M K -TYPE ESTIMATES FOR MULTILINEAR COMMUTATOR OF SINGULAR INTEGRAL OPERATOR WITH GENERAL KERNEL. Guo Sheng, Huang Chuangxia and Liu Lanzhe

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1 Acta Universitatis Apulensis ISSN: No. 33/203 pp M K -TYPE ESTIMATES FOR MULTILINEAR OMMUTATOR OF SINGULAR INTEGRAL OPERATOR WITH GENERAL KERNEL Guo Sheng, Huang huangxia and Liu Lanzhe Abstract. In this paper, we prove the M k -type inequality for ultilinear coutator related to generalized singular integral operator. By using the M k -type inequality, we obtain the weighted L p -nor inequality and the weighted estiate on the generalized Morrey spaces for the ultilinear coutator Matheatics Subject lassification: 42B20, 42B25.. Introduction and Preliinaries Let b BMO(R n ) and T be the alderón-zygand operator. coutator defined by [b, T ](f) = bt (f) T (bf). onsider the As the developent of singular integral operators(see [5][6]), their coutators have been well studied. In [4][3][4][5], the authors prove that the coutators generated by the singular integral operators and BM O functions are bounded on L p (R n ) for < p <. hanillo (see [2]) proves a siilar result when singular integral operators are replaced by the fractional integral operators. In this paper, we will study soe singular integral operators as following (see [][8]). Definition. Let T : S S be a linear operator such that T is bounded on L 2 (R n ) and there exists a locally integrable function K(x, y) on R n R n \ {(x, y) R n R n : x = y} such that T (f)(x) = K(x, y)f(y)dy R n for every bounded and copactly supported function f, where K satisfies: there is a sequence of positive constant nubers { k } such that for any k, ( K(x, y) K(x, z) + K(y, x) K(z, x) )dx, 2 y z < x y 3

2 and ( ) /q ( K(x, y) K(x, z) + K(y, x) K(z, x) ) q dy 2 k z y x y <2 k+ z y k (2 k z y ) n/q, where < q < 2 and /q + /q =. Suppose b j (j =,, ) are the fixed locally integrable functions on R n. The ultilinear coutator of the singular integral operator is defined by T b (f)(x) = R n (b j (x) b j (y))k(x, y)f(y)dy. Note that the classical alderón-zygund singular integral operator satisfies Definition with j = 2 jδ (see [5][6]). Also note that when =, T b is just the coutator what we entioned above. It is well known that ultilinear operator are of great interest in haronic analysis and have been widely studied by any authors (see [3-4]). In [5], Pérez and Trujillo-Gonzalez prove a sharp estiate for the ultilinear coutator. The purpose of this paper has two-fold, first, we establish a M k -type estiate for the ultilinear coutator related to the generalized singular integral operators, and second, we obtain the weighted L p -nor inequality and the weighted estiates on the generalized Morrey space for the ultilinear coutator by using the M k -type inequality. Definition 2. Let ϕ be a positive, increasing function on R + and there exists a constant D > 0 such that ϕ(2t) Dϕ(t) for t 0. Let w be a non-negative weight function on R n and f be a locally integrable function on R n. Set, for p <, f L p,ϕ (w) = sup x R n, d>0 ( /p f(y) w(y)dy) p, ϕ(d) (x,d) where (x, d) = {y R n : x y < d}. The generalized weighted Morrey space is defined by L p,ϕ (R n, w) = {f L loc(r n ) : f L p,ϕ (w) < }. If ϕ(d) = d δ, δ > 0, then L p,ϕ (R n, w) = L p,δ (R n, w), which is the classical Morrey spaces (see [][2]). If ϕ(d) =, then L p,ϕ (R n, w) = L p (w), which is the weighted Lebesgue spaces (see [5]). 32

3 As the Morrey space ay be considered as an extension of the Lebesgue space, it is natural and iportant to study the boundedness of the operator on the Morrey spaces (see [3][6][7][9][0]). Now, let us introduce soe notations. Throughout this paper, will denote a cube of R n with sides parallel to the axes. For any locally integrable function f, the sharp axial function of f is defined by (f) # (x) = sup f(y) f dy, x where, and in what follows, f = f(x)dx. It is well-known that (see [5][6]) (f) # (x) sup inf x c f(y) c dy. We say that f belongs to BMO(R n ) if f # belongs to L (R n ) and define f BMO = f # L. Let M(f)(x) = sup x For 0 < p <, we denote M p f(x) by f(y) dy. M p (f)(x) = [M( f p )(x)] /p. For k N, we denote by M k the operator M iterated k ties, i.e. M (f)(x) = M(f)(x) and M k (f)(x) = M(M k (f))(x) when k 2. Let Φ be a Young function and Φ be the copleentary associated to Φ, we denote that the Φ-average by, for a function f, f Φ, = inf { λ > 0 : Φ and the axial function associated to Φ by ( fy λ M Φ (f)(x) = sup f Φ,. x ) } d(y) The Young functions to be using in this paper are Φ(t) = t( + logt) r and Φ(t) = exp(t /r ), the corresponding average and axial functions denoted by L(logL) r,b, 33

4 M L(logL) r and expl /r,b, M expl/r. Following [3][4], we know the generalized Hölder s inequality: f(y)g(y) dy f Φ, g Φ,. And we can also obtain the following inequalities: f L(logL) /r, M L(logL) /r(f) M L(logL) (f) M + (f), b b expl r, b BMO, b 2 k+ b 2 k b BMO. for r, r j, j =, 2,, with /r = /r + /r 2 + /r, and b BMO(R n ). Given a positive integer and j, we denote by j the faily of all finite subsets σ = {σ(),, σ(j)} of {,, } of j different eleents and σ(i) < σ(j) when i < j. For σ j, set σc = {,, } \ σ. For b = (b,, b ) and σ = {σ(),, σ(j)} j, set b σ = (b σ(),, b σ(j) ), b σ = j b σ(i) and b σ BMO = j b σ(i) BMO. and A p = i= We denote the Muckenhoupt weights by A p for p < (see [5]), that is { w : sup A = {w : M(w)(x) w(x), a.e.} ( ) ( } p w(x)dx w(x) dx) /(p ) <, < p <. 2. Theores and Proofs Now we give soe theores as following. Theore.Let T be the singular integral operator as Definition, the sequence {k k } l, q s <, 0 < r <, k +, k N and b j BMO(R n ) for j =,,. Then there exists a constant > 0 such that for any f 0 (Rn ) and any x R n, (T b (f)) # r ( x) b BMO M k (f)( x) + M k (T (f))( x) + M bσ c s(f)( x). σ j i= 34

5 Theore 2.Let T be the singular integral operator as Definition, the sequence {k k } l, q p <, w A p and b j BMO(R n ) for j =,,. Then T b is bounded on L p (w). Theore 3.Let T be the singular integral operator as Definition, the sequence {k k } l, q p <, w A and b j BMO(R n ) for j =,,. Then, if 0 < D < 2 n, T b (f) L p,ϕ (w) b BMO f L p,ϕ (w). In order to better proof of the theore above, we need the following leas Lea.Let < r < and b j BMO(R n ) with j =,, k and k N. Then, we have k k b j (y) (b j ) dy b j BMO, /r k k b j (y) (b j ) r dy b j BMO. Siilarly, for σ k,when k and N, we have: (b(y) (b j ) ) σ dy b σ BMO and ( /r (b(y) (b j ) ) σ dy) r b σ BMO. In fact, we just need to choose p j > and q j >, where j k, such that /p + + /p k = and r/q + + r/q k =. After that, using the Hölder s inequality with exponent /p + +/p k = and r/q + +r/q k =. respectively, we ay get the results. Lea 2.([5, p.485])let 0 < p < q < and for any function f 0. We define that, for /r = /p /q f W L q = sup λ>0 λ {x R n : f(x) > λ} /q, N p,q (f) = sup fχ E L p/ χ E L r, E where the sup is taken for all easurable sets E with 0 < E <. Then f W L q N p,q (f) (q/(q p)) /p f W L q. 35

6 Lea 3.(see [5])Let 0 < p, η < and w r< A r. Then M η (f) L p (w) f # η (f) L p (w). Lea 4.Let < p <, q < p and w A. Then, if 0 < D < 2 n, M q (f) L p,ϕ (w) f L p,ϕ (w). Proof. Let f L p,ϕ (R n, w). Note that q < p and for any w A, M q (f)(y) p w(y)dy f(y) p w(y)dy. R n R n For a cube = (x, d) R n, we get M q (f)(y) p w(y)dy M q (f)(y) p M(wχ )(y)dy R n f(y) p M(wχ )(y)dy R [ n ] = f(y) p M(wχ )(y)dy + f(y) p M(wχ )(y)dy k=0 2 k+ \2 k [ ] f(y) p w(y)dy + f(y) p w(y) k=0 2 k+ \2 k 2 k+ dy [ f(y) p w(y)dy + f(y) p M(w)(y) ] dy k=0 2 k+ 2n(k+) [ f(y) p w(y)dy + f(y) p w(y) ] dy k=0 2 k+ 2nk f p L p,ϕ (w) 2 nk ϕ(2 k+ d) k=0 f p L p,ϕ (w) (2 n D) k ϕ(d) k=0 f p L p,ϕ (w) ϕ(d), thus M q (f) L p,ϕ (ω) f L p,ϕ (w). 36

7 Lea 5.Let < p <, 0 < D < 2 n, w A. Then, for f L p,ϕ (R n, w), M(f) L p,ϕ (w) f # L p,ϕ (w). Lea 6.Let T be the bounded linear operators on L q (R n, w) for any < q < and w A. Then, for < p <, w A and 0 < D < 2 n, T (f) L p,ϕ (w) f L p,ϕ (w). The proofs of two Leas are siilar to that of Lea 4, we oit the details. Proof of Theore. It suffices to prove for f 0 (Rn ) and soe constant 0, the following inequality holds: ( /r T b (f)(x) 0 dx) r b BMO M k (f)( x) + M k (T (f))( x) bσ. σ j c Fix a ball = (x 0, d) and x, we write f = fχ 2 and f 2 = fχ (2) c. Following [20], we will consider the cases = and >, and choose 0 = T (((b ) 2 b )f 2 )(x 0 ) and 0 = T ( (b j (b j ) 2 )f 2 )(x 0 ), respectively. We first consider the ase =. For 0 = T (((b ) 2 b )f 2 )(x 0 ), we write Then T b (f)(x) = (b (x) (b ) 2 )T (f)(x) T ((b (b ) 2 )f)(x). T b (f)(x) 0 = (b (x) (b ) 2 )T (f)(x) + T (((b ) 2 b )f)(x) T (((b ) 2 b )f 2 )(x 0 ) (b (x) (b ) 2 )T (f)(x) + T (((b ) 2 b )f )(x) + T (((b ) 2 b )f 2 )(x) T (((b ) 2 b )f 2 )(x 0 ) = A(x) + B(x) + (x). For A(x), we get ( ) /r A(x) r dx A(x) dx (b (x) (b ) 2 )T (f)(x) dx b (b ) 2 exp L,2 T (f) L(log L),2 b BMO M 2 (T (f))( x). 37

8 For B(x), by the weak type (,) of T and Lea 2, we obtain ( ) /r B(x) r dx B(x) dx = T (((b ) 2 b )f )(x) dx ( ) /p T ((b (b ) 2 )fχ 2 )(x) p dx 2 = T ((b p (b ) 2 )fχ 2 ) L p T ((b (b ) 2 )fχ 2 ) W L ((b (b ) 2 )fχ 2 L b (x) (b ) 2 f(x) dx 2 b (b ) 2 expl,2 f L(log L),2 b BMO M 2 (f)( x). For (x), recalling that s > q, taking < p <, < t < s with /p+/q+/t =, by the Hölder s inequality, we have, for x, = T ((b (b ) 2 )f 2 )(x) T ((b (b ) 2 )f 2 )(x 0 ) (b (y) (b ) 2 )f(y)(k(x, y) K(x 0, y))dy (2) c K(x, y) K(x 0, y) f(y) b (y) (b ) 2 dy k= 2 k x x 0 y x 0 <2 k+ x x 0 ( k= ( k= K(x, y) K(x 0, y) q dy 2 k x x 0 y x 0 <2 k+ x x 0 ) /p ( b (y) (b ) 2 p dy y x 0 <2 k+ x x 0 ( k 2 k+ /p+/t (2 k d) n/q k b BMO 2 k+ ) /q f(y) t dy y x 0 <2 k+ x x 0 2 k+ f(y) s dy ) /s ) /t 38

9 b BMO k k M s (f)( x) k= b BMO M s (f)( x), thus ( /r (x) dx) r b BMO M s (f)( x). Now, we consider the ase 2. we have, for b = (b,, b ), T b (f)(x) = (b j (x) b j (y))k(x, y)f(y)dy = = = = R n j=0 σ j R n [(b j (x) (b j ) 2 ) (b j (y) (b j ) 2 )]K(x, y)f(y)dy ( ) j (b(x) (b) 2 ) σ R n (b(y) (b) 2 ) σ ck(x, y)f(y)dy (b j (x) (b j ) 2 ) K(x, y)f(y)dy + ( ) R n R n (b j (y) (b j ) 2 )K(x, y)f(y)dy + ( ) j (b j (x) (b j ) 2 ) σ (b j (y) (b j ) 2 ) σ ck(x, y)f(y)dy σ R j n (b j (x) (b j ) 2 )T (f)(x) + ( ) T ( (b j (b j ) 2 )f)(x) + σ j ( ) j ((b j (x) (b j ) 2B ) σ T (b j (b j ) 2B ) σ c(f)(x) thus, recall that 0 = T ( (b j (b j ) 2B )f 2 )(x 0 ), T b (f)(x) T ( (b j (b j ) 2B )f 2 )(x 0 ) (b j (x) (b j ) 2 )T (f)(x) + T ( (b j (b j ) 2 )f )(x) + σ j ((b j (x) (b j ) 2 ) σ T (b j (b j ) 2 ) σ c(f)(x) 39

10 For I (x), we get, + T ( (b j (b j ) 2 )f 2 )(x) T ( (b j (b j ) 2 )f 2 )(x 0 ) = I (x) + I 2 (x) + I 3 (x) + I 4 (x). ( ) /r I (x) r dx I (x) dx (b j (x) (b j ) 2 ) T (f)(x) dx (b j (b j ) 2 ) exp L /r j,2 T (f) L(log L) r,2 b j BMO M + (T (f))( x) b BMO M k (T (f))( x). For I 2 (x), by the boundness of T on L p (R n ) and siilar to the proof of B(x), using Lea 2, we get ( ) /r I 2 (x) r dx I 2 (x) dx = T ( (b j (y) (b j ) 2 )f )(x) dx /p T ( (b j (b j ) 2 )f )(x) p dx = T ( (b j (b j ) 2 )f ) L p p T ( (b j (b j ) 2 )f ) W L ( (b j (b j ) 2 )f ) L (b j (x) (b j ) 2 ) f (x) dx B 40

11 (b j (b j ) 2 ) exp L /r j,2 f L(log L) r,2 b BMO M + (f)( x) b BMO M k (f)( x). For I 3 (x), by Lea 2, ( I 3 (x) r dµ(x) σ j σ j σ j σ j ) /r I 3 (x) dx (b j (x) (b j ) 2 ) σ T (b j (b j ) 2 ) σ c(f)(x) dx (b j (x) (b j ) 2 ) σ exp L /r j,2 T (b j (b j ) 2 ) σ c(f) L(log L) r,2 b σ BMO M + (T bσ c (f))( x) b BMO M k (T bσ c (f))( x). For I 4 (x), siilar to the proof of (x) in the ase =, for < p <, < t < s with /p + /q + /t =, we have T (( (b j (b j ) 2 )f 2 )(x) T (( (b j (b j ) 2 )f 2 )(x 0 ) (K(x, y) K(x 0, y)) f(y) (b j (y) (b j ) 2 ) dy k= 2 k x x 0 y x 0 <2 k+ x x 0 ( ) /q K(x, y) K(x 0, y) q dy k= 2 k x x 0 y x 0 <2 k+ x x 0 /p ( ) /t (b j (y) (b j ) 2 p dy f(y) t dy y x 0 <2 k+ x x 0 y x 0 <2 k+ x x 0 2 k+ /p+/t ( ) k (2 k= k /s b j BMO d) n/q 2 f(y) s dy 2 k+ b BMO k k M s (f)( x) k= 4

12 b BMO M s (f)( x), thus ( /r I 4 (x) dx) r b BMO M s (f)( x). This copletes the proof of the theore. Proof of Theore 2. hoose q < s < p in Theore, by the L p (w)-boundedness of M k and M s, we ay obtain the conclusion of Theore 2 by induction. Proof of Theore 3. We first consider the case =. hoose q < s < p in Theore, by Theore and Lea 4-6, we obtain T b (f) L p,ϕ (w) M(T b (f)) L p,ϕ (w) (T b ) # r (f) L p,ϕ (w) b BMO ( M k (f) L p,ϕ (w) + M k (T (f)) L p,ϕ (w) + M s (f) L p,ϕ (w) ) b BMO ( f L p,ϕ (w) + T (f)) L p,ϕ (w) + f L p,ϕ (w) ) b BMO ( f L p,ϕ (w) + f L p,ϕ (w) b BMO f L p,ϕ (w). When 2, we ay get the conclusion of Theore 3 by induction. This copletes the proof of Theore 3. ) References [] D.. hang, J. F. Li and J. Xiao, Weighted scale estiates for alderón- Zygund type operators, onteporary Matheatics, 446(2007), [2] S. hanillo, A note on coutators, Indiana Univ. Math. J., 3(982), 7-6. [3] F. hiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood axial function, Rend. Mat., 7(987), [4] R. oifan, R. Rochberg and G. Weiss, Factorization theores for Hardy spaces in several variables, Ann. of Math., 03(976), [5] J. Garcia-uerva and J. L. Rubio de Francia, Weighted Nor Inequalities and Related Topics, North-Holland Math., 6, Asterda, 985. [6] G. Di FaZio and M. A. Ragusa, outators and Morrey spaces, Boll. Un. Mat. Ital., 5-A(7)(99), [7] G. Di Fazio and M. A. Ragusa, Interior estiates in Morrey spaces for strong solutions to nondivergence for equations with discontinuous coefficients, J. Func. Anal., 2(993), [8] Y. Lin, Sharp axial function estiates for alderón-zygund type operators and coutators, Acta Math. Scientia, 3(A)(20),

13 [9] L. Z. Liu, Interior estiates in Morrey spaces for solutions of elliptic equations and weighted boundedness for coutators of singular integral operators, Acta Math. Scientia, 25(B)()(2005), [0] T. Mizuhara, Boundedness of soe classical operators on generalized Morrey spaces, in Haronic Analysis, Proceedings of a conference held in Sendai, Japan, 990, [] J. Peetre, On convolution operators leaving L p,λ -spaces invariant, Ann. Mat. Pura. Appl., 72(966), [2] J. Peetre, On the theory of L p,λ -spaces, J. Func. Anal., 4(969), [3]. Pérez, Endpoint estiate for coutators of singular integral operators, J. Func. Anal., 28(995), [4]. Pérez and G. Pradolini, Sharp weighted endpoint estiates for coutators of singular integral operators, Michigan Math. J., 49(200), [5]. Pérez and R. Trujillo-Gonzalez, Sharp Weighted estiates for ultilinear coutators, J. London Math. Soc., 65(2002), [6] E. M. Stein, Haronic analysis: real variable ethods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton NJ, 993. Guo Sheng, Huang huangxia and Liu Lanzhe ollege of Matheatics hangsha university of Science and Technology hangsha, 40077, P. R. of hina eail:lanzheliu@63.co 43