AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/695617 695617 Research Article A Proof of Łojasiewicz’s Theorem http://orcid.org/0000-0002-3961-0186 Kim Namhoon Barbatis Gerassimos Department of Mathematics Education Hongik University 72-1 Sangsu-dong, Mapo-gu, Seoul 121-791 Republic of Korea hongik.ac.kr 2014 1192014 2014 28 04 2014 18 08 2014 11 9 2014 2014 Copyright © 2014 Namhoon Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Let D=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 fD is also denoted by f(x), and the dual pairing between fD and a test function ϕD 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 , a distribution fD has the value cC at x0 if (1)f(ax+x0)c in D as a0. If such a value c exists at x0, we will say that f is evaluable at x0 and write f(x0)=c. For f to be evaluable at x0, it suffices for lima0f(ax+x0) to exist in D, as the limit can only be a constant. We can equivalently require that there exists cC such that lima0+f(ax+x0)=c, as this entails lima0-f(ax+x0)=c. Simply requiring the existence of lima0+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 distribution f 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 distribution f 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 .

In view of the simplicity and naturality of Theorem 1, 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.

Theorem 5 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 . 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 Theorem <xref ref-type="statement" rid="thm1">1</xref>

In order to fix our notation, we briefly recall the following elementary notions . Suppose we have a continuous family of distributions {fu}uI depending on a parameter u in an interval I, meaning that fu,ϕ is continuous in u for each ϕD. If fu,ϕ is differentiable at u0I for each ϕD, we say that {fu}uI is differentiable with respect to u at u0 and define ufu|u=u0 by (3)ufu|u=u0,ϕ=ufu,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,bI, we define abfudu by (4)abfudu,ϕ=abfu,ϕdu, 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}uI and {Fu}uI are continuous families of distributions with uFu|u=u0=fu0 for all u0I, then, for any a,bI, (5)abfudu=Fb-Fa.

Let us note that, for any distribution f(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 a0, then {f(ax)}aR becomes a continuous family of distributions if we define f(0)=c. Our argument uses this simple observation.

Proof of Theorem <xref ref-type="statement" rid="thm1">1</xref>.

Let f=F in D and suppose f is evaluable at x0=0. As seen above, {f(ax)}aR is a continuous family of distributions and so is the family {xf(ax)}aR. 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 uR and gives a continuous family as u ranges over the real line, and thus, taking the limit u0+ on both sides, we see that F(ux)L(x) as u0+ 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 if F is a primitive of a distribution f such that f(x0)=c, the family {F(ax+x0)}aR 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 that f is evaluable at x0 was not entirely necessary. Let us say that fD is integrable from  x0 if the following two conditions hold.

For u0>0,uu0xf(ax+x0)da converges in D as u0+.

af(ax+x0)0 in D as a0+.

By the same argument, this definition gives a necessary and sufficient condition for a primitive F of f to be evaluable at x0. Indeed, if we set x0=0, then (i) is equivalent to the existence of L(x)limu0+F(ux). In this case, since uf(ux)L(x) as u0+, (ii) is equivalent to L(x) being a constant. We summarize this as follows.

Proposition 2.

Let F be a distribution and let f=F. Then F is evaluable at x0 if and only if f is integrable from x0.

We denote by Dx0 the space of all distributions integrable from x0. For fDx0, we define a distribution x0x+x0f by the formula (7)x0x+x0flimu0+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 u0+, 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 fDx0 with basepoint x0 by (10)x0xflimu0+u1(x-x0)f(ax-ax0+x0)da. It follows that we have x0xf=F(x)-F(x0) and xx0xf=f(x). We also note that x0xf is evaluable with value 0 at x0.

It is easy to see that if fn is a sequence in Dx0, then fnf in D for some fDx0 does not imply x0xfnx0xf in general. In order to remedy this, we introduce the following notions.

Suppose fn is a sequence in Dx0. We say fn is bounded at x0 if, for each ϕD,fn(ax+x0),ϕ(x) is bounded independently of n as well as of a(0,1]. Let us say fn converges boundedly to fDx0 if fnf in D and fn-f is eventually bounded at x0. Finally, we say fn converges uniformly to fDx0 if, for each ϕD,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 sequence fn in Dx0 converges boundedly (resp., uniformly) to fDx0, then the sequence x0xfn converges boundedly (resp., uniformly) to x0xf.

Proof.

Suppose fn in Dx0 converges to 0 with fn bounded at x0, and let Fn(x)=x0xfn. We have (11)Fn(ax+x0),ϕ(x)=limu0+u1axfn(abx+x0),ϕ(x)db=limu0+auafn(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 write FnF 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:Dx0Dx0 the map that sends f(x) to x0xf.

Lemma 4.

Let fn,f be distributions in Dx0. If fn converges boundedly to f, then, for every bounded open neighborhood Ω of x0, there exists an integer k0 such that (x0x)kfn(x0x)kf on Ω.

Proof.

Let I be a compact interval containing Ω. We can find k0 and a sequence of continuous functions Fn,F on I such that fn=kFn,f=kF with FnF on I (see ). Thus, (x0x)kfn and Fn (resp., (x0x)kf and F) differ by polynomials Pn (resp., P) of degree <k on I. By Lemma 3, fnf boundedly implies (x0x)kfn(x0x)kf in D, and since FnF in D(Ω), we have PnP in D(Ω), which is the case only when PnP 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).

Let fD. Then f(ax+x0)c as a0 if and only if f=kF for some k0, where F is a continuous function near x0 such that limxx0(F(x)/(x-x0)k)=c/k!.

Proof.

Let x0=0. If f=kF and F is continuous near 0 with limx0F(x)/xk=c/k!, then F(ax)/akcxk/k! in D as a0, and applying xk we obtain f(ax)c in D. Conversely, suppose f(ax)c in D as a0. Letting fa(x)=f(ax), it is easily observed that fa(x) converges boundedly (in fact, uniformly) to c as a0. By Lemma 4, there exist a neighborhood Ω of 0 and k0 such that (0x)kfa(0x)kc=cxk/k! on Ω. As (0x)kfa=F(ax)/ak if F(0x)kf, we have F(ax)/akcxk/k! as a0. For any fixed x0 in Ω, we have F(ax)/(ax)kc/k! as a0; namely, lima0F(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 . 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 . 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, nN,

fn(x)=0 for |x|an,

fn=1,

|fn| is bounded independently of nN.

We say that a distribution TD(Rd) has δ-value cC at x0Rd if (12)T,τx0fnc 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  (see also  for a proof based on ideas from nonstandard analysis), TD(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 a0+. 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. A δ-sequence (fn) of the form (14)fn(x)=an-df(xan), where fD(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,τx0fnc 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 distributions Dp,x0 for 1p< such that, for any 1qp<, (15){Distributionswith  δ-valueat  x0}Dq,x0Dp,x0{Distributions  withŁojasiewicz’s  valueat  x0} with analogous structural results involving Lp functions. These classes of distributions can be defined sequentially in a natural way.

Definition 6.

Let 1p< 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, nN,

fn(x)=0 for |x|an,

fn=1,

and(p-1)|fn|p is bounded independently of nN.

A distribution TD(Rd) is said to have δp-value  cC at  x0Rd if (16)T,τx0fnc 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 any 1qp, we have (e.g., ) that if ΩRd is a nonempty open subset of finite measure μ(Ω) and if fLp(Ω), then fLq(Ω) and (17)fLq(Ω)μ(Ω)1/q-1/pfLp(Ω). Let 1qp<, and suppose (fn) is a δp-sequence, with a contracting sequence (an)0. From (17) we obtain (18)an-d/q(|fn|q)1/qan-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<pq, (20)a-d/pΨLp(Bx0(a))a-d/qΨLq(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 1p<, if TD(Rd) has δp-value c at x0Rd for some p, then it has the same value c at x0 in the sense of Łojasiewicz.

For a nonempty open set ΩRd, 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 ΨD(Ω), we define (23)ΨDp(Ω)=supfDp1(Ω)|Ψ,f|,ΨDp(Ω)+=supfDp1(Ω)+|Ψ,f|, taking values in [0,]. We then have the following simple estimate.

Lemma 8.

We have (24)ΨDp(Ω)+ΨDp(Ω)4ΨDp(Ω)+.

Proof.

The first inequality follows trivially since Dp1(Ω)+Dp1(Ω). In order to see the second inequality, suppose fDR(Ω). We can write f=f++f-, where f+(x)=max{f(x),0} and f-(x)=min{f(x),0} for xΩ. As f+ and f- are compactly supported continuous functions, we can find f1D(Ω)+ (resp., f2D(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|ΨDp(Ω)+(f1Lp(Ω)+f2Lp(Ω)), since f±Lp(Ω)fLp(Ω), we have (26)|Ψ,f|2ΨDp(Ω)+fLp(Ω). By (26), if fD(Ω), then since we have Re(f)Lp(Ω)fLp(Ω) and Im(f)Lp(Ω)fLp(Ω), (27)|Ψ,f||Ψ,Re(f)|+|Ψ,Im(f)|4ΨDp(Ω)+fLp(Ω).

Suppose ΨDp(Ω)< for some 1p<. 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) . We thus have ΨLp(Ω) and ΨDp(Ω)=ΨLp(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 nN. Then and(p-1)|fn+gn|p is bounded independently of nN.

Proof.

By multiplying by and(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 of δp-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.

Let TD(Rd). Then, the following statements are equivalent.

T has δp-value cC at x0Rd.

T,τx0fnc as n for all δp-sequences (fn) such that fn0.

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/pΨLp(Bx0(a))0 as a0+, 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 case x0=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/pΨDp(B0(a))+0 as a0+. Otherwise, for some ε0>0, we can find a sequence of positive real numbers (an)0 and functions gnDp1(B0(an))+ such that an-d/p|Ψ,gn|ε0 for all nN. We note (31)and(p-1)(an-d/pgn)p=and(p-1)an-d(p-1)gnp=1, and, in particular, it is bounded independently of nN. Applying inequality (19) to the functions an-d/pgn (with q=1), we obtain (32)an-d/pgn1 for all nN. 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/pgn+bnhn, where bn=1-an-d/pgn[0,1). Observe that fn0 with fn=1, and applying Lemma 9 to the sequences an-d/pgn 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/pΨ,gn+bnΨ,hn, the fact that (bnΨ,hn)0  implies (an-d/pΨ,gn)0, a contradiction. Hence, (30) follows, which implies (iii) by Lemma 8 and the paragraph following it.

Lastly, we assume that (iii) holds for x0=0. Let (fn) be a δp-sequence with a contracting sequence (an)0. By Hölder’s inequality, (35)|Ψ,fn|ΨLp(B(an))fnLp(B(an))=an-d/pΨLp(B(an))(and(p-1)|fn|p)1/p0 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 Theorem 5, and the above theorem gives some others.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author wishes to thank the referees for insightful comments and valuable advice on the presentation.

Łojasiewicz S. Sur la valeur et la limite d'une distribution en un point Studia Mathematica 1957 16 1 36 MR0087905 Estrada R. Vindas J. A general integral Dissertationes Mathematicae 2012 483 1 49 10.4064/dm483-0-1 MR2952047 Vindas J. Pilipović S. Structural theorems for quasiasymptotics of distributions at the origin Mathematische Nachrichten 2009 282 11 1584 1599 10.1002/mana.200710090 MR2573468 2-s2.0-76849113967 Gel'fand I. M. Shilov G. E. Generalized Functions. Vol. 1: Properties and Operations 1964 New York, NY, USA Academic Press Antosik P. Mikusiński J. Sikorski R. Theory of Distributions. The Sequential Approach 1973 Amsterdam, The Netherlands Elsevier Scientific MR0365130 Łojasiewicz S. Sur la fixation des variables dans une distribution Studia Mathematica 1958 17 1 64 MR0107167 Shiraishi R. Itano M. On the multiplicative products of distributions Journal of Science of the Hiroshima University A-I: Mathematics 1964 28 223 235 MR0218896 Itano M. On the multiplicative products of distributions Journal of Science of the Hiroshima University A-I: Mathematics 1965 29 51 74 MR0184079 Shiraishi R. On the value of distributions at a point and the multiplicative products Journal of Science of the Hiroshima University A-I: Mathematics 1967 31 89 104 MR0218895 Cohen M. D. On the value of a distribution at a point Mathematische Zeitschrift 1971 122 101 103 10.1007/BF01110083 MR0291797 ZBL0213.40003 2-s2.0-34250460122 Mikusiński J. On the value of a distribution at a point Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 1960 8 681 683 MR0131753 Vernaeve H. Vindas J. Characterization of distributions having a value at a point in the sense of Robinson Journal of Mathematical Analysis and Applications 2012 396 1 371 374 10.1016/j.jmaa.2012.06.023 MR2956970 ZBL1276.46032 2-s2.0-84864855058 Trèves F. Topological Vector Spaces, Distributions and Kernels 1967 New York, NY, USA Academic Press MR0225131