤p<∞,
(15){Distributionswithδ-valueatx0}⊆Dq,x0⊆Dp,x0⊆{DistributionswithŁojasiewicz’svalueatx0}
with analogous structural results involving Lp functions. These classes of distributions can be defined sequentially in a natural way.Definition 6.
Let1≤p<∞ be fixed. A sequence (fn) in D(Rd) is called a δp-sequence if there exists a sequence of positive real numbers (an)→0 such that, ∀n∈N,(i)
fn(x)=0 for |x|≥an,(ii)
∫‍fn=1,(iii)
and(p-1)∫‍|fn|p is bounded independently of n∈N.
A distributionT∈D′(Rd) is said to have δp-value  c∈C  at  x0∈Rd if
(16)〈T,τx0fn〉⟶c
as n→∞ for all δp-sequences (fn).Remark 7.
In the above definition, we will say that(an) is a contracting sequence of (fn).For any1≤q≤p≤∞, we have (e.g., [13]) that if Ω⊆Rd is a nonempty open subset of finite measure μ(Ω) and if f∈Lp(Ω), then f∈Lq(Ω) and
(17)∥f∥Lq(Ω)≤μ(Ω)1/q-1/p∥f∥Lp(Ω).
Let 1≤q≤p<∞, and suppose (fn) is a δp-sequence, with a contracting sequence (an)→0. From (17) we obtain
(18)an-d/q(∫‍|fn|q)1/q≤an-d/p(∫‍|fn|p)1/p,
and multiplying both sides by and gives
(19)(and(q-1)∫‍|fn|q)1/q≤(and(p-1)∫‍|fn|p)1/p,
which shows that (fn) is also a δq-sequence. Therefore, if a distribution T has δq-value c at x0, then it has the same δp-value at x0. This will also follow from Theorem 10 (iii), as we have, since 1<p′≤q′≤∞,
(20)a-d/p′∥Ψ∥Lp′(Bx0(a))≤a-d/q′∥Ψ∥Lq′(Bx0(a))
by (17). Hence, the condition of a distribution having δp-value at a point becomes less restrictive as p increases. As any model sequence is a δp-sequence for all 1≤p<∞, if T∈D(Rd) has δp-value c at x0∈Rd for some p, then it has the same value c at x0 in the sense of Łojasiewicz.For a nonempty open setΩ⊆Rd, we let DR(Ω)⊆D(Ω) be the subspace of all real valued test functions and let D(Ω)+⊆DR(Ω) (resp., D(Ω)-⊆DR(Ω)) be the subset of all nonnegative (resp., nonpositive) test functions. Let
(21)Dp1(Ω)⊆D(Ω)
be the subset consisting of f such that ∫‍|f|p=1, and let
(22)Dp1(Ω)+=Dp1(Ω)∩D(Ω)+.
For Ψ∈D′(Ω), we define
(23)∥Ψ∥Dp′(Ω)=sup⁡f∈Dp1(Ω)|〈Ψ,f〉|,∥Ψ∥Dp′(Ω)+=sup⁡f∈Dp1(Ω)+|〈Ψ,f〉|,
taking values in [0,∞]. We then have the following simple estimate.Lemma 8.
We have(24)∥Ψ∥Dp′(Ω)+≤∥Ψ∥Dp′(Ω)≤4∥Ψ∥Dp′(Ω)+.Proof.
The first inequality follows trivially sinceDp1(Ω)+⊆Dp1(Ω). In order to see the second inequality, suppose f∈DR(Ω). We can write f=f++f-, where f+(x)=max⁡{f(x),0} and f-(x)=min⁡{f(x),0} for x∈Ω. As f+ and f- are compactly supported continuous functions, we can find f1∈D(Ω)+ (resp., f2∈D(Ω)-) that is as close as we want to f+ (resp., f-) in the Lp-norm, such that f=f1+f2. Hence, from
(25)|〈Ψ,f〉|≤|〈Ψ,f1〉|+|〈Ψ,f2〉|≤∥Ψ∥Dp′(Ω)+(∥f1∥Lp(Ω)+∥f2∥Lp(Ω)),
since ∥f±∥Lp(Ω)≤∥f∥Lp(Ω), we have
(26)|〈Ψ,f〉|≤2∥Ψ∥Dp′(Ω)+∥f∥Lp(Ω).
By (26), if f∈D(Ω), then since we have ∥Re(f)∥Lp(Ω)≤∥f∥Lp(Ω) and ∥Im⁡(f)∥Lp(Ω)≤∥f∥Lp(Ω),
(27)|〈Ψ,f〉|≤|〈Ψ,Re(f)〉|+|〈Ψ,Im⁡(f)〉|≤4∥Ψ∥Dp′(Ω)+∥f∥Lp(Ω).Suppose∥Ψ∥Dp′(Ω)<∞ for some 1≤p<∞. Since D(Ω) is dense in Lp(Ω),Ψ extends to a continuous functional on Lp(Ω) and lies in the strong dual of Lp(Ω), which is isometric to Lp′(Ω) [13]. We thus have Ψ∈Lp′(Ω) and ∥Ψ∥Dp′(Ω)=∥Ψ∥Lp′(Ω).Lemma 9.
Let(an) be a sequence of positive real numbers such that (an)→0. Suppose we have two sequences (fn) and (gn) in D(Rd) such that both and(p-1)∫‍|fn|p and and(p-1)∫‍|gn|p are bounded independently of n∈N. Then and(p-1)∫‍|fn+gn|p is bounded independently of n∈N.Proof.
By multiplying byand(p-1) on both sides of Minkowski’s inequality for fn and gn, we obtain
(28)and(p-1)∫‍|fn+gn|p≤((and(p-1)∫‍|fn|p)1/p+(and(p-1)∫‍|gn|p)1/p)p,
from which the lemma follows.We can now give a structure theorem on our notion ofδp-value of a distribution. The only tricky part of the following argument seems to be that our definition is unaffected even if we only restrict ourselves to real nonnegative δp-sequences (Theorem 10 (ii)).Theorem 10.
LetT∈D′(Rd). Then, the following statements are equivalent.(i)
T has δp-value c∈C at x0∈Rd.(ii)
〈T,τx0fn〉→c as n→∞ for all δp-sequences (fn) such that fn≥0.(iii)
T=c+Ψ, where Ψ can be represented as an Lp′-function in some open ball Bx0(a) of radius a>0 around x0, and
(29)a-d/p′∥Ψ∥Lp′(Bx0(a))⟶0
as a→0+, where p′=p/(p-1)∈(1,∞] is the Hölder conjugate of p.Proof.
As the implication (i)⇒(ii) is immediate, it only remains to show (ii)⇒(iii)⇒(i).
Let us assume (ii). It suffices to consider the special casex0=0. Let T be a distribution such that (〈T,fn〉)→c for all nonnegative δp-sequences (fn). For Ψ=T-c, since ∫‍fn=1 and 〈T-c,fn〉=〈T,fn〉-c, we have (〈T,fn〉)→c if and only if (〈Ψ,fn〉)→0. We now claim that
(30)a-d/p′∥Ψ∥Dp′(B0(a))+⟶0
as a→0+. Otherwise, for some ε0>0, we can find a sequence of positive real numbers (an)→0 and functions gn∈Dp1(B0(an))+ such that an-d/p′|〈Ψ,gn〉|≥ε0 for all n∈N. We note
(31)and(p-1)∫‍(an-d/p′gn)p=and(p-1)an-d(p-1)∫‍gnp=1,
and, in particular, it is bounded independently of n∈N. Applying inequality (19) to the functions an-d/p′gn (with q=1), we obtain
(32)∫‍an-d/p′gn≤1
for all n∈N. Let (hn) be any fixed nonnegative δp-sequence of which (an) is a contracting sequence, such as a nonnegative model sequence. We let
(33)fn=an-d/p′gn+bnhn,
where bn=1-∫‍an-d/p′gn∈[0,1). Observe that fn≥0 with ∫‍fn=1, and applying Lemma 9 to the sequences an-d/p′gn and bnhn, we see that (fn) is in fact a nonnegative δp-sequence. Thus, we must have
(34)(〈Ψ,fn〉)⟶0.
But as 〈Ψ,fn〉=an-d/p′〈Ψ,gn〉+bn〈Ψ,hn〉, the fact that (bn〈Ψ,hn〉)→0  implies (an-d/p′〈Ψ,gn〉)→0, a contradiction. Hence, (30) follows, which implies (iii) by Lemma 8 and the paragraph following it.
Lastly, we assume that (iii) holds forx0=0. Let (fn) be a δp-sequence with a contracting sequence (an)→0. By Hölder’s inequality,
(35)|〈Ψ,fn〉|≤∥Ψ∥Lp′(B(an))∥fn∥Lp(B(an))=an-d/p′∥Ψ∥Lp′(B(an))(and(p-1)∫‍|fn|p)1/p⟶0
as n→∞, and (i) follows.It is often useful to relate a given notion of a value at a point, usually defined through the pairing of a distribution with test functions, to a statement revealing the internal structure of the distribution. One such result is Theorem5, and the above theorem gives some others.



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