LIPSCHITZ ESTIMATES FOR MULTILINEAR COMMUTATOR OF MARCINKIEWICZ OPERATOR

Transcription

1 Reseh d ouiios i heis d hei Siees Vo. Issue Pges -46 ISSN Puished Oie o Deee 7 Joi Adei Pess h://oideiess.e IPSHITZ ESTIATES FOR UTIINEAR OUTATOR OF ARINKIEWIZ OPERATOR DAZHAO HEN Dee o Siee d Ioio Siee Shog Uivesi Hu Shog 4 P. R. hi e-i: hedzho7@si.o As I his e we wi sud he oiui o uiie ouo geeed iiewiz oeo d o Tiee-izoi se Hd se d Hez-Hd se whee he uio eogs o ishiz se. The iiewiz oeo Sei [4]. Sei oved h. Ioduio (iodued eow) ws is deied is ouded o ( R ) o whe is oiuous d sisies i ( S )( ) odiio. uhos hve ioved he esu. O he ohe hd Tohis d Wg (see [5]) osideed he oudedess o he ouo heis Sue ssiiio: 4B 4B5. Kewods d hses: uiie ouo Tiee-izoi se Hez-Hd se Hez se ishiz se. ouied iu zhe. Reeived Augus

2 DAZHAO HEN o. The oved he ouo [ ] is ouded o ( w) BO w A whe is oiuous d sisies i ( S )( ) odiio. Foowig he he i uose o his e is o disuss he oudedess o uiie ouo geeed iiewiz oeo d o Tiee-izoi se Hd se d Hez-Hd se whee i. o. Peiiies d Deiiios Thoughou his e ( ) wi deoe he Hd-iewood i uio o d wie ( ) ( ( )) o wi deoe ue o d R wih sides e o he es. e # d su d. Deoe he Hd ses ( R H ). I is we ow h H ( R )( ) he oi deoosiio heizio (see [] [8] []). Fo > hs d > e F e he hoogeeous Tiee-izoi se. The ishiz se i ( R ) is he se o uios suh h i su R. e (See []). Fo we hve F su d su i d.

3 IPSHITZ ESTIATES FOR UTIINEAR e (See []). Fo we hve su i d su d. e (See []). Fo d > e ( )( ) su d suose h d he ( ). e 4 (See [4]). I B( ) d ( B( ) ). The ( ) ( ). γ γ Deiiio. e. A uio o R is ed H -o i () Su B( ) o soe d o soe > (o o soe ); () B( ) ; () d. R

4 4 DAZHAO HEN e 5 (See [7] []). e. A disiuio o R is i H ( R ) i d o i e wie s i he disiuio sese whee eh is H - o d eh is os. oeove H i wih he iiu e ove deoosiios o s ove. Deiiio. e R B { R } E B \ B d χ χ o Z. whee E () The hoogeeous Hez se is deied K o ( R ) { ( R \{ }) : } K K χ. whee () The ohoogeeous Hez se is deied K o ( R ) { ( R ) : } K K ( R ) χ χb. Deiiio. e R. () The hoogeeous Hez e Hd se is deied HK ( R ) { S ( R ) : G( ) K ( R )}

5 IPSHITZ ESTIATES FOR UTIINEAR 5 d G( ). HK K () The ohoogeeous Hez e Hd se is deied ( R ) { S ( R ) : G( ) K ( R )} HK d G( ) HK K whee G ( ) (see []) is he gd i uio o. The Hez e Hd ses hve he oi deoosiio heizio. Deiiio 4. e R. A uio o R is ed e ( ) -o (o e ( ) -o o esi e) i () Su B( ) o soe > (o o soe ); () B( ) ; () d R η o η [ ( ) ]. e 6 (See [9]). e d ( ). A eee disiuio eogs o HK ( R )( o HK ( R )) i d o i hee eis e ( ) -os (o e ( ) -os o esi e) suoed o B B( ) d oss suh h ( o ) i he S ( R ) sese d HK ( o HK ) ~.

6 6 DAZHAO HEN Deiiio 5 (See []). e R. () A esue uio is sid o eog o hoogeeous we Hez se ( R W ) i K WK > su ( { E : > } ). () A esue uio is sid o eog o ihoogeeous we Hez se ( R WK ) i WK : > su ( { E > } { B : > } ). Deiiio 6. e γ d e hoogeeous o degee zeo o R suh h ( ) d( ). S Assue h i ( S ) γ h is hee eiss os > suh h o S γ ( ). The iiewiz uiie ouo is deied whee Se F d F ( ) ( ) ( ) d. F ( ) ( ) d

7 we so deie h IPSHITZ ESTIATES FOR UTIINEAR 7 d F whih is he iiewiz oeo (see []). e H e he se H h : h ( ) h d. The i is e h ( ) F ( ) d ( ) F ( ). Noe h whe is us he ode ouo. I is we ow h ouos e o ge iees i hoi sis d hve ee wide sudied uhos (see [5] [6] [9] [] []). Ou i uose is o esish he oudedess o he uiie ouo o Tiee-izoi se Hd se d Hez-Hd se. Give osiive iege d we se i d deoe i he i o iie suses { () () } o { } o diee eees is he eee ue o. Fo se { } \. Fo ( ) d { () ()} se ( () ()) () () d (). i i i. Theoes d Poos Theoe. e i( γ ) ( ) wih i ( R ) o d e he uiie ouo o iiewiz oeo s i Deiiio 6. The

8 8 DAZHAO HEN () () is ouded o ( R ) o F ( R ). is ouded o ( R ) o ( R ) o d >. Poo. () Fied ue ( ) d ~. Se ( ( ) ) whee ( ) d. Wie whee χ χ we hve R \ ( ) F ( ) ( ) ( ) R ( ( ) ) ( ( ) ) F ( ) ( ) F (( ( ) ) ( ( ) ) ) ( ) ( ) d ( ) ( ) d ( ( ) ) ( ( ) ) F ( ) ( ) F (( ( ) ) ( ( ) ) ) ( ) F (( ( ) ) ( ( ) ) ) ( ) ( ) F (( ) )

9 IPSHITZ ESTIATES FOR UTIINEAR 9 he (( ) ( ) ) (( ) ( ) ) F F ( ) ( ) F ( ) ( ) F (( ) ( ) ) F (( ) ( ) ) F (( ) ( ) ) F 4 I I I I hus ( ) ( ) ) d d I d I d I d I 4 IV. III II I Fo I usig e we hve d I su d i

10 DAZHAO HEN ( ). ~ i Fi. Fo II usig he Höde s ieui d he oudedess o o d e we ge ( ) ( ) d II ( ) d ( ) d ( ) d ( ) d i i d. ~ i Fo III Höde s ieui we hve (( ) ( ) ) d III

11 IPSHITZ ESTIATES FOR UTIINEAR ( ( ) ) R d ( ) d i d. ~ i Fo IV sie o we hve (( ) ( ) ) F I 4 (( ) ( ) ) F ( ( ) d ( ) ) d d ( ) > d d ( ) > d d ( ) d d. J J J

12 DAZHAO HEN Fo J sie ( ) d d J ( ) d ( ) d ( ) d \ d i i ~. ~ i Fo J sii o J we hve. ~ i J Fo J usig es 4 d γ ( ) d d J ( ) d d γ γ

13 IPSHITZ ESTIATES FOR UTIINEAR ( ) ( ) d γ γ \ ( ) d i γ γ ( ) i ~ γ. ~ i Thus. ~ IV i We u hese esies ogehe usig e d ig he sueu ove suh h we oi. i F This oee he oo o (). () B soe gue s i he oo o () we hve (( ) ( ) ) d d I d I d I d I 4 ( ( ) ) i hus ( ) ( ( ) ). # i

14 4 DAZHAO HEN B usig e d he oudedess o we hve ( ) ( ( )) # i ( ( ( )) ( ) ( ) ). This oee he oo o (). Theoe. e i( γ ) ( ) ( ) wih ( i R ) o. The is ouded o H ( R ) o ( R ). Poo. B e 5 i suies o show h hee eiss os > suh h o eve H -o Wie ( ). ( ) ( ) d ( ) > d I II. Fo I hoose d suh h. B R he oudedess o o o ( R ) (see Theoe ) he size odiio o d Höde s ieui we ge I ( ) ( ) i i.

15 IPSHITZ ESTIATES FOR UTIINEAR 5 Fo II sie > we hve ( ( ) ) d d ( ( ) ) d d J. J Oseve h o > d B B i oows h. ~ ~ B he iowsi ieui d γ S S i we oi d d J R d B i d B i d B i. i Noie h o d B i oows we oi ( ( ) ) d d J

16 DAZHAO HEN 6 ( ( ) R ( ) ) d d ( ( ) ( R ) ) d d ( ) B ( ) d d ( B ) d d γ γ d B i ( ) i γ γ i

17 IPSHITZ ESTIATES FOR UTIINEAR 7 ( ). i γ γ Theeoe sie > > > d >. γ > i d II > i d > i d > i d γ > γ. i oiig he esies o I d II he eds o he desied esu. I is we-ow h he du se o R H is. R BO Fo his d Theoe du gue we esi dedue he oowig ousio: oo. e. i R i γ The s R oiuous io. R BO Theoe. e wih R i o. The is ouded o R K H o. K

18 DAZHAO HEN 8 Poo. B e 6 e R HK d B B su e e -o d. The we hve K χ χ I. I Fo I he oudedess o o (see Theoe ) i is es o vei h i I ( ) i i ( )( ). i

19 Fo I oe h IPSHITZ ESTIATES FOR UTIINEAR 9 ( ) d ( ) ( ( ) d ) Whe ( ) d ( ( ) d ) E J J. d wih i oows o h ~ ~. The he iowsi ieui J R d d ( ) i R d i ( ) d B ( ) [( ) ]. i B he ehod o he esie o J i he oo o he Theoe we ge ( ) J i ( ) i ( ) γ γ ) ( sie ( ) he ( ) χ J J

20 4 i DAZHAO HEN [ ( ( ) ( ) ) ] d E i E [ ( ( ) ( ) ) d ] i [ ( E ( ) d ) ] γ γ ( ) [ ( ) ] i d E i ( ) (( ) ) ( ) (( ) ) ( ) i ( ) (( ) ) ( ) ( ) i (( γ ) ) γ ( ) ( ) i ( ) ( ) ( ( ) ) i i ( ) ( )( ( ) γ) i. i Thus I [ ( i ( )( ( ) ) ) ( )( ( ) ) ) (

21 IPSHITZ ESTIATES FOR UTIINEAR 4 ( ( ( )( ( ) ) ) ( )( ( ) γ ) ) ]. Whe I ( )( ( ) ) [ i ( )( ( ) ) ( )( ( ) ) ( )( ( ) γ) ] i. Whe > I [ ( i ( )( ( ) ) ) ( ( )( ( ) ) ) ( ( )( ( ) ) ( )( ( ) ) )( )

22 4 DAZHAO HEN ( ( )( ( ) ) ( )( ( ) ) )( ) ( ( )( ( ) γ) ( )( ( ) γ) )( ) i. The esies o I d I ed o ( ) K i ( ) d desied esie oows o ig iiu ove deoosiios o. Whe ( ) his id o oudedess is. I [5] u d Xu ove i whe. Now we give esie o we e. Theoe 4. e i( γ ) ( ) wih ( i R ) o. The ( ) s HK ( ) ( R ) oiuous io WK ( R ). Poo. We wie whee eh is e ( ( ) ) o suoed o B d. Wie ( ( ) ) ( ) WK { su > { } E : > }

23 IPSHITZ ESTIATES FOR UTIINEAR 4 ( ( ) ) su { > 4 { } E : > } G G. B he ( ) oudedess o I i Theoe we ge d esie sii o h o G ( ( ) ) ( ) χ. i To esie G e us ow use he esie ( ) ( ) ( ) i ( ) i ( ) ( ) i γ γ ( ) ( ) i whih we ge i he oo o Theoe. Noe h whe E ( ) 4 4 ( ) [ ( ) i ( )

24 44 DAZHAO HEN ( ) ( ) ( ) γ γ ( ) ( ) ] 4 i 4 [( ) ( ) 4 ( ) ( ) ( ) 4 ( ) ( ) ( ) 4 γ γ ( ) ( ) ( ) ] 4 ( ) (( ) ) i ( ( ) ) i ( ) o > e e he i osiive iege sisig he i > we hve ( ) i 4 { E : ( ) > }.

25 So we oi IPSHITZ ESTIATES FOR UTIINEAR 45 G su { > ( ( ) ) su { ( ) > ( ) su > ( ) } ( ) } i ( ). Now oiig he ove esies o G d G we oi ( ) ( ) WK i ( ). Theoe 4 oows ig he iiu ove e oi deoosiios. Reeees [] S. hio A oe o ouos Idi Uiv. h. J. (98) 7-6. [] W. G. he Besov esies o ss o uiie sigu iegs A h. Sii 6 () [] R. A. Devoe d R.. Sh i uios esuig soohess e. Ae. h. So. 47 (984). [4] J. Gi-uev d. J.. Heeo A heo o Hd ses ssoied o Hez ses Po. odo h. So. 69 (994) [5] G. E. Hu S. Z. u d D.. Yg The weed Hez ses J. Beiig No. Uiv. (N. Si.)(hi) (997) 7-4. [6] S. Jso e osiio d ouos o sigu ieg oeos A. h. 6 (978) 6-7. [7]. Z. iu Boudedess o uiie oeo o Tiee-izoi ses Ie. J. h. & h. Si. 5 (4) 59-7.

26 46 DAZHAO HEN [8]. Z. iu The oiui o ouos o Tiee-izoi ses Ieg Euios d Oeo Theo 49 (4) [9] S. Z. u. Wu d D.. Yg Boudedess o ouos o Hd e ses Si. i hi (Se. A) 45 () [] S. Z. u d. F. Xu Boudedess o ouos eed o iiewiz iegs o Hd e ses Asis i Theo d Aiios (4) 5-. [] S. Z. u d D.. Yg The deoosiio o he weighed Hez ses d is iios Si. i hi (Se. A) 8 (995) [] S. Z. u d D.. Yg The weighed Hez e Hd ses d is iios Si. i hi (Se. A) 8 (995) []. Puszsi heizio o he Besov ses vi he ouo oeo o oi Roheg d Weiss Idi Uiv. h. J. 44 (995) -7. [4] E.. Sei Hoi Asis: Re-Vie ehods Ohogoi d Osio Iegs Pieo Uiv. Pess Pieo 99. [5] A. Tohis d S. Wg A oe o he iiewiz ieg oo. h. 6/6 (99) 5-4. g