СИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ

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1 S e MR ISSN СИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ Siberia Electroic Mathematical Reports Том 2, стр (2005) УДК MSC 42B20 THE CONTINUITY OF MULTILINEAR SINGULAR INTEGRAL OPERATORS WITH VARIABLE CALDERÓN ZYGMUND KERNEL ON HARDY AND HERZ SPACES LIU LANZHE Abstract. We prove the cotiuity of some multiliear operators geerated by sigular itegral operators with variable Calderó-Zygmud kerel ad Lipschitz fuctios o some Hardy ad Herz-type spaces. 1. Itroductio Let b BMO(R ) ad T be the Calderó-Zygmud operator. The commutator b, T geerated by b ad T is defied by b, Tf(x) = b(x)tf(x) T(bf)(x). By the classical result of Coifma, Rochberg ad Weiss 8, wthe commutator b, T is bouded o L p (R ) for 1 < p <. However, it was observed that b, T is ot bouded, i geeral, from H p (R ) to L p (R ) for 0 < p 1. But, the boudedess hold if b belogs to the Lipschitz spaces Lip β (R ) 15. This show the differece of b BMO(R ) ad b Lip β (R ). I 13 ad 19 it is proved that if b is a Lipschitz fuctio the the commutators are L p (p > 1)-bouded. I 1 Calderó ad Zygmud itroduced some sigular itegral operators with variable kerel ad discuss their boudedess. I 10 the authors obtaied the boudedess for the commutators geerated by the sigular itegral operators with variable kerel ad BMO fuctios. I 18 the authors proved the boudedess of the multiliear oscillatory sigular itegral operators geerated by the operators ad BM O fuctios. The purpose of this paper is to study the cotiuity properties of the multiliear operators geerated by the sigular itegral operators with variable kerel ad Lipschitz fuctios o some Hardy ad Herz-type spaces. Liu L.Z., The cotiuity of multiliear sigular itegral operators with variable Calderó Zygmud kerel o Hardy ad Herz Spaces. c 2005 Liu L.Z. Received April 18, 2005, published September 2,

2 THE CONTINUITY OF MULTILINEAR SINGULAR INTEGRAL OPERATORS 157 First, let us itroduce some otatios (see11-12,16-17, 19). Throughout this paper, Q will deote a cube of R with side parallel to the axes. For a cube Q ad a locally itegrable fuctio f, let f Q = Q 1 f(x)dx. Deote the Hardy spaces Q by H p (R ). It is well kow that H p (R )(0 < p 1) has the atomic decompositio characterizatio (see 9,16). For β > 0, the Lipschitz space Lip β (R ) is the space of fuctios f such that (see15) f Lipβ = sup f(x h) f(x) / h β <. x,h R, h>0 Defiitio 1. Let 0 < p, <, α R. For k Z, defie B k = {x R : x 2 k } ad C k = B k \ B k 1. Deote by χ k the characteristic fuctio of C k ad χ 0 the characteristic fuctio of B 0. (1) The homogeeous Herz space is defied by where K (R ) = {f L loc (R \ {0}) : f K < }, f K = 2 kαp fχ k p L 1/p (2) The ohomogeeous Herz space is defied by where K (R ) = {f L loc (R ) : f K < }, f K = 1/p 2 kαp fχ k p L fχ 0 p L. k=1 Defiitio 2. Let α R, 0 < p, <. (1) The homogeeous Herz type Hardy space is defied by ad ad H K (R ) = {f S (R ) : G(f) f H K = G(f) K ; ; K (R )}, (2) The ohomogeeous Herz type Hardy space is defied by HK (R ) = {f S (R ) : G(f) K (R )}, f HK = G(f) K ; where G(f) is the grad maximal fuctio of f. The Herz type Hardy spaces have the atomic decompositio characterizatio. Defiitio 3. Let α R, 1 < <. A fuctio a(x) o R is called a cetral (α, )-atom (or a cetral (a, )-atom of restrict type), if 1) Suppa B(0, r) for some r > 0 (or for some r 1), 2) a L B(0, r) α/, 3) a(x)x γ dx = 0 for γ α (1 1/).

3 158 LIU LANZHE Лемма 1 (17). Let 0 < p <, 1 < < ad α (1 1/). A temperate distributio f belogs to H K (R )(or HK (R )) if ad oly if there exist cetral (α, )-atoms(or cetral (α, )-atoms of restrict type)a j supported o B j = B(0, 2 j ) ad costats λ j with j λ j p < such that f = λ ja j (or f = j=0 λ ja j )i the S (R ) sese, ad f H K ( or f HK ) 1/p λ j p. j 2. Mai results I this paper we study a class of multiliear operators related to the sigular itegral operators with variable kerel, whose defiitios are as follows. Defiitio 4. Let k(x) = Ω(x)/ x : R \ {0} R. k is said to be a Calderó- Zygmud kerel if (a) Ω C (R \ {0}); (b) Ω is homogeeous of degree zero; (c) Σ Ω(x)xα dσ(x) = 0 for all multi-idices α (N {0}) with α = N, where Σ = {x R : x = 1} is the uit sphere of R. Defiitio 5. Let k(x, y) = Ω(x, y)/ y : R (R \ {0}) R. k is said to be a variable Calderó-Zygmud kerel if (d) k(x, ) is a Calderó-Zygmud kerel for a.e. x R ; (e) max γ 2 γ γ y Ω(x, y) L (R Σ) = M <. Let m j be the positive itegers(j = 1,, l), m 1 m l = m ad A j be the fuctios o R (j = 1,, l). Deote that R mj1(a j ; x, y) = A j (x) 1 γ! Dγ A j (y)(x y) γ γ m j ad Q mj1(a j ; x, y) = R mj (A j ; x, y) 1 γ! Dγ A j (x)(x y) γ. γ =m j The multiliear sigular itegral operator with variable Calderó-Zygmud kerel is defied by T A Ω(x, x y) l (f)(x) = R x y m R mj1(a j ; x, y)f(y)dy, where Ω(x, y)/ y is a variable Calderó-Zygmud kerel. We also defie that Ω(x, x y) T(f)(x) = R x y f(y)dy, which is the sigular itegral operator with variable Calderó-Zygmud kerel (see 1). We also cosider the variat of T A, which is defied by T A Ω(x, x y) l (f)(x) = R x y m Q mj1(a j ; x, y)f(y)dy.

4 THE CONTINUITY OF MULTILINEAR SINGULAR INTEGRAL OPERATORS 159 Note that whe m = 0, T A is just higher order commutator of the operators T ad A (see 10,13,19), while whe m > 0, it is o-trivial geeralizatios of the commutator. It is well kow that multiliear operators are of great iterest i harmoic aalysis ad have bee widely studied by may authors whe A has derivatives of order m i BMO(R )(see 2,4-7,9). I 2 the L p (p > 1)-boudedess of multiliear sigular itegral operators geerated by some sigular itegrals operators ad Lipschitz fuctios are obtaied. The mai purpose of this paper is to prove the cotiuity properties of the multiliear sigular itegral operators with variable Calderó-Zygmud kerel o Hardy ad Herz-type spaces. We shall prove the followig theorems i Sectio 3. Theorem 1. Let D γ A j Lip β (R ) for all γ with γ = m j ad j = 1,, l. (a) If 0 < β 1, /( β) < p 1 ad 1/p 1/ = lβ/, the T A maps H p (R ) cotiuously ito L (R ); (b) If 0 < β 1/l, the T A maps H /(lβ) (R ) cotiuously ito L 1 (R ). Theorem 2. Let D γ A j Lip β (R ) for all γ with γ = m j ad j = 1,, l. (i) If 0 < β 1, 0 < p <, 1 < 1, 2 <, 1/ 1 1/ 2 = lβ/ ad (1 1/ 1 ) α < (1 1/ 1 ) lβ, the T A maps H K 1 (R ) cotiuously ito K 2 (R ); (ii) If 0 < β 1/l, 0 < p 1, 1 < 1, 2 < ad 1/ 1 1/ 2 = lβ/, the T A (1 1/1)lβ,p maps H K 1 (R (1 1/1)lβ,p ) cotiuously ito K 2 (R ). Remark. Theorem 2 also hold for the ohomogeeous Herz ad Herz type Hardy space. We begi with a prelimiary lemma. 3. Proofs of mai results Лемма 2 (6). Let A be a fuctio o R such that D γ A L loc (R ) for γ = m ad some >. The R m (A; x, y) C x y ( 1/ m 1 Q(x, D γ A(z) dz), y) Q(x,y) γ =m where Q(x, y) is the cube cetered at x ad havig side legth 5 x y. Proof of Theorem 1. (a) It suffices to show that there exists a costat C > 0 such that for every H p -atom a, there is T A (a) L C. Without loss of geerality, we may assume l = 2. Let a be a H p -atom, that is that a supported o a cube Q = Q(x 0, d), a L Q 1/p ad a(x)x η dx = 0 for η (1/p 1). We write ( ) R T A (a)(x) dx = x x 0 2d x x 0 >2d T A (a)(x) dx = I 1 I 2. For I 1, a aalog of the proof i 14, it follows that T A (f)(x) C I 2β ( f )(x), γ j =m j

5 160 LIU LANZHE where I µ is the fractioal itegral operator of order µ, thus T A is bouded from L r (R ) to L s (R ) for ay r, s with 1 < r < /2β ad 1/r 1/s = 2β/ by 20. Takig 1 > ad 1 < p 1 < /2β such that 1/p 1 1/ 1 = 2β/, by Hölder s ieuality ad the (L p1, L 1 ) -boudedess of T A, we have I 1 C T A (a) L 1 2Q 1 /1 C a L p 1 Q 1 /1 C. To estimate I 2, we eed to estimate T A (a)(x) for x (2Q) c. By 3, we kow that g k T A Y hk(x y) (f)(x) = a hk (x) R x y m R mj1(a j ; x, y)f(y)dy k=1 h=1 g k = k=1 h=1 a hk (x)u A hk (f)(x), where g k Ck 2, a hk L Ck 2, Y hk (x y) Ck /2 1 ad Y hk (x y) x y Y hk(x x 0 ) Ck/2 x 0 y / 1 for > 2 x 0 y > 0. Let Ãj(x) = A j (x) γ =m j 1 γ! (Dγ A j ) Q x γ. The R mj (A j ; x, y) = R mj (Ãj; x, y) ad D γ Ã j = D γ A j (D γ A j ) Q for γ = m j. We write, by the vaishig momet of a, Uhk A (a)(x) = Yhk (x y) R x y m Y hk(x x 0 ) x x 0 m R m1 (Ã1; x, y)r m2 (Ã2; x, y)a(y)dy Y hk(x x 0 ) R x x 0 m R m 1 (Ã1; x, y) R m1 (Ã1; x, x 0 )R m2 (Ã2; x, y)a(y)dy Y hk(x x 0 ) R x x 0 m R m 2 (Ã2; x, y) R m2 (Ã2; x, x 0 )R m1 (Ã1; x, x 0 )a(y)dy Yhk (x y)(x y) γ2 x y m Y hk(x x 0 )(x x 0 ) γ2 m γ 2 =m 2 γ 2 =m 2 γ 1 =m 1 γ 1 =m 1 γ 1 =m 1, γ 2 =m 2 R R m1 (Ã1; x, y)d γ2ã 2(y)a(y)dy Y hk(x x 0 )(x x 0 ) γ2 R x x 0 m R m1 (Ã1; x, y) R m1 (Ã1; x, x 0 ) R D γ2ã 2(y)a(y)dy Yhk (x y)(x y) γ1 x y m Y hk(x x 0 )(x x 0 ) γ1 m R m2 (Ã2; x, y)d γ1ã 1(y)a(y)dy Y hk(x x 0 )(x x 0 ) γ1 R x x 0 m R m1 (Ã2; x, y) R m2 (Ã2; x, x 0 ) R D γ1ã 1(y)a(y)dy Yhk (x y)(x y) γ1γ2 x y m Y hk(x x 0 )(x x 0 ) γ1γ2 m D γ1ã 1(y)D γ2ã 2(y)a(y)dy.

6 THE CONTINUITY OF MULTILINEAR SINGULAR INTEGRAL OPERATORS 161 By Lemma 2 ad the followig ieuality b(x) b Q 1 b Lipβ x y β dy b Lipβ ( d) β, Q we get Q R mj (Ãj; x, y) γ =m j D γ A j Lipβ ( x y d) mjβ ; O the other had, by the formula (see 6): R mj (Ãj; x, y) R mj (Ãj; x, x 0 ) = 1 η! R m j η (D η Ã j ; x 0, y)(x x 0 ) η, η <m ote that x y for y Q ad x R \ 2Q, we obtai Uhk A (a)(x)(x) Ck/2 i=j γ j =m j Q y x 0 1 2β y x 0 β β y x 0 2β Ck /2 γ j =m j Q 1/1 1/p Q β/1 1/p Q 2β/1 1/p 1 2β β a(y) dy Thus g k T A (f)(x) C a hk (x) k /2 k=1 h=1 γ j =m j Q 1/1 1/p Q β/1 1/p 1 2β C k 2/2 2 k=1 γ j =m j Q 1/1 1/p Q β/1 1/p 1 2β C γ j =m j Q 1/1 1/p Q β/1 1/p 1 2β ; β Q 2β/1 1/p β Q 2β/1 1/p β Q 2β/1 1/p ad recall that /( β) < p 1, 1/p 1/ = lβ/, we obtai I 2 T A (a)(x) dx i=1 2 i1 Q\2 i Q,

7 162 LIU LANZHE C 2 i(1/p (1)/) 2 i(1/p (β)/) γ j =m j i=1 C C, γ j =m j which together with the estimate for I 1 yields the desired result. (b) Without loss of geerality, we may assume l = 2. It is oly to prove that there exists a costat C > 0 such that for every H /(2β) -atom a supported o Q = Q(x 0, d), there is T A (a) L 1 C. We write R T A (a)(x) dx = x x 0 2d For J 1, by the followig euality Q m1 (A; x, y) = R m1 (A; x, y) we get T A (a)(x) C x x 0 >2d γ =m T A (a)(x) dx := J 1 J 2. 1 γ! (x y)γ (D γ A(x) D γ A(y)), I 2β ( a )(x), γ j =m j thus, T A is bouded from L r (R ) to L s (R ) for ay r, s with 1 < r < /2β ad 1/r 1/s = 2β/ by 20. We get, for 1 < p < /2β ad 1/ = 1/p 2β/, we J 1 C T A (a) L 2Q 1 1/ C a L p Q 1 1/ C. Let us obtai the estimate for J 2. Put à j (x) = A j (x) 1 γ! (Dγ A j ) 2Q x γ. γ =m j The Q mj (A j ; x, y) = Q mj (Ãj; x, y) ad Q mj1(a j ; x, y) = R mj (A j ; x, y) 1 γ! Dγ A j (x)(x y) γ. γ =m j By 3, we kow that T A (f)(x) = g k k=1 h=1 g k = k=1 h=1 Y hk(x y) a hk (x) R x y m a hk (x)v A hk (f)(x), Q mj1(a j ; x, y)f(y)dy we write, by the vaishig momet of a ad for x (2Q) c, R Vhk(a)(x) A = Yhk (x y) x y m Y hk(x x 0 ) m R m1 (Ã1; x, y)r m2 (Ã2; x, y)a(y)dy

8 C THE CONTINUITY OF MULTILINEAR SINGULAR INTEGRAL OPERATORS 163 Y hk(x x 0 ) R x x 0 m R m 1 (Ã1; x, y) R m1 (Ã1; x, x 0 )R m2 (Ã2; x, y)a(y)dy m R m 2 (Ã2; x, y) R m2 (Ã2; x, x 0 )R m1 (Ã1; x, x 0 )a(y)dy C C R Y hk(x x 0 ) C γ 2 =m 2 γ 2 =m 2 C γ 1 =m 1 γ 1 =m 1 γ 1 =m 1, γ 2 =m 2 R Yhk (x y)(x y) γ2 x y m Y hk(x x 0 )(x x 0 ) γ2 m R m1 (Ã1; x, y)d γ2ã 2(x)a(y)dy Y hk(x y)(x x 0 ) γ2 R x x 0 m R m1 (Ã1; x, y) R m1 (Ã1; x, x 0 ) R D γ2ã 2(x)a(y)dy Yhk (x y)(x y) γ1 x y m Y hk(x x 0 )(x x 0 ) γ1 m R m2 (Ã2; x, y)d γ1ã 1(x)a(y)dy Y hk(x y)(x x 0 ) γ1 R x x 0 m R m2 (Ã2; x, y) R m2 (Ã2; x, x 0 ) R D γ1ã 1(x)a(y)dy Yhk (x y)(x y) γ1γ2 x y m Y hk(x y)(x x 0 ) γ1γ2 m D γ1ã 1(x)D γ2ã 2(x)a(y)dy. The as i the proof of (a) we obtai Vhk(a)(x) A Ck /2 γ j =m j Ck /2 γ j =m j Q y x 0 a(y) dy 1 2β Q (1 2β)/ 1 2β, thus T A (a)(x) C k 2/2 2 Q (1 2β)/ x x k= β γ j =m j C Q (1 2β)/ x x 0 1 2β, γ j =m j ad J 2 C 2 i(2β 1) C, γ j =m j i=1 which together with the estimate for J 1 yields the desired result. This completes the proof of Theorem 1. Proof of Theorem 2.

9 164 LIU LANZHE (i) Without loss of geerality, we may assume l = 2. Let f H 1 (R ) ad f(x) = λ ja j (x) be the atomic decompositio for f as i Lemma 1. We write p k 3 T A (f) 2 p K kαp λ j T A (a j )χ k L 2 2 p 2 kαp λ j T A (a j )χ k L 2 = L 1 L 2. j=k 2 For L 2, by the (L 1, L 2 ) boudedess of T A, we have L 2 C 2 kαp λ j a j L 1 j=k 2 C λ j p ( j2 2(k j)αp ), 0 < p 1 C λ j p ( j2 2(k j)αp/2 )( j2 2(k j)αp /2 C λ j p C f p H K 1. p ) p/p K, p > 1 For L 1 as i the proof of Theorem 1 (a) for x C k ad j k 3 we obtai ( T A Bj 1/ (a j )(x) C x 1 2β B j β/ x β B j 2β/ ) x a j (y) dy R ( 2 j(1(1 1/ 1) α) ) C x 1 2β 2j(β(1 1/1) α) x β, thus T A (a j )χ k L 2 C2 kα ( 2 (j k)(1(1 1/1) α) 2 (j k)(β(1 1/1) α)) ; To be simply, deote W(j, k) = 2 (j k)(1(1 1/1) α) 2 (j k)(β(1 1/1) α) ad recall that α < (1 1/ 1 ) β, the p k 3 L 1 C λ j W(j, k) C λ j p k=j3 W(j, k)p, 0 < p 1 C λ j p k=j3 W(j, k)p/2 k=j3 W(j, k)p /2 C These yield the desired result. λ j p C f p H K 1. p/p, p > 1

10 THE CONTINUITY OF MULTILINEAR SINGULAR INTEGRAL OPERATORS 165 K (1 1/1)2β,p (ii) Without loss of geerality, we assume l = 2. Let f H 1 (R ) ad f(x) = λ ja j (x) be the atomic decompositio for f as i Lemma 1. Write p T k 3 A (f) p K 2 kp((1 1/1)2β) λ (1 1/ 1 )2β,p j T A (a j )χ k L kp((1 1/1)2β) j=k 2 λ j T A (a j )χ k L 2 For M 2, by the (L 1, L 2 ) boudedess of T A, we get M 2 C 2 kp((1 1/1)2β) C C ( j2 λ j p j=k 2 p λ j a j L 1 2 (k j)p((1 1/1)2β) ) λ j p C f p H K. (1 1/ 1 )2β,p 1 For M 1, as i the proof of Theorem 1 (b), we obtai T A (a)(x) C B j 1/ x 1 2β a j (y) dy R for x C k ad j k 3. Thus M 1 C C C 2 kp((1 1/1)2β) λ j p k=j3 C 2j(1 2β) x 1 2β. k 3 2 p(1 2β)(j k) λ j p C f p H K. (1 1/ 1 )2β,p 1 λ j p 2j(1 2β) 2 k(1 2β) These yield the desired result ad fiish the proof of Theorem 2. = M 1 M 2. p p 2 kp/2 Refereces 1 A. P. Calderó ad A. Zygmud, O sigular itegrals with variable kerels, Appl. Aal., 7(1978), W. G. Che, Besov estimates for a class of multiliear sigular itegrals, Acta Math. Siica, 16(2000), F. Chiareza, M. Frasca ad P. Logo, Iterior W 2,p -estimates for odivergece elliptic euatios with discotiuous coefficiets, Ricerche Mat., 40(1991), J. Cohe, A sharp estimate for a multiliear sigular itegral o R, Idiaa Uiv. Math. J., 30(1981),

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