1. Introduction. Let P be the orthogonal projection from L2( T ) onto H2( T), where T = {z e C z = 1} and H2(T) is the Hardy space on

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1 Tohoku Math. Journ. 30 (1978), ON THE COMPACTNESS OF OPERATORS OF HANKEL TYPE AKIHITO UCHIYAMA (Received December 22, 1976) 1. Introduction. Let P be the orthogonal projection from L2( T ) onto H2( T), where T = {z e C z = 1} H2(T) is the Hardy space on T, that is, k = -1, -2,... }. For a function cp e H2(T) satisfying the Hankel operator L0 is defined by Lfr) = P(cp ), ~/r e H2( T) (1 L~( T), where the bar denotes complex conjugation. Nehari [5], Hartman [3], Coif man, Rochberg weiss [1] considered some properties of the Hankel operators. In this paper we are concerned with the following theorems. THEOREM A ([5], [2]). L0 is a bounded operator from H2 to H2 if only if cp e BMO. Furthermore the operator norm L~ is equivalent to cp IBMO' THEOREM B ([3], [7]). L0 is a compact operator if only if cp e CMO. The definitions of BMO, BMO-norm CMO will be given at the end of Section 1. We note that more general situations are considered in [1]. In the following all the functions considered will be real valued functions defined on Rn. For a measurable function b we define B(f) = b f. As pointed out in [1] for the one dimensional case the study of [H, B] = HB - BH, where H is the Hilbert transform, is often essentially equivalent to that of L~. Suppose that K is a Calderon-Zygmund singular integral operator with smooth kernel. That is, there is an Q(x) which is homogeneous of degree zero, which satisfies 9 ~ 0 ~ 2(x) -.ic(y) I c I x - y when x = y = 1, that THEOREM A' ([1]). I f b is in BMO, then [K, B] is a bounded map of L2(Rn) to itself, 1 < p < oc, with operator norm [K, B] (p) <CK,P b IBMO. Conversely, if [B, R2], where R1, R27..., Rn are the Riesz transforms,

2 Q 164 A. UCHIYAMA are bounded on L(R) for some p, l <p< c i = 1,.., n then b is in BMO We shall improve Theorem A' in Section 2 extend Theorem B on Rn in Section 3. In the latter case we shall find some difficulties in the functions of CMO over Rn which do not occur in the unit circle case. To avoid it we shall use the characterization of CMO over R~ which is announced in Neri [6]. NOTATION. i, j,k m mean always integers. A dyadic cube is a cube of the form {x==(x1,.., xn) E R~ 11~ti2' < xi < (2 +1)2~ for i=1,.., n}. For a measurable set E, I El, m(f, E), E XE mean the Lebesgue measure of the closure of E the characteristic function of E respectively. For a cube Q in Rn, M(f, Q) means Rp R(x, a, b) mean {x e Rn II x2i <2 for i =1,.., n} {yer1a n < Ix - I <b} respectively. DEFINITION. For f E Lio~(Rn), Hf HBM0 will denote sup{m(f, Q) I Q is a cube in Rn}. Identifying functions which differ by a constant, the set of functions satisfying I If ( IBMO < is a Banach space under the norm IBMO we call this space BMO. The BMO-closure of.', where, f is the set of C~-functions with compact support, is denoted by CMO. [See [6], p 186.] 2. THEOREM 1. Let 1 < p < oc b E U>1 qlga~(rn). Then I b BbJO<_ A(p, K) II[K, B]I (p) PROOF. In this proof for i = 1,.., 10 Ai is a positive constant depending only on K, p A;(1 < <i). We may assume I1[K, B]II (p) = 1. We want to prove (*) sup M(b, Q)~A(p, K). Since li[k, B] I ( p) = H[K, Br, xo] I I (p) for every xo E Rn r E R+, where Br,xo(f)(x) = b(r-1x + xo) f (x), it suffices to prove the inequality (*) for for j =1,..., n}. Let M= M(b, Q1) = IQ1L1 qr be such that Since [K, B -- a0] = [K, B], we may assume ao = 0. IkkIlL-1, supp "k c Q1, Let p

3 COMPACTNESS OF OPERATORS OF HANKEL TYPE 165 (x)b(x) >_ 0 Let a closed subset of Al, a positive number, be such that - where m is the measure on which is induced from the Lebesgue measure on Rn, Q(x) - Q(y) I < 2~1Q(x) for every every satisfying x - y <A1. Then for xeg={xerniixi >A2=2A~1+1 IfK, B1/c(x) I? I K(b)(x)I - I b(x)k()(x) -n- ~ A3M Ix ~-" - A4 I U(x)I IxI1. Let F= {x E G I I b(x) I> (MA3/2AJ I C I ~ x l < MP'/}, where p-' + p`-1=1, then Thus Fl Let g(x) = (sgn(b(x)k(x)))xf(x), A6Mp' - A2 >_ A614'.l p'/2 if M > (2A2 A61)' '. then for x e Ql where Since [K*, B] is the adjoint operator of [K, B], II[K*, B]{I(p,) =1. Thus A1oM>_ IIgII' n? I[K*, B]9 In~ Then, GUI <-- A(K, p). COROLLARY. For f 2n H1(R)

4 166 A. UCHIYAMA For the definition of H'(Rn) we refer to [2]. The corollary will be proved in the same way as in Theorem II of [1] using Theorem A' Theorem LEMMA. Let f E BMO. Then f E CMO if only if f satisfies the following three conditions. (1) lira sup M(f, Q) = 0. ago iqi=a (ii) lira sup M(f, Q) = 0. ato IQI=a (iii) lira M(f, Q + x) = 0 for each Q. x-~co This lemma, which seems to be due to Herz, Strichartz Sarason, is announced in Neri [6] without proof. PROOF. In this proof A is a positive constant depending only on n. From the definition of CMO, it is trivial that CMO satisfies (i) (ii) (iii). In the following we prove that if f satisfies (i) (ii) (iii), then for any s > 0 there exists ge E BMO such that (1) inf I g~ - h BMO < As. h~11 (2) I I g~ _ f I (BRIO < As. From (1) (ii) there exist ie k~ such that sup {M(f, Q) i I Q I <_ 2'"E} < 5 Sllp {M(f, 'W) I I'4' I ~ 2} kf} <s. From (i), (ii) (iii) there exists je such that je > i~, k. sup {M(f, Q) I Q n R~~ = o } < s We define Qx as follows. If x e R;~, Qx means the dyadic cube of side length 2z~ that contains x. If x E Rm'Rm_i where j E <m, Qx means the dyadic cube of side length 22E+m We set g(x) = m(f, Qx). From (ii) there exists m> 3E such that sup{i g(x) - g(y) x, y e R,, \Rmfi-i} <5. If x E R,, we define g(x) = g(x) if x E RmE, we define g(x) =

5 COMPACTNESS OF OPERATORS OF HANKEL TYPE 167 m(f, Rm\Rm_1). Note the fact that (3) if Q, (1 Q, ~ 0, diam Qx < 2 diam Q,. Then by the definition (4) g(x) - g(y) I < As. of i, j, m, if Qx (1 Q,, ~ 0 or x, y e RmE_1f then Thus (1) is obvious. From the definition of ie je (5) for every x e Rm. Let Q be an arbitrary cube in Rn. First we consider the case such that Q c Rm ~ max {diam Qx Q, (1 Q ~ O } > 4 diam Q. Then by (3) the number of Qx such that Qx (1 Q ~ 0 is bounded by A, if Q (1 R; E ~ 0, I Q i is less than 2"e. Thus from (4) the definition of ie j f, M(f - g~, Q) < As. Second if Q C Rm,E max {diam Qx I Qx n Q ~ O } < 4 diam Q, by (5). Third if Qc Rm_1, by the definition of m M(f -ge,q)~m(f,q)+a~<(1+a)s. Lastly we consider the case Q (1 Rm~ 0 Q (1 RmE_1 ~ 0. Let pq be the smallest integer satisfying Q C RpQ, then M(f - gf, Q) < AM(f - g~, RpQ). Since me > ke, I m(f, Rq) - m(f, Rq_1) ( <A s for every integer q such that m< < q. Then Thus (2) is proved. THEOREM 2. Let b e Uq>i Llo~(Rn). Then [K, B] is a compact operator from L2 to itself, 1 < p <00, if only if b e CMO. PROOF. If [K, B] is a compact operator, then from Theorem 1 b e BMO. Thus we may assume I 1 b I IBM0 =1. First suppose that b does

6 168 A. UCHIYAMA not satisfy (i) of the previous lemma. Then there exist > 0 a sequence of cubes {Q},~ 1 such that (11) M(b,Q)>d for every j lim;~. q; = 0 where q; is the diameter of Q;. In the following for i = 20,..., 36 Al is a positive constant depending only on K, p, 3 Aj(20 < j < i). Let b; be a real number such that as follows (12) f~(b - b;) >_ 0, (13) supp f, C Q~ (14) (15) If() - I Q~ +1/p x; the center of Q;, we define f; for every y e Q;. Note that [K, B] f = K((b - b;) f) - (b - b;)k(f ). From (13) (15) (16) K((b b;)f3)(y) I C A2n { Q; 11-1/p x; -- y I -n for y A21Q;. By (11), (12) the continuity of the kernel (17) K((b - b,)f~)(y) I >_ A22~ IQI'"', -Ix; - y1-n for is as in the proof of Theorem 1. On the other h, by (14) the smoothness of the kernel (18) (b(y) - bg)k(f;)(y) I A23 I b(y) - b; I I x; - y I'q7 -n-i for y A21Qd. Since 11 b I l BMO - 1, Q; 1-1/p [See for example [2][4].] Thus if a > A21

7 COMPACTNESS OF OPERATORS OF HANKEL TYPE 169 Then from (17), for /3> a > A21 So from (16) there exist A29, A30 A31 satisfying (19) A29 < A30, (20) By the result of [2] [4], (21) I {y I b(y) ~~ > u T A32} 11 R(x~, A29g9, A30q~) I ~--~ A33 I Q9 I e-a31u. Let EC R(x;, A29q, A30q;) be an arbitrary measurable set. Then by (16), (18), (21) I b I BMO =1 Thus there exists A39 such that for every measurable set E satisfying E c R(xd, A29q;, A30g9) I El < /t If we select a subsequence {Q)} ;(ksatisfying (22) q~(k+1)i q~ck) < A361 A30 7 then for m > 0 using (19), (20) (22) we get

8 170 A. UCHIYAMA Thus {[K, B] f;};, is not relatively compact in L~, i.e., [K, B] is not compact. Quite similarly we can prove that if b does not satisfy (ii) or (iii) of the previous lemma, [K, B] is not a compact operator. Conversely, suppose that b e CMO. Then for any > 0 there exists b, e f such that 1 b -- b, IIBMO < ~. By Theorem A' [K, B] - [K, B,]11(P) < Thus for the proof of the converse part it suffices to prove that [K, B] is a compact operator for b e?". In the following for i = 40,.., 48 Ai is a positive constant depending only on b, p, K A; (40 < <i). It is clear that (31) [IC, B]J (x) I '--~ A40 I I J I D I xi I -n for x > A41 from Theorem A' (32) II[.K, B]f I I, ~ A,: I I f I!o Take an arbitrary 2-'> > 0 z e Rn. Then, (33) The first term of (33) is dominated by where The second term is dominated by The last two terms are dominated by

9 COMPACTNESS OF OPERATORS OF HANKEL TYPE 171 Note that K (f) I I A47 I I f I ro [see [3], p42] that b is uniformly continuous. Then by taking z sufficiently small depending on E, we can get (34) Thus from (31), (32), (34) the theorem of Frechet-Kolmogorov ([9], p275), [K, B] is a compact operator. REFERENCES [1] R. R. COIFMAN, R. ROCHBERG AND G. WEISS, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), [2] C. FEFFERMAN AND E. M. STEIN, Hp spaces of several variables, Acta Math. 129 (1972), [3] P. HARTMAN, On completely continuous Hankel matrices, Proc. Amer. Math. Soc. 9 (1958), [4] F. JOHN AND L. NIREMBERG, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), [5] Z. NEHARI, On bounded bilinear forms, Ann. of Math. 65 (1957), [6] U. NERI, Fractional integration on the space H1 its dual, Studia Math., LIII (1975), [7] D. SARASON, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), [8] E. M. STEIN, Singular Integrals Differentiability Properties of Functions, Princeton Univ. Press [9] K. YOSIDA, Functional Analysis, Springer, MATHEMATICAL INSTITUTE TOHOKU UNIVERSITY SENDAI, JAPAN