EXISTENCE AND BOUNDEDNESS OF gλ -FUNCTION AND MARCINKIEWICZ FUNCTIONS ON CAMPANATO SPACES

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1 Scieiae Mahemaicae Jaoicae Olie, Vol. 9, 3), EXISTENCE AND BOUNDEDNESS OF gλ -FUNCTION AND MARCINKIEWICZ FUNCTIONS ON CAMPANATO SPACES KÔZÔ YABUTA Received Decembe 3, Absac. Le gf), Sf), gλ f) be he Lilewood-Paley g fucio, Lusi aea fucio, ad Lilewood-Paley gλ fucio of f, esecively. I 99 Che Jiecheg ad Wag Silei showed ha if, fo a BMO fucio f, oe of he above fucios is fiie fo a sigle oi i R, he i is fiie a.e. o R, ad BMO boudedess holds. Recely, Su Yogzhog exeded his esul o he case of Camaao saces i.e. Moey saces, BMO, ad Lischiz saces). We imove his gλ esul fuhe. His assumio is λ>3/. We show his is elaxed o λ>max, /) / α<), λ> α</), ad λ>α/ / α<). We also ea geealized Macikiewicz fucios µ ρ f), µ ρ S f) ad µ,ρ λ f).. Iocio I his oe we su he exisece ad boudedess oey of squae fucio oeaos, such as Lilewood-Paley s gλ -fucio ad Macikiewicz fucios, o Camaao saces. Fis, we ecall he defiiio of Lilewood-Paley s fucios geealized oes) i he -dimesioal Euclidea sace R. Defiiio. A coiuous fucio ψ o R is called a LP fucio, if hee exis osiive cosas C, C, δ, η ad γ such ha i) ψ L R ) ad R ψx) dx =; ii) ψx) C x ) δ ; iii) ψx h) ψx) C h γ x ) η fo h x /. Fom ii) ad iii) i follows iii ) R ψx h) ψx) dx h γ fo h R. I fac, we have ψx h) ψx) dx R x h ψx h) ψx) dx x < h ψx h) ψx) ) dx h γ x ) η dx C mi h, x ) δ dx) h γ. R R Fo a LP fucio we defie Lilewood-Paley s g ad Lusi s aea fucios as follows. Hee ad heeafe, f x) deoes fx/). ψ fx) ) gf)x) = d, ) d Sf)x) = ψ fy), Γx) Mahemaics Subjec Classificaio. 4B5. Key wods ad hases. Lilewood-Paley fucios, Macikiewicz fucio, aea fucio, Camaao sace, Moey sace, Lischiz sace, BMO.

2 6 Kôzô YABUTA whee Γx) ={, y) R ; x y <}. gλf)x) = R x y ) λ ψ fy) d whee λ>. L boudedess of hese oeaos ae kow like as he classical Lilewood- Paley s g-fucios. Tha is, g ad S ae L bouded fo <<, ad gλ is L bouded fo << if λ>max, ) see fo examle Tochisky [, ]). Hee ad heeafe, he lee C deoes a cosa deedig o mai aamees ad may vay a each occuece. Sei s geealizaio of he Macikiewicz fucio is as follows [8]: Le Ωx) bea fucio o R which saisfies he followig wo codiios: i) Ωx) is homogeeous of degee ad coiuous o he ui shee S, ad saisfies fo some <β Ωx ) Ωy ) C x y β, x,y S. ii) Ωx ) dσx ) =, whee dσ is he suface Lebesgue measue o S. S Defie µf)x) by ψ fx) ) µf)x) = d, whee ψx) = Ωx) x χ { x }. I hei wok o Macikiewicz iegal, A. Tochisky ad S. Wag [3] ioced he Macikiewicz fucios µ S f) ad µ λ f) coesodig o he S fucio ad g λ fucio. O he ohe had, i he coecio of µf) a aameized Macikiewicz fucio µ ρ f) was cosideed by L. Hömade [3]. I coesods o he case ψx) = Ωx) x ρ χ { x }. Thus, we have cosideed i [5] aameized µ ρ S f) ad µ,ρ λ f), whee ψx) = Ωx) x ρ χ { x }, fo ρ C wih Re ρ>. L boudedess fo hese oeaos ae well discussed i [7, 5], ad will be used i his ae. We ecall also he defiiio of Camaao saces [5]. Defiiio. Fo < ad / α, he Camaao sace E α, is defied by he se of fucios fo which / f E α, = su su fx) f x R B B α/ B dx) <, B B whee B moves ove all balls ceeed a x, ad f B is he aveage of f ove B,/ B ) B f) d. I is kow ha fo < α, E α, = Li α : he Baach sace of Lischiz coiuous fucios of exoe α, ad he oms ae equivale. If α =, E α, coicides wih BMO: he sace of fucios of bouded mea oscillaio. Ad if α <, E α, is equivale o he Moey sace L,α. I is also easily checked ha f α, C su B if a C B α/ B B fx) a dx ) / / α ), ad hece hese oms ae equivale. We oe ha balls ca be elaced by cubes wih sides aallel o he coodiae axes ad he oms ae equivale. I [7, 4] we have deced he boudedess fom he exisece of Macikiewicz fucios o a se of osiive measue. Recely, Su Yogzhog [] gives he followig esuls, exedig he BMO esuls by Wag ad Che [8]. )

3 gλ -FUNCTION AND MARCINKIEWICZ FUNCTIONS 6 Theoem. Le << ad / α<mi,δ,γ,η). If f E α, ad gf)x ) is fiie fo a oi x R, he gf)x) < a.e. o R, ad hee is a cosa C ideede of f, such ha gf) E α, f E α,. Theoem. Le << ad / α<max mi,δ), miδ, γ, η)). If f E α, ad Sf)x ) is fiie fo a oi x R, he Sf)x) < a.e. o R, ad hee is a cosa C ideede of f, such ha Sf) E α, f E α,. He shows he above esuls i he case ψx), x ) ψx) C x ) δ =, η = ad γ = ), bu i is easily see ha his esuls hold i he above cases. He also gives he coesodig esul fo g λ fucio. I his ae, we fuhe imove his esul o g λ as follows. Theoem 3. Le <<, / α<mi,δ) ad λ>λ, whee λ = max, /) / α<), λ = α</), ad λ =α/ / α<mi,δ,γ,η). If f E α, ad gλ f)x ) is fiie fo a oi x R, he gλ f)x) < a.e. o R, ad hee is a cosa C ideede of f, such ha gλ f) E α, f E α,. Su s assumio is λ>3 see also Wag ad Che [8] i he case α = ). Ou esul also imoves he auho s oe i [4], he assumio was λ> i he case α<. As fo Macikiewicz fucios, we ca imove ou esuls i [5] as follows. Theoem 4. Le σ>, << ad / α<β. The, if f E α, ad µ ρ f)x ) is fiie fo a oi x R, he µ ρ f)x) < a.e. o R, ad hee is a cosa C ideede of f, such ha µ ρ f) E α, f E α,. Theoem 5. Le σ>, max, σ ) <<, ad / α<max, miβ,σ)). The, if f E α, ad µ ρ S f)x ) is fiie fo a oi x R, he µ ρ S f)x) < a.e. o R, ad hee is a cosa C ideede of f, such ha µ ρ S f)x) E α, f E α,. Theoem 6. Le σ>, max, σ ) <<, λ>λ, ad / α<max, miβ,σ)). The, if f E α, ad µ,ρ λ f)x ) is fiie fo a oi x R, he µ,ρ λ f)x) < a.e. o R, ad hee is a cosa C ideede of f, such ha µ,ρ λ f) E α, f E α,, whee λ = max, /) / α<), λ = α</), ad λ =α/ / α<). To ove he above heoems we use he followig wo key lemmas. Lemma. Le <. Ifδ> ad / α<mi,δ/), he hee exiss C> such ha fo ay ball B = Bx, ) ad ay f E α, R fy) f B y x ) δ α δ f E α,. This ca be oved easily by modifyig he oof of Lemma.3 i []. )

4 6 Kôzô YABUTA Lemma. Le α, β >. Suose ϕx) x ) α) ad ψx) x ) β). The, ϕ ψ x) C 3 x ) miα,β)) Poof. Sice ϕ ψ x) =ϕ ψ) x), we may assume =. Noe ha if y x x /, he y x /. So, we have ϕy)ψx y) y ) α) ψx y) Ad, y x x / y x x / y x x / x /) α) R ψx y) ψ x /) α). ϕy)ψx y) Hece, we obai he coclusio. y x x / ϕy) x y ) β) x /) β) R ϕy) ϕ x /) β). Fially i his secio, we meio some examles of LP fucios. Le P, x) =c x ) ad Qx) = P, x) =. The, Qx) is a LP fucio saisfyig he codiios i Defiiio wih δ =,η =,γ =. Le Rx) =Qx) cos x. The, Rx) isa LP fucio saisfyig he codiios i Defiiio wih δ =,η =,γ =.. Poof of Theoem 3 fo gλ -fucios Le ψx) be a LP fucio. Followig he ocee of he oof by Su, we use fis he followig: Lemma 3. Le λ>, < ad / α<. The hee exiss C> such ha fo ay ball B = Bx,), ay x B ad ay f E α, g λ, f )x)g λ,, f )x) α f E α,, whee f x) =fx) f 4B )χ 4B ad gλ, f )x) := ψ x u y)f y) ) ) λ d, R R gλ,, f )x) := ψ x u y)f y) ) λ d 8 R Poof. Sice ψ is bouded, we have ) gλ, f ψ )x) R R f y) ) ) λ d ψ fy) f 4B ) λ d 4B R α d f E α, ) ) ) α f E α,.

5 gλ -FUNCTION AND MARCINKIEWICZ FUNCTIONS 63 Now fo x B, y 4B ad 8, we have x u y x x x y 3 8, ad hece ψ x u y)f y) ) λ d 8 R ) fy) f 4B 8 4B x u y ) δ ) λ d δ ) fy) f 4B 8 4B δ ) λ d ) δλ d δ λ fy) f 4B 8 4B δλ δ λ α f E α, α f E α,. Nex usig Lemmas ad, we have Lemma 4. Le < ad / α<mi,δ). The hee exiss C> such ha fo ay ball B = Bx,), ay x B ad ay f E α, g λ, f 3)x) α f E α,, ovided λ> i he case α =ad λ>max, ) i he case α<, ad g λ,, f 3)x) α f E α,, ovided λ> i he case α<, whee f 3 x) =fx) f 4B )χ 4B) c, gλ,f 3 )x) := ψ x u y)f 3 y) R R gλ,,f 3 )x) := ψ x u y)f 3 y) 8 R ) ) λ d, ad ) λ d Poof. i) The case, λ> ad α<miδ, λ )/. By he Hölde iequaliy q = ) ad he Mikowski iequaliy / ), g λ,f 3 )x) ψ q ψ q ). ψ x u y) f 3 y) ) ) λ d R R ψ x u y) ) ) λ d f3 R R y) By Lemma we have ψ x u y) ) λ d R x y δ) δ) mi,λ) mi,λ) d x y ). ) mi δ),λ) d δ) mi,λ). δ) mi x y,λ)

6 64 Kôzô YABUTA We have used hee λ>.fo x x <ad x y > 4, we have x y 3 4 x y 4 x y ). Hece, by Lemma R gλ,f miδ, λ) fy) f 4B ) 3 )x) x y ) miδ, λ) miδ, λ) )/ α miδ, λ) )/ f E α, α f E α,. We have hee used α<miδ, λ )/. ii) The case <<, λ> ad α<. The coclusio i his case follows fom i) fo = ad he fac f E α, f E α, fo. iii) The case α =. I his case, i is kow ha E α, om is equivale o he usual BMO om fo evey <. Hece he coclusio follows fom i) fo =. iv) The case <α<. I his case, i is kow ha E α, om is equivale o he usual Lischiz om Li α fo evey <. So, fo y x > 4 we have fy) f 4B y x α 4) α ) f Liα y x α f Liα. Hece gλ,,f 3 )x) = ψ x u y)fy) f 4B ) 8 4B) c y x α ) f Liα y x >4 x u y ) δ y x α ) f Liα 4< y x < x u y ) δ < 8 < 8 y x y x α f Liα x u y ) δ ) ) λ d ) ) λ d ) λ d ) ) λ d. Fo, x B, y/ 4B, we have x y u x y x x x y x y, ad hece So, 4B) c I := y x α x u y ) δ α δ R y x >4 y x >4 C y x α y x /) δ δ y >4 y x α ) x u y ) δ ) λ δ d = C α δ y δ α = C δ α δ. ) λ d R ) λ δ d = C α δ δ = C α. Fo 8, x B ad y x we have x u y x y x x 4 y x, ad hece as above y x α ) I 3 := < 8 y x x u y ) δ ) λ d α.

7 gλ -FUNCTION AND MARCINKIEWICZ FUNCTIONS 65 Now ake b> so ha <b<λ )/. The y x α ) I := < 8 4< y x < x u y ) δ α ) < 8 4< y x < x u y ) b α < 8 4< y x < b x u y b = C α λ b dv d λ v b < 8 ) λ d ) λ d ) ) λ d v < = C α λ b λ b = C α. Thus, we have gλ,,f 3 )x) I I I 3 ) f Liα α f Liα B α f E α,. ) Lemma 5. Le < ad / α<mi,δ). The hee exiss C > such ha fo ay ball B = Bx,) ad ay f E α, saisfyig gλ, f 3)x ) <, i holds gλ, f 3)x) < fo ay x B ad gλ, f 3)x) gλ, f 3)x ) C α f E α, fo ay x B, ovided λ>max, ) i he case α<, λ>i he case α<, ad λ> α i he case α<miδ, γ, η), whee f 3x) =fx) f 4B )χ 4B) c. Poof. By seig v = x u we ge gλ, f 3 )x) = ψ v y)f 3 y) v x ) ) λ d. R R Hece, if we ca show I := ψ v y)f 3 y) v x ) λ v x ) ) λ d R R α f E α, fo x B, he we have by Mikowski s iequaliy gλ, f 3)x) I gλ, f 3)x ) α f E α, gλ, f 3)x ) < fo x B, ad gλ, f 3 )x) gλ, f 3 )x ) C α f E α, fo x B. So, we will esimae I. By he mea value heoem we have v x ) λ v x ) λ = x l x l ) x v ) λx θx x ) ) dθ x l l= x θx x ) v ) λ dθ

8 66 Kôzô YABUTA Hece I ψ v y)f 3 y) x θx x ) v ) ) λ dvddθ. R R i) The case, λ> ad α<miδ/, λ )). By Hölde s iequaliy / /q = ) we have ) ) ψ v y)f 3 y) q ψ v y) ψ v y) f 3 y) R R R ) = C ψ v y) f 3 y). R Hece by Mikowski s iequaliy I C ψ v y) x θx x ) v ) ) ) λ dvddθ f3 R R y). By Lemma we ge ψ v y) x θx x ) v ) λ dv R v y ) δ) x θx x ) v ) λ dv R x θx x ) y ) mi δ),λ). Fo y x > 4 ad x x <we have x θx x ) y y x x x > 3 4 y x. So, seig η = mi δ),λ ), we ge I y x ) ) ) η d fy) f4b. y x >4 Sice y x ) η d y x η y x η d d y x η = η y x y x η we have by usig Lemma [ I fy) f 4B y x η y x >4 ) y x, ) η fy) f 4B ) ] y x >4 y x [ α η )/ η α )/] f E α, α f E α,. we have used hee α</ ad α< η i.e. α< λ ) ad α< δ ). ii) The case <<, λ> ad α<. The coclusio i his case follows fom i) fo = ad he fac f E α, f E α, fo. iii) The case α =. I his case, i is kow ha E α, om is equivale o he usual BMO om fo evey <. Hece he coclusio follows fom i) fo =.

9 gλ -FUNCTION AND MARCINKIEWICZ FUNCTIONS 67 iv) The case <α< ad λ>. I his case, Eα, om is equivale o he usual Lischiz om Li α fo evey <. Hece, i he case <α< δ, he coclusio follows fom i) fo =. So, we ea he case <δ<. Puig u = x θx x ) v we ge I R R R ψ x θx x ) u y)f 3 y) ) λ ddθ R ) fy) f 4B 4B) c x u yθx x) ) δ ) λ ddθ y x α ) f Liα 4B) c x u yθx x) ) δ ) λ ddθ ) ) ). Fo, x B, y/ 4B, we have x y u θx x ) x y x x x y x y, ad hece akig δ > wih α <δ < mi, δ) we have 4B)c y x α x y uθx x) ) δ y x >4 δ y x α δ = C δ y x >4 y x α x y y >4 ) δ y α δ = C δ α δ. So, 4B) c y x α x y uθx x) δ α δ d ) λ α δ δ d ) ) δ ) λ d α δ δ d α δ δ α. Fo he iegal o >we oceed as follows fy) f 4B 4B) c x u yθx x) ) δ fy) fx u θx x )) fx u θx x )) fx u) 4B) c x u yθx x) ) δ 4B) c x u yθx x) ) δ fx u) f 4B 4B) c x u yθx x) ) δ

10 68 Kôzô YABUTA x u y θx x ) α f Liα 4B) c x u yθx x) ) δ C 4B) c x u yθx x) x u y θx x ) α f Liα R x u yθx x ) ) δ α f Liα x u yθx x ) R R 4B) c x x α f Li α x u yθx x) ) δ fx u) f 4B R α ) δ x u yθx x ) ) δ ) δ fx u) f 4B y α y ) δ Cα f Liα y ) fx u) f 4B R Now we ge α d > ) λ R Similaly we ge α d > ) λ α R Ad by chage of vaiable = s ad usig Lemma = ) we have fx u) f 4B d ) λ > > > δ ). R y ) δ ) λ α d α. fx u) f 4B d ) λ fx u) f 4B R ) λ d α. ds ) λs s fu) f 4B u x ) α f E α,. Alogehe, we have I α f Liα α f E α,. v) The case α<miδ, γ, η) ad λ> α. Sice by Mikowski s iequaliy we ge gλ, f 3)x) gλ, f 3)x ) ) ψ x u y) ψ x u y) f 3 y) ) ) λ d R R f Liα, ad sice f 3 y) = fy) f 4B χ 4B) c y x α f Liα, i suffices o show ) J := ψ x u y) ψ x u y) y x α R y 4B) c ) ) λ d f Liα α f E α,.

11 gλ -FUNCTION AND MARCINKIEWICZ FUNCTIONS 69 Fo ad y x > 4, we have x u y y x 3 4 y x 3 3 x x. So, fo we have by he assumio iii) fo ψ ad usig α<η ψ x u y) ψ x u y) y x α y 4B) c y 4B) c x x ) γ y x α ) γ R v α ) η v ) η dv α γ α γ. y x Fo >ad y x > 4 we have x u y y x 3 4 y x > 3 3 x x. So, like as above, we have fo > ψ x u y) ψ x u y) y x α γ α γ. y x >4 Ad fo he iegaio o 4 < y x 4, we have, usig he oey iii ) of ψ ψ x u y) ψ x u y) y x α 4< y x 4 α ψ x u y) ψ x u y) 4< y x 4 R α ψ x u y) ψ x u y) α x x ) γ γ γ α. Thus we have γ α γ γ α γ α γ γ ) ) d J ) λ > ) λ f Liα R γ α γ α γ γ ) ) d ) λ > ) λ f Liα γ d R R ) λ αγ α ) d ) λ α γ α f Liα We have used hee α<γad λ α > i.e. λ> α. γ α γ ) f Liα α f E α,. Lemma 6. Le λ>, < ad <α<mi,δ). The hee exiss C>such ha fo ay ball B = Bx,) ad ay f E α, saisfyig gλ,, f 3)x ) <, i holds gλ,, f 3)x) < fo ay x B ad gλ,, f 3)x) gλ,, f 3)x ) C α f E α, fo ay x B, whee f 3 x) =fx) f 4B )χ 4B) c ad gλ,, f 3)x) := ψ x u y)f 3 y) ) ) λ d. >8 R Poof. By seig v = x u we see gλ,, f 3 )x) = ψ v y)f 3 y) v x >8 R v x ) ) λ dvd.

12 7 Kôzô YABUTA Hece, fo x B we have gλ,, f 3 )x) v x >8 ψ v y)f 3 y) R ψ v y)f 3 y) R v x 9 v x ) ) λ dvd v x ) λ dvd ). We see by Lemma 4 is vaia elaced 8 by 9) ha he secod em i he igh-had side of he above iequaliy is bouded by C α f E α,. Hece, we have gλ,, f 3 )x) α f E α, gλ,, f 3 )x ) ψ v y)f 3 y) v x ) λ v x ) ) λ dvd v x >8 R = C α f E α, gλ,, f 3 )x )I, say. By he mea value heoem we ge I ψ v y)f 3 y) R x x v x θx x ) ) ) λ dvd dθ y x α ) f Li α v x >8 4B) c v y ) δ v x >8 v x θx x ) ) λ dvd ) dθ. We ake b> so ha b <λ. The oig α<δad v y y x x v y x fo y x v x, we have y x 4 y x α v y ) δ 4 y x < v x v y <3 v x y x α v y ) b v x α b v y b C y x v x y x v x y x α v y δ y x δ α ) δ b v x αb C δ v x α δ

13 gλ -FUNCTION AND MARCINKIEWICZ FUNCTIONS 7 Hece oig b <λ ad α<δwe have I v x αb λ b v x λ v x >8 λ b d >8 v x α δ ) v x λ λδ dvd f Liα λ α b λδ d >8 ) λ αδ f Liα λ b λαb λδ λα δ ) f Liα α f E α,. Thus, we have g λ,, f 3)x) g λ,, f 3)x )C α f E α, fo ay x B. Revesig he oles of g λ,, f 3)x ) ad g λ,, f 3)x), we have ad hece we have g λ,, f 3 )x ) g λ,, f 3 )x)c α f E α, fo ay x B, g λ,, f 3 )x) g λ,, f 3 )x ) C α f E α, fo ay x B, Poof of Theoem 3. We follow he idea by Kuz [4]. Le >ad B = Bx,). Se f = f 4B, f =f f 4B )χ 4B ad f 3 =f f 4B )χ 4B) c. The, f = f f f 3 ad gλ f )=. i) The case <α<. By assumio, gλ f)x ) <. So, we have gλ, f)x ) gλ,, f)x ) gλ f)x ) <. Usig Lemma 3 we have gλ, f 3)x )gλ,, f 3)x ) gλ, f)x )gλ,, f)x )gλ, f )x )gλ,, f )x ) <. Hece by Lemmas 4, 5 ad 6 we have fo x B gλf 3 )x) gλ,,f 3 )x)gλ,, f 3 )x)gλ, f 3 )x) 3C α f E α, gλ,, f 3)x )gλ, f 3)x ) <, ad gλ f 3)x) gλ f 3)x ) gλ,, f 3)x) gλ,, f 3)x ) gλ,, f 3)x) gλ,, f 3)x ) gλ, f 3 )x) gλ, f 3 )x ) 4C α f E α,. Usig L -boudedess of gλ we have g λ f ) L f L, ad fom his i follows ha gλ f )x) < fo almos all x B. Thus, we have gλ f)x) g λ f )x)gλ f 3)x) < fo almos all x B. Sice is abiay, we see ha gλ f)x) < fo almos all x R. Le E = {x R ; gλ f)x) < }. We have oly o show ha fo ay ball B = Bx,) wih cee x E, ) gλf)x) gλf)) B dx B α f E α,. B

14 7 Kôzô YABUTA Se f = f f f 3 as above. Noig gλ f ) = gλ, f ) = gλ, f) =, ad usig gλ f ) L f L B α f E α, ad he above iequaliy fo gλ f 3), we have B B gλ f)x) g λ f)) B dx gλ B f)x) g λ f 3)x ) dx B = gλ B f f 3 )x) gλ f 3)x)gλ f 3)x) gλ f 3)x ) dx B gλ B f )x) dx gλ B B f 3)x) gλ f 3)x ) dx B ) f x) dx C α f E α, α f E α,. B 4B ii) The case α. I his case, he oof is simle ha he case i). We have oly o use gλ, ad g λ,, Lemmas 3, 4 ad 5. So, we leave he deailed oof o he eade. This comlees he oof of Theoem Poofs of Theoems 4, 5 ad 6 We oceed as i he oof of Theoem 3. Fo a ball B = Bx,) ad a fucio f we se always f = f 4B, f =fy) f 4B )χ 4B ad f 3 =fy) f 4B )χ 4B) c. Lemma 7. Le Ω L S ), Ωx) dσx) =, S α<, ad ρ = σ iτ σ >,τ R). The, if f E α, ad µ ρ f)x ) < fo some x R, hee exiss C> such ha fo ay ball B = Bx,) µ ρ f )x ) µ ρ f)x ) Ω α f E α,). Poof. By assumio we have Ωy x ) ρ fy) ) d y x y x ρ µ ρ f)x ) <. Hece, fo some we ge Ωy x ) ρ fy) y x y x ρ µρ f)x ). Sice, i he above iegal, he iegaio domai is coaied i y x 4, we see, usig he cacellaio oey of Ω, ha he above iegal is equal o Ωy x ) y x y x ρ fy) f 4B)χ 4B. Hece Ωy x ) y x y x ρ fy) f 4B)χ 4B σ µ ρ f)x ). Thus fo >we have Ωy x ) y x y x ρ f y) Ωy x ) y x y x ρ f y) Ω fy) f 4B < y x <mi,4) y x σ σ µ ρ f)x )C Ω σα f E α,.

15 gλ -FUNCTION AND MARCINKIEWICZ FUNCTIONS 73 Theefoe we have µ ρ f )x )= ρ y x σ µ ρ f)x ) Ω α f E α,) Ωy x ) y x ρ f y) d d σ ) ) µ ρ f)x ) Ω α f E α,). As fo µ ρ S, f ) ad µ,ρ λ, f ) we have Lemma 8. Le Ω L S ), ρ = σ iτ σ >,τ R), max, σ ) <<, ad α<. The, fo ay f Eα,, ay ball B = Bx,) ad ay x R µ ρ S, f )x) = Ωu y)f y) u x ρ y u u y ρ ) d α f E α,. Lemma 9. Le Ω L S ), ρ = σ iτ σ >,τ R), λ>. Suose α ad saisfiy a) max, σ ) << ad α< o b) < ad α<. The, fo ay f E α,, ay ball B = Bx,) ad ay x R ) µ,ρ λ, f )x) = Ωu y)f y) R ρ y u u y ρ ) λ d α u x f E α,. Sice we ca ove Lemmas 8 ad 9 i simila ways, we oly ove Lemma 9. R Poof. i) The case <σ<ad max, σ ) <<. Fis we see easily Ωu y)f y) < y u u y ρ Ω σ fy) f 4B ασ f E α,. y x 4 Hece Ωu y)f y) R ρ < y u u y ρ u x ) ) λ d ασ u x ) ) λ d f E σ α, So, we eed oly o show I := Sice > R σ ρ, we have σ) y u ασ Ωu y)f y) u y ρ σ d σ u x ) ) = σ σ) So, we ake = mi,) ad choose a eal umbe a so ha =, σ) >a> σ). ) f E α, α f E α,. ) ) λ d α f E α,. ) >. σ

16 74 Kôzô YABUTA The, oig < σ) a) <we have by Hölde s iequaliy Ωu y)f y) y u u y ρ ) ) f y) Ω y u u y σ) a) y u u y σ)a σ) a) y u f y) u y σ)a ). Hece by Mikowski s iequaliy ) ad by usig a σ) <we ge ) λ I σ) a) u x ) ) d f y) ) 4B y u u y σ)a σ ) σ) a) σ)a ii) The case σ. I his case we see easily µ,ρ λ, f )x) R R fy) f 4B 4B ) d σ σ) α f E α, σ α f E α,. ) σ σ Ω f y) y u u x u x ) ) λ d ) ) ) λ d fy) f 4B 4B α f E α, α f E α,. iii) The case α =. I his case E α, = BMO < ), ad he oms ae equivale. So, ake = i he above i) ad ii). iv) The case <α<. I his case E α, = Li α < ), ad he oms ae equivale. So, ake = i he above i) ad ii). As fo µ,ρ λ f ), we eed Lemma. Le Ω L S ), ρ = σ iτ σ >,τ R), λ>, < ad α<. The, fo ay f Eα,, ay ball B = Bx,) ad ay x B µ,ρ λ,, f )x) = u x >8 ρ y u Ωu y)f y) u y ρ u x ) ) λ d =. Poof. Fo y x 4, y u ad x x, we have u x u y y x x x 6, ad hece he iegaio u-domai of he above iegal is emy. Nex we ivesigae µ ρ f 3 ), µ ρ S f 3) ad µ,ρ λ f 3). Lemma. Le Ω L S ), ρ = σ iτ σ >,τ R), < ad α<. The, fo ay f E α,, fo ay ball B = Bx,) ad ay x B µ ρ f 3)x) = Ωx y)f 3 y) ρ x y ρ ) d =, y x

17 gλ -FUNCTION AND MARCINKIEWICZ FUNCTIONS 75 ad µ ρ S, f 3)x) = u x ρ y u Ωu y)f 3 y) u y ρ ) d =. Poof. Fo x x ad x y, we have x y, ad hece he iegaio domai wih esec o y has o iesecio wih he suo of f 3 i he exessio of µ ρ f 3). So, we have µ ρ f 3) = fo x B. Fo x x, u x ad u y, we have x y x x x u u y 3, ad hece he iegaio domai wih esec o y has o iesecio wih he suo of f 3 i he exessio of µ ρ S, f 3). So, we have µ ρ S, f 3) = fo x B. Lemma. Le Ω L S ), ρ = σ iτ σ >,τ R), λ>. Suose α, λ ad saisfiy a) max, σ ) <<, λ>max, ) ad α< o b) <, λ> α ad α<. The, fo ay f Eα,, fo ay ball B = Bx,) ad ay x B ) µ,ρ λ, f 3)x) = Ωu y)f 3 y) R ρ y u u y ρ ) λ d α u x f E α,. Poof. i) The case <σ<ad max, σ ) <<. Take ad a as i he oof of Lemma. The, by Hölde s iequaliy we have Ωu y)f 3 y) y u u y ρ ) ) f 3 y) Ω y u u y σ) a) y u u y σ)a ) σ) a) f 3 y). y u u y σ)a Hece usig Mikowski s iequaliy ) ad he oig u x y x y u x x > 4 y x ) fo u y, y x > 4 ad x x, we have µ,ρ λ, f 3)x) R y u ) f 3 y) u y σ)a u x ) ) λ σ) a) σ d ) χ y u ) 4B R u y σ)a u x ) f3 λ y) ) χ y u f 3 y) 4B R u y σ)a y x ) λ ) fy) f 4B 4B y x ) σ)a λ λ ) λ σ) σ α d λ ) f E α, We have used hee λ >, α< λ ) ad Lemma. ) σ) a) σ d ) ) λ σ) a) σ d σ) a) σ d ) λ ) α λ f E α, α f E α,.

18 76 Kôzô YABUTA ii) The case σ. I his case, we ake = mi,) ad a =. The he easoig i he se i) sill woks. iii) The case α =. I his case E α, = BMO < ), ad he oms ae equivale. So, ake = i he above i) ad ii). iv) The case <α<. I his case E α, = Li α < ), ad he oms ae equivale. So, akig = = i he above i) ad ii) ad oig λ> α imlies α< λ ), we have he desied iequaliy. Lemma 3. Le Ω L S ), ρ = σ iτ σ >,τ R), λ>, < ad <α<. The, fo ay f E α,, fo ay ball B = Bx,) ad ay x B µ,ρ λ,, f 3)x) α f E α,, whee µ,ρ λ,, f 3)x) = u x 8 ad f 3 x) =fx) f 4B )χ 4B) c. ρ y u Ωu y)f 3 y) u y ρ u x ) ) λ d, Poof. I his case E α, = Li α < ), ad he oms ae equivale. So, fo y 4B) c we have f 3 y) = fy) f 4B fy) fx ) fx ) f 4B f Liα y x α α ) f Liα y x α. Fo x x, u y ad u x 8, we have y x y u u x x x, ad fo x x, u y ad x 4B) c we have u x y x u y x x > y x >. Hece we have µ,ρ λ,, f 3)x) Ω < u x 8 < u x 8 σ σ y u 4< y x y u α u y y x α u y σ σ α λ λ d λ λ d ) f Liα ) ) λ d f Liα ) f Liα α λ λ ) f Liα α f E α,. Now we eae hee moe lemmas. Lemma 4. Le Ω Li β S )<β ), ρ = σ iτ σ >,τ R), <<, ad / α<β. The hee exiss C>such ha fo ay ball B = Bx,) ad ay f E α, saisfyig µ ρ f 3 )x ) <, i holds µ ρ f 3 )x) < fo ay x B ad µ ρ f 3 )x) µ ρ f 3 )x ) C α f E α, fo ay x B, whee f 3 x) =fx) f 4B )χ 4B) c. Lemma 5. Le Ω Li β S )<β ), ρ = σ iτ σ >,τ R), max, σ ) < <, ad / α</ o / α<miβ,σ). The hee exiss C > such ha fo ay ball B = Bx,) ad ay f E α, saisfyig µ ρ S, f 3)x ) <, i holds µ ρ S, f 3)x) < fo ay x B ad µ ρ S, f 3)x) µ ρ S, f 3)x ) C α f E α, fo ay x B,

19 gλ -FUNCTION AND MARCINKIEWICZ FUNCTIONS 77 whee f 3 x) =fx) f 4B )χ 4B) c. Lemma 6. Le Ω Li β S )<β ), ρ = σ iτ σ >,τ R), max, σ ) < <, ad / α</ o / α<miβ,σ). The hee exiss C > such ha fo ay ball B = Bx,) ad ay f E α, saisfyig µ,ρ λ, f 3)x ) <, i holds µ,ρ λ, f 3)x) < fo ay x B ad µ,ρ λ, f 3)x) µ,ρ λ, f 3)x ) C α f E α, fo ay x B, ovided λ> i he case α<, λ> α i he case α< ad λ>max, ) i he case α<, whee f 3x) =fx) f 4B )χ 4B) c. We ca ove he above hee lemmas modifyig he oofs i he cube seig see, Ha [], Qiu [6], Yabua [4] ad Sakamoo ad Yabua [5]. We give hee a way o use he cube seig diecly i he case of Lemma 6. Le Q be a cube wih cee x ad side legh, Q be he cube wih cee x ad side legh 6. Le B be he ball wih cee x ad adius. Le f 3 x) =fx) f 4B )χ 4B) c ad f 4 x) =fx) f Q )χ Q ) c. The we have Lemma 7. Le Ω L S ) ad Ωx) dσx) =.Leρ = σ iτ σ >,τ R), S <, ad / α<. Le x, B, Q, Q, f 3 ad f 4 be as above. The, hee exiss C> such ha fo ay x B R ρ y u Ωu y) f 3 y) f 4 y) ) u y ρ u x ) ) λ d α f E α,. Poof. Le I be he lef had side of he above iequaliy i he saeme of Lemma 7. The by he assumio S Ωx) dσx) = we see ha I = Ωu y) fy) f Q ) χq χ 4B) c) R ρ y u u y ρ ) λ d u x Ω fy) f Q R σ ) ) λ d y u u y σ u x 4< y x <8 fy) f Q R σ ) ) λ d y u u y σ u x 4< y x <8 C fy) f Q σ ) ) λ d u y σ u x R y u 4< y x <8 =: I I. I α f E α, ca be oved i a way quie simila o he oof of Lemma, ad I α f E α, ca be oved i a way quie simila o he oof of Lemma 9. Usig his lemma, we ca use he coesodig esul o Lemma 6 i he cube seig. We oe hee ha i Sakamoo ad Yabua [7,. 37 4], hey eally oved µ,ρ λ, f 3)x) µ,ρ λ, f 3)x ) C α f E α, i he case α< ad λ> α. Poofs of Theoems 4, 5 ad 6. Usig Lemmas 7 6 ad L boudedess esuls i [7], we ca ove hese heoems i he same way as i he oof of Theoem 3, ad so we leave he deailed oofs o he eade. )

20 78 Kôzô YABUTA Refeeces [] E. B. Fabes, R. L. Johso ad U. Nei, Saces of hamoic fucios eeseable by Poisso iegals of fucios i BMO ad L,λ, Idiaa Uiv. Mah. J., 5 976), [] Ha Yog sheg, O some oeies of s-fucio ad Macikiewicz iegals, Aca Sci. Nau. Uiv. Pekiesis, 5 987), 34. [3] L. Hömade, Taslaio ivaia oeaos, Aca Mah., 4 96), [4] D. S. Kuz, Lilewood-Paley oeaos o BMO, Poc. Ame. Mah. Soc., ), [5] J. Peee, O he heoy of L,λ saces, J. Fuc. Aal., 4 969), [6] Qiu Sigag, Boudedess of Lilewood-Paley oeaos ad Macikiewicz iegal o E α,, J. Mah. Res. Exosiio, 99), 4 5. [7] M. Sakamoo ad K. Yabua, Boudedess of Macikiewicz fucios, Sudia Mah., ), 3 4 [8] E. M. Sei, O he fucios of Lilewood-Paley, Lusi, ad Macikiewicz, Tas. Ame. Mah. Soc., ), [9] E. M. Sei, Sigula iegals ad diffeeiabiliy oeies of fucios, Piceo Uiv. Pess, Piceo, N.J., 97. [] E. M. Sei ad G. Weiss, Iocio o Fouie Aalysis o Euclidea Saces, Piceo Uiv. Pess, Piceo, N.J., 97. [] Su Yogzhog, O he exisece ad boudedess of squae fucio oeaos o Camaao saces, ei. [] A. Tochisky, Real-Vaiable Mehods i Hamoic Aalysis, Academic Pess, Sa Diego, Calif., 986. [3] A. Tochisky ad Shili Wag, A oe o he Macikiewicz iegal, Colloq. Mah., 6/6 99), [4] K. Yabua, Boudedess of Lilewood-Paley oeaos, Mah. Jaoica, ), [5] K. Yabua, Some emaks o Macikiewicz fucios, Kwasei Gakui Uiv. Na. Sci. Rev., 6 ), 9 5. [6] Wag Shili, Boudedess of he Lilewood-Paley g-fucio o Li α R )<α<), Illiois J. Mah., ), [7] Wag Silei, Some oeies of he Lilewood-Paley g-fucio, Coem. Mah., 4 985), 9. [8] Wag Silei ad Che Jiecheg, Some oes o squae fucio oeao, Aals of Mahemaics Chiese), Seies A, 99), School of Sciece ad Techology, Kwasei Gakui Uivesiy Gakue - Sada, Hyogo , JAPAN yabua@kwasei.ac.j