A Proof of Łojasiewicz’s Theorem
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.
Let be the topological -vector space of complex valued compactly supported test functions on , and let be the space of complex valued distributions on . In the following discussion, a distribution is also denoted by , and the dual pairing between and a test function is denoted by either or . On the other hand, the letter will always denote a point.
According to Łojasiewicz , a distribution has the value at if in as . If such a value exists at , we will say that is evaluable at and write . For to be evaluable at , it suffices for to exist in , as the limit can only be a constant. We can equivalently require that there exists such that , as this entails . Simply requiring the existence of does not suffice, as the limit may in general be of the form , where is the Heaviside step function.
One interesting consequence of this definition is the following.
Theorem 1 (Łojasiewicz). If a distribution is evaluable at , then any primitive of is also evaluable at .
This result is useful in various circumstances. For instance, if a distribution is evaluable at and , then so is any primitive of , and we may define a definite integral of as 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 1
In order to fix our notation, we briefly recall the following elementary notions . Suppose we have a continuous family of distributions depending on a parameter in an interval , meaning that is continuous in for each . If is differentiable at for each , we say that is differentiable with respect to at and define by Evidently is a distribution as it is the limit of distributions given by the difference quotients. Similarly, for , we define by 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 and are continuous families of distributions with for all , then, for any ,
Let us note that, for any distribution , both and are continuous families of distributions. If is evaluable at , namely, if as , then becomes a continuous family of distributions if we define . Our argument uses this simple observation.
Proof of Theorem 1. Let in and suppose is evaluable at . As seen above, is a continuous family of distributions and so is the family . It is trivial to verify that the family is differentiable with respect to with . By (5), for , The left-hand side is well defined for and gives a continuous family as ranges over the real line, and thus, taking the limit on both sides, we see that as for some . Applying gives , but clearly . We conclude that is a constant.
It also follows that if is a primitive of a distribution such that , the family is differentiable with respect to and we have . In particular, .
3. Distributions Integrable from a Basepoint
In the preceding proof, it is clear that the assumption that is evaluable at was not entirely necessary. Let us say that is integrable from if the following two conditions hold.(i)For converges in as .(ii) in as .
By the same argument, this definition gives a necessary and sufficient condition for a primitive of to be evaluable at . Indeed, if we set , then (i) is equivalent to the existence of . In this case, since as , (ii) is equivalent to being a constant. We summarize this as follows.
Proposition 2. Let be a distribution and let . Then is evaluable at if and only if is integrable from .
We denote by the space of all distributions integrable from . For , we define a distribution by the formula Let be a primitive of . For any , and if is integrable from , taking the limit , we have as exists by Proposition 2. Replacing with , we define the indefinite integral of with basepoint by It follows that we have and . We also note that is evaluable with value at .
It is easy to see that if is a sequence in , then in for some does not imply in general. In order to remedy this, we introduce the following notions.
Suppose is a sequence in . We say is bounded at if, for each is bounded independently of as well as of . Let us say converges boundedly to if in and is eventually bounded at . Finally, we say converges uniformly to if, for each converges to uniformly in . Clearly, uniform convergence implies bounded convergence.
Lemma 3. If a sequence in converges boundedly (resp., uniformly) to , then the sequence converges boundedly (resp., uniformly) to .
Proof. Suppose in converges to with bounded at , and let . We have which shows is bounded independently of and of , and by taking the limit in under the integral sign, we see that converges boundedly to . If in fact converges uniformly to , the uniform convergence of is also apparent from the same expression.
Let us write on to mean that and are continuous functions on such that converges to uniformly on . Let us also denote by the map that sends to .
Lemma 4. Let be distributions in . If converges boundedly to , then, for every bounded open neighborhood of , there exists an integer such that on .
Proof. Let be a compact interval containing . We can find and a sequence of continuous functions on such that with on (see ). Thus, and (resp., and ) differ by polynomials (resp., ) of degree on . By Lemma 3, boundedly implies in , and since in , we have in , which is the case only when on . Hence on .
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 several times, and the converse is obtained by applying several times.
Theorem 5 (Łojasiewicz). Let . Then as if and only if for some , where is a continuous function near such that .
Proof. Let . If and is continuous near with , then in as , and applying we obtain in . Conversely, suppose in as . Letting , it is easily observed that converges boundedly (in fact, uniformly) to as . By Lemma 4, there exist a neighborhood of and such that on . As if , we have as . For any fixed in , we have as ; namely, .
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 in a -sequence if there is a sequence of positive real numbers such that, ,(i) for ,(ii), (iii) is bounded independently of .
We say that a distribution has -value at if as for all -sequences , where . In fact, we can restrict this condition to real nonnegative -sequences (which are called -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), has -value at if and only if it can be represented as an -function near which is continuous at with value . Thus, we have , with as . 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 -sequence of the form where with and with , is called a model sequence. One sees that a distribution has Łojasiewicz’s value at if and only if for all model sequences. A structural result given by Theorem 5 tells us that the condition imposed on is much weaker.
In this section we find a continuous family of classes of distributions for such that, for any , with analogous structural results involving functions. These classes of distributions can be defined sequentially in a natural way.
Definition 6. Let be fixed. A sequence in is called a -sequence if there exists a sequence of positive real numbers such that, ,(i) for ,(ii),
(iii) is bounded independently of .
A distribution is said to have -value at if as for all -sequences .
Remark 7. In the above definition, we will say that is a contracting sequence of .
For any , we have (e.g., ) that if is a nonempty open subset of finite measure and if , then and Let , and suppose is a -sequence, with a contracting sequence . From (17) we obtain and multiplying both sides by gives which shows that is also a -sequence. Therefore, if a distribution has -value at , then it has the same -value at . This will also follow from Theorem 10 (iii), as we have, since , by (17). Hence, the condition of a distribution having -value at a point becomes less restrictive as increases. As any model sequence is a -sequence for all , if has -value at for some , then it has the same value at in the sense of Łojasiewicz.
For a nonempty open set , we let be the subspace of all real valued test functions and let (resp., ) be the subset of all nonnegative (resp., nonpositive) test functions. Let be the subset consisting of such that , and let For , we define taking values in . We then have the following simple estimate.
Lemma 8. We have
Proof. The first inequality follows trivially since . In order to see the second inequality, suppose . We can write , where and for . As and are compactly supported continuous functions, we can find (resp., ) that is as close as we want to (resp., ) in the -norm, such that . Hence, from since , we have By (26), if , then since we have and ,
Suppose for some . Since is dense in extends to a continuous functional on and lies in the strong dual of , which is isometric to . We thus have and .
Lemma 9. Let be a sequence of positive real numbers such that . Suppose we have two sequences and in such that both and are bounded independently of . Then is bounded independently of .
Proof. By multiplying by on both sides of Minkowski’s inequality for and , we obtain from which the lemma follows.
We can now give a structure theorem on our notion of -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 -sequences (Theorem 10 (ii)).
Theorem 10. Let . Then, the following statements are equivalent.(i) has -value at .(ii) as for all -sequences such that .(iii), where can be represented as an -function in some open ball of radius around , and as , where is the Hölder conjugate of .
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 . Let be a distribution such that for all nonnegative -sequences . For , since and , we have if and only if . We now claim that as . Otherwise, for some , we can find a sequence of positive real numbers and functions such that for all . We note and, in particular, it is bounded independently of . Applying inequality (19) to the functions (with ), we obtain for all . Let be any fixed nonnegative -sequence of which is a contracting sequence, such as a nonnegative model sequence. We let where . Observe that with , and applying Lemma 9 to the sequences and , we see that is in fact a nonnegative -sequence. Thus, we must have But as , the fact that implies , a contradiction. Hence, (30) follows, which implies (iii) by Lemma 8 and the paragraph following it.
Lastly, we assume that (iii) holds for . Let be a -sequence with a contracting sequence . By Hölder’s inequality, as , 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.
The author wishes to thank the referees for insightful comments and valuable advice on the presentation.
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