LANZHE LIU. Changsha University of Science and Technology Changsha , China

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1 Известия НАН Армении. Математика, том 45, н. 3, 200, стр SHARP AND WEIGHTED BOUNDEDNESS FOR MULTILINEAR OPERATORS OF PSEUDO-DIFFERENTIAL OPERATORS ON MORREY SPACE LANZHE LIU Chagsha Uiversity of Sciece ad Techology Chagsha 40077, Chia Abstract. The paper proves boudedess of the multiliear operators related to some pseudo-differetial operators o the geeralized weighted Morrey spaces usig the sharp estimate of the multiliear operators. MSC2000 umber: 42B20, 42B25 Keywords: multiliear operator; pseudo-differetial operator; Morrey space; BMO; A -weight.. Prelimiaries ad statemets of mai results Throughout this paper, φ deotes a positive, icreasig fuctio o R + ad it is assumed that there exists a costat D > 0 such that φ2t) Dφt) for t 0. Let w be a weight fuctio o, that is a oegative locally itegrable fuctio, ad f be a locally itegrable fuctio o. Defie that, for p <, /p f L p,φ w) = fy) wy)dy) p, φd) sup x, d>0 Bx,d) where Bx, d) = {y : x y < d}. The geeralized weighted Morrey spaces are defied by L p,φ, w) = {f L loc ) : f L p,φ w) < }. If φd) = d δ, δ > 0, the L p,φ, w) = L p,δ, w), which is the classical Morrey space see [6], [7]). As the developmet of the Calderó-Zygmud sigular itegral operators, their commutators ad multiliear operators have bee well studied see [3] - [6], [9]). I [4], Hu ad Yag proved a versio sharp estimate for the multiliear sigular itegral operators. I [8], [9], C. Pérez, G. Pradolii ad R. Trujillo-Gozalez obtaied a sharp weighted estimates for the sigular itegral operators ad their commutators. The boudedess of the pseudo-differetial operators was studied by may authors 57

2 58 LANZHE LIU see [], [7], [2], [5], [20] - [2]). I [20], the boudedess of the commutators associated to the pseudo-differetial operators are obtaied. The mai purpose of this paper is to study the multiliear pseudo-differetial operators as follows. We say a symbol σx, ξ) belogs to the class Sρ,δ m, if µ ν x µ σx, ξ) ξν µ,ν + ξ ) m ρ ν +δ µ, x, ξ, where µ, ν are multi-idices ad µ = µ µ. A pseudo-differetial operator with symbol σx, ξ) Sρ,δ m is defied by T f)x) = e 2πix ξ σx, ξ) ˆfξ)dξ, where f is a Schwartz fuctio ad ˆf deotes the Fourier trasform of f. It is kow see []) that there exists a kerel Kx, y) such that T f)x) = Kx, x y)fy)dy, where Kx, y) = e 2πix y) ξ σx, ξ)dξ. I [2] the boudedess of the pseudo-differetial operators with symbol σ S β θ,δ β < θ/2, 0 δ < θ) are obtaied. I [5] the boudedess of the pseudodifferetial operators with symbol of orders 0 ad is proved. I [] some sharp estimate of the pseudo-differetial operators with symbol σ S θ/2 θ,δ 0 < θ <, 0 δ < θ) are obtaied. I [20] the boudedess of the pseudo-differetial operators ad their commutators with symbol σ S θ/2 θ,δ 0 < θ <, 0 δ < θ) are obtaied. Our study are motivated by these papers. Assumig that T is a pseudo-differetial operator with symbol σx, ξ) S m ρ,δ that m j j =,..., l) are some positive itegers such that m m l = m ad b j are fuctios give o, we set R mj+b j ; x, y) = b j x) α! Dα b j y)x y) α, j m. α m j The multiliear operator associated to T is defied by l j= T b f)x) = R m j+b j ; x, y) x y m Kx, x y)fy) dy. Note that for m = 0, T b is just the multiliear commutator geerated by T ad b see [8], [9]), while for m > 0, T b is otrivial 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 see [3] - [6]). Besides, the Morrey space ca be cosidered as a extesio of the Lebesgue space, sice the Morrey

3 SHARP AND WEIGHTED BOUNDEDNESS FOR MULTILINEAR OPERATORS space L p,λ becomes Lebesgue space L p for λ = 0). Hece, it is atural ad importat to study the boudedess of multiliear sigular itegral operators o the Morrey spaces L p,λ with λ > 0 see [2], [0], []). The purpose of this paper is twofold. First, we establish a sharp iequality for multiliear pseudo-differetial operator T b with symbol σ S θ/2 θ,δ 0 < θ <, 0 δ < θ). The, we use this sharp iequality to prove the boudedess for the multiliear operators o the geeralized weighted Morrey spaces. Now, we itroduce some otatios. Deote by a cube i with sides parallel to the coordiate axes. For ay locally itegrable fuctio f, its sharp fuctio is defied by f # x) = sup fy) f dy, x where, ad i what follows, f = fx) dx. It is well-kow see [3]) that f # x) sup if fy) c dy. x c C We say that f belogs to BMO ) if f # L ) ad deote f BMO = f # L. Let M be the Hardy-Littlewood maximal operator Mf)x) = sup fy) dy, 0 < p <. x We set M p f) = Mf p )) /p ad deote by A the class of Muckehoupt weights see [3]): A = {0 < w L loc ) : Mw)x) wx), a.e.}. The followig theorem is the mai result of this paper. Theorem. Let T be a pseudo-differetial operator with a symbol σ S θ/2 θ,δ 0 < θ <, 0 δ < θ) ad let 2 < p <, 0 < D < 2, w A, ad D α b j BMO ) for all α with α = m j ad j =,..., l. The l T b f) L p,φ w) f L p,φ w). j= α j =m j 2. Proof of the Theorem To prove the theorem, we eed the followig lemmas.

4 60 LANZHE LIU Lemma. [3]) Let b be a fuctio o ad D α b L q ) for all α with α = m ad some q >. The, for ay x y, R m b; x, y) x y m α =m x, y) /q D α bz) dz) q, x,y) where is the cube cetered at x with the side legth 5 x y. Lemma 2. []) Let T be a pseudo-differetial operator with a symbol σ S θ/2 θ,δ 0 < θ <, 0 δ < θ). The, for every p, < p <, T f) L p p f L p, f L p ). Lemma 3. []) Let σ S θ/2 θ,δ 0 < θ <, 0 δ < θ) ad K be the kerel of a pseudo-differetial operator T with a symbol σ. The, for x 0 x d < ad k, ) /2 Kx, x y) Kx 0, x 0 y) 2 dy 2 k d) θ y x 0 <2 k+ d) θ x 0 x θ)m /2) 2 k d) m θ), provided m is a iteger such that /2 < m < /2 + / θ). Lemma 4. []) Let σ Sρ,δ 0 0 < ρ < ) ad Kx, w) = e 2πiw ξ σx, ξ) dξ. The, for w /4 ad ay iteger N, Kx, w) N w 2N. Lemma 5. Let < p <, 0 < D < 2, w A. The, for ay fuctio f L p,φ, w) a) Mf) L p,φ w) f # L p,φ w); b) M q f) L p,φ w) f L p,φ w) for < q < p. Proof. a) Let f L p,φ, w). The Mwχ B ) A for ay ball B = Bx, d) see [8]). Therefore, usig the iequality see [3]) Mf)y) p uy)dy f # y) p uy)dy, which is true for ay u A, we get Mf)y) p wy)dy Mf)y) p Mwχ B )y)dy B R f # y) p Mwχ B )y)dy

5 [ [ SHARP AND WEIGHTED BOUNDEDNESS FOR MULTILINEAR OPERATORS... 6 B B f # y) p Mw)y)dy + f # y) p Mw)y)dy + [ B [ f # y) p wy)dy + B f # y) p wy)dy + 2 k+ B\2 k B 2 k+ B\2 k B ) ] f # y) sup p wz)dz dy y B ) ] f # y) p 2 k+ wz)dz dy B B ] f # y) p Mw)y) dy 2k+) ] f # y) p wy) dy 2k 2 k+ B 2 k+ B f # p L p,φ w) 2 k φ2 k+ d) f # p L p,φ w) 2 D) k φd) f # p L p,φ w) φd). Thus, Mf) L p,φ w) f # L p,φ w). The iequality b) is proved by a argumet similar to that i the proof of a), ad we omit the details. Key Lemma. Let T be a pseudo-differetial operator with a symbol σ S θ/2 θ,δ 0 < θ <, 0 δ < θ) ad let D α b j BMO ) for all α with α = m j j =,, l). The there exists a costat C > 0 such that for ay f C0 ), 2 < r < ad x, T b f)) # x) l M r f) x). α j =m j j= Proof. It suffices to prove that for ay f C 0 ) ad some costat C 0, the followig iequality holds: T b f)x) C 0 dx l M r f) x). α j =m j j= Without loss of geerality, we ca assume l = 2. Fix a cube = x 0, d) ad x. We cosider two cases. Case. d. Let be the cocetric with cube with side legth d θ, = 5 ad bj x) = b j x) α! Dα b j ) x α. α =m j

6 62 LANZHE LIU The R mj b j ; x, y) = R mj b j ; x, y) ad D α b j = D α b j D α b j ) for α = m j. Cosequetly, for f = fχ + fχ R \ = f + f 2 we obtai 2 j= T b f)x) = R m j+ b j ; x, y) R x y m Kx, x y)fy)dy = 2 j= = R m j b j ; x, y) R x y m Kx, x y)f y)dy R m2 b 2 ; x, y)x y) α D α b y) α! α R x y m Kx, x y)f y)dy =m α 2! α 2 =m 2 + α!α 2! α =m, α 2 =m 2 Therefore, + C + C + C + α =m, α 2 =m 2 R m b ; x, y)x y) α2 D α2 b2 y) x y m Kx, x y)f y)dy+ x y) α+α2 D α b y)d α2 b2 y) x y m Kx, x y)f y)dy+ 2 j= R m j+ b j ; x, y) x y m Kx, x y)f 2 y) dy. T b f)x) T bf 2 )x 0 ) dx 2 j= R m j b j ; x, y) R x y m Kx, x y)f y)dy dx+ R m2 b 2 ; x, y)x y) α α R x y m D α b y)kx, x y)f y)dy =m dx+ R m b ; x, y)x y) α2 α R x y m D α 2 b2 y)kx, x y)f y)dy 2 =m 2 dx+ x y) α+α2 D α b y)d α2 b2 y) R x y m Kx, x y)f y)dy dx+ + T bf 2 )x) T bf 2 )x 0 ) dx =: I + I 2 + I 3 + I 4 + I 5. To estimate the quatities I, I 2, I 3, I 4 ad I 5, first, for x ad y, we use Lemma ad obtai R m b j ; x, y) x y m α j =m. Now, we suppose σx, ξ) = σx, ξ) ξ θ/2 ξ θ/2 = qx, ξ) ξ θ/2. The qx, ξ) S θ,δ 0. Therefore, deotig the pseudo-differetial operator with symbol qx, ξ) by S

7 SHARP AND WEIGHTED BOUNDEDNESS FOR MULTILINEAR OPERATORS ad applyig the Hardy-Littlewood-Soboleve fractioal itegratio theorem ad the L 2 -boudedess of S see []), we obtai that for /p = /2 θ/2, I T f )x) dx j= α j =m j ) /p T f )x) p dx j= α j =m j /p Sf )x) j= α R 2 dx j =m j ) /2 /p f x) j= α R 2 dx j =m j /2 ) /2 /p fx) 2 dx j= α j =m j ) /r fx) r dx j= α j =m j M r f) x). j= α j =m j ) /2 For I 2, we use Lemma ad Hölder s iequality ad obtai that for /r +/r = /2, I 2 α 2 =m 2 D α2 b 2 BMO α 2 =m 2 D α2 b 2 BMO α 2 =m 2 D α2 b 2 BMO α =m α =m /p α =m /p D α2 b 2 /2 BMO /p α 2 =m 2 fx) r dx) /r α =m j= T D α b /p f )x) dx) p SD α b ) /2 f )x) 2 dx ) /2 D α b x)f x) 2 dx ) /r D α b x) D α b j ) r dx M r f) x). α =m j

8 64 LANZHE LIU Similarly, for I 3, we get I 3 M r f) x). α =m j j= For I 4, takig r, r 2 > such that /r + /r + /r 2 = /2, we obtai /p I 4 T D α b D α 2 b2 f )x) dx) p α =m, α 2 =m 2 α =m, α 2 =m 2 /p α =m, α 2 =m 2 /p /2 /p α =m, α 2 =m 2 j= fx) r dx) /r To estimate I 5, observe that T bf 2 )x) T bf 2 )x 0 ) = + ) /2 SD α b D α 2 b2 f )x) 2 dx D α b x)d α2 b ) /2 2 x)f x) 2 dx j= /rj D α j bj x) r j dx) M r f) x). α =m j ) Rm2 b 2 ; x, y) R m b ; x, y) R m b ; x 0, y) Kx, x y) x y m Kx ) 0, x 0 y) x 0 y m R mj b j ; x, y)f 2 y)dy j= R x 0 y m Kx 0, x 0 y)f 2 y)dy ) Rm b ; x 0, y) + R m2 b 2 ; x, y) R m2 b 2 ; x 0, y) R x 0 y m Kx 0, x 0 y)f 2 y)dy α! α R =m D α b y)f 2 y)dy α 2! α 2 =m 2 D α2 b 2 y)f 2 y)dy + α!α 2! α =m, α 2 =m 2 D α b y)d α 2 b2 y)f 2 y)dy [ R m2 b 2 ; x, y)x y) α x y m Kx, x y) R m 2 b 2 ; x 0, y)x 0 y) α x 0 y m Kx 0, x 0 y) [ R m b ; x, y)x y) α2 x y m Kx, x y) R m b ; x 0, y)x 0 y) α2 x 0 y m Kx 0, x 0 y) =: I ) 5 + I 2) 5 + I 3) 5 + I 4) 5 + I 5) 5 + I 6) 5. [ x y) α +α 2 x y m Kx, x y) x ] 0 y) α+α2 x 0 y m Kx 0, x 0 y) ] ]

9 SHARP AND WEIGHTED BOUNDEDNESS FOR MULTILINEAR OPERATORS By Lemma ad the followig iequality see [3]) b b 2 log 2 / ) b BMO, which is true whe 2, imply R m b; x, y) x y m D α b BMO + D α b) x,y) D α b) ) α =m k x y m α =m D α b BMO, for x ad y x 0, 2 k+ d) θ )\x 0, 2 k d) θ ). Therefore, otig that x y x 0 y for x ad y \, we obtai I ) 5 k 2 2 Kx, x y) Kx 0, x 0 y) k d) θ y x 0 <2 k+ d) θ x y m + k 2 R mj b j ; x, y) fy) dy j= Kx 0, x 0 y) 2 k d) θ y x 0 <2 k+ d) θ R mj b j ; x, y) fy) dy j= x y m x 0 y m C k 2 fy) 2 dy j= α =m j y x 0 <2 k+ d) θ Kx, x y) Kx 0, x 0 y) 2 dy 2 k d) θ y x 0 <2 k+ d) θ +C k 2 fy) 2 dy j= α =m j y x 0 <2 k+ d) θ ) /2 x 0 x 2 2 k d) θ y x 0 <2 k+ d) x θ 0 y 2 Kx 0, x 0 y) 2 dy, ) /2 ) /2 ) /2 for the secod term above, arguig as i the proof of Lemma 2. of [], we obtai ) /2 x 0 x 2 2 k d) θ y x 0 <2 k+ d) x θ 0 y 2 Kx 0, x 0 y) 2 dy x 0 x θ)m /2), 2 k d) m θ)

10 66 LANZHE LIU thus, by Lemma 3 ad for /2 < m, we get I ) 5 j= α =m j ) /2 k 2 d θ)m /2) fy) 2 dy 2 k d) m θ) y x 0 <2 k+ d) θ k 2 2 k θ)/2 m) j= α =m j k= ) /r x 0, 2 k d) θ fy) r dy ) x 0,2 k d) θ ) k 2 2 k θ)/2 m) M r f) x) j= α =m j k= M r f) x). j= α =m j To estimate I 2) 5, by the equality see [3]): R m b; x, y) R m b; x 0, y) = ad Lemma, we get thus R m b; x, y) R m b; x 0, y) β <m β <m α =m β! R m β D β b; x, x0 )x y) β x x 0 m β x y β D α b BMO, I 2) 5 j= α =m j k x x 0 2 k d) θ y x 0 <2 k+ d) x θ 0 y Kx 0, x 0 y) fy) dy k2 k θ)/2 m) j= α =m j k= /r x 0, 2 k d) θ fy) dy) r ) x 0,2 k d) θ ) M r f) x). j= α =m j

11 SHARP AND WEIGHTED BOUNDEDNESS FOR MULTILINEAR OPERATORS Similarly, we obtai I 3) 5 M r f) x). j= α =m j For I 4) 5, as for I) 5 ad I 2) 5, we get that for /r + /r = /2 I 4) 5 x y) α α =m R x y m x 0 y) α x 0 y m R m 2 b 2 ; x, y) Kx, x y) D α b y) f 2 y) dy+ +C R m2 b 2 ; x, y) R m2 b 2 ; x 0, y) x 0 y) α α R x =m 0 y m Kx, x y) Dα b y) f 2 y) dy+ +C Kx, x y) Kx 0, x 0 y) x 0 y) α α =m R x 0 y m R m 2 b 2 ; x 0, y) D α b y) f 2 y) dy D α b 2 BMO k2 k θ)/2 m) α =m 2 α =m k= /2 x 0, 2 k d) θ fy)d α b y) dy) 2 ) x 0,2 k d) θ ) /r D α b 2 BMO k2 k θ)/2 m) x 0, 2 k d) θ fy) dy) r ) α =m 2 k= x 0,2 k d) θ ) ) /r x 0, 2 k d) θ D α b y) D α b ) ) r dy ) α =m x 0,2 k d) θ ) k 2 2 k θ)/2 m) M r f) x) j= α =m j k= M r f) x). j= α =m j Similarly, I 5) 5 M r f) x). j= α =m j For I 6) 5, as for I) +C 5, we get, that for /r + /r + /r 2 = /2, x y) α+α2 x y m x 0 y) α+α2 x 0 y m I 6) 5 α =m, α 2 =m 2 Kx, x y) D α b y) D α 2 b2 y) f 2 y) dy+ Kx, x y) Kx 0, x 0 y) x 0 y) α+α2 R x 0 y m α =m, α 2 =m 2

12 68 LANZHE LIU Thus x 0, 2 k d) θ ) α =m, α 2 =m 2 k= j= D α b y) D α 2 b2 y) f 2 y) dy 2 k θ)/2 m) α =m, α 2 =m 2 k= x 0,2 k d) θ ) 2 k θ)/2 m) x 0, 2 k d) θ ) fy)d α b y)d α 2 b2 y) 2 dy) /2 x 0,2 k d) θ ) /rj x 0, 2 k d) θ D αj b j y) D αj b j ) ) dy) rj x 0,2 k d) θ ) M r f) x). j= α =m j I 5 M r f) x). α =m j j= Case 2. d >. I this case, let = 5 ad bj x) = b j x) α =m j α! Dα b j ) x α. The R mj b j ; x, y) = R mj b j ; x, y) ad D α bj = D α b j D α b j ), α = m j. fy) r dy) /r Hece, for f = fχ + fχ R \ = f + f 2, we have T b f)x) dx 2 j= R m j b j ; x, y) R x y m Kx, x y)f y)dy dx + C R m2 b 2 ; x, y)x y) α α =m R x y m D α b y)kx, x y)f y)dy dx + C R m b ; x, y)x y) α2 α R x y m D α 2 b2 y)kx, x y)f y)dy 2 =m 2 dx + C x y) α+α2 D α b y)d α2 b2 y) R x y m Kx, x y)f y)dy dx α =m, α 2 =m 2

13 SHARP AND WEIGHTED BOUNDEDNESS FOR MULTILINEAR OPERATORS T bf 2 )x) dx =: J + J 2 + J 3 + J 4 + J 5. As for I, I 2, I 3 ad I 4, by the L p < p < )-boudedess of T see Lemma 2), we get J ) /r T f )x) r dx j= α j =m j ) /r /r f x) j= α j =m R r dx j ) /r fx) r dx j= α j =m j M r f) x); j= α j =m j J 2 D α2 b 2 BMO T D α b ) /p f )x) p dx α 2 =m 2 α =m D α2 b 2 BMO /r D α b ) /p x)f x) p dx α 2 =m 2 α =m ) /r D α2 b 2 BMO D α b x) D α b ) r dx α 2 =m 2 α =m /r fx) dx) r M r f) x); j= α =m j J 3 M r f) x); j= α =m j J 4 T D α b D α2 b ) /r 2 f )x) r dx α =m, α 2 =m 2 /r α =m, α 2 =m 2 /r α =m, α 2 =m 2 j= j= D α b x)d α 2 b2 x)f x) r dx ) /rj D α j bj x) r j dx M r f) x). α =m j ) /r ) /r fx) r dx

14 70 LANZHE LIU To estimate J 5, observe that 2 T bf j= 2 )x) = R m j b j ; x, y) R x y m Kx, x y)f 2 y)dy R m2 b 2 ; x, y)x y) α α! α R x y m Kx, x y)d α b y)f 2 y)dy =m R m b ; x, y)x y) α2 α 2! α R x y m Kx, x y)d α 2 b2 y)f 2 y)dy 2 =m 2 x y) α+α2 + α!α 2! α =m, α 2 =m R x y m Kx, x y)dα b y)d α 2 b2 y)f 2 y)dy. 2 Hece, we use Lemma 4 ad similar to I 5, we get T bf 2 )x) k 2 j= α =m j +C α =m 2 D α b 2 BMO +C +C α =m D α b BMO α =m, α 2 =m 2 α =m α 2 =m 2 2k+ \2 k d α =m j j= +C α =m 2 D α b 2 BMO d +C α =m 2 k 2 k α =m D α b BMO d +C α 2 =m 2 2 k 2 k α =m, α 2 =m 2 d 2k+ \2 k x y 2 fy) dy k x y 2 D α b y) fy) dy 2k+ \2 k k x y 2 D α 2 b2 y) fy) dy 2k+ \2 k k= x y 2 D α b y) D α2 b 2 y) fy) dy k= ) /r k 2 2 k 2 k fy) r dy 2 k k2 k 2 k 2 k D α b y) D α b ) ) r dy k= ) /r fy) r dy ) /r ) /r k2 k 2 k fy) r dy 2 k D α2 b 2 y) D α b 2 ) ) r dy k= ) /r ) /r 2 k 2 k fy) r dy 2 k

15 SHARP AND WEIGHTED BOUNDEDNESS FOR MULTILINEAR OPERATORS... 7 ) /rj 2 j= k D α j b j y) D α j b j ) r j dy 2 k k 2 2 k M r f) x) j= α =m j k= M r f) x). j= α =m j Thus, J 5 M r f) x). j= α =m j This completes the proof of Key Lemma. Proof of Theorem. Takig 2 < r < p i Key Lemma, by Lemma 5, we obtai T b f) L p,φ w) MT b f)) L p,φ w) T b f)) # L p,φ w) M r f) L p,φ w) j= α =m j f L p,φ w). j= α =m j This fiishes the proof. Ackowledgemet. The author would like to express gratitude to the referee for his commets ad suggestios. Список литературы [] S. Chaillo ad A.Torchisky, Sharp fuctio ad weighted L p estimates for a class of pseudodifferetial operators, Ark. Math., 24, ). [2] F. Chiareza ad M. Frasca, Morrey spaces ad Hardy-Littlewood maximal fuctio, Red. Mat., 7, ). [3] J. Cohe, A sharp estimate for a multiliear sigular itegral o, Idiaa Uiv. Math. J., 30, ). [4] J. Cohe ad J. Gosseli, O multiliear sigular itegral operators o, Studia Math., 72, ). [5] J. Cohe, J. Gosseli, A BMO estimate for multiliear sigular itegral operators, Illiois J. Math., 30, ). [6] R. Coifma, Y. Meyer, Wavelets, Calderó-Zygmud ad Multiliear Operators, Cambridge Studies i Advaced Math. 48, Cambridge Uiversity Press, Cambridge 997). [7] R. Coifma, Y. Meyer, Au delá des opérateurs pseudo-différetiels, Astérisque, ). [8] R. Coifma, R. Rochberg, Aother characterizatio of BMO, Proc. Amer. Math. Soc., 79, ). [9] Y. Dig, S. Z. Lu, Weighted boudedess for a class rough multiliear operators, Acta Math. Siica, 7, ). [0] G. Di FaZio, M. A. Ragusa, Commutators ad Morrey spaces, Boll. U. Mat. Ital., 7)5-A, ).

16 72 LANZHE LIU [] G. Di Fazio, M. A. Ragusa, Iterior estimates i Morrey spaces for strog solutios to odivergece form equatios with discotiuous coefficiets, J. Fuc. Aal., 2, ). [2] C. Fefferma, L p bouds for pseudo-differetial operators, Israel J. Math., 4, ). [3] J. Garcia-Cuerva, J. L. Rubio de Fracia, Weighted orm iequalities ad related topics, North-Hollad Math.6, Amsterdam 985). [4] G. Hu, D. C. Yag, A variat sharp estimate for multiliear sigular itegral operators, Studia Math., 4, ). [5] N. Miller, Weighted Sobolev spaces ad pseudo-differetial operators with smooth symbols, Tras. Amer. Math. Soc., 269, ). [6] J. Peetre, O covolutio operators leavig L p,λ -spaces ivariat, A. Mat. Pura. Appl., 72, ). [7] J. Peetre, O the theory of L p,λ -spaces, J. Fuc. Aal., 4, ). [8] C. Pérez, G. Pradolii, Sharp weighted edpoit estimates for commutators of sigular itegral operators, Michiga Math. J., 49, ). [9] C. Pérez, R. Trujillo-Gozalez, Sharp weighted estimates for multiliear commutators, J. Lodo Math. Soc., 65, ). [20] M. Saidai, A. Lahmar-Beberou, S. Gala, Pseudo-differetial operators ad commutators i multiplier spaces, Africa Diaspora J. of Math., 6, ). [2] M. Sugimoto, N. Tomita, Boudedess properties of pseudo-differetial ad Calderó-Zygmud operators o modulatio spaces, J. of Fourier Aal. Appl., 4, ). [22] M. E. Taylor, Pseudo-differetial Operators ad Noliear PDE, Birkhauser, Bosto 99). Поступила 7 апреля 2009