446 EIICHI NAKAI where The f p;f = 8 >< sup sup f(r) f(r) L ;f (R )= jb(a; r)j jf(x)j p dx! =p ; 0 <p<; ess sup jf(x)j; p = : x2 ( f0g; if 0<r< f(r) =

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1 Scietiae Mathematicae Vol. 3, No. 3(2000), A CHARACTERIATION OF POINTWISE MULTIPLIERS ON THE MORREY SPACES EIICHI NAKAI Received April 28, 2000; revised October 4, 2000 Dedicated to the memory of Professor Hiroai Taehaa Abstract. Let L pi ;f i (i =; 2; 3) be Morrey spaces. A fuctio g is called a poitwise multiplier from L p ;f to L p2,ifthepoitwise product ;f 2 fg belogs to L p2 ;f 2 for each f 2 L p ;f. We deote by PWM(L p ;f ;L p2 ;f 2 ) the set of all poitwise multipliers from L p ;f to L p2. A sufciet coditio o ;f 2 p i ad f i (i = ; 2; 3) for PWM(L p ;f ;L p2 ;f 2 )=L p3 ;f 3 was give i [9]. I this paper, we give a ecessary coditio. I coectio with these coditios, we alsogive sufciet coditios for PWM(L p ;f ;L p2 )=f0g. ;f 2. Itroductio The theory of the Morrey spaces has bee developed by may authors, Peetre [0], [], Adams [], Chiareza ad Frasca [3], Mizuhara [4], Arai ad Mizuhara [2], etc. We ivestigate poitwise multipliers o the Morrey spaces. Let E ad F be spaces of real- or complex-valued fuctios deed o a set X. A fuctio g deed o X is called a poitwise multiplier from E to F, if the poitwise product fg belogs to F for each f 2 E. We deote by PWM(E;F) the set of all poitwise multipliers from E to F. L p -spaces (0 <p») o a measure space X are complete quasi-ormed liear spaces. If» p», the they are Baach spaces. If X is ff-ite, ad if =p +=p 3 ==p 2, the it is ow that (.) PWM(L p (X);L p2 (X)) = L p3 (X) ad g Op = g L p 3; where g Op is the operator orm of g 2 PWM(L p (X);L p2 (X)), i.e. g Op = iff >0: fg L p 2» f L p for all f 2 L p (X)g: O some assumptios the equalities i (.) were geeralized to the Morrey spaces L p;f (X) i [9], where 0 < p», f : X (0; +)! (0; +) ad X is a space of homogeeous type. I this paper, we cosider the ecessity of the assumptios i the case of f :(0; +)! (0; +) adx = R. May authors studied i this case. For a 2 R ad r>0, let B(a; r) be the ball fx 2 R : jx aj <rg. For a measurable set E ρ R,we deote by jej the Lebesgue measure of E. For 0 <p»ad f :(0; +)! (0; +), let L p;f (R )=ff 2 L p loc (R ):f p;f < +g ; 2000 Mathematics Subject Classicatio. Primary 42B5, 46E30; Secodary 46E35, 42B35. Key words ad phrases. multiplier, poitwise multiplier, Morrey space. This research was partially supported by the Grat-i-Aid for Scietic Research (C), No , 999, the Miistry of Educatio, Sciece, Sports ad Culture, Japa.

2 446 EIICHI NAKAI where The f p;f = 8 >< sup sup f(r) f(r) L ;f (R )= jb(a; r)j jf(x)j p dx! =p ; 0 <p<; ess sup jf(x)j; p = : x2 ( f0g; if 0<r< f(r) =0; L (R ); if 0<r< f(r) > 0: If if 0<r< f(r) > 0, the f ;f = f L =(if 0<r< f(r)). For f(r) =r ( )=p (0 <p<, 0< <), let L p; (R )=L p;f (R ). The L p; (R ) is the classical Morrey spaces itroduced i [5]. If f(r) =r =p, the L p;f (R )=L p (R ). If f(r), the L p;f (R )=L (R ). The fuctio space L p;f (R ) is a complete quasi-ormed liear space. If» p», the it is a Baach space. A fuctio :(0; +)! (0; +) is said to be almost icreasig (almost decreasig) if there exists a costat C>0suchthat (r)» C (s) ( (r) C (s)) for r» s. If if t»r f(t) = 0 for some r>0, the L p;f (R )=f0g. Let if t»r f(t) > 0forevery r>0 ad (r) = if t»r f(t). The is decreasig ad L p;f (R ) = L p; (R ) with equivalet orms. If if t r f(t)t =p = 0 for some r>0, the L p;f (R )=f0g. Let if t r f(t)t =p > 0 for every r>0ad(r) =r =p if t r f(t)t =p. The (r)r =p is icreasig ad L p;f (R )= L p; (R )withequivalet orms. To cosider poitwise multipliers from L p;f (R )tol p2;f 2 (R ), we may assume that f i (i =; 2) are almost decreasig ad f i (r)r =pi (i =; 2) are almost icreasig. If f is almost decreasig ad f(r)r =p is almost icreasig, the f satises (.2) A» f(s) f(r)» A for 2» s r» 2; where A>0 is idepedet of r;s>0. I the case of f :(0; +)! (0; +) ad X = R, Theorems 2., 2.2 ad 2.3 i [9] ca be stated as follows: Theorem.. Let 0 <p 2» p»ad =p +=p 3 ==p 2. Suppose that f i (i =; 2) are almost decreasig. Let f 3 = f 2 =f. If f (r)r =p ad f 2 p 2=p =f (f 2 =f whe p = p 2 =, =f whe p 2 <p = ) are almost icreasig, ad if lim if r!0 f 3(r) > 0, the (.3) PWM(L p;f (R );L p2;f 2 (R )) = L p3;f 3 (R ): Moreover, the operator orm of g 2 PWM(L p;f (R );L p2;f 2 (R )) is comparable to g p3;f 3. If f 2 p 2=p =f =, the the operator orm is equal to g p3;f 3. Remar.. If f (r)r =p p ad f 2=p 2 =f are almost icreasig, so are f 2 (r)r =p2 ad f 3 (r)r =p3. I this paper, we show that the almost icreasigess of f 2 p 2=p =f is a ecessary ad sufciet coditio for (.3) whe p 2» p, ad that if p 2 >p or if lim if r!0 f 3 (r) =0 the PWM(L p;f (R );L p2;f 2 (R )) = f0g: It turs out that Theorem. holds without the coditio lim if r!0 f 3 (r) > 0 i the case of f :(0; +)! (0; +).

3 A CHARACTERIATION OF POINTWISE MULTIPLIERS ON THE MORREY SPACES 447 We state mai results i the ext sectio. Sectio 3 is for the prelimiaries. I sectio 4wegive proofs of the results. The letters C will always deote a positive costat, ot ecessarily the same oe. For fuctios ;» :(0; +)! (0; +), we deote (r) ο»(r) if there exists a costat C>0 such that C (r)»»(r)» C (r); r>0: 2. Mai results Theorem 2.. Let 0 <p 2» p». Suppose that f i (i =; 2) are almost decreasig ad that f i (r)r =pi (i =; 2) are almost icreasig. The PWM(L p;f (R );L p2;f 2 (R )) = L p3;f 3 (R ); if ad oly if f 2 p 2=p =f (f 2 =f whe p = p 2 =, =f whe p 2 <p = ) is almost icreasig, where =p 3 ==p 2 =p ad f 3 = f 2 =f. I this case, the operator orm of g 2 PWM(L p;f (R );L p2;f 2 (R )) is comparable to g p3;f 3. Remar 2.. Whe p = p 2 =, f i (r) =f i (r)r =pi (i =; 2) are almost decreasig ad almost icreasig, i.e. f ο f 2 ο adf 2 =f is almost icreasig. Whe p 2 <p =, f (r) = f (r)r =p is almost decreasig ad almost icreasig, i.e. f ο ad =f is almost icreasig. Remar 2.2. I geeral, from Lemma 3. it follows that PWM(L p;f (R );L p2;f 2 (R )) ff L p3;f 3 (R ): Theorem 2.2. Suppose that f is almost decreasig, that f (r)! + as r! 0 ad that f (r)r =p is almost icreasig. If 0 <p <p 2 <, the PWM(L p;f (R );L p2;f 2 (R )) = f0g: Lemma 2.3. Let 0 <p ;p 2 <. Suppose that f is almost decreasig ad that f (r)r =p is almost icreasig. If lim if r!0 f 2 (r)=f (r) =0,the Remar 2.3. If PWM(L p;f (R );L p2;f 2 (R )) = f0g: PWM(L p;f (R );L p2;f 2 (R )) = f0g; the there is a fuctio f 2 L p;f (R )suchthatf =2 L p2;f 2 (R ), sice a costat fuctio is ot a poitwise multiplier. For the classical Morrey spaces L p; (R ), we have the followig.

4 448 EIICHI NAKAI Corollary 2.4. Let 0 <p i < ad 0 < i <(i =; 2). The PWM(L p; (R );L p2; 2 (R )) 8 >< = f0g; p <p 2 ; = f0g; p = p 2 ad < 2 ; = L (R ); p = p 2 ad = 2 ; % f0g; p = p 2 ad > 2 ; = f0g; p >p 2 ad +( )p 2 =p < 2 ; = L (R ); p >p 2 ad 2 = +( )p 2 =p ; = L p3; 3 (R ); p >p 2 ad» 2 <+( )p 2 =p ; % L p3; 3 (R ); p >p 2 ad p 2 =p < 2 < ; % L p3 (R ); p >p 2 ad 2 = p 2 =p ; % f0g; p >p 2 ad 2 < p 2 =p ; where p 3 = p p 2 =(p p 2 ) ad 3 =(p 2 p 2 )=(p p 2 ). We state lemmas to prove the theorems. 3. Prelimiaries Lemma 3. ([9]). Let 0 <p 2» p», =p +=p 3 ==p 2 ad f f 3 = f 2. The fg p2;f 2»f p;f g p3;f 3 : Lemma 3.2 ([9]). Let 0 <p<. Suppose that f satises (.2) ad f(r)r =p is almost icreasig. If supp f ρ B(a; r) ad if sup B(b;s)ρB(a;3r) f(s) jb(b; s)j B(b;s) jf(x)j p dx! =p» M; the f 2 L p;f (R ) ad f p;f» CM; where C>0 idepedet of f, B(a; r) ad M. Lemma 3.3. Let 0 <p<, f be almostdecreasig ad f(r)r =p be almost icreasig. If f is the characteristic fuctio of the ball of radius r>0, the f p;f ο f(r) : Proof. For all balls B(b; s) ρ B(a; 3r), we have! =p jf(x)j p dx» f(s) jb(b; s)j f(s)» C f(r) : B(b;s) By Lemma 3.2 we obtai the desired result.

5 A CHARACTERIATION OF POINTWISE MULTIPLIERS ON THE MORREY SPACES 449 Lemma 3.4. Let 0 <p<, f be almost decreasig ad f(r)r =p be almost icreasig. Let fb(a j ;r j )g j=0 beballs ad ff jg j= be fuctios such that 8 >< B(a j ;r j ) ρ B(a 0 ;r 0 ); supp f j ρ B(a j ;r j =3); f j L p» c f(r 0 )r j =p ; f j p;f» c 2 ; for j =; 2; ; B(a i ;r i ) B(a j ;r j )=; for i; j =; 2; ; i 6= j: The f = P j= f j is i L p;f (R ) ad f p;f» C(c + c 2 ). Proof. For ay ballb(a; r) ρ B(a 0 ; 3r 0 ), let ad J = fj 2 N : B(a; r) B(a j ;r j =3) 6= ;;r j =3» rg; J 2 = fj 2 N : B(a; r) B(a j ;r j =3) 6= ;;r j =3 >rg; 0 I i f(r) jb(a; r)j X j2j i f j (x) If j 2 J, the B(a j ;r j =3) ρ B(a; 3r). It follows that Hece I» f(r) [ j2j B(a j ;r j =3) ρ B(a; 3r) jb(a; r)j X ad jf j (x)j j2j B(a p dx j;r j=3) A =p p X» c f(r 0 ) f(r) dx A =p ; i =; 2: j2j (r j =3)» (3r) : X r j jb(a; r)j j2j If j 2 J 2, the B(a; r) ρ B(a j ;r j ), i.e. J 2 has oly oe elemet. Hece I 2» f(r) jb(a; r)j By Lemma 3.2 we have f p;f» C(c + c 2 ). jf j (x)j p dx! =p»f j p;f» c 2 : A =p» Cc : First, we state a proof of Lemma Proofs Proof of Lemma 2.3. There are positive umbers fr j g j= such that r j! 0 ad f 2 (r j )=f (r j )! 0 as j! +: If g 2 PWM(L p;f (R );L p2;f 2 (R )), the the closed graph theorem shows that g is a bouded operator. For ay a 2 R, let f be the characteristic fuctio of the ball B(a; r j ).

6 450 EIICHI NAKAI By Lemma 3.3 we have jb(a; r j )j B(a;r j) Therefore g(a) =0a.e. a 2 R. jg(x)j p2 dx! =p2» f 2 (r j )fg p2;f 2» f 2 (r j )f p;f g Op» C f 2(r j ) f (r j ) g Op! 0 as j! +: Let I(a; r) be the cube fx 2 R : jx i a i j»r=2;i=; 2; ;g whose edges have legth r ad are parallel to the coordiate axes. Proof of Theorem 2.. If lim if r!0 f 3 (r) > 0, the the sufciecy follows from Theorem.. If lim if r!0 f 3 (r) = 0, the L p3;f3 (R ) = f0g. From Lemma 2.3 it follows that PWM(L p;f (R );L p2;f2 (R )) = L p3;f3 (R ): If f 2 p 2=p =f is ot almost icreasig, the, for all 2 N, there are positive realumbers r ad s such that s <r ad f 2 (s ) p2=p =f (s ) 4 =p2 f 2 (r ) p2=p =f (r ): Let» r f 2 (r ) p2= m = s f 2 (s ) p2= +; where [ff] deotes the iteger part of the positive real umber ff. By almost icreasigess of f 2 (r)r =p2 ad by almost decreasigess of f 2,wehave m ο r f 2 (r ) p2= (4.) s f 2 (s ) p2= ; r (4.2) > 6c 0 s for some c 0 > 0: m Let O 2 R be the origi. We divide the cube I(O;r )ito m sub-cubes I(b ;j ;r =m ): j =; 2; ;m,i.e. m[ I(O;r )= I(b ;j ;r =m ); j= I(b ;i ;r =m ) f I(b ;j ;r =m ) f = ; for i 6= j; where I(b; r) f is the iterior of I(b; r). Let ( f 3 (c 0 s ); x 2 B(b ;j ;c 0 s ); g ;j (x) = 0; x=2 B(b ;j ;c 0 s ); We show We ote that mx g = g ;j ; g = j= X =! =p2 2 (g ) p2 : g 2 PWM(L p;f (R );L p2;f 2 (R )) L p3;f 3 (R ): supp g ;j ρ B(b ;j ;r =(6m )); B(b ;j ;r =(2m )) ρ I(b ;j ;r =m ):

7 A CHARACTERIATION OF POINTWISE MULTIPLIERS ON THE MORREY SPACES 45 For all f 2 L p;f (R ), by Hölder's iequality, (.2) ad (4.) we have (4.3) fg ;j L p 2» 8 >< R R B(b ;j ;c 0s ) jf(x)jp dx =p R B(b ;j ;c 0s ) jg ;j(x)j p3 dx =p3 whe p >p 2 =p B(b ;j ;c 0s ) dx jf(x)jp f3 (c 0 s ) whe p = p 2 ( f (c 0 s )jb(b ;j ;c 0 s )j =p f p;f» f 3 (c 0 s )jb(b ;j ;c 0 s )j =p3 whe p >p 2 f (c 0 s )jb(b ;j ;c 0 s )j =p f p;f f 3 (c 0 s ) whe p = p 2 By Lemma 3. ad Lemma 3.3 we have = f 2 (c 0 s )jb(b ;j ;c 0 s )j =p2 f p;f ο f 2 (s )s =p 2 f p;f ο f 2 (r )(r =m ) =p2 f p;f : fg ;j p2;f 2»f p;f g ;j p3;f 3» Cf p;f : By Lemma 3.4 we have fg is i L p2;f2 ad fg p2;f2» C 0 f p;f. This implies jf(x)g (x)j p2 dx»jb(a; r)j (f(r)c 0 f p;f ) p2 for each ball B(a; r): From Beppo-Levi's theorem it follows that X = coverges a.e. x 2 B(a; r) ad 2 jf(x)g (x)j p2 = jf(x)j p2 X = 2 g (x) p2 jf(x)g(x)j p2 dx»jb(a; r)j (f(r)c 0 f p;f ) p2 for each ball B(a; r): Hece we have ad fg p2;f 2» C 0 f p;f ; g 2 PWM(L p;f (R );L p2;f 2 (R )): O the other had, sice supp g ρ B(O; p r =2), f 3 ( p r =2) jb(o; p jg(x)j p3 dx r =2)j ο f 3 ( p r =2) B(O; p r =2) jb(o; p r =2)j B(O; p r =2)! =p3 2 g (x) =p2 p3 dx! =p3 f 3(s )(m s ) =p3 2 =p2 f 3 (r )(r ) = m s f 2 (s ) p2= =p3 f 2 (s ) p2=p =f (s ) =p3 r f 2 (r ) p2= 2 =p2 f 2 (r ) p2=p =f (r ) 2 =p2 for all 2 N; whe p >p 2 ;

8 452 EIICHI NAKAI ad f 3 ( p r =2) ess supφ g(x) :x 2 B(O; p Ψ r =2) This shows g =2 L p3;f 3 (R ). f 3 ( p r =2) ess sup ρ 2 =p2 g (x) :x 2 B(O; p r =2) ο f 3(s ) 2 =p2 f 3 (r ) = f 2(s )=f (s ) 2 =p2 f 2 (r )=f (r ) 2 =p2 for all 2 N; whe p = p 2 : Remar 4.. If fr g = i the above proof is bouded, the the support of g is compact. Proof of Theorem 2.2. Let fs g = be positive real umbers such that s > s + ( = ; 2; )ads! 0as! +. Let h l =» f (s ) p= s +; m = f (s ) p=i +: The, by the almost icreasigess of f (r)r =p ad the almost decreasigess of f,we have Hece (4.4) (4.5) l ο f (s ) p= s ; m ο f (s ) p= : f (=(l m )) p =m ο f (s ) p s l ο f (=(l m )) p2 =m ο f (s ) p2 s l ο f (s ) p2 p! + as! +: For ay xed a 0 2 R, we divide the cube I(a 0 ; ) ito l sub-cubes I(b ;j ; =l ): j = ; 2; ;l,i.e. I(a 0 ; ) = l[ I(b ;j ; =l ); j= I(b ;j ; =l ) f I(b ;j 0; =l ) f = ; for j 6= j 0 : We divide the cube I(O; =l )ito m sub-cubes I(e ;i ; =(l m )): i =; 2; ;m, i.e. The 8 >< m[ I(O; =l )= I(e ;i ; =(l m )); i= I(e ;i ; =(l m )) f I(e ;i 0; =(l m )) f = ; for i 6= i 0 : m[ I(b ;j ; =l )= I(b ;j + e ;i ; =(l m )); i= I(b ;j + e ;i ; =(l m )) f I(b ;j + e ;i 0; =(l m )) f = ; for i 6= i 0 ; ff j =; 2; ;l :

9 Let First we show A CHARACTERIATION OF POINTWISE MULTIPLIERS ON THE MORREY SPACES 453 f ;j;i (x) = ( f (=(l m )); x 2 I(b ;j + e ;i ; =(l m )); 0; x =2 I(b ;j + e ;i ; =(l m )); Xl f ;i = f ;j;i : j= (4.6) f ;i p;f» C; for i =; 2; ;m ; ad for all 2 N: We ote that B(b ;j + e ;i ; =(2l )) ρ I(b ;j + e ;i ; =l ) ρ I(a 0 ; 2) ρ B(a 0 ; p ); B(b ;j + e ;i ; =(2l )) B(b ;j 0 + e ;i ; =(2l )) = ; for j 6= j 0 : Sice f (s )! + as!,wemay assume m 3 p. Hece supp f ;j;i = I(b ;j + e ;i ; =(l m )) By (4.4) we have ρ B(b ;j + e ;i ; p =(2l m )) ρ B(b ;j + e ;i ; =(6l )): f ;j;i L p» f (=(l m ))(=(l m )) =p ο (=l ) =p : By Lemma 3.3 we have f ;j;i p;f» C: Hece, by Lemma 3.4 we have (4.6). If g 2 PWM(L p;f (R );L p2;f2 (R )), the the closed graph theorem shows that g is a bouded operator. Hece f (=(l m )) f 2 ( p ) This is = jb(a 0 ; p )j f 2 ( p ) Sice supp f ;i = 8>< we have by (4.5) I(a 0 ; ) = supp f ;i jg(x)j p2 dx jb(a 0 ; p )j! =p2 B(a p jf ;i (x)g(x)j p2 dx 0; ) supp f ;i jg(x)j p2 dx» Cf (=(l m )) p2 g Op p 2 : l[ j= [ m I(b ;j + e ;i ; =(l m )); supp f ;i ; i= (supp f ;i ) f (supp f ;i 0) f = ; for i 6= i 0 ;! =p2»f ;i g p2;f 2» Cg Op : for all 2 N; jg(x)j p2 dx» Cm f (=(l m )) p2 p g 2 Op! 0 as! +: I(a 0;) Therefore g = 0 a.e.

10 454 EIICHI NAKAI Refereces [] D. R. Adams, A ote o Riesz potetials, DueMath. J. 42 (975), [2] H. Arai ad T. Mizuhara, Morrey spaces o spaces of homogeeous type ad estimates for Λ b ad the Cauchy-Szego projectio, Math. Nachr. 85 (997), [3] F. Chiareza ad M. Frasca, Morrey spaces ad Hardy-Littlewood maximal fuctio, Red. Mat. 7 (987), [4] T. Mizuhara, Boudedess of some classical operators o geeralized Morrey spaces, Harmoic Aalysis (Sedai, 990), 83 89, ICM-90 Satell. Cof. Proc., Spriger, Toyo, 99. [5] C. B. Morrey, O the solutios of quasi-liear elliptic partial differetial equatios, Tras. Amer. Math. Soc. 43 (938), [6] E. Naai, Poitwise multipliers for fuctios of weighted bouded mea oscillatio, Studia Math. 05 (993), [7], Hardy-Littlewood maximal operator, sigular itegral operators ad the Riesz potetials o geeralized Morrey spaces, Math. Nachr. 66 (994), [8], Poitwise multipliers o the Loretz spaces, Mem. Osaa Kyoiu Uiv. III Natur. Sci. Appl. Sci. 45 (996), 7. [9], Poitwise multipliers o the Morrey spaces, Mem. Osaa Kyoiu Uiv. III Natur. Sci. Appl. Sci. 46 (997),. [0] J. Peetre, O covolutio operators leavig L p; spaces ivariat, A. Mat. Pura Appl. 72 (966), [] J. Peetre, O the theory of L p; spaces, J.Fuct. Aal. 4 (969), Departmet of Mathematics, Osaa Kyoiu Uiversity, Kashiwara, Osaa , Japa address: eaai@cc.osaa-yoiu.ac.jp