Analysis in Theory and Applications Volume 15 Issue 4 1999 [Doi 10.1007%2Fbf02848671] Liu Zongguang -- Weak Type Inequalities for Fractional Maximal Operator on Weighted Orlicz Spaces

8/16/2019 Analysis in Theory and Applications Volume 15 Issue 4 1999 [Doi 10.1007%2Fbf02848671] Liu Zongguang -- Weak…

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App rox . Th eory ~ i t s Appl . 15 :

4 1 9 9 9 6 4 - - 7 0

W E K T Y P E IN E Q U L IT I E S F O R

F R C T I O N L M X I M L O P E R T O R O N

W E I G H T E D O R L I C Z S P C E S

I n M e m o r y o f P r o f e s s o r M T C h e n g

L i u Z o n g g u a n g

(Huaihua T ,~cher s Co l lege , Ch ina)

A b s t r a c t

Th e fra ct ional ma xim al operator on homogeneous space ( X , d , l~ is de f in ed as

r ~ - ~ .p l a (B (x , r ) ) , _ , . , . , . , [ f ( y ) l d lz , 0 • a < •.

In this pa pe r, the suf f icien t and necessary conditions fo r the r to be o f xeeah ty pe an d extra vz~ealt yp e

be g i~ . . .

1 I n t r o d u c t i o n

A h o m o g e n e o u s s p a c e ( X , d , p ) is a s e t X t o g e t h e r w i t h a q u a s i - m e t r i c d a n d d o u b l i n g

m e a s u r e ~ W e r ec al l t h a t a q u a s i - m et r i c i s a m a p p i n g d : X X X - ~ [ O , o o ) w h i c h s at i sf i es h e

q u a s i - t r i a n g l e i n e q u a l i t y , d ( x , y ) ~ K [ d ( x , z ) + d ( z , y ) ] f o r a l l x , y , z E X , w h e r e K > ~ I i s a

c o n s t a n t i n d e p e n d e n t o n x,y,z. F o r a ll h a l l B = B ( x , r ) = { y E X . d ( x , y ) < r} , w e d e n o t e b y

c B t h e h a l l B(x,cr). W e a l s o r ec al l h a t a d o u b l i n g m e a s u r e / ~ o n X i s a n o n n e g a t i v e m e a s u r e

on Bore l subse t s o f X sa t i s f ing the inequa l i ty [ 2 B I , ~ A [ B ] , for all balls B C X , w h i c h I B I ,

d e n o t e s t h e / ~ - m e a s u r e o f t h e b a l l B .

L e t X . = X X R + = { ( x , t ) : x E X , t > f O } , B = ] ~ x , r ) = { ( y , t ) E X + : y E B ( x , r ) , O


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Liu Zon ggua ng t W eak T yp e Inequa li ti es for Frac tiona l M ax im al Opera tor 65

I n t h e sp ec ia l c a s e, . . . K ' ~ f ( x , O ) = M f ( x ) , w h e r e M is th e w e ll -k n o w n ce n tr al H a r d y - L i t t l e -

wood maximal opera to r .

A nonneg a t ive Bore l measu re ~ on X+ i s said to be a quas i -dobu ling measure i f fo r each

ball B = B ( x , r ) o n X , t h e r e e x i st s a c o n s ta n t c > O s u c h t h a t - p ( a B ) ~ c a - p ( t ~ ) , where c i s a

cons tan t indepen den t on ba l l B and

a .

Let w (x ) be a we igh t ( tha t i s , a nonega t ive and local ly in teg rab le func t ion) on X and

be a quas i-doub l ing mea sure on X + , the w e igh ted max imal opera to r i s de f ined as

~ ~ . , ~ f ( x , t ) = sup 1 I I f ( Y ) I w ( y ) d p .

,;~, -~ (B (x ,r ) ) B(~,,)

Th e a im of th i s paper i s to ob ta in th e su f f i c ien t and necessa ry cond i tions fo r . .~" to be o f

the we ak type and ex t ra -w eak typ e on the we igh ted Orl icz space . Now we reca ll the con cep t o f

weighted Orl icz space.

L e t ~ ( t ) b e a Y o u n g ' s f u n c t i o n , t h e w e l l -k n o w n Y o u n g ' s i n eq u a li ty is th e f o ll o w i n g

st / O ,

w h e r e ~ ( t ) is c alle d Y o u n g ' s c o m p l e me n t a r y f u n c ti o n o f ~ ( t ) , i . e . ,

~ ;( t ) = s u p ( t r - ~ ( r ) ) .

r>0

L e t w ( x ) b e a w e i g h t o n X , w e ca ll L t . , ( X ) = { f ~ II f II t . . < o o / a s a w e igh ted Or li cz space ,

W e reca ll the w e igh ted Ho lde r inequa l i ty on the we igh ted Or l icz space as

I I x f ( X ) g ( x ) w ( : c ) d l< ~ I I f I I t . . I I g I I

~ ~

In th e fo l lowing sec t ions , we need the fo l lowing lemmas .

L e m m a 1 [ tj Let E be a d-bounded subset o f X and assume that for each x E E there exis ts

r ( x ) > 0 , then there exists a sequence {Bi} o f d i sjoint ba ll s such tha t E C U 5K~B j , where B i=

i - - I

B ( x j , r ( x i ) ) a n d 5 K ZB j = B ( x j , 5 K t r ( x t ) ).

B y L e m m a 1 we can get the fol lowing lemma easi ly .

L e m m a 2 Let w (x ) be a we ight on X and ~ be a quasi-doubl ing measure on X + , then

~a((~.,, is o f we ak ty p e (1 ,w l l ,~ ) , i .e . , there exis ts a constant c> O such that

~ ( { ( : r , t ) E X + : c C ~ Z . . . f ( x , t ) > 2}) ~< 7 x I f ( y ) I w ( y ) d #

for each ,~>0.

Let ~ ( t ) be a Young ' s func t ion , we in t roduce fo l lowing func t ions on R+ 9

r S t ( t ) -~ ~ ( t ) f or t > 0 a n d / ~ ( 0 ) = S t ( 0 ) = 0 .

/ ~ t ) = t t


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66

App rox . Theory & i ts Ap p l .

1 5 :

4 , 1999

t>~o

Le mm a 3 [z]

Let ~ ( t ) be a Youn g' s funct ion, then the fo llmz , ng es t imates hold fo r every

~ 5 ( R , ( t ) ) ~ ~ ( t ) ~ ~ 5 ( 2 R ~ ( t) ) ;

9 ( S ~ ( t ) ) ~ ( t ) ~

~ ( 2 S , ( t ) ) ~

S , ( R ~ ( t ) ) ~ t ~

2 S , ( 2 R ~ ( t ) ) ;

9 ( )~S.( t ) ) ~ c) ,~ ( t ) ,

for 0 ~ 2 ~ 1.

a)

( b )

( c )

( d )

2 W eak type inequal i ty

T h e o r e m

1 L e t X , d , p ) b e a

homogeneou~ spa ce, -~ be a qu asi-doubling mea sure

o n X + ,

w ( x ) be a w eigh t on X and ~ ( t ) , ~ ( t ) be Young 's functions, then the fo l low ing condit ion are

equivalent :

( 1 )

There exs it s c~ 'O such tha t f o r each f ( x ) am 1

2 > 0 :

~ ( { ( x , t ) E X + ; ~ / ' f ( x , t ) > •} Ir ~ C f x ~ ( C i f ( x )

( 2 ) ( ~ , w ) E A ~ . ~ ,,

i .e . , there exists r such that

P r o o f .

( 1 ) = ~ ( 2 ) .

F o r e a c h b a l l

B ~ X

a n d f ( x ) , b y t h e d o u b l i n g c o n d i ti o n o f / ~ , t h e r e e x is t s a c o n s t a n t co

> 0 d e p e n d { n g o n l y o n t h e c o n s t a n ts

K , A

a n d a , s u c h t h a t

1 9

/~ ~ { ( x , t ) E

X + ; c g * f ( x , t ) >

Co/a(~3)~_

I [ f y ) , d l a } .

B y 1 ) , w e

g e t

7 a(] 3)~ I c o t t ( 1 ) ~ - . I n ' f l d # ) ~ I x ~ ( c l f ( x ) l ) w ( x ) d l~ .

F or fixed r and k E N , le t f ( x ) = c - l S . ( ~ x ) l , E , = { x E B . w ( x ) ~ l / k } .

D e n o t e

f ~ (x ) = f ( x ) Z E , ( x ) ,

T h e l as t in e q u a l it y a n d ( b ) o f L e m m a 3 te ll u s t h a t

1

~ ( b ) ~ ( C o l 2 (B ) l- .f J ' d t z ) ~ c IE l l r( S , ( ~ ) ) w ( x ) d l l

1

c f f~d/~ ,

< /

w h i c h i n t u r n i m p l i e s t h a t


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Liu Zon ggua ng: W ea k Ty pe Inequalities for Fractional M axim al Operator 67

1 1

Since g ( B \ U E k )= 0 and the con s tant on the r igh t hand s ide of the last inequali ty does not de-

i 1

pend on k , le t k- -*oo, we get

sup sup,>o o/~(B)-' ' '~ 'r R. [ ,(B-~l-or176 ( 1 ) )

S , d • = c < o o ,

w h e r e r = (coc)-~.

(2 ) =~(1 )

F o r e ac h e > 0 a n d

B = B ( x , r ) C X ,

Young's inequali ty and A~,~,condi tion give tha t

.r ~r162 ~ I ~

( ~ ( . v ) ) ~ ( y , d u

< p ( B ) ' - ' I . ( [ f ( Y ) t )w ( y ) d t z + _ _ r )"

V

e t ~ = e ~ g ( B ) 1-

g o ( I f ( y ) l ) w ( y ) d ~ , then

~ , ( A , - . ; I f e y ) i d a ~ < , + - -1~ f ~ o ~ I , ~ S ' ( I f ( y ) l ) ~ ( Y ) d l ' )

T a k i n g t h e su p re m u m i n r f o r

r > t

w e g e t

J:.f x.t) 2r ) ~ ~ ( { ( x , t ) : R ~ ( 1 . . d : ; . . g r ( l f l ) ( x , t ) ) ~ r162 )

~ ( { ( x , t ) : ~ . ~ ' ; . . , ~ ( f i ) ( x , t ) > eL/~(t0E,) })

c f ~ t " ( [ f ( y ) l ) w ( y ) d t ~

C ~ o r


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68 Ap pro x . Theory ~ i ts Ap p l . 15 : 4 , 1999

c I O I f x ) I )w x)d /z , a )

( { ( x , t ) E X + . . ~ ( ' f ( z , t ) > ~ l} ) ~ ~ - - ~ x

su p su p ~ . , / Z ( / ~ ) i _ . I n , ( ~ - ~ ) d / z ) < o o . ( b

T h i s i s T h e o r e m 1 i n E 4 3 , w h e r e i t i s a s s u m e d t h a t ~ ( t ) s at is fie s A ~ c o n d i t i o n , i . e . , ( 2 t ) ~

cr for all t > 0 .

3 Extra-w eak type inequal i ty

Th e o r e m 2 L e t ( x , d , / a ) be a homog eneous space with a quasi-doubling measure ~ on

X + , w ( x ) a we igh t on X and ~ ( t ) a Young 's func t ion , then the fo l lowing condi tions are e -

quvialent :

( 1 ) There ex i st s c > O such that for each 2 > 0 a n d f ( x )

2 ) ( ~ , w ) E E '~ , i . e . , there exi st s r such that

/ z B ) ~ _ ~ x ) - - E , ~ o /z = c <

~

P r o o f . ( 1 ) ~ ( 2 )

F o r e a c h b a l l B C X a n d e a c h i n te g e r k > 0 , w e p u t E l -- - { x E B : w ( x ) > l / k } a nd g j ( x ) =

S e / Z ( B ) t_ - ~ w ( x ) ] Z ~, ( x ) , w h e r e r ., i s a p o s it iv e n u m b e r w h i c h w i l l b e c h o s e n l a t e r . S i m i l a r t o

t h e p ro o f o f ( 1 ) = } ( 2 ) o f T h e o r e m 1 i n w 2 , w e o b ta i n

[ d

O n t h e o t h e r h a n d , w e h a v e

= ~ ( - - ~ - T J C ~ / z ( B ) - - 7 : ~ ( : , , ) I d / z

~ ~ f - , x ) d . .

S

f / z ~ ) ~ _ . sg~ x ) d / z > t co c , by d) o f Lem ma 3 , i t f o llows t h a t

~ j / ~ 0 ~ ~ )

- , x ) d , .< / z ~ i / , x ) d / zc ~ o i f f r

/ Z ( ~ ( x ) ] w ( x )d /z .


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Liu Z onggu ang. W eak T yp e Inequali ti es for Frac tional M axima l O perator 69

9

f /x( /~) ~_ I g , (x )d/ x ~ coc , t hen

r l ( x ) d

T h u s

I ~

~(B)l-'w(x)r ) w ( x ) d t ~ c o c ~ Iz ~ ( I~(B)a_.w(x)~(]~) } w (x )d l~ + coc~ -~(~3).

Choosing r smal l enou gh such tha t c ,c r we then obta in

f ~ e , ~ C B ) I w x ) . I c oc o,

Since the r igh t hand s ide of the las t inqual i ty does not depend on ~ , thus w e have

s up ~ ~ { , ~ ( 2 ~ )

I w ( x ) , -

l J ( B - 7 ~ w ( x ) ) ~ ( - ~ a l ~ < . o o.

i . e . , ( ~ ,w ) E ~ .

( 2 ) = ~ ( 1 )

For each ba l l

B = B ( x , r ) ,

b y Y o u n g ' s in e qu a lit y a n d ~ , w e g e t

w ( y ) . j _ 1 ~ r ~w ( y ) .

/ j ( B ) a - ' w ( x )

~< ~ ( I f ( y ) I ) w ( y ld / ~ + to

s

T a k i n g t h e s u p r em u m ov r r ~ t , we conclude tha t

~ r ~ ~r - I- r

and

{ ( x , t ) E X + : ~ ' R ' ~ ~ 1b_.....Ac} C { ( x , t ) E X + . . . ~ ' ; . o ~ ( l f l ) ( x , t ) ~ 1}.

F ~

Since . . ~ r ; ., i s o f weak type (1 ,w ;1 ,~ ) , we ge t

~ ( { ( x , t ) E X + . . ~ ' . f ( x ,D > i } ) = ~ { ( x , t ) E X+ 88 r ) ( x , t ) > ~o I

~ < ~ ( < x , t ) E X + , .~ '~ , .~ ( 1 ~ + I1 1 /; x , t ) > 1 )1

~ I x ~ P ( l q ' c ~ I f ( x )

I ) w (x)dt~ .

Th i s com ple te s t he p roo f o f Theorem 2 .

R e m a r k 2 . I f ~ ( t ) = t ( l q - lo g + t ) l / P , f l > O , t hen the conc lus ion o f ou r Th eorem 2 i s t he

sam e a s t ha t o f Theorem 2 in [4 ] .

Acknotoledgnu, t . Th is w ork wa s f in ished dur ing my vis i ting a t Hangzhou U niv ers i ty ,

the au thor expresses h is gra t i tude to Pro f . W ang Si le i for h is enthusiast ic guidance .


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70 Approx. Theory ~ its Appl. 15. 4 1999

eferneces

[ ]

Col{man, R. R. and Weiss, G. , Extension of Hardy Space and Their use in Analysis, Bull.

Amer. Math. Sor , 83(1977), 569-645.

[2 ] Kokilashvill, V. and Krbec, M. , Weighted Inequalities in Lorentz and Orlicz Space, World Scien-

tific Pres, 1991.

[ 3] Pick, L. , Two-weight Weak Type Maximal Inequalities on Orlicz Space, Studia Math. , 100

(1991), 207-218.

[ 4] Lai Qinsheng, Weighted Weak Type Inequalities for the Fractional Maximal Operator on Space of

Homogeneous Space (in Chinese), Acta Math. Siniea, 32(1989), 448- 456.

Department of Mathematics

Huaihua Tea cher s College,

Hunan, 418008

PRC

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Analysis in Theory and Applications Volume 15 Issue 4 1999 [Doi 10.1007%2Fbf02848671] Liu Zongguang -- Weak Type Inequalities for Fractional Maximal Operator on Weighted Orlicz Spaces