Proceedings of the American Mathematical Society, Vol. 88, No. 1. (May, 1983), pp

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1 Approximate Identities and H1(R) Akihito Uchiyama; J. Michael Wilson Proceedings of the American Mathematical Society, Vol. 88, No. 1. (May, 1983), pp Stable URL: Proceedings of the American Mathematical Society is currently published by American Mathematical Society. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org. Wed Sep 26 11:42:

2 PROCEEDINGS OF THE AMERICAN 1vLATHEMATICALSOCIETY Volume 88. Number I. May 1983 APPROXIMATE IDENTITIES AND H '(R) AKIHITO UCHIYAMA AND J. MICHAEL WILSON ABSTRACT.Let q(x) t L'(R) n LX(R)be a real-valued function with jrq d.x f 0. For? > 0, let q,(.x)= y-'q(x/,v). Forj(x) L'(R) define We investigate the space ~ 1;= (j L'(R): j,",* LL'(R)) 1. Introduction. If cp is the Poisson kernel, then H; is defined to be HI. Fefferman and Stein [2] showed that H; = H 1 for any cp that is smooth and dies quickly at infinity; e.g. cp can be in the Schwartz class, or Lipschtz continuous (of any order) and compactly supported. However, it is easy to show that H; = (0) if cp = x,~,,, (see [3]),where X, is the characteristic function of a set E. G. Weiss asked whether there was an H; that was nontrivial but not HI. In this note, we show the following two results. THEOREM1. If H; # {0),then a(x) E H;, where 1 O<x<l, -1-1<~<0, 0 otherwise. THEOREM2. There exists cp(x) 3 0 such that H; # {O}, H: # H' As a corollary of Theorem 1, we get COROLLARY1. If H; # (0)' then Hi n HI is dense in HI Comment on notation. To distinguish the "y" in rp,(x) (= y-'cp(x/y)) from the other subindices, in the following we write ( T)~instead of cp,. The letter C denotes various constants. 2. Proof of Theorem 1. For f E H; define 1 1 f 1 1 ~4 = 1 1 f: 1 1 This norm makes H; a Banach space. We use two simple facts about ,;. FACT1.Iff E H ;,~E L', thenf *g E with /If * gilh;< Ilf lih~llgll~~. Received by the editors February 16, Muthematics Subject Ckussi/icution. Primary 42B30; Secondary 46J15. Key words und phrases. HI, BMO, maximal functions. C1983 Amerlcan Mathemat~calSociety /82/ /$02 25

3 54 AKIHITO UCHIYAMA AND J. M. WILSON FACT2. If y > 0 and f E Hi, then Il(f 11 H$ = ( If \Ilf$. Let f E Hi,f r 0 and fix f. In the following part of this section, the constants c depend on this function f. We may assume that f is real-valued (since cp is real-valued). We shall construct functions p,, g, (- co < n < m) satisfying (I) 11 PI,11 H; cj a = 2 Pt, * g, 9 where the convergence is in H;. This implies the theorem. The construction ofp,, and g,,. Since f E 0 and since f is real-valued, we may assume there exist r > 1 and E > 0 such that ~f([)i > E on [-r, -r-'1 U [r-', r]. Let +(x) E S(R) be a real-valued even function such that m supp 4 c [-r, -r-'1 u [r-i, r], 4 (r")2 = 1 for any [ # 0. Now we invoke Wiener's Lemma: Letf,(x), f2(x) E L'(R). If there exist an E > 0 and an interval I C R for which Ifl(t) I > E, [ E I, and suppf2 C I, then there is an h(x) E L'(R) such thatfl;(() = h^(<)f,((). Applying Wiener's Lemma to f(x) and (~X~,~,,)V, we get h,(x) E L1(R) such that 4 (E)X{~.,)(S)= h^l(~)f(t). Set h^(t) = + zl(-[). Then 4(5) = h^([)f([), and (4) l l + l l H ; ~ ~ ~ h ~ ~ L ~llfll~;. ~ ~ l f We now define P,,(.) = (+)r4x>, g,,(x) = a * (+)r.(x). Then (1) follows from (4). By taking Fourier transforms, we see that a = B:=-,p, * g,, in 5'. To estimate II g,, /I,I, we divide into two cases. Case 1. n 2 0.We write

4 APPROXIMATE IDENTITIES AND 1l1(R) where R(x ) = sup,, - / $'( y ) /. Therefore, Case 2. n < 0. We distinguish three subcases. Subcase 1. 1 x / > 3. ($ is rapidly decreasing). Thus, jlxl,,i gtl(x)(g cr3". Subcase 2. 1 x I 4 3, min(/ x I, / x + 1 /, / x - 1 1) rn/*. These x's are away from the discontinuities of a(x). We have The second term can be estimated as in the first subcase. The first term equals zero or it equals I jl,l,r-j,,/z$(t) dt I (because j$(t) dt = 0). This is dominated by crn, since $ is rapidly decreasing. Subcase 3. min(l x I, I x + 1 1,I x - 1 1) < rnl2. Here the best we can do is I a * +,(x) I G c. But the measure of this set is G 6rn/*. Combining the three subcases yields for n < 0, I g, I I, G crni2. We therefore have (2). 3. Proof of Corollary 1. It is well known that the dual space of HI is the space BMO (see [2]).This is the space of locally integrable functions h(x) that satisfy The supremum is over all intervals I C R; hi denotes the average of h(x) over I. Clearly a(x) E HI. Also HI and H; are closed under translations and dilations. If H; n HI is not dense, then there is an h E BMO such that I h 11, = 1 but jh(x)g(x) dx = 0, for any g E H; n HI. The same must hold for any dilation or translation of a(x). Ths implies that h is constant and ll h 11, = Proof of Theorem 2. An examination of the proof of Theorem 1 shows that it works because of the relative smoothness of a(x). In this section, we exhbit an H; that is not trivial or HI, by building functions b(x) E HI and cp(x), each of which has "large" high frequency terms in its Fourier series. The hlgh frequencies of cp(x) almost cancel out when cp(x) is convolved with a(x), but they match up with those of b(x) to make H;. For n = 1, 2,3,...,define k= l We estimate / a * (pn),(x) I as follows.

5 56 AKIHITO UCHIYAhlA AND J M. WILSON Case 1. y < 1 Case 2. y > 2" Case3. 1 Gy G 2". I a * (p,,), (x) I< C ( / ) ( / + 2 (l/y)(2'/~) C/Y. logz i <kgri I~k~log, 1 Now observe that a * (p,,),.(t)= 0 if y G (t - 1)/2 or y < (-t - 1)/2. Thus This yields I l a,*,,ll,i G Cn. If LY > 1, then by and by similar observations as above, we get (5) lla:,!(a, I 1 I, where C does not depend on LY > 1. Define * (P,,((Y.)),(t) = LY-la * t~,,),/a(t)* G Cn. where E~ > 0 is a small number. Then, by (5) we have Let E > 0 be a small number. Define From the fact that b E L~, jb dx = 0 and supp b C [-2, -11: it follows that b E H1 (see [11> We claim that for 12 > N, and 0 G i G n/2,

6 This is because the left-hand side equals APPROXIMATE IDENTITIES AND H'(R) The first integral equals The second integral is no larger than " (k- i)-'+, z CEne k=i+l i (since jb dx = 0).Thus if (7) O<i<n/2 and n>n,. Therefore, if (7) holds, 2 c,'(~'~~)-'~-~-eo+e. Thus, b:(x) z ~,'(2'a,)-'n-~-'0+' on En,,= {x: 2'-'a, <(x (<2'an - (2' - 1)). Thus, L.,b; dx z c:n-'-'oie, which yields, upon summing for 0 < i < n/2, Therefore if e0 < E. Take v(x) E S such that v(x)+ q(x)2 0 for any x E R. Then the kernel cp = v + 7 is nonnegative and a$ E L' and L', by (6) and (8). Thus H;#{o) and H;#H'

7 AKIHITO UCHIYAMA AND J. M. WILSON 1. R. Coifman and G. Weiss, Extensrons of Hard,: spuces urld thew uses in unulpsis, Bull. Amer. Math. Soc. 83 (1977) C. Fefferman and E. M. Stein, HP spuces of seoerul ouriuhles, Acta Math. 129 (1972) G. Weiss, Son~e prohlents in the theon' of Hurdv spuces, Proc. Sympos. Pure Math., vol. 35, Amer. Math. Soc., Providence, R. I., 1979, pp DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF CHICAGO,CHICAGO, ILLINOIS (Current address of J. M. Wilson) Current uddress (Akihito Uchiyama): Department of Mathematics, College of General Education, Tbhoku University, Sendai, Miyagi-ken, 980, Japan