AN APPLICATION OF THE PROCESS OF REGULARIZATION TO THE ANALYSIS OF DISTRIBUTIONS Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY CHARLES RICHARD DIMINNIE 1970 LIBRARY Michigan Saw University JHags This is to certify that the thesis entitled his [thP‘LICL—k'TICN C1" TEE PROCESS OF R EGULARJ ZAT] CATS 'I‘C' TE. E A] TALE/€118 CF L] STR 3 jfiUTl 017.8 presented by Charles Richard Piwihnie has been accepted towards fulfillment of the requirements for Ph D degree in l‘at‘n‘fiia’t‘ics cf SWAT Major professor 4 Date nly 27, 3“?“ (- 0-169 BIN‘OING BY "BAG 5 snu3'" nuox mom mc. LIBRARY BINDERSI, ABSTRACT AN APPLICATION OF THE PROCESS OF REGULARIZATION TO THE ANALYSIS OF DISTRIBUTIONS BY Charles Richard Diminnie This work represents an attempt to apply certain classical techniques of real analysis to the study of dis- tributions. Historically, mathematicians such as Denjoy and Clarkson have employed the sets EaB g {X1 0 < f(x) < B} to study the behavior of derivatives of functions. In the pre- sent case, a similar approach is used to analyze distributions. Let .D and 3' denote the Spaces of test functions and distributions, respectively, as defined by L. Schwartz. Choose ¢ 6.9 satisfying the following conditions: 1. ¢(x) 2 O on R; 2. IR¢(x)dx = 1; 3. ¢(x) = ¢(-x) for all x; 4. the support of ¢ = [-1,1]; and 5. ¢'(x) > 0 on ]-1,0[, while ¢'(x) < O on ]O,1[. Next, define ¢A(x) = A ¢(xx) for all x 6 R. Then, the net {¢A} con- verges to the Dirac measure 5 in .D', which implies that the net {T*¢A} converges to T in .D' for each distribu- tion T. Hence, each distribution T may be represented by the net of regularizations {T*¢A}. Further, it is demonstrated in Chapter II that the function FT[(x,x)] = T*¢A(X) is con- “ + tinuous on the Space R2 ‘ {(X,A)3 A 2 ll' Charles Richard Diminnie These considerations indicate that it might be possible to examine each distribution T by means of the sets T,N «B N. In Chapter IV, it is shown that these sets satisfy the E ‘ {(x,l): a < T*¢A(X) < B, A 2 N} for large values of following conditions. Theorem 4.10. For all real numbers 0:8 such that a < a, exactly one of the following cases occurs: 1. TSQI or T23. 2. For all N, m (ET’N) > 0 but m (ET’N) 4 0 as N 2.”, 2 as 2 GB T,N a6 Theorem 4.10 leads to the definition of a series of 3. For all N, m2(E ) = a. classes of distributions. Definition 5A. A distribution T is in Class 0-8 if for all a and B: exactly one of the following is satisfied: 1. T s a or T 2 a. 2. There is a set E c R and a number N' such that T,N m1(E) > 0 and (E X [N,m[) G E08 for all N 2 N'. Definition SB. T is in Class O-W if for all a and B, exactly one of the following is satisfied: 1. T s a or T 2 a. 2. m2(E:éN) = a for all N. Definition 5c. T is in Class 9 for e > 0, if for all a and B: exactly one of the following is satisfied: 1. T s a or T 2 B- T N 9 2. There exist numbers N', M such that m2(Eaé ) 2 M(1/N) for N 2 N'. Charles Richard Diminnie It is easily proven that Class 0-8 C Class O-W : Class 9, for e > 0. Also, for 91 < 62, Class 91 c Class 92. The major results concerning these classes are given in Chapter V. If f is an ordinary derivative, then f may be used to define a distribution Tf in this way: 3 P fRf(x)¢(x)dx for all it E .D, where the notation "Rf" denotes Perron integration. Theorem 5.2 states that all distributions Tf, where f is an ordinary derivative, are included in Class 0-8. The main result for the remaining (n)T classes is given by Theorem 5.7. The notation D is used to denote the nth distributional derivative of T. Theorem 5.7. If T = D(n)g, where g is locally bounded, then T 6 01833 n, (Class O-W if n = 0); if T = D(n)g, where g is a locally Lp function for p 2 1, then T G Class (n + 1/p); if T = D(n)p for some measure u, then T E Class (n + 1). Finally, Chapter VI gives examples to illustrate the following distinctions between the e-classes: 1. Class 0-8? Class O-W. 2.1 Class O-WE Class 9, for any a > O. 3. For any“ v satisfying 0 < v < 2, U Class 9i: Class v. 9 0, where m denotes one- either E ‘ Q ’or a 1 a ml(EaB dimensional Lebesgue measure. Since distributions are usually considered to be "generalized functions", it is only natural to ask whether they can be studied by means of a similar approach. It was in the course of developing this problem that the results of this paper were formulated. Since it is impossible to define a set similar to EGB for a distribution, we use an alternate approach. For any distribution T, there is a family of infinitely differentiable functions {ix} such that VA d T in the sense of distribu- tions. With this in mind, we study T using the sets {x: a.< ¢x(x) < a} for large values of x. More specifically, we consider the relationship of the two-dimensional Lebesgue measure of the sets {(x,x): a.< fixix) < e, A 2 N} to powers of (1/N) as the criterion for defining classes of distribu- tions. The major result obtained by using these classes is that for most distributions, membership in the appropriate class is dependent solely on the character of the primitives of these distributions. Thus, a result which is somewhat analogous to that of Denjoy and Clarkson may be obtained by our method. In Chapter II, we give a further justification for the approach described above and we discuss the type of family {wk} that we will choose to "represent" the distribution T. Chapter III is devoted to the development of a certain collec- tion of distributions which are not usually discussed in the literature. A thorough analysis of the nature of the sets {(x,x): a < ¢x(x) < B, A 2‘N} is given in Chapter IV. Chapter V describes the classes of distributions defined by means of these sets and gives the main results concerning these classes. Chapter VI consists of examples which lend more weight to the results of Chapter V. Finally, a short discussion of conclusions and open questions is found in Chapter VII. Before we proceed to the work at hand, it might be advisable to include a brief discussion of the space of dis- tributions and a statement of some of the results which are used in this paper. The development described here is basically that of Laurent Schwartz, with minor adjustments to meet the requirements of our problem. The reader is referred to either [3], [9], or [11] for a more detailed account of the theory of distributions. We begin by considering CC(R), the Space of continuous, real-valued functions with compact support. Since the usual topology assigned to CC(R) is somewhat complicated, we will not Specify it completely. Instead, we will make note of the definition of convergence in CC(R). Vn'fi 0 in CC(R) if 1. there is a compact set R such that the support of in is contained in K for all n; 2. ¢n(x) A 0 uniformly. If .H is a continuous, real-valued, linear functional on CC(R), then u is called a measure. (The relationship between these measures and the usual notion of set functions is given by the famous Reisz representation theorem.) The following are Specific examples of measures: 1. The Dirac delta measure 5 is defined by <59W> = ¢(0) for all W E CC(R). 2. If f is a function which is Lebesgue integrable on each finite interval, then f may be used to define a measure in this way: - IRf(x)v(x)dx for all w E CC(R). Note that it is conventional to identify the function f with the functional it defines and to use f to denote both concepts. The Space of measures forms a first generalization to the notion of function. It includes most functions as well as certain other objects, such as the Dirac measure, which have been (incorrectly) used as functions in classical physics and mathematics. However, for the needs of differential equations and certain other aSpects of mathematics, it becomes necessary to enlarge the space of measures. This is accomplished by reducing the set of objects on which the functionals are to be applied. The following reduction of the Space CC(R) leads us to the expansion of the space of measures to the Space of distributions. Let C:(R) denote the space of infinitely differentiable, real-valued functions with compact support. We use the symbol .9 to describe C:(R) endowed with the topology which has the following definition of convergence: tin-.0 in .8 if 1. there is a compact set R such that the support of in is contained in K for all n; 2. for each integer k 2 0, ¢:R)(x) a O uniformly, where wik) denotes the kth ordinary derivative of -¢n. We will frequently refer to the elements of ,D as "test functions". Next, we define the Space of distributions,.&', to be the space of continuous, real-valued, linear functionals on 3. Since ,0 forms a topological vector Subspace of CC(R), it is clear that .D' is an enlargement of the Space of measures. The usual topology given to .3' is such that Tn-a0 in .9' if -oO for all i6). A11 measures and locally Lebesgue integrable functions may now be considered as distributions. Again, we will not distinguish between the measure p and the distribution H or between the function f and the distribution f. The concept of distributions has an immediate advantage over that of measures since there is a convenient method for defining differentiation in .fi'. We define the nth distribu- tional derivative of a distribution T, D(n)T, to be that dis- tribution satisfying: (1.1) = (-1) (n) for all ‘I E .9. It is clear from the definition of convergence in .3, that the linear functional D(n)T is continuous on .3 and hence defines a distribution. Further, under the above definition of differentiation, every distribution has derivatives of all orders. If f is a function which is n-times continuously differentiable, then for all V 6.3, we have = (-1)“<£,¢(“)> = (-1)“ij(x)¢(“)(x)dx. Using integration by parts n-times, we obtain = IRf(n)(x)¢(x)dx =‘ for all T 6.8. (n)f is the same as the functional Thus, the functional D defined by f(n). According to our convention, we identify D(n)f and f(n) and say that the nth distributional derivative of f is the same as the nth ordinary derivative of f. A distribution T is said to be positive, denoted T 2 0, if for every t 6.5 such that ¢(x) 2 0 on R, we have I 2 0. For two distributions S and T, S 2 T if (S-T) 2 O. The following result relates positive dis- tributions to the subSpace of measures (c.f. [9], Chapter I, Theorem.V.). Theorem II. If T is a positive distribution, then T is a positive measure. If it 6.8, then we may also consider t as an element of 3'. For any T E 3', we define the convolution product T*¢ to be the distribution given by the function T*¢(x) = , where the subscript t on T is used to indicate that the functional T is operating on ¢(x-t) considered as a function of t. It is easily seen that in the case where T is a locally Lebesgue integrable function f, the above notion of convolution agrees with the classical definition. It can be shown that T*¢(x) is an infinitely differentiable function of x, whose derivatives are given by _ (n) (1.2) '(‘r*¢>(“) = (D‘n’nwa) - w (x). The convolution product T*¢ is also called the regularization Of T by i. The concept of convolution can be generalized to certain other distributions, but the conditions on the distributions involved are somewhat complicated. Since we will make little use of this notion, we will not consider it further. In [8], Schwartz gives examples of families of test functions {¢A} which converge to 6 in .3'. Further, he gives sufficient conditions for a family of test functions to converge to 6 in .D' (c.f. [8], Chapter II, Theorem 13). If {¢A} is a family of test functions which converges to 5 in .D', it can be Shown that for any distribution T, the family {T*¢A} converges to T in .D' (c.f. [8], Chapter III, Theorem 7). This justifies our earlier statement that every distribution T is the limit in .D' of a family of infinitely differentiable functions. CHAPTER II DISTRIBUTIONS AS IlMITS 0F INFINITELY DIFFERENTIABLE FUNCTIONS As mentioned in Chapter I, we are going to study each distribution T by means of a net of infinitely differentiable functions {WA} which converges to T in .D' as A a m. We will demonstrate that this is an appropriate method by briefly relating the approach used by Mikusinski in [6] and Temple in [10] to arrive at an alternate definition of the Space 3'. Basically, they define the Space of distributions as a completion of the set of infinitely differentiable functions. The approach is essentially the same as that used by Cantor in the construction of the real number system from the rationals. To begin, we define a sequence of functions {in} to be regular if 1. each in is infinitely differentiable; 2. for each n 6.9, IR¢n(x)N(x)dx converges to a limit, which we will denote by ; 3. this limit I is continuous on .D, i.e., -« 0 whenever 'nn-oO in .9. Two regular sequences {an} and {on} are said to be £333: valent if for each ‘n E .6, we have IRC¢n(x) - °n(x)lR(X)dx a 0 as n «.m. 8 Since this notion is obviously an equivalence relation, it partitions the set of regular sequences into equivalence classes which are designated as_ggneralized functions. Thus, each of these generalized functions may be specified by any of the regular sequences in its class. In particular, the Space of generalized functions is a completion of the set of infinitely differentiable functions. Each such function i may be considered as the generalized function represented by the sequence {in}, where in - i for all n. Operations on generalized functions are defined by means of the correspond- ing operations on the regular sequences which represent these functions. For example, if g denotes the generalized func- tion represented by the regular sequence {nu}, then the mth (m) g derivative of g, D , is defined to be the generalized (“0} function represented by the sequence {in The preceding paragraphs present a basic description of the construction of these generalized functions. Rather than complete the development of this theory, we will now address ourselves to the work of reconciling these concepts to the distributions of Schwartz and ultimately to the task of applying these ideas to the problem before us. First of all, it is easy to see that if the generalized function g is represented by the regular sequence {in}, it may be associated with the functional T on .8 defined by . - lim‘IR*n(x)N(x)dx. The continuity of T then follows nqw directly from condition 3 of the definition of regular 10 sequence. Thus, the generalized functions of Mikusinski and Temple are included in the Space of Schwartzian distributions. To Show the correspondence in the other direction, we will require the services of an auxiliary function ¢ 6.3 having the following properties: 1. ¢(x) 2 0 for all x E R; 2. IR¢(x)dx = l; 3. the support of ¢(x) = [-l,l]; (2.1) 4. ¢(x) = ¢(-x) for all x; 5. ¢'(x) > 0 for x 6 ]-l,0[ and ¢'(x) < 0 for x E ]O,1[; 6. max ¢(x) = 05(0). xER (Actually, 2.1.6 follows from 2.1.3 and 2.1.5, but it will be used so often in this work that we will list it as a separate prOperty.) A particular example of such a function is the approximating function ¢* defined by: * O for |x\ 2 l ¢(X)={ 2 C exp[-1/(1-x )] for \x‘ < 1, where .C is chosen so that IR¢*(x)dx = 1. For ¢ satisfying conditions (2.1), we will make the following definitions: (2.2) 1. For A > 0, ¢A(t) = k¢(kt)- 2. For A 2 O and x E R, ¢x,x(t) = mxix‘t) ' X¢EX(X‘C)]- It can be Shown that the net {¢R} converges to 6 in .3' as x.» m and hence, for any distribution T, the net ll {T*¢A} converges to T in .3' as A a o (c.f. [8], Chapter 11, Theorem 13 and Chapter III, Theorem 7). In particular, since each of the regularizations T*¢A is an infinitely differentiable function, we see that the sequence {T*¢n}, where ¢n(t) = n¢(nt), is an appropriate regular sequence to represent T as a generalized function. Thus, the theory of generalized functions of Mikusinski and Temple is in complete accord with Schwartz' theory of distributions. For our purposes, it is even more important to notice that their theory illustrates that it is entirely natural to study a distribution T by means of the sequence of regularizations [T*¢n(x)}, since they are essentially the same object. Also, we note that all of the methods of Mikusinski and Temple may be applied through the use of regular nets instead of regular sequences. The reasons why we choose to use the regular net {T*¢A} to represent T will be made more apparent later in this chapter. The remainder of this chapter will be devoted to dis- playing the continuity properties of the regularizations T*¢x(x) considered as a function of both x and A- We begin with the following two lemmas. Lemma 2.1. If (xn,>.n) .. (mi) in I?»2 (two dimensional Euclidean space) and A as well as all the kn are positive, then ¢x converges to ¢x A in .9- , n’xn Proof: Since the numbers ‘xn-x‘ and ‘1/(xn)-1/x‘ are bounded independent of n and the support of each ¢x ,x n n 12 [xn-l/(xn), xn+l/(An)], it is clear that there is a compact set R such that the support of each 9x A is contained in K. The compactness of K implies that there is a constant C Such that max ‘t-x‘ s C. Then, for any t E K, we obtain t€K Iln(Xn-t)->.(x-t)| S inlxn-xl + Iin-in-q s xn‘xn-xl + \xn-Mc. Therefore, xn(xn-t) a X(x-t) uniformly on K since xn « x and )‘n --o A' Let a > 0 be given. By the uniform continuity of ¢, there is a number A depending only on e such that ltl-tzl < A implies ‘¢(t1)-¢(t2)| < e/(Zk)- If we denote max ‘¢(t)| by M, our next Step is to choose N large enough that for n 2 N, we obtain both [An-kl < e/(ZM) and max ll (x -t)-x(x-t)‘ < A. Then, for t E K and n 2 N, tEK n n |¢x ’An(t)-¢X:A(t)' Ixnalin(xn-t)] - i¢[i(x-t)]I n S Ixn-XII¢Exnll + il¢lin(xn=t)] - ¢[i(x-t)]I s lin-iIM + iI¢£in(xn-t>] - ¢[i(x-t)]\ < e/Z'l'e/Z = 3. Since xn(xn-t) a x(x-t) uniformly on K, and A was de- pendent only on as our choice of N is uniform for all t 6 K. ‘Thus, ¢x :R « ¢x,x uniformly on K which implies n n that ¢ 4 ¢x,x uniformly on R. x n’xn 13 By noting that the only properties of ¢ used in the above argument were its uniform continuity and its boundedness, we see that a similar proof yields that ¢:k)k a ¢éki uniformly n’ n ’ on R for all positive integers k. Hence ¢x :l a ¢Xal n n in .3. Q.E.D. Lemma 2.2. If ¢x A a y in .3 and there is a positive -__—_- 3 n n number 10 such that An 2 A0 for all n, then there is an element (x,x) E R2 such that (xn’xn) a (x,x) in R2 and further, ¢ = ¢x A. 3 Proof: Since ¢ 4 W in .3, we have that ¢x l a W n’xn n’ n uniformly on R. Hence, |max Rx ,A (t) - max ¢x A (t)‘ a O t€R n n tER m’ m as m,n a m which implies that lin¢(0)-xm¢(0)| a 0 as m,n as, by (2.1.6) and (2.2.2). Thus, ‘ln'lml -. o as m,n a 0 Since ¢(0) >»O. Therefore, {An} is a Cauchy sequence in R and there is a A 6 R Such that An a l- Further, since each kn 2 No, we have A 2 A0 > 0. Again, using the fact that ¢x a w uniformly on n’xn R, we obtain that max A (t) a max ¢(t). However, Since tER n’ n t€R 1 a = 0 for each n and a A: we a so “an“ ing“) in tER n n have that max ¢x A (t) a l¢(0) which implies that 3 n t6R n . max ¢(t) = A¢(0)- Thus, the fact that y 6.3 yields the tER existence of an x 6 R such that x¢(0) = max ¢(t) - ¢(x). tER We will Show that xn a x and hence that (anln) * (Xal) in R2. Since (t) » ¢(t) pointwise on R, then in ¢xn"‘n particular, ¢Xn’xn(x) « ¢(x) 8 l¢(0)- Thus, 14 in¢lxn(xn-x)] —~ i¢(0) as n -. a which implies that (*) $[xn(xn-x)] a ¢(0) as n a m since Kn a A and each An 2 A0 > 0. Now, suppose rxn A x. Then, there is an e > 0 and a subsequence {xnk} of {xn} such that ‘xn -x\ 2 e for all k. Thus, for all k, xnk‘Xnk-x‘ 2 Take 2 x05. Also, by- (2.1.5) and (2.1.6), we obtain both ¢(xoe) < ¢(O) and ¢[xn (xn -x)] S ¢(xoe) for all k. Therefore, k k lim sup ¢[x (x -x)] s ¢(l e) < ¢(0), which is impossible by n n o k—aco k k * -v -o ' , ( ). Hence, xn x and (xn’xn) (x,x) in R2 Since A as well as all the Rn are positive, we may apply Lemma 2.1 to obtain that ¢ ~ ¢ in .3. xnahn xsh Thus, V = ¢ A and the proof is completed. Q.E.D. X, Note that the proof of these lemmas relied heavily on the restriction of A to values which were bounded away from zero. Since we will be primarily interested in working with the functions ¢x X when A is a large positive number, 3 we will impose a lower bound on the values of l for all that follows. With this in mind, we will use the symbol + R2 to designate the Space {(x,x): x 6 R and A 2 1}. By combining the previous lemmas, we obtain the follow- ing theorem: + . Theorem 2.3. The map p: R2 —9 fi deflned by p[ (XSX)] = ¢x’x is a homeomorphism from R: into .3. Using the definition of the functions ¢x A, we may 3 express each of the regularizations T*¢X(x) as follows: 15 (2.3) T*¢x(x) =‘ -‘' Thus, by Theorem 2.3, we may associate each T 6.3' with a . . + . . unique continuous map FT from R2 to R in this way: (2-4) FT[(X,>.)] = T o p[(x,).)] = = = <1" Rx»)? g T*¢>.(x)' It is this continuity of the expression T*¢x(x) with respect to both x and A which leads us to consider the net {T*¢A}’ rather than the sequence {T*¢n], as the suit- able representation for T. CHAPTER III DISTRIBUTIONS DEFINED BY PERRON INTEGRALS In this chapter, we digress briefly to consider a certain collection of distributions which are not usually mentioned in the standard references. This family, which includes the subspace of locally Lebesgue integrable func- tions, will furnish us with a number of explicit examples of distributions other than the obvious Specimens ordinarily cited. Moreover, we will utilize them to generalize a well known property of the regularizations of Lebesgue integrable functions. We will base our approach on the generalized integral developed by Perron. Due to the complicated nature of this theory, we will not enter into a complete discussion of the Perron integral, except to mention that it is loosely founded on the notion of defining integration as the inverse Operation of differentiation. The reader is referred to [5], Chapter VIII, for a full development of Perron integration. Instead, we will state some of the main reSults con- cerning the Perron integral. For the most part, these theorems are in slightly weaker form than the versions given in the above source. We will use the notation "PI" to denote Perron integration, while if" will pertain to Lebesgue integration. 16 17 Theorem A. If f1 and f2 are Perron integrable on the interval [a, b] and k1, k2 are constants, then k1f1+k2t2 is Perron integrable on [a, b] and PI: [k1f1(x) + k2f2(x)]dx -= kIPJ”: f1(x)dx + kZPJ”: f2(x)dx. Theorem B. If the function F is continuous on [a, b] and F' is defined and finite on ]a, b[, then F' is Perron integrable on [a, b] and Pr: F'(t)dt = F(x)-F(a) for all x 6 [a, b]. Theorem C. If f is Lebesgue integrable on [a, b], then f is also Perron integrable on [a, b] and PI: f(x)dx = = I: f(x)dx. Theorem D. If f is Perron integrable on [a, b] and F(x) = PT: f(t)dt, then F is continuous. Theorem E. If f is Perron integrable on [a, b] and F(x) = Pf: f(t)dt, then F'(x) = f(x) a.e. on [a, b]. (The notation "a.e.” denotes "almost everywhere", i.e., except for a set of Lebesgue measure 0). Theorem F. (Integration by Parts). If f is Perron integrable on [a, b] and y is of bounded variation on [a, b], then ft is Perron integrable on [a, b] and Pf: f(x)¢(x)dx = F(b)¢(b) - I: F(x)d¢, where F(x) = Br: f(t)dt and f: F(x)d¢ is a Stieltjes integral. With these preliminaries over, we now proceed to use Perron integrals to define distributions. Let f be a func- tion which is Perron integrable over every finite interval. By Theorem B, finite derivatives of continuous functions provide 18 examples of Such functions. For any test function y 6.3, Theorem F tells us that ft is also Perron integrable over every finite interval. Further, since fl has compact support, we may use the symbol ITRhf(x)$(x)dx without confusion. Using Theorem A, we see that the functional Tf defined by df,¢> = FIR f(x)¢(x)dx is linear on .3. To prove that Tf is also continuous on .3, we make use of the function G(x) - If; f(t)dt. Since Theorem D states that G is a con- tinuous function, we may also consider G as a distribution. Therefore, we use Theorem F to obtain the following: = PPR f(x)w(x)dx = -IR [G(x)+C]¢'(x)dx, where C is a constant. However, since W 6.3 implies that IR¢'(x)dx = 0, we may write (3,1) = 'i3 G(x)¢'(x)dx = . Hence, the continuity of the functional DC on .3 insures the continuity of Tf. Further, (3.1) tells us that the dis- tribution Tr is the distributional derivative of G, i.e., f (3.2) T = DG. Let us consider the case where g is a continuous function having a finite derivative g' everywhere on R. Then, the above arguments may be combined with Theorem B to obtain the following: (3.3) T3 - Dg. In this way, we observe that for a continuous function g with a finite derivative g', the notions of distributional l9 derivative and ordinary derivative are basically the same even though g' is not necessarily locally Lebesgue integrable. We now address ourselves to the consideration of the net {Tf*¢l} of regularizations of Tf by the family {¢A: A 2 1} described in Chapter II. Recalling the definition of convolu- tion product given in Chapter I, we see that (3.4) Tf‘k (x) = = pf f(t)¢ (x-t)dt. Cb. t i R K It is a well known classical result for the case where f is locally Lebesgue integrable that f*¢x(x) 4 f(x) a.e. on R. These next two results will show that the same statement is true when f is locally Perron integrable. Lemma 3.1. Let F be locally Lebesgue integrable. If F is differentiable at x0, then F*¢i(xo) a F'(xo) as A «>0. Proof: If we set v(t) = F(xO)-F(xo-t)-tF'(xo), then |v(t)/t| d 0 as t a 0. Hence, if e > O is given, there is a A > 0 such that ‘v(t)| { eltl whenever ‘t‘ < A. Using (2.1.2), integration by parts, and the fact that 11 l/y ¢x(t)dt ‘ 0 we obtain the following: 1/). . fix 1/, v(t)o,'\(t)dt= F(x 01)f 1/). ¢x(t)dt -.j_1/A F(xO-t)¢)\(t)dt -F' (x fi)f l/l t¢i(t)dt xo+1lx g PIX 'l/X o = F'(xo) - F*¢'(xo ). 1/ F(t)¢i(xo-t)dt + F'(xo)j_l>x ¢A(t)dt Thus, for all i. IF*¢'(xo )-F' (x o)| sfl" 1/). [v(t)H¢;‘(t)‘dt. In particular, for A > l/A, 20 I I 1/ , 1 |F*¢x(xo)-F (xo)| < 3I_1>x|t||¢ mm = -c‘[_]/.>xt¢;\(t)dt by (2.1.5) .1 I Therefore, F*¢i(xo) a F'(xo) as A 41m. Q.E.D. Theorem 3.2. If f is locally Perron integrable on R, then 1im Tf*¢k(x) 8 f(x) a.e. on R. In particular, this result X—m is true for all x such that F'(x) = f(x), where F(x) 3 9]"; f (t )dt. Proof: By Lemma 3.1, F*¢i(x) a F'(x) whenever F'(x) exists. Using (1.2) and (3.2), we obtain that F*¢i(x) = DF*¢X(X) = Tf*¢x(x) for all x. Therefore, Tf*¢x(x) a F'(x) whenever F'(x) exists, which implies that Tf*¢x(x) a f(x) a.e. on R by Theorem E. Q.E.D. We conclude this chapter by noting that an analogous result is also possible in the case of a measure u, in that the net {u*¢x(x)} converges a.e. to a function which is closely related to u. Although this result is not useful to us here, the reader may find it instructive to refer to the discussion of the Poisson integral given in [7], Chapter 11. The methods used there may be adapted to prove Theorem 3.2 if we use ¢x(x-t) in place of the Poisson kernel and Substitute integration by parts for the use of Fubini's Theorem in (Rudin's) Lemma 11.9. CHAPTER IV THE SETS ET AND ET’N as as We are now ready to set up the machinery for our analysis of individual distributions by means of their reg- ularizations by the net {¢A’ K 2 1]. As was indicated in Chapter I, we will attempt to apply a Denjoy-type approach to each T 6.3' by examining the sets {x: a < T*¢k(x) < B} for large values of x. However, in order to make the maximum use of the continuity of the expression T*¢X(X) with reapect to both x and A, we will find it more advan- tageous to consider the sets {(x,x): a < T*¢K(X) < B: A 2 N} + in R2, as N becomes large. With this in mind, we begin with the following definitions: Definition 4A. If T 6.3' and oz< B, then T , * 1 1. E018 -= {(x,).). a< T ¢l(x) < B, I. 2 } 2. 152:1 = {(x,l): at < my) < a, i 2 N}- T -1 . . . Note that according to (2.4), EaB = FT (]a,e[), which implies that E28 is an open set in R:. Further, it is clear that ET’N+1 : ET’N : ET for all N 2 1. a8 dB «8 The first series of results will deal with the essen- tials of the set ET . The symbol m2 will be used throughout as 21 22 to denote two-dimensional Lebesgue measure. This first lemma follows directly from the fact that each Egg is an open set in R3. Lemma 4.1. For all real numbers a,5 such that a < B, we have either E26 3 Q or m2(E:B) > 0. A similar argument, using the properties of PT restricted to {(x,x): A 2 N}, yields this corollary: Corollary 4.2. If N 2 1, then for all 0:5 such that a < a, we have either EZSN ' Q or m2(E:éN) > 0. This next lemma, originally given by S. Lojasiewicz in [4], serves to Show the relationship between a positive distribution T and the point functions T*¢A(X) which re- present T. More specifically, it demonstrates that if a dis- tribution is not positive, it fails because eventually the regularizations T*¢x(x) take on negative values. Although this result is used only Sparingly in this chapter, we will find it to be crucial for many of the results of Chapter V. Lemma 4.3. If there is a test function y 6.3 such that ¢(x) 2 0 on R but -< 0, then there is a number A such that correSponding to each A 2 A, there is an xx in the support of V for which T*¢A(xl) < 0- Proof: Since T*¢A 4T in .3' as A a m, we observe that .‘IR[T*¢A(X)]W(x)dx ~1 as A d 9- Thus, '< 0 implies that there is a A such that .IR[T*¢x(X)]¢(X)dx < 0 for all A 2 A. The conclusion then follows from the fact that ¢(x) 2 0 on R. Q.E.D. 23 We now produce two lemmas which will enable us to establish an interesting correspondence between the sets 5:3 and the statements ”T s a", "T 2 B"- Note that these state- ments make sense in that any constant y may be considered as both the constant function y(t) a y and the constant dis- tribution defined by - yJ'Rtht for all y e .9. T + Lemma 4.4. EaB = 6 iff either FT 2 B on R2 or FT 5 a on R:. Proof: The sufficiency is obvious, since E35 = F£1(]a38[). + -1 -1 -1 Necessity: sz‘ FT (]-w,a]) U FT (1a,B[) U FT ([B>“[)- -l T Therefore, if FT (]a,B[) 2 E08 = 6, we have + - - . . R2 = PT1(]-w,a]) U FT1([B,Q[). The continuity of FT and + -1 the connectedness of R then imply that either FT (]-,a]) = a 2 or T;1([B,m[) = Q from which the conclusion follows. Q.E.D. Lemma 4.5. E28 = 6 iff T s a or T 2 a. T + Proof: By Lemma 4.4, Ea = 6 iff T S a on p(R2) or B T 2 B on 9(R3), (see Theorem 2.3 and the relation (2.4)). + Now, assume T s a on .3. Then, for each (x,x) 6 R2, ' ' t , >s° tdt, ¢X2A(t) 2 O on R which implies the 0. 2. T S a. 3. T23. The remainder of this chapter will be devoted to prov- ing that the properties of E: are essentially determined by B the sets Ei’N, N 2 1. T . . T,N* Lemma 4.7. E B = 6 iff there is an N* such that Ed - Q. "“"“" a Proof: The necessity is true for all N 2 1 since each T N T E ’ c E . as 018 T ,N* = 9 Sufficiency: Suppose there is an N* such that Ea Then, since FT restricted to {(x,A): A 2 N*} is still a continuous function, we may use a proof similar to that of Lemma 4.4 to obtain that T s a on p({(x,A): A 2 N*}) or T 2 B on p([(x,A): A 2 N*]). Again, Lemma 4.3 tells us that this is enough to insure that T s a or T 2 B~ The con- clusion follows from Lemma 4.5. Q.E.D. By applying Lemma 4.1, Corollary 4.2, and Lemma 4.7, we obtain the following corollary: T,N . Corollary 4.8. m2(E:B) > 0 iff m2(EmB ) > 0 for all N 2 l. Before presenting the main result of Chapter IV, we will need some notational conventions and an additional lemma + concerning subsets of R2. 25 Definition 43. If G is a measurable subset of R;, then a“ = {(x,A) e c: y. en}. Definition 4C. For all N 2 1, we define HN = {(x,A): N s A1< N+l}. Lemma 4.9. If G is a measurable subset of R:, then either m2(GN) = a for all N or m2(GN) a 0 as N a a. Proof: It is clear from Definition 4C that for each N, we Q Q have a“ = u (c n "1) and m2(GN) .. z m2(G n HK). Thus, if K=N N* KFN ¢ there is an N* such that m (G ) < a, then 2 m (G n H ) < w 2 KFN* 2 K fl which implies that lim m (GN) = 1im [ 2 m (G n HR)] = 0. Q.E.D. 2 2 New Nam KFN Finally, we present the result which will form the foundation for the definition of the classes to be studied in Chapter V. This theorem follows directly from Theorem 4.6, Corollary 4.8, and Lemma 4.9. Theorem 4.10. For all real numbers 0:8 such that a < e, exactly one of the following cases occurs: T l. E = Q, i.e., T s a or T 2 6. QB T N T,N a d 2. For all N, m2(Eaé ) > 0, but m2(Ea8 ) O as N m. ,N B )=«:. T 3. For all N, m2(Ea CHAPTER V THE e-CIASSES It is obvious that condition 1 of Theorem 4.10 cannot occur for all a and 5- Further, there are numerous dis- tributions for which this case never occurs. In particular, Theorem II of Chapter I tells us that any distribution which is not a measure can never satisfy this condition. For these reasons, we will rely on conditions 2 and 3 of Theorem 4.10 for our analysis of distributions. With this in mind, we make the following definitions: Definition 5A. A distribution T is said to be in Class 0-8 if for all a,B such that a < a, exactly one of the follow- ing occurs: 1. T S a or T 2 B. 2. There is a measurable set E<: R and a number N' such that m1(E) > 0 and (E X [N,m[) G E:;N for all N 2 N'. Definition SB. A distribution T is in Class 04W if for all a,8 such that a.< B, exactly one of the following occurs: 1. T s a or T-2 e. 3 8 Definition 5C. A distribution T is in Class E for e > 0, )=-. T 2. For each N, m2(Ea if for all 0:3 such that a < a, exactly one of the follow- ing occurs: 1. Tsa or T25. 26 27 ,N B 2. There exist numbers N' and K such that m2(E: ) 2 K(l/N)e for all N 2 u'. We may make some immediate observations about the above definitions. Remark 1. Suppose T 6 Class 0-8 and condition 2 of Defini- tion 5A is satisfied. Then, there is a set E G R and a T,N GB for all N 2 N'. In particular, for N 2 N', we apply Fubini's number N' > 0 such that m1(E) > 0 and E X [N,m[ C E 3N g G a a > 2 mm x £N.~[> jN m1dx e. l since m1(E) > 0. Further, since EE’N G ET”N 6 0:8 N sN', we have that m2(E:éN) = on for all N. Thus, every Theorem to obtain m2(E: for all T in Class 0-8 is also in Class 04W, i.e., Class O-S : Class O-W. .Remark II. Clearly, Class O-W C Class 9 for all e > 0. Remark III. Suppose that O < 91 < 92. If T 6 Class 91 and condition 2 of Definition SC is satisfied, then there 6 T,N I exist numbers N', K such that m2(EdB ) 2 K(1/N) for 9 92 all N 2 N'. However, since (l/N) 1 z (l/N) , we have that m2(E:éN) 2 K(1/N)62 also for N 2 N'. Thus, T 6 Class 92 which implies that Class 91 C Class 92 when- ever 0 < 91 < 92. The remainder of this chapter will be devoted to the establishment of significant sufficient conditions for member- ship in the various classes. We begin by proving a result based on Theorem 3.2 of Chapter III. Lemma 5.1. Let h be a locally Perron integrable function and a < B. If m1({x: a.< h(x) < 3}) > 0, then there is a 28 set E C R and a number N' such that m1(E) > O and h,N (E X [N,co[) C EO’B denote the distribution Th as well as the point function for all N 2 N'. (Here, we use h to h(x) in order to alleviate the notational problem). Proof: Since {x: a < h(x) < a} = ,G'Hi’ where Hi = [x: a + l/i s h(x) s B-l/i}, there isla set H and a number n > 0 such that O < m1(H) < m and a + n s h(x) S B-fl for all x E H. To simplify matters, let hx(x) = h*¢x(x). Then, since hk(X) a h(x) a.e. on H and m1(H) < a, we apply Egoroff's Theorem to find a subset L of H such that m1(L) < [m1(H)]/2 and hx(x) a h(x) uniformly on H\L, (the complement of L in H). The conclusion follows if we set E = H\l and choose N' large enough that max ‘hx(x)-h(x)‘ < M2 for all x 2N'. Q.E.D. xEE Definition 5D. A locally Perron integrable function h will be called admissible if for all numbers a,B such that a < B, we have m1({x: a < h(x) < 5}) = 0 iff either h(x) s a a.e. or h(x) 2 B a.e. If h is a finite derivative, then h satisfies the Darboux condition, i.e. h(C) is a connected set whenever C is con- nected. This condition, together with Theorem B of Chapter III and Theorem I of Chapter I, tells us that finite derivatives furnish examples of admissible functions. Definition 5D provides us with a family of distributions which belong to Class 0-8. 29 Theorem 5.2. If h is an admissible function, then the dis- tribution Th is in Class 0-8. Proof: If h(x) s a a.e., then it is a well-known result that h is locally Lebesgue integrable, (c.f. [5], Chapter VIII, Theorem 62.1). Thus, for any test function W 6.5 such that ¢(x) 2 0 on R, we have = RTRh(x)W(x)dx = IRh(x)¢(x)dx S qIR¢(x)dx, or Th 5 a. Similarly, if h(x) 2 B a.e., then Th 2 5. Finally, if neither of these cases occurs, we have m1({x: a < h(x) < 5}) > 0 which implies condition 2 of Definition 5A by Lemma 5.1. Q.E.D. In particular, Theorem 5.2 tells us that all continuous functions and finite derivatives are included in Class 0-8. To bring out our main result, we will depend on a series of lemmas which concern the local properties of the regulariza- tions of T. For the rest of the chapter, we will use fx(x) to denote T*¢A(X). Recall also, that we are assuming that X 2 l in each case. Lemma 5.3. Suppose I = D(n)g, where g is a locally bounded function. Then, for each finite interval [a, b], there is a +1 constant K such that ‘fi(x)‘ s Kin for all x 6 [a, b]. Proof: By (1.2), we have fi(x) = T*¢i(x) = D(n)g*¢i(x) = g*¢§é+l)(x) for all x 6 R. In particular, if ‘g(x)‘ S M a.e. on [a-l, b+l], we obtain the following for all x 6 [a, b]. 1 +1 ’. ( +1) ‘fi(x)‘ S I:ti;;\g(t)||¢in+l)(x-t)‘dt s ”I:-1;:‘¢xn (x-t)‘dt +1 -- Mxn+1J'}i[¢(n+1)(u)|du = 10,“ . Q.E.D. 30 lemma 5.4. Suppose T = D(n)g, where g is a locally Lp function for p > 1. Then, for each finite interval [a, b], + +1 there is a constant K such that \fi(x)‘ s Kin 1 /p for all x E [a, b]. (n+1) k Now, if q is the number for which 1/p+l/q - l, we apply Proof: As in Lemma 5.3, fi(x) = g*¢ (x) for all x E R. Holder's Inequality to derive the following for all x E [a, b]. \f;\ s.IZTi§§|s\I¢{“*1’ldt .+1/ 1/ +1/x ( +1) q l/q s {J:-1,§1s|pdti p{5:-1,,\¢,“ \ dc} [A - 1 1 1 {Igfii8(t)ipdt}1/pkn+2 1/q{f_1‘¢(n+ )(u)‘qdu} /q n+l+l/p Q E D (n) Lemma 5.5. Suppose T = D p, where p is a measure. Then =Kx for each finite interval [a, b], there is a constant K such + that ‘fi(x)‘ s Kxn 2 for all x 6 [a, b]. _ (n+l) Proof: As in the previous lemmas, fi(x) - u*¢x (x) for (n+1) all x E R. Then, if M = max ‘¢ (x)‘ and \p‘ denotes xéR the total variation of the measure n, we have the folloW1ng for all x e [a, b]. n+2.x+l/x ‘fi(x)‘ s I:t:;:‘¢;n+l)(x-t)‘d\u(t)| 5 MI Jx_1/xd\u(t)‘ .2 s mlu\([a-1, b+1])xn+2 = xxn+ . Q.E.D. Lemma 5.6. Suppose that T is not a constant distribution. If T ¢ a and T i a, then there is a finite interval [a, b] and numbers a', 5', and N' such that: 31 1. a > a- Also, T 2 5 implies that there is a $2 E.D such that ¢2(x) 2 O on R, IR¢2(x)dx = l, and '< 3. Further, since T is not con- stant, $1 and $2 may be chosen so that ’* . Now, choose a' and 3' in such a way that max[a, min(, )] < 01' < B. < min[B: max(<1‘,¢1>, d3¢2>)]° Since either < a' or ’< a', we apply Lemma 4.3 to find a A1 Such that for each X 2 A1 there is an XX 6 (support of $1) U (support of $2) for which fk(xk) < a'. Using similar reasoning for 5', we obtain a A2 such that for each X 2 A2, there is an x)\ 6 (support of $1) U (support of $2) for which fk(xk) > 5'. Choose N' z max(A1,A2) and a and b such that (support of $1) U (support of $2) C [a, b]. 1 2 Then, for each X 2 N', there exist two elements xk’xk 6 [a,b] 2 . . such that fk(x:) < a' while fk(xk) > 3'. Finally, Since fx(x) is continuous in x, for all X 2 N' there is an interval 1 c [a, b] such that fk(x) takes on the values A a' and B' at the endpoints of 11 while a' < fk(x) < B' on the interior of I . Q.E.D. A 32 Now, we use these last four lemmas to prove the main result of this work. Theorem 5.7. If T = D(n)g, where g is locally bounded, then T 6 Class n, (Class O-W if n = 0); if T - D(n)g, where g is a locally Lp function for p 2 1, then T 6 Class (n+1/p); if T = D(n)u for some measure u, then T 6 Class (n+1). Proof: If T is a constant distribution, then T 6 Class 0-8 by Theorem 5.2. Therefore, we will Suppose that T is not constant. If T t a and T i B, we apply Lemma 5.6 to obtain a finite interval [a, b] and numbers a', 3', and N' Such that: 1) a < a' < B' < B3 and 2) for each A 2 N', there is an interval 11‘: [a, b] such that fk(x) takes on the values a' and B' at the endpoints of 11 While a' < fk(x) < 6' on the interior of 1%. Therefore, for N 2 N', T N T N -m u u . ’ ’ = : < f x < d (5 1) m2(E,B > 2 m2 a then we use Lemma 5.4 and arguments similar to those of case +'+ l to obtain that m1(Ix) 2 M'(1/),n 1 l/p) for 1 2 N'. iHence, +1+1/ n+l/ for all N 2 N', m2(E:éN) 2 MKI; dx/xn p ‘ M(1/N) p: where M = M'/(n+1/p). Again, this implies that T 6 Class (n+1/p). Case 3. If' T = D(n)u, Lemma 5.5 tells us that m1(1l) 2 T,N n+2) for A 2 N'. Thus, for N 2 N', m2(EaB ) 2 M'(l/‘), Mdf; dX/xn+2 = M(1/N)n+l, where M = M'/(n+l). This is enough to insure that T E Class (n+1). Finally, the case where T = D(n)g, where g is locally L1, may be handled as part of Case 3. Q.E.D. As a result of'TheoremSS.2 and 5.7, we can place a large segment of the Space of distributions in the appropriate 9-Classes. In particular, continuous functions and finite derivatives belong in Class O-S; locally bounded functions fit into Class 04W; locally Lp functions, with p 2 1, are -settled into Class (l/p); and measures are located in Class 1. 34 Further, if T is any distribution covered by the above cases and T 6 Class 9', we see that D(n)T is in Class (e'+n). In Chapter VI, we will discuss some examples to show that these results are significant. We will also include some comments and conjectures in Chapter VII. CHAPTER VI EXAMPLES In this chapter, we give some examples to illustrate that our previous results were not trivial. For instance, Theorem 5.7 would be completely useless if there were no distinctions between the various e-classes. At present, we are able to give examples to show that the e-classes are dis- tinct for e < 2. It seems likely that similar examples exist for the higher classes also, but we are unable to exhibit them at this time. Some discussion of these cases is included at the end of the chapter. We begin with an example to show that Class O-S is a proper subset of Class O-W. Example 6.1. Consider the Heaviside function defined as follows: 0 for x s 0, H(X) = l for x > 0. If we set HX(X) = H*¢x(x), then Ix+1/l (x-t)dt = l for x 2 1/k3 x-l/x ¢x HX(X) 3 I:+l/x ¢X(x-t)dt = $5? ¢(u)du for -1/k < x < l/X; 0 for x s -1/x. 35 36 Our next step is to define F(x) = If1¢(u)du for all x 6 [-1,1]. Then, F is strictly increasing, F(-1) . 0, and F(l) a 1. Further: Hx(X) = F(XX) for all x E ]-1/1, 1/x[. If we choose a and B such that 0 < a < B < 1, then [x: a < Hx(x) < a} : ]-1/x, 1/x[ for all x 2 1. For A 2 1, we have a < Hk(x) < B iff a.< F(xx) < 3 iff (l/x)F-1(a) < xi< (l/x)F-1(B). Therefore, for l 2 l, {x: a < Hl(x) < B} 2 Jl/xp'1(a), l/xF-1(B)[ which implies that m1({x: a < Hk(x) < 5}) a l/XEF-1(B) - F-1(a)]. Since m1({x: a < HX(X) < 6}) a 0 as K d a, we see that H cannot satisfy condition 2 of Definition 5A when a and B satisfy 0 < a < B < 1. Further, H can- not satisfy condition 1 for these values of a and B because each H1 takes on all values between 0 and 1. Hence, we have exhibited values of a and B for which H satisfies neither of the conditions of Definition 5A. We complete the example by noting that since H is locally bounded, H 6 Class O-W by Theorem 5.7. Up to this time, the only conditions we have placed on the function ¢ were those listed in (2.1). Unfortunately, in order to perform the computations required for these next examples, we are forced to add an additional restriction on ¢. (6.1) There is an element a E ]0, 1[ such that ¢m(x) = 0 for all x 6 [-a, a], and ¢m has exactly one zero in 1a, 1[. Combining (6.1) with the properties of ¢ listed in (2.1), we see that the first three derivatives of ¢ may be pictured as in the illustrations on the following page. 37 ._ ' l l l l 4 -1 l -1 -b -a a b 1 Figure 6.1 - as Figure 6.2 - (5' l l l l 4 t P 1 -l-c-b-a abcl -1-c -F a Figure 6.3 - ¢" Figure 6-4 ' om 38 We will use the information Shown in these figures to prove a series of lemmas which are necessary for the remaining examples. Lemma 6.2. For 0 < v < l, the following are true: 1. '3 t-v¢(t)dt > 0. 2. .IS t-v¢'(t)dt < O. 3. I; t'v¢"(t)dt < 0. 4. I; t-V¢(n)(t)dt > O for n 2 3. Proof: 1 and 2 follow easily from the properties of ¢ and ¢ . 1 - 'b ' u 1 " n [0 t V¢"(t)dt = JO t V¢ (t)dt +'lb t V¢ (t)dt - - 1 < b YIB ¢”(t)dt + b vjb ¢"(t)dt = 0 since ¢'(1) = ¢'(O) = 0. 1 -v Therefore, f0 t ¢"(t)dt < O. - 1 - m 13 t W = y, t v. e... - .1 _ = I: t v¢’”(t)dt +-JC t v¢'"(t)dt > °-Yl: ¢m(t)dt + c-VI: ¢m(t)dt = C‘V[-¢"(a)] >0. Therefore, I; t-V¢m(t)dt > O. For n > 3, we use integration by parts (n-3) times to obtain I; t-\)Q3(n)(l’-)dt =,f: t-V¢(n)(t)dt 1 - - - 3 III = v(v + 1)‘°°(v + n - 4)J'a t V (n )¢ (t)dt. 39 Now, for any number m > 0, we have .1 t“m MI (1 _ C -m III d 1 -m III la ¢ (t) t - la t ¢ (t) t +DIc t ¢ (t)dt > C-mJo:¢/Il(t)dt + C-mJO: ¢lfl(t)dt = c'm[-¢"(a)] > 0. Therefore, for n >13, Iét-v¢(n)(t)dt > O. Q.E.D. We will omit the proof for this next lemma since it is almost identical to that of Lemma 6.2. Lemma 6.3. The following inequalities hold: 1. I; In t ¢(t)dt < O. 2. I; In t ¢'(t)dt > O. 3. .I3 In t ¢"(t)dt > 0. 4. I; In t ¢(n)(t)dt < O for n 2 3. Lemma 6.4. Suppose h is locally Lebesgue integrable and 1 6.3. Then, if one of these functions is odd, while the other is even, the convolution product h*¢(x) is an odd function. Proof: Choose n large enough that the support of ¢ is contained in [-n, n]. Then, for all x, we have that h*w(x) = j:f: h(t)¢(x-t)dt. Now, if we assume that h is odd and W is even, we obtain the following: h*¢(x) = ::f: h(-t)¢(x + t)dt = -I::T: h(t)¢(-x-t)dt 3 -h*¢(~x). 40 Thus, when h is odd and V is even, h*¢(x) is an odd func- tion. The case where h is even and W is odd is proven similarly. Q.E.D. Our second example of this chapter will serve to illustrate that 01: Class 9 EClass v, when v satisfies v 0 < v < 1. Example 6.5. We begin by introducing an auxiliary function g defined as follows: -(~x)-v for x < O; g(x) ={ -v x for x > 0. Note that g is locally Lp for l s p < l/v. We will prove a number of properties concerning g and then proceed to apply these to a second distribution related to g. It is this second distribution which will provide the desired example. , .x+l d First of all, we let G(x) = g*¢(x) = Jx-l g(t)¢(x-t) t. In particular, for x E j-l,l[, we have: (6.2) G(x) j‘o -(-t)-v¢(x-t)dt +I3+1t-v¢(x-t)dt x-l ‘y: (U-X)-v¢(u)du +-If1 (x-u)-V¢(u)du. Now for x 2 l, G(x) > 0 since g(t) > O on ]x - l, x + l[. a - .+1- For x 6 10, 1[, G(x) = -I:_1(-t) v¢(x-t)dt +-Jg t V¢(x-t)dt - _ +1 -v = I0x+1t V[¢(x-t)-¢(x+t)]dt +Djfx+1t ¢(X't)dt - +1 2 I: (u-x) V[¢(2x-u)-¢(u)]du + Ifx+lt-v¢(x-t)dt. 41 Since x > O, we have |2x - u‘ < u for all u E jx, 1[ which implies that ¢(2x - u) 2 ¢(u) for u E ]x, l[. Therefore, for x 6 10, l[, G(x) > 0 which assures us that G(x) > O for all x > 0. By Lemma 6.4, G is an odd function since g is odd and ¢ is even. Thus, using the previous information on G and its continuity, we obtain the following: > 0 for x > O, 0 for x =0, (6.3) G(x) < 0 for x < 0. Next, we consider the behavior of G'(x) for x near 0. Note that ¢' is an odd function and g¢' is even. c'<0> = g*¢'<0> j}, g(t)¢'(-t)dt ejll g(t>¢'(t)dt -2fé g(t)¢'(t)dt -2j; t'V¢'(t)dt > 0 by condition 2 of Lemma 6.2. Since G'(x) is continuous, there is an h > 0 Such that (6.4) G'(x) > O on ]-h, b[, i.e., G is strictly increasing on J-h, h[. Now, using the fact that g¢" is an odd function, we obtain: 1 .. G“(0) = g*¢"(0) =JE1 g(t)¢”(-t)dt = L1 g(t)¢ (t)dt = 0- Further, using condition 4 of Lemma 6.2 and an argument similar to that used for G'(0), we have that Gm(0) < 0. By the continuity of Gm, there is a k > 0 such that Gm(x) < 0 on ]-k, k[. Combining this with the fact that 42 G"(O) = 0, we see that (6.5) G"(x) < O on ]O, k[, i.e., G' is strictly decreasing on 10, k[. We will make one final observation about G. If we let (6.6) w = min(h, k) and v = min G(x), then by (6.3), we x€]w,l[ have v > 0. With all the preliminaries completed, we are now pre- pared to describe the desired distribution. Let f be the function defined as follows: (6.7) f(x) = ( Ll for x 2 1. By Theorem 5.7, the fact that f is locally Lp for 1 S pi< l/v implies that f 6 Class (l/p) for v < 1/p S 1. We will demonstrate that the distribution f is in Class v, but f é Class 9 for any 9 < v. Following our earlier convention, we will set fx(X) = f*¢x(x). Then, for l 2 2, we have 1X - l/k, x + llkl G ]-1, 1[ whenever x E 1-1/x, l/Kl: and 1 o - fk(x) = l:il;§ f¢x(x‘t)dt = lx-l/x-('t) v¢x(x-t)dt cx+1/x -v _ +-JO t ¢x(x t)dt = -f;x[(u-xx)/x]-v¢(u)du +-j§:[(XX-u)/x]-v¢(u)du 1 _ .- = 1"H )x (“‘1") V0301)“ + it‘l‘l“) WWW]- 43 (6.8) Therefore, by (6.2), we obtain that fx(x) = xVG(xx) whenever k 2 2 and x E j-l/x, 1/x[; Henceforth, we will assume that X 2 2. Then, for +1/;, .x+1/), 'l/X f(t)¢x(x-t)dt 2 Jx'l/X ¢l since f(t) 2 l on ]x-l/x, x+1/x[. Further, for x E ]0, l/lL, x 2 1/;,, we have f)‘(x) = 1‘: (x-t)dt = 1, we use (6.3) and (6.8) to obtain that fx(x) > 0. Using this information, Lemma 6.4, and the continuity of f, we may write the following statement: 2 1 for x 2 l/x, _> O for O < x < 1/l, (6.9) fx(x) = 0 for x = 0, < 0 for -l/x < x < 0, S -l for x s -1/x. Let 0 < a < B < 1. It is obvious by (6.9) that f does not satisfy condition 1 of Definition SC for these values of a and 5. Also, (6.9) tells us that when a < fx(x) < B, we must have x 6 ]O, 1/k[- Now, choose N* greater than max {[B/G(k)]1/", (e/v)1/V, 2}. Then, for 1, 2 N*, a < fx(x) < 8 iff an," < G(xx) < an," and x 610, 1/;,[. Using (6.6), we see that when x 2 N*, and G(xx) < B/kv. we have that 0.< XX < h. Therefore, a.< fx(x) < 5 iff (l/l)G-1(a/Av) < x < (l/x)G-1(B/xv). Hence, for k 2 N*, {xz a < fx 6's} = unmade/1"). (Meade/1M which - -1 implies that Ifilm: a < fk(x) < 5}) .. 1mg 1(s/m-c (st/Tm. 44 f For N 2 N*, m2(EaéN) ='I; m1([x: a < fx(x) < 8})dk =J'§£G'11d1/1. (6.10) Further, when T 2 N*, we have 0 < G-1(a/xv) < c-1(a/iv) < k. By the mean value theorem for derivatives, there is a tK E JG-1(a/xv), G-1(e/xv)[ such that G'(tx) = [(B-a)/1V]/[G'1(e/i”) - G'1(a/1V)], or c'l(e/1“> - G-1(a/1v) = (e-o>/[1”c'(t,)]. Using (6.10) and (6.5), we obtain that G'[G-1(B/xv)] s G'(tx)'s G'[G-1(a/xv)]. Therefore, for all k 2 N*, (6 11) (e-o)/{1”G'[G'13} s G'1 - G'1(a/1”) s (e-a)/{1”G'[G'1(e/xv)]}. In particular, for N 2 N*, m -1 m2(E:éN) s (e-aofN 1/{1vc'ic (3/1”)]}d1/1. If we make the change of variables u = 6-1(B/xv), then G(u) = 53/),v and G'(u)du = (avg/Av+l)dx. Hence, for N 2 N*, f N G-1(e/N”) ' m2(Eaé ) s [(a-o)/vs]j0 l/G'(u) G (u)du = [(B-a)/vejc'1(e/NV). To complete the proof that f é Class 9 for any 9 < v, we must Show that for any 9 < v and for any M > 0, we have G-1(B/NV) < M,(l/N)e as N becomes large. Since 9 < v, we may write 9 = yv for some y E ]0, 1[. Then, we consider the function n defined as follows: 45 n(y) = c'lm - (14/5th for y e [0, G(w)[. By (6.3), n(O) = 0. Further, for y 6 J0, G(w)[, we have n'(y) = 1/G'[6-1(y)] - (vM)/(BYy1-¥) by the inverse function theorem. Since G'(O) > 0 and G-1(O) = 0, we obtain that iig G'[G-l(y)]/y1-Y = c. Therefore, there is a j > 0 such that G'[<:.'1(y)]/y1'Y > BY/(YM) for all y e )0, j[, i.e., n'(y) < O on 10, j[. Combining this with the fact that n(O) e 0, we see that n(y) < O for all y E ]0, j[, or G-l(y) < (M/BY)yY for y 6 10, j[. l /v, we have B/NV < j which implies Thus, for N > (B/j) that c'1(e/N") < (M/BY)(B/NV)Y -= 1~i(1/N)W -= M(1/N)e. In summation, we have demonstrated values of a and B such that for any M > 0 and any 9 < v, we have m2(E:éN) < M(1/N)e for large N. Together with our previous remarks, this shows that f é Class 9 for any 9 < v. To complete the example, we must show that f 6 Class v. It is clear that f fails to satisfy condition 1 of Definition SC for any a and 5' Hence, we must show that f satisfies condition 2 of Definition 5C for all a and B- If 0 < a < B < 1, then by the left side of (6.11) and a series of steps similar to those used directly after (6.11), we obtain that for N 2 N*, m2(E:éN) 2 [(B-a)/av]G-1(a/Nv). Choose Mi< 1/G'(0) and consider the function q(y) = G-1(y)-My for y E [0, G(w)[. Using the same methods as needed for the function n(y), we see that there is an s > 0 such that q(y) > 0 for y E 10, s[, i.e., G-1(y) > My for y E 10, s[. 46 Hence, for large values of N, we obtain the following: m2 (EiéN) 2 [(B-a) /o!]G-1(a/NV) > [(8-01) /oz]M(a/NV)=(B-a)M(1/N)v- Thus, for 0 < a < 5 < l, f satisfies condition 2 of Defini- tion 5C. When 1 s a < 5 < a, we have that m1({x: a < f(x) < 5}) > 0. By using local versions of Theorem 3.2 and Lemma 5.1, it can ,N B be shown that this is sufficient to insure that m2(E: ) - o for all N. Any of the other possible cases for O s a < 5 < a may be handled as subcases of the two mentioned above. Finally, since fx(x) is an odd function for all X, we see that symmetric results are obtained when -m < a4< 5 s 0. There- fore, when -l s a < 5 s 1, there are numbers N* and M 2(EiéN) 2 M(1/N)V for N 2 N*. For all other f,N possible cases, we have m2(EmB f E C1638 v and the example is completed. Such that m ) = a for all N. Thus, The next example illustrates that U Class 5 is a 5<1 proper subset of Class 1. Since the actual details of the example are as intricate as those of the previous example, we will give a brief outline of the steps involved. Where possible, we will refer back to similar procedures used in Example 6.5. Example 6.6. In this case, we begin by defining an auxiliary distribution 8. Let g(x) = In ‘x‘ for all x # 0. Then, since g is locally Lebesgue integrable, we may consider the 47 distribution S = Dg. (Note: this distribution is usually referred to as "Pv(1/x)" and is defined by 1 - pNIR¢(X)/x dx, where "pv denotes Cauchy principal value. For our purposes, it is more convenient to consider S in the above form, A more thorough discussion of this distribu- tion is given in [8], pp. 84-85.). The next Step is to define G(x) = S*¢(x) and to Show that G satisfies conditions similar to (6.3), (6.4), and (6.5). In these steps, we have to make use of Lemmas 6.3 and 6.4. With this accomplished, we define a function f(x) as follows: ln x for 0 < x s l, f(X) = In (-x) for -1 s x < 0, -x - l for x s -1. Finally, we prove that the distribution T ‘ Df provides the desired example. To do so, we set fx(x) = T*¢X(X) and Show that f satisfies conditions similar to those listed in (6.9). Further, it is easily shown that fx(x) = xC(xx) when x 6 1-1/x, 1/x[. The remainder of the steps are exactly the same as those of the preceding example. In the end, it can be shown that for 0 < a1< B < 1, we have T é a, T 2 5, and for any a < 1 and any constant M, m2(ET’N) < M(1/N)e for large values of N. This demonstrates 06 that T é U Class 5. Also, following the same procedure as 5