WAVELET DECOMPOSITION OF CALDERÓN-ZYGMUND OPERATORS ON FUNCTION SPACES

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1 J. Aust. Math. Soc. 77 (2004), WAVELET DECOMPOSITION OF CALDERÓN-ZYGMUND OPERATORS ON FUNCTION SPACES KA-SING LAU and LIXIN YAN (Received 8 December 2000; revised March 2003) Communicated by A. H. Dooley Abstract We mae use of the Beylin-Coifman-Rohlin wavelet decomosition algorithm on the Calderón- Zygmund ernel to obtain some fine estimates on the oerator and rove the T./ theorem on Besov and Triebel-Lizorin saces. This extends revious results of Frazier et al., and Han and Hofmann Mathematics subject classification: rimary 42B20, 46B30. Keywords and hrases: Calderón-Zygmund oerator, Besov sace, Triebel-Lizorin sace, wavelet, BCR algorithm.. Introduction In recent years, there has been significant rogress on the roblem of boundedness of generalized Calderón-Zygmund oerators on various function saces. The oerators in uestion can be described as follows. Let K.x; y/ be a continuous function defined on.r n R n /\{x = y} and let T : D D be the linear oerator associated with the ernel K.x; y/,thatis, T '; = K.x; y/'.y/.x/ dydx R n R n where ', D are C -test functions on R n with disjoint suorts. For convenience, we write K.x; x ; y; y / = K.x; y/ K.x ; y / + K.y; x/ K.y ; x / : Both authors are artially suorted by a direct grant from the Chinese University of Hong Kong. The second author is also artially suorted by NSF of China and NSF of Guangdong Province. c 2004 Australian Mathematical Society /04 $A2:00 + 0:00 29

2 30 Ka-Sing Lau and Lixin Yan [2] It is customary to assume that K.x; y/ satisfies the following ointwise conditions: (.) (.2) K.x; y/ C x y n ; and K.x; x ; y; y/ C x x x y n for x y 2 x x ; where 0 <. In their celebrated aer [4], David and Journé characterized the tye of ernel K.x; y/ for which T is a bounded oerator on L 2. ThisisnowcalledtheT./ theorem. They roved that under conditions (.) and (.2) on K.x; y/, T extends to a bounded oerator on L 2 if and only if both T./ and T./ are BMO functions, and T has the following wea boundedness roerty (WBP): For ', D with diam.su '/, diam.su / t, (.3) T '; t n( ' + t ' )( + t ) : Later, Meyer [] imroved the theorem by relacing the ointwise assumtion with the following integral assumtion on K.x; y/: ( ) (: ) K.x; y/ + K.y; x/ dy ; and (:2 ) su r>0 B./ = r x y <2r su r>0 u + v r. + /B./ < ; with =0 ( 2 r x y <2 + r ) K.x + u; x; y + v; y/ dy : The T./ theorem has also been considered by Lemarié on the Besov saces [0], Frazier et al on the Triebel-Lizorin saces [7], and Han and Hofmann on both classes of saces [8]. The definitions of such saces can be stated as follows (see [3]): Let S.R n / be the sace of temered test functions. Let ' S.R n / with su ˆ' {¾ R n : =2 ¾ 2} and ˆ'.¾/ c > 0 for 3=5 ¾ 5=3; ut ' j.x/ = 2 jn '.2 j x/ and Q j. f /.x/ = ' j f.x/. For Þ R and 0 <, <, the Besov saces Ḃ Þ; is the collection of all f S =P (the temered distributions modulo olynomials) satisfying ( ) = ( ) f Ḃ Þ; = 2 jþ Q j f < : The Triebel-Lizorin sace is defined analogously, all f S =P such that ( ( f Ḟ Þ; = 2 jþ Q j f ) ) = j j Ḟ Þ; < : being the collection of

3 [3] Wavelet decomosition of Calderón-Zygmund oerators 3 In this aer, we will rove the following two theorems: THEOREM.. Suose T satisfies the WBP (.3), the ernel K.x; y/ satisfies (i) (: ) and (:2 ). If T./ = T./ = 0, then T is bounded on Ḃ 0;,, <. (ii) (: ) and =0 2Þ B./ <. If T./ = 0, then T is bounded on Ḃ Þ;, 0 <Þ<and, <. THEOREM.2. Suose T satisfies the WBP (.3), the ernel K.x; y/ satisfies (i) (: ) and. + =0 /2 = B./ <. If T./ = T./ = 0, then T is bounded on Ḟ 0;,, <. (ii) (: ) and =0 2Þ B./ <. If T./ = 0, then T is bounded on Ḟ Þ;, 0 <Þ<and, <. We remar that the two theorems extend the results of Han and Hofmann [8]; they need to assume that B./ 2 ž for 0 <Þ<žin both theorems. For <Þ<0, Theorem. and Theorem.2 also hold by interchanging the role of T./ and T./ because of the duality (the dual of Ḃ Þ; Note that Ḟ 0;2 is Ḃ Þ; Þ; and similarly for Ḟ ). is of secial interest because it is the Hardy sace H when = and is L when >. For the Hardy sace H, the ernel condition in Theorem.2 is. + /3=2 B./ <. In[5], T is roved to be bounded on L 2 under the ernel condition. + /=2 B./ <. By the interolation theorem, a direct alication of the theorem yields the following result, which is stronger than the corresonding case stated in (i). COROLLARY.3. Suose T satisfies the WBP (.3), the ernel K.x; y/ satisfies (: ) and. + /=2+2 = =2 B./ <. If T./ = T./ = 0, then T is bounded on L, < <. The main tool used in roving the theorems is wavelets, initiated in [2] and [5]. This is uite different from the aroaches in [7, 8, 0, ]. The roof of Theorem. deends on the Beylin-Coifman-Rohlin wavelet decomosition of the oerator T. For Theorem.2, we first rove the boundedness of T on Ḟ Þ; using an atomic decomosition on this sace. This, together with an interolation on Ḟ Þ;.= Ḃ Þ; /, yields the boundedness of T for the other case. The aer is organized as follows. In Section 2 we will give some reliminaries on wavelets and the BCR decomosition of T. We also set u the roof in terms of wavelet terminology. The T./ theorem on the Besov saces is roved in Section 3 and on the Triebel-Lizorin saces in Section 4.

4 32 Ka-Sing Lau and Lixin Yan [4] 2. Preliminaries For simlicity, we only consider the one dimensional case. The higher dimensional case is similar. Let us recall the concet of multiresolution analysis in L 2.R/ [2]: it is an increasing seuence of closed linear subsaces {V j } j Z L 2.R/ with the following roerties: (i) V j Z j ={0}, V j Z j is dense in L 2.R/; (ii) For every j Z and f L 2.R/, f V j f.2 / V j+ ; (iii) There exists a ' in V 0 such that '.x /, Z, is an orthonormal basis for V 0. The above ' is called a scaling function. Note that by adjusting a normalization constant,.x / = for all x R [3]. For each j Z, wedefine ' j.x/ = 2 j=2 '.2 j x /, Z. The seuence {' j } Z forms an orthonormal basis for V j. From ' we can construct a wavelet function. Then { j } Z forms an orthonormal basis for W j, the orthogonal comlement to V j inside V j+, that is, V j+ = V j W j. It follows that { j } j; Z is an orthonormal basis for L 2.R/. In this aer, we assume that the wavelets are comactly suorted, say su ', su [0; M] for some integer M. Also we assume that they have the desirable degree of smoothness whenever needed. We need the following characterizations of the Besov and Triebel-Lizorin saces [6, 2]. PROPOSITION 2.. Suose C is a comactly suorted wavelet and { j } j; Z forms an orthonormal basis of L 2.R/. Letf be locally integrable and write f.x/ = j Þ.j; / j.x/ formally. (i) For 0 <Þ<,, <, f Ḃ Þ; j if and only if ( 2. =+Þ+=2/ j Þ.j; / ) = = < : (ii) For 0 <Þ<,, <, f Ḟ Þ; if and only if ( = A. f /.x/ = 2.Þ+=2/ j Þ.j; /.2 j x /) L.R/; j; where denotes the characteristic function of [0; /. In this case, f Ḟ Þ; A. f /. Let P j : L 2.R/ V j be the orthonormal rojection and Q j = P j+ P j. Then Q j : L 2.R/ W j is the corresonding orthonormal rojection. In [2], Beylin,

5 [5] Wavelet decomosition of Calderón-Zygmund oerators 33 Coifman and Rohlin give a decomosition of T in terms of P j and Q j : (2.) T = P j TQ j + Q j TP j + Q j TQ j : The corresonding distribution ernel is (2.2) K.x; y/ = jl a. j; ; l/' j.x/ jl.y/ + jl b. j; ; l/ j.x/' jl.y/ + jl c. j; ; l/ j.x/ jl.y/; where a. j; ; l/ = T jl ;' j = K;'j jl ; b. j; ; l/ = T ' jl ; j = K; j ' jl ; c. j; ; l/ = T jl ; j = K; j jl : We call such a. j; ; l/; b. j; ; l/; c. j; ; l/ the BCR-coefficients. It is easy to show that PROPOSITION 2.2. Suose T satisfies the conditions in Theorem. or Theorem.2, then T./ = 0 imlies that for any j; l Z, a. j; ; l/ = 0; similarly T./ = 0 imlies that for any j; Z, l b. j; ; l/ = 0. PROOF. Assuming that T./ = 0 and using '.x / =, we have T jl ;' j = T jl ; 2 j=2 = jl; 2 j=2 T./ = 0: a. j; ; l/ = The second art can be roved similarly. PROPOSITION 2.3. Suose T satisfies the conditions in Theorem. or Theorem.2. Let ( ) A.m/ = su a. j; ; l/ + a. j; l; / + b. j; ; l/ + b. j; l; / j;l : :2 m l <2 m+ Then there exists C such that A.m/ B.m/ for all m 0. Moreover we have ( ) (2.3) c. j; ; l/ + c. j; l; / < : su j;l

6 34 Ka-Sing Lau and Lixin Yan [6] PROOF. We first observe that A.m/ < for each m 0. This comes directly from the WBP in (.3) and the exressions for a. j; ; l/ and b. j; ; l/. By using this, it suffices to rove the ineuality for 2 m > M. Let y 0 = 2 j l, then a. j; ; l/ = K.x; y/' j.x/ jl.y/ dx dy Hence for 2 m > M, (K = 2 j.x; y + y0 / K.x; y 0 / ) '.2 j x /.2 j y/ dx dy su y [0;2 j M] K.x; y + y 0 / K.x; y 0 / dx: [2 j ;.+M/2 j ] :2 m l <2 m+ a. j; ; l/ C su y [0;2 j M] E K.x; y + y 0 / K.x; y 0 / dx; where E ={x R : 2 m j x y 0 2 m+ j + 2 j M}. According to the definition of B.m/, wehave :2 m l <2 a. j; ; l/ B.m/ for 2 m > M. The same m+ argument alies to the other terms in A.m/, which comletes the roof for the first assertion. The roof of (2.3) is essentially the same. We consider { su j;l J ( c. j; ; l/ + c. j; l; / ) } : It is bounded if J ={ : l M}; and it is B.m/ if J ={ : 2 m l < 2 m+ } if 2 m+ > M. This imlies (2.3). 3. T () Theorem on Besov saces In view of Proosition 2.2 and Proosition 2.3, we will rove the following theorem in terms of the BCR-coefficients, which imlies Theorem.. THEOREM 3.. Let T : D D be a Calderón-Zygmund oerator with the wavelet decomosition as in (2.), (2.2) and satisfying (: ), (2.3). (i) If.m + /A.m/ < and a. j; ; l/ = m=0 l b. j; ; l/ = 0, then T is a bounded oerator on Ḃ 0;,, <. (ii) If m=0 2Þm A.m/ < and l b. j; ; l/ = 0 for any j, Z, then T is bounded on Ḃ Þ;, 0 <Þ<,, <.

7 [7] Wavelet decomosition of Calderón-Zygmund oerators 35 Let us rewrite T = P j TQ j + Q j TP j + Q j TQ j = T./ + T.2/ + T.3/ : We will first consider the term T.2/. Its distributional ernel is K.2/.x; y/ = jl b. j; ; l/ j.x/' jl.y/: Let J m ={.; l/ : 2 m l < 2 m+ },and (3.) b. j; ; l/.; l/ J m ; b m. j; ; l/ = n:.;n/ J m b. j; ; n/ l = ; 0 otherwise, where m = 0; ; 2;:::. The definition imlies that b l m. j; ; l/ = 0 for each j, Z. Since l b. j; ; l/ = 0 by assumtion, we have b. j; ; / = Hence K.2/.x; y/ can be decomosed as m=0 l:.;l/ J m b. j; ; l/: (3.2) K.2/.x; y/ = b. j; ; / j.x/' j.y/ j + b. j; ; l/ j.x/' jl.y/ j l:l = = ( ) b. j; ; l/ j.x/' j.y/ j m=0 l:.;l/ J m + b. j; ; l/ j.x/' jl.y/ m=0 j l:.;l/ J m = b m. j; ; l/ j.x/' jl.y/ = K.2/ m.x; y/: m=0 jl m=0 Let T.2/ m denote the oerator with distributional ernel K.2/ m.x; y/. Then we can decomose T.2/ as: T.2/ = T.2/.2/ m=0 m. We call each T m a bloc oerator. The following lemma together with the assumtion on A.m/ will imly that T.2/ is a bounded oerator on Ḃ Þ;.

8 36 Ka-Sing Lau and Lixin Yan [8] LEMMA 3.2. Under the hyothesis of Theorem 3., let ; <, 0 Þ<. Then T.2/ m is a bounded oerator on ḂÞ;, and the oerator norm satisfies T.2/ m where C is indeendent of m. { C.m + /A.m/ Þ = 0; C2 Þm A.m/ 0 <Þ<; PROOF. Let f.y/= Þ.j; / j j.y/ be in Ḃ Þ;, and let g.x/= þ.j; / j j.x/ be in the dual sace Ḃ Þ;. Noting that ' jl; j = 0 imlies that j > j and 2 j j l 2 j j. + M/ + M (recall that ', have comact suorts contained in [0; M]), one can write T.2/ f; g m = Þ.j ; /b m. j; ; l/þ. j; / ' jl ; j j = jl j; j :0< j j m + = I + II: j; j : j j m+ ( Þ.j ; /b m. j; ; l/þ. j; / ) ' jl ; j l ( Þ.j ; /b m. j; ; l/þ. j; / ) ' jl ; j l Let 0 mj s l = b m. j + s; ; l/ ' j +sl; j. By using Proosition 2. and the Hölder ineuality, we obtain I m Þ.j ; /b m. j + s; ; l/þ. j + s; / ' j +sl; j j l ( m Þ.j ; / ) = þ.j + s; / 0 mj s l = s= s= j l m s= j 2.= +Þ =2/s f Ḃ Þ; 2. = Þ+=2/. j +s/ l þ.j + s; /0 mj s l = = :

9 [9] Wavelet decomosition of Calderón-Zygmund oerators 37 We will mae a searate estimation of. Note that 0 mj s l 2 s=2 b m. j + s; ; l/ '.x l/;.2 s x / l l 2 s=2 A.m/ '.x l/;.2 s x / l 2 s=2 A.m/; (the last ineuality holds because ' has comact suort and l '.x l/ is bounded) and 0 mj s l 2 s=2 b m. j + s; ; l/ '.x l/ ;.2 s x / l l Hence (3.3) It follows that 2 s=2 A.m/: þ.j + s; /0 mj s l l ( þ.j + s; / )( ) mj 0 s = l 0 mj s l l l ( þ.j + s; / ) 0 mj s (2 s=2 A.m/ ) = I C l ( þ.j + s; / ) (2 s=2 A.m/ )( 2 s=2 A.m/ ) = l = C2 s. = /=2 A.m/ þ.j + s; / : m s= 2.= +Þ =2/s 2 s.= = /=2 A.m/ f Ḃ Þ; ( 2. = Þ+=2/. j +s/ þ.j + s; / ) = j m 2 Þm A.m/ f Ḃ Þ; g Ḃ Þ; s= C.m + /A.m/ f Ḃ Þ; g Ḃ Þ; Þ = 0; C2 Þm A.m/ f Ḃ Þ; 0 <Þ<: g Ḃ Þ; =

10 38 Ka-Sing Lau and Lixin Yan [0] We now estimate the exression II. For convenience, we use the same notation C to denote the different constants in the different lace. Define g j.x/ = x l b m. j; ; l/'.y l/dy and 0 j s = g j +s.x/;.2 s x / : Then II = = C j; j : j j m+ s=m+ j = s=m+ j Þ.j ; /b m. j; ; l/þ. j; / ' jl ; j 2 3s=2 Þ.j ; /þ. j + s; / 0 j s 2 3s=2 f Ḃ Þ;. = Þ+=2/ j 2 = þ.j + s; / 0 j s = : We claim that (3.4) þ.j + s; / 0 j s 2.s+m/. / 2 m A.m/ þ.j + s; / : In fact, the condition l b m. j; ; l/ = 0 imlies that su g j [ 2 m+ ; +2 m+ M]. Then 0 j s g j +s.x/ ;.2 s x / g j +s.x/ ; 2 m A.m/: On the other hand, for s = j j since su.2 s x / [2 s ; 2 s. + M/],we now that 2 s 2 s M + 2 m + M 2 s. It follows that 0 j s g j +s.x/ ;.2 s x / 2 s+m A.m/:

11 [] Wavelet decomosition of Calderón-Zygmund oerators 39 Combining these two estimates we have þ.j + s; / 0 j s ( þ.j + s; / )( ) 0 j s = 0 j s 2.s+m/. / 2 m A.m/ þ.j + s; / : This roves the claim. We return to the estimate of II II C s=m+ 2.= +Þ 2/s f Ḃ Þ; ( 2.s+m/. / 2 m A.m/ 2. = Þ+=2/. j +s/ þ.j + s; / ) = ( ) 2.Þ /s 2 m A.m/ f Ḃ Þ; g Ḃ Þ; s=m+ 2 Þm A.m/ f Ḃ Þ; g Ḃ Þ; : j By the estimates of I and II we conclude that T.2/ is a bounded oerator on Ḃ Þ;, ; <, 0 Þ< with the oerator norm as secified. = LEMMA 3.3. Under the hyothesis of Theorem 3., let ; <, 0 Þ<. Then T./ is a bounded oerator on Ḃ Þ;. PROOF. We can write T./ f; g = jj a. j; ; l/þ. j ; /Þ. j; l/ ' j ; j with j > j and 2 j j l 2 j j. + M/+ M as in Lemma 3.2. For the case Þ = 0, we have by assumtion that a. j; ; l/ = 0, and we can aly the same roof as in Lemma 3.2 (by relacing l b. j; ; l/ = 0). For the case 0 <Þ<, we have not assumed that a. j; ; l/ = 0 and we need

12 40 Ka-Sing Lau and Lixin Yan [2] to modify the roof by searating out the diagonal term (as in (3.2)): T./ f; g þ.j ; /a. j; ; /Þ. j; / ' j ; j j; j : j j >0 + þ.j ; /a. j; ; l/þ. j; l/ ' j ; j : m=0 j; j : j j >0.;l/ J m By using the same argument as in Lemma 3.2, we can show that the first term is bounded by C s= 2 Þs f Ḃ Þ; g Ḃ Þ; and the second term is bounded by CA.m/ s= 2 Þs f Ḃ Þ; g Ḃ Þ; (note that in here the term s= 2 Þs converges and we only need to use the estimation for I without recoursing to the estimation II in the last roof). This roves the lemma. LEMMA 3.4. Under the hyothesis of Theorem 3. let ; <, 0 Þ<. Then T.3/ is a bounded oerator on the Besov saces Ḃ Þ;. PROOF. We can write T.3/ f; g = jl c. j; ; l/þ. j; /þ. j; l/; where f, g are defined as in Lemma 3.2. By using the duality and condition (2.3), it can be checed as in Lemma 3.2 that T.3/ f; g f Ḃ Þ; g Ḃ Þ; Theorem 3. follows directly from Lemmas T() Theorem on Triebel-Lizorin saces In view of Proosition 2.2 and Proosition 2.3, we will rove the following theorem in terms of the BCR-coefficients, which imlies Theorem.2. THEOREM 4.. Let T : D D be a Calderón-Zygmund oerator with the wavelet decomosition as in (2.), (2.2) and satisfying (: ), (2.3). (i) If.m + m=0 /2 = A.m/ <, and a. j; ; l/ = l b. j; ; l/ = 0, then T is a bounded oerator on Ḟ 0;,, <. (ii) If m=0 2Þm A.m/ < and l b. j; ; l/ = 0 for any j; Z, then T is bounded on Ḟ Þ;, 0 <Þ<,, <. We will first rove the theorem for Ḟ Þ;, then aly an interolation theorem

13 [3] Wavelet decomosition of Calderón-Zygmund oerators 4 on Ḟ Þ; and Ḟ Þ;.= Ḃ Þ; /, and a duality argument to conclude the theorem. We need the following notion of atom which can be found in [9]. DEFINITION. Let a.x/ = Þ.j; / j j.x/ be a locally integrable function. We say that a.x/ is an.þ; ; /-atom if there is a dyadic cube I R such that (i) su a.x/ I; (ii) a.x/ dx = 0; I (iii) A.a/ I =, where A.a/ is defined by (4.) ( = A.a/.x/ = 2.Þ+=2/ j Þ.j; /.2 j x /) : j; For j; Z,letI j denote the interval [2 j ; 2 j. + /]. Let (4.2) a j.x/ = j ; :I j I j Þ.j ; / j.x/: Note that the sum is actually adding all j ; with j j ; [2 j j ; 2 j j. + //. It follows that the suort of a j.x/ is contained in [2 j ; 2 j. + M/]. By the definition, we now that a j.x/ is an.þ; ; /-atom if a.þ;;/ j. / j := 2 2. =+Þ+=2/ j Þ.j ; / j ; :I j I j = < : LEMMA 4.2. Let a j.x/ be the atom as in (4.2), and let T m.2/ Lemma 3.2. Then we have be defined as in T.2/ m a j Ḟ Þ; { C.m + / 2 = A.m/ a j Þ = 0;.Þ;;/ C2 Þm A.m/ a.þ;;/ j 0 <Þ<; where C is indeendent of m; j;. PROOF. Without loss of generality we consider a 00.y/ and denote it by a.y/ for simlicity, that is, a.y/ = Þ.j; / 0 j 0 <2 j.y/. Noting that ' j jl ; j =0

14 42 Ka-Sing Lau and Lixin Yan [4] imlies that j > j and 2 j j l 2 j j. + M/ + M. One can write ( T.2/ a) m.x/ = Þ.j ; /b m. j; ; l/ ' jl ; j j.x/ jl j = ( Þ.j ; /b m. j; ; l/ ) ' jl ; j j.x/ j m j l + ( Þ.j ; /b m. j; ; l/ ) ' jl ; j j.x/ j>m j :0< j j m l + ( Þ.j ; /b m. j; ; l/ ) ' jl ; j j.x/ j>m j : j j >m l = a.x/ + a 2.x/ + a 3.x/: Since Ḃ Þ; = Ḟ Þ; Ḟ Þ;, <, it follows that there exists a constant C such that f Ḟ Þ; f ḂÞ;. Using Proosition 2. and the Hölder ineuality, we obtain a Ḟ Þ; Þ.j ; /b m. j; ; l/ ' jl ; j j j m j l Ḃ Þ; 2.Þ =2/ j Þ.j ; /b m. j; ; l/ ' jl ; j j m j l A.m/ 2. j j/=2 2.Þ =2/ j 0 j m 0< j < j 0 <2 j ( ) = A.m/ 2 Þj 2 Þj ( j 0 j m 0< j <m. =+Þ+=2/ j 2 Þ.j ; / ) = Þ.j ; / { C.m + / 2 = A.m/ a.þ;;/ Þ = 0; C2 Þm A.m/ a.þ;;/ 0 <Þ<: To estimate a 2 Ḟ Þ;, we first observe that the condition 0 < 2 j imlies that 0 2 j l M. By the exression of a 2.x/, we now that E j = [ 2 m+ ; 2 m+ + 2 j + 2 j M], and hence su a 2.x/ [2 j ; 2 j. + M/] [ ; 2M]: j>m E j

15 [5] Wavelet decomosition of Calderón-Zygmund oerators 43 Let 0 mjs l = b m. j; ; l/ ' jl ; j s. Then a 2 Ḟ Þ; Þ.j ; /b m. j; ; l/ ' jl ; j j.x/ j>m j :0< j j m l Ḟ Þ; m Þ.j s; /0 mjs l j s= j=m+ l 0 <2 j s Ḟ Þ; m ( = 2.Þ+=2/ j Þ.j s; /0 mjs l.2 j x /) dx s= R j=m+ l { m } = 2 j 2.Þ+=2/ j Þ.j s; /0 mjs l : s= j=m+ Similar to the estimate (3.3) in Lemma 3.2,wehave Þ.j s; /0 mjs l 2 s. = /=2 A.m/ Þ.j s; / : l It follows that { m a 2 Ḟ Þ; A.m/ 2 j+.þ+=2/ j 2 s. = /=2 Þ.j s; / } = A.m/ s= s= j=m+ { m 2 Þs j=m+ l 2. =+Þ+=2/. j s/ Þ.j s; / } = { C.m + /A.m/ a.þ;;/ Þ = 0; C2 Þm A.m/ a.þ;;/ 0 <Þ<: We now turn to estimate the term a 3.x/. Lie a 2.x/, su a 3.x/ [ ; 2M]. Let g j.x/ be defined as in the estimate of II of Lemma 3.2 and 0 js = g j.x/;.2 s x /. Then a 3.x/ Ḟ Þ; j>m j : j j >m l s=m+ s=m+ 2 3s=2 Þ.j ; /b m. j; ; l/ ' jl ; j j=m+ 2 3s=2 { j=m+ Þ.j s; / 0 js j.x/ Ḟ Þ; j.x/ Ḟ Þ; } = 2 j+.þ+=2/ j Þ.j s; / 0 js :

16 44 Ka-Sing Lau and Lixin Yan [6] By the same estimate as (3.4) in Lemma 3.2, we obtain Þ.j s; / 0 js 2 s+m 2 m= A.m/ Þ.j s; / : Hence a 3 Ḟ Þ; s=m+ A.m/ 2 3s=2 { s=m+ j=m+ 2 Þm A.m/ a.þ;;/ : 2 j+.þ+=2/ j 2 s+m 2 m= Þ.j s; / } = 2 m 2.Þ /s { j 2. =+Þ+=2/. j s/ Þ.j s; / } = We have hence roved Lemma 4.2 by combining the estimates of a.x/; a 2.x/ and a 3.x/. Now we state the following atomic decomosition of Ḟ Þ;,whichisgivenin[] for Ḟ 0;2.= H / and in [6] and [9] for the general case. For comleteness we modify their roof and setch it here. LEMMA 4.3. Let f.x/ = j Þ.j; / j.x/ be in Ḟ Þ;. Then there exists a seuence {h sn.x/} s;n of.þ; ; /-atoms and {½ sn } R such that (4.3) f.x/ = s;n ½ sn h sn.x/ and C f Ḟ Þ; s;n ½ sn C 2 f Ḟ Þ; for some fixed C ; C 2 > 0 indeendent of f. PROOF. By Proosition 2., f Ḟ Þ; has an euivalent norm given by f Ḟ Þ; A. f /. If 2 s, s Z, is a given threshold, we define s ={x : A. f /.x/ >2 s }. This allows us to write s = Q n N sn, where each Q sn is a maximal dyadic interval in s. The intervals Q sn, being dyadic and maximal, are either identical or disjoint. For each s Z, n N, consider the family F sn of all dyadic intervals I j such that I j Q sn which are contained in no Q s+ for any. By the above construction, we can write s = : s s;n I j F sn I j

17 [7] Wavelet decomosition of Calderón-Zygmund oerators 45 For each s Z, n N,let (4.4) h sn.x/ = ½ sn I j F sn Þ.j; / j.x/; where ½ sn = Q sn ={ I j F sn 2. =+Þ+=2/ j Þ.j; / } =. Then, h sn.x/ is an.þ; ; /-atom and (4.5) f.x/ = j; Þ.j; / j.x/ = s;n ½ sn h sn.x/ gives an atomic decomosition of f. For the details we refer the reader to [, 6, 9]. PROOF OF THEOREM 4.. By Lemma 4.4 we can write f.x/ Ḟ Þ; as an atomic decomosition f.x/ = ½ s;n snh sn.x/, where each h sn.x/ is an.þ; ; /-atom defined in (4.4). For each h sn.x/, we can rewrite it as the form in (4.2) by assigning Þ.j; / = 0 for I j Q sn but I j F sn. Using Lemma 4.2 and Lemma 4.3,wehave T.2/ f m Þ; Ḟ { C.m + / 2 = A.m/ s;n ½ sn Þ = 0; C2 Þm A.m/ s;n ½ sn 0 <Þ<; where C is indeendent of m; s; n. It follows that T.2/ is bounded on Ḟ Þ;. Similarly as in Lemma 3.3, and Lemma 3.4, we can show that both T./ and T.3/ are also bounded oerators on Ḟ Þ;. Hence, we have roved that T is bounded on Ḟ Þ;,and T.Ḟ Þ; ;Ḟ Þ; / { C m=0.m + /2 = A.m/ Þ = 0; C m=0 2Þm A.m/ 0 <Þ<; where 0 Þ<, <. Since T is bounded on Ḃ Þ;.= Ḟ Þ; /, the interolation theorem [3] imlies that T is bounded on Ḟ Þ;,0 Þ<,. Similarly as in the roof of Theorem 3. and Theorem 4., we can show, by interchanging the role of T./ and T./, that T is bounded on both Ḟ Þ; and Ḟ Þ; with <Þ 0; <. Again, alying the interolation theorem and the duality, T is bounded on Ḟ Þ;,0 Þ<,. This finishes the roof of Theorem 4.. PROOF OF COROLLARY.3. With the above notation, we have T.2/ m.h ;H / C.m + / 3=2 A.m/ and T.2/ m.l 2 ;L 2 /.m + / =2 A.m/ for some C > 0 indeendent of m. For < < 2, by the interolation theorem we have T m.2/.l ;L / C.m + / =2+2.= =2/ A.m/. Using the duality argument, for 2 < < we have T m.2/.l ;L /.m + / =2+2 = =2 A.m/. It follows from the assumtion on A.m/ that T.2/ is bounded on L. Similarly, T./ and T.3/ are bounded oerators on L ;soist. This comletes the roof of Corollary.3.

18 46 Ka-Sing Lau and Lixin Yan [8] Acnowledgement We than the referee for valuable comments and suggestions. References [] J. Aguirre, M. Escobedo, J. C. Perel and Ph. Tchamitchian, Basis of wavelets and atomic decomositions of H.R n / and H.R n R n /;, Proc. Amer. Math. Soc. (99), [2] G. Beylin, R. Coifman and V. Rohlin, Fast wavelet transforms and numerical algorithms, Comm. Pure Al. Math. 44 (99), [3] I. Daubechies, Ten lectures on wavelets, CBMS NSF Regional Conference Series in Al. Math. 6 (SIAM, Philadelhia, 992). [4] G. David and J. L. Journé, A boundedness criterion for generalized Calderón-Zygmund oerators, Ann. of Math. 20 (984), [5] D. G. Deng, L. X. Yan and Q. X. Yang, Blocing analysis and T./ theorem, Science in China 4 (998), [6] M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley theory and the study of functions, CBMS Regional Conference Series in Mathematics 79 (AMS, Providence, RI, 99). [7] M. Frazier, R. Torres and G. Weiss, The boundedness of Calderón-Zygmund oerator on the saces Ḟ Þ;, Rev. Mat. Iberoamericana 4 (998), [8] Y. Han and S. Hofmann, T./ theorem for Besov and Triebel-Lizorin saces, Trans. Amer. Math. Soc. 237 (993), [9] Y. Han, M. Paluszynsi and G. Weiss, A new atomic decomosition for the Triebel-Lizorin saces, Contemorary Math. 89 (995), [0] P. G. Lemarié, Continuité sur les esaces de Besov and oeratéurs definis ar des intégrales singulières, Ann. Inst. Fourier (Grenoble) 35 (985), [] Y. Meyer, La minimalité de l esace de Besov Ḃ 0 et la continuité des oérateurs definis ar des integrales singulières, Monografias de Matematicas, 4 (Univ. Autonoma de Madrid, 986). [2], Ondelettes et oérateurs, Vols I, II (Hermann, Paris, 990). [3] H. Triebel, Theory of function saces (Birhäuser, Basel, 983). Deartment of Mathematics The Chinese University of Hong Kong Shatin, NT, Hong Kong slau@math.cuh.edu.h Deartment of Mathematics Zhongshan University Guangzhou, P. R. China lixin@ics.m.edu.au