TANGENTIAL BOUNDARY BEHAVIOR OF THE POISSON INTEGRALS OF FUNCTIONS IN THE POTENTIAL SPACE WITH THE ORLICZ NORM

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1 Scieniae Mahemaicae Japonicae Online, Vol. 9, (23), TANGENTIAL BOUNDARY BEHAVIOR OF THE POISSON INTEGRALS OF FUNCTIONS IN THE POTENTIAL SPACE WITH THE ORLICZ NORM EIICHI NAKAI AND SHIGEO OKAMOTO Received Februar 4, 23; revised June 27, 23 Absrac. Nagel, Rudin and Shapiro (982) invesigaed he angenial boundar behavior of he Poisson inegrals of funcions in he poenial space L p K (Rn )=K F : F L p (R n )} wih a kernel K : R n \} [, + ) which is posiive, inegrable, radial and decreasing. In his paper, we exend he resul o L Φ K (Rn )=K F : F L Φ (R n )}, where L Φ (R n ) is he Orlicz space. Moreover we inroduce Ω R -limi for a coninuous increasing funcion R :[, + ) [, + ) wihr() as. The angenial approach region is defined b he funcion R. We give a relaion beween R() forwhich all funcions in L Φ K (Rn )haveheω R -limi and he L eφ -norm of P K, whereφishe e complemenar funcion of Φ, and calculae R() precisel.. Inroducion I is well known ha, for f L p (R n ), is Poisson inegral u(x, ) =P f(x), x R n, >, converges nonangeniall o f(x) a.e. when ends o. I is also well known ha, for general f L p (R n ), convergence fails when he approach regions have a cerain degree of angeniali. The angenial boundar behavior of he Poisson inegrals of funcions in subspaces of L p (R n ) was sudied b Nagel, Rudin and Shapiro [4], Nagel and Sein [5], Dorronsoro [], ec. Nagel, Rudin and Shapiro [4] invesigaed he poenial space L p K (Rn )=K F : F L p (R n )} wih a kernel K : R n \} [, + ) which is posiive, inegrable, radial and decreasing. We noe ha L p K (Rn ) is a subspace of L p (R n ). The [4] gave he relaion beween he geomeric properies of approach regions on which he angenial limi of P f exiss for all f L p K and he -norm of P Lp K wih K/ L p, where /p +/p =. In his paper, we exend he resul in [4] o L Φ K (Rn )=K F : F L Φ (R n )}, where L Φ (R n ) is he Orlicz space, and give he relaion beween he geomeric properies of approach regions and he L eφ -norm of P K wih K / L eφ, where is he complemenar funcion of Φ. However, he L eφ -norm of P K is no simple. So we inroduce Ω R -limi for a coninuous increasing funcion R :[, + ) [, + ) wih R() as (see Definiion 3.2). The angenial approach region is defined b he funcion R. We give a relaion beween R() for which he Poisson inegrals of all funcions in L Φ K (Rn )havehe Ω R -limi and he L eφ -norm of P K, and calculae R() precisel for kernels K of he form K(x) =K ρ (x) = ρ( x ) x n, 2 Mahemaics Subjec Classificaion. Primar 3B25, Secondar 46E3. Ke words and phrases. angenial limi, Poisson inegral, poenial, Orlicz space.

2 88 EIICHI NAKAI AND SHIGEO OKAMOTO where he funcion ρ :(, + ) (, + ) saisfies ha ρ(r)/r n is decreasing and + ρ(r) dr < +. r If K L eφ (R n ), hen K F is coninuous for ever F L Φ (R n ). In his case, he angenial limi of he Poisson inegral of f L Φ K exiss riviall. So we are ineresed in he case K/ L eφ (R n ). The Bessel kernel J α,<α<n, is he funcion on R n whose Fourier ransform is Ĵ α (ξ) =(+ ξ 2 ) α/2, ξ R n. Then J α (x) K ρ (x) for small x wih ρ(r) =r α for small r>. This case was sudied in [4] and [5]. If ρ(r) =r α for small r>wih < α < n/p, hen he Hard-Lilewood-Sobolev heorem shows ha L p K ρ (R n ) L p (R n ) L q (R n ), where n/p + α = n/q. In his case he Poisson inegrals of all funcions f L p K ρ have he Ω R -limi wih R() = αp/n. As α is bigger, he Ω R -limi ges more angenial. If α = n/p, hen L p K ρ (R n ) L p (R n ) BMO(R n ). In his case R() = (log(/)) (p )/n (see Theorem 3.7 and Example 3.). We noe ha here is a larger class of funcions han L p J α (R n ) such ha all funcions in he class have he Ω R -limi wih he above R. (see Dorronsoro []). If α > n/p, hen L p K ρ (R n ) L p (R n ) Lip β (R n ), where β = p/n + α. In his case, he angenial limi of he Poisson inegral of f L p K ρ exiss riviall. One of he auhors [7, 8, 9] showed ha, if hen Φ ( r n ) r ρ() d CΨ ( r n ) L Φ K ρ (R n ) L Φ (R n ) L Ψ (R n ). If φ :(, + ) (, + ) is increasing and ( ) r Φ ρ() r n d Cφ(r) for r>, hen for L Φ K ρ (R n ) L Φ (R n ) BMO φ (R n ). r>, If φ, hen BMO φ is he usual BMO. In hese cases, see Theorem 3.8, Remark 3.4 and Example 3.2. Le Φ 2. Then, for all funcions F such ha Φ(k F (x) ) is inegrable for all k>, he Poisson inegrals of f = K ρ F have he Ω R -limi wih a leas R() =/ ( (ρ(r)/r) dr) /n, for ever kernel K ρ (see Theorem 3.9). Noaions and definiions of funcion spaces are in he nex secion. We sae main resuls and examples in Secion 3 and proofs of main resuls in Secions 4 6. The auhors would like o hank he referee for his man commens.

3 TANGENTIAL BOUNDARY BEHAVIOR OF THE POISSON INTEGRALS Noaions and definiions In his secion we sae some noaions and definiions. We also sae properies of he N-funcion and he Orlicz space. 2.. R n is he n-dimensional Euclidean space, wih norm x = x 2 i, x =(x,..., x n ) and R n+ + = (x, ) :x R n, >}. Le B(a, r) be he ball x R n : x a <r} wih cener a and of radius r>. We denoe he measure of a measurable se E R n b E. Leσ n = B(, ). Then B(,r) = σ n r n A funcion Φ : [, + ) [, + ) is called an N-funcion if Φ is coninuous, convex, sricl increasing, lim r + Φ(r)/r = and lim r + Φ(r)/r =+. For an N-funcion Φ, he complemenar funcion is defined b (r) =suprs Φ(s) :s }, r. Then is also an N-funcion, = Φ, and, (2.) r Φ (r) (r) 2r A funcion Φ : [, + ) [, + ) is said o saisf he 2 -condiion, denoed Φ 2, if Φ(2r) CΦ(r), r, for some C >. This condiion is also called he doubling condiion. A funcion Φ : [, + ) [, + ) is said o saisf he 2 -condiion, denoed Φ 2,if Φ(r) Φ(kr), r, 2k for some k>. Le Φ is an N-funcion. Then Φ 2 if and onl if For an N-funcion Φ, le } L Φ (R n )= f L loc(r n ): Φ(ɛ f(x) ) dx < + for some ɛ>, R n } M Φ (R n )= f L loc(r n ): Φ(k f(x) ) dx < + for all k>, R ( n ) } f(x) f Φ =inf λ>: Φ dx. R λ n Then L Φ (R n ) is a Banach space wih he norm L Φ. M Φ (R n ) is a closed subspace of L Φ (R n ). If and onl if Φ 2, hen L Φ (R n )=M Φ (R n ). Le C comp (R n ) be he se of all coninuous funcions wih compac suppors. Then C comp (R n ) is dense in M Φ (R n ). We have Hölder s inequali for Orlicz spaces: (2.2) f(x)g(x) dx 2 f Φ g eφ. R n We also have he following equivalence: } (2.3) f Φ sup f(x)g(x) dx : ( g(x) ) dx 2 f Φ. R n R n

4 9 EIICHI NAKAI AND SHIGEO OKAMOTO If and onl if Φ 2, hen ( ) f(x) (2.4) Φ dx = for all f L Φ wih f Φ. R f n Φ Le f j } j L Φ.Iff j inl Φ as j +, hen (2.5) Φ( f j (x) ) dx as j +. R n IfandonlifΦ 2, he converse is rue The leer K and he word kernel denoe a nonnegaive L -funcion on R n which is radial and decreasing; i.e., K(x) =K(x )if x = x and K(x) K(x )if x x. Also, K() = + (we are no ineresed in bounded K), and we usuall normalize so ha K =. Le L Φ K = L Φ K(R n )=f = K F : F L Φ (R n )}, MK Φ = MK(R Φ n )=f = K F : F M Φ (R n )} The Poisson kernel for R n+ + is c n P (x) = (x R n, >), ( x ) (n+)/2 where c n =Γ ( ) n+ 2 π (n+)/2 is so chosen ha P =for<<+. (P K)(x) is he harmonic exension of K o R n+ +. For f L Φ K, he Poisson inegral u = P [f] means ha u(x, ) =P [f](x, ) =(P f)(x) =(P K F )(x), for some F L Φ (R n ) For a funcion φ :(, + ) (, + ), le BMO φ (R n )= f L loc(r n ): f BMOφ < + }, where f BMOφ = sup f(x) f B dx, B=B(a,r) φ(r) B B and f B = f(x) dx. B B If φ(r), hen BMO φ (R n )=BMO(R n ). If φ(r) =r α,<α, hen i is known ha BMO φ (R n )=Lip α (R n ). All funcions in BMO φ are coninuous, if and onl if φ() (2.6) d < + (see [2] and [6]) For funcions θ, κ : (, + ) (, + ), we denoe θ(r) κ(r) if here exiss a consan C>such ha C θ(r) κ(r) Cθ(r), r >. A funcion θ :(, + ) (, + ) is said o be almos increasing (almos decreasing) if here exiss a consan C> such ha θ(r) Cθ(s) (θ(r) Cθ(s)) for r s. The leer C shall alwas denoe a consan, no necessaril he same one.

5 TANGENTIAL BOUNDARY BEHAVIOR OF THE POISSON INTEGRALS 9 3. Main resuls and examples For an N-funcion Φ and for a ball B, le ( ) } f(x) f Φ,B =inf λ>: Φ dx. B B λ For a kernel K and for an N-funcion Φ, le k(r, ) = B(,r) P K eφ,b(,r), r >, >, where is he complemenar N-funcion of Φ. Le <β<+. We define he approach region Ω Φ K,β(x )=(x, ) R n+ + : k( x x,) <β}, x R n, and he associaed maximal funcion (M[Ω Φ K,β]f)(x )=sup u(x, ) :(x, ) Ω Φ K,β(x )}, where u = P [f]. We assume ha Φ 2 and K/ L eφ (R n ). Le (3.) τ β () =supr >:k(r, ) <β}. Then <τ β () < +, τ β is coninuous and increasing, (3.2) τ β () as, k(τ β (),)=β, k(r, ) <β r<τ β (). Hence we can wrie Ω Φ K,β(x )=(x, ) R n+ + : k( x x,) <β} = (x, ) R n+ + : x x <τ β ()}, x R n. Moreover, he approach region Ω Φ K,β (x ) is angenial o he boundar of R n+ +, i.e. (3.3) τ β ()/ c > for all >, τ β ()/ + as. The properies (3.2) and (3.3) will be proved in he nex secion. If K L eφ and F L Φ, hen f = K F is coninuous, so ha u = P [f] is coninuous on he closure of R n+ +. In his case he angenial limi of u exiss riviall. Therefore we are ineresed in he case K/ L eφ. Our main resuls are following. Theorem 3.. Le Φ be an N-funcion and Φ 2. Then here exiss a consan C> such ha, for all f = K F L Φ K and for all >, x R n :(M[Ω Φ K,β]f)(x) >} ( ) C(β + K ) F (x) Φ dx, R n where C is independen of K, β, F and. Definiion 3.. A funcion u on R n+ + is said o have Ω Φ K -limi L a a poin x R n if i is rue for ever <β<+ ha u(x, ) L as (x, ) (x, ) wihin Ω Φ K,β (x ).

6 92 EIICHI NAKAI AND SHIGEO OKAMOTO Remark 3.. Since k(r, ) is increasing wih respec o r (see (4.6)), τ β () is increasing wih respec o β. Hence Ω Φ K,β ΩΦ K,β if β β. Theorem 3.2. Le Φ be an N-funcion and Φ 2. If f MK Φ and u = P [f], hen, for almos all x R n, he Ω Φ K -limi of u exiss a x and equals f(x ). Corollar 3.3. Le Φ be an N-funcion and Φ 2 2.Iff L Φ K and u = P [f], hen, for almos all x R n, he Ω Φ K -limi of u exiss a x and equals f(x ). The nex proposiion shows ha Theorem 3.2 is opimal wih regard o he size of he approach regions. To formulae his precisel, compare Ω Φ K,β(x )=(x, ) R n+ + : x x <τ β ()} wih anoher region (3.4) Ω(x )=(x, ) R n+ + : x x <ω()}, where ω is some posiive coninuous funcion. Le (3.5) where u = P [f]. (M[Ω]f)(x )=sup u(x, ) :(x, ) Ω(x )}, x R n, Proposiion 3.4. Le Ω and M[Ω] be as in (3.4) and (3.5), respecivel. If here exiss c > such ha, for all f = K F L Φ K (Rn ) and for all >, ( ) x R n c F (x) (3.6) :(M[Ω]f)(x) >} Φ dx, R n hen here exiss β> such ha, for all >, ω() τ β (). Theorems 3., 3.2, Corollar 3.3 and Proposiion 3.4 are generalizaion of he resuls of Nagel, Rudin and Shapiro [4]. However he definiion of τ β is no simple and τ β is difficul o calculae. To invesigae he geomeric properies of approach regions, we inroduce Ω R -limi. Definiion 3.2. For R :(, + ) (, + ) andfor<b<+, le Ω R,b (x )=(x, ) R n+ + : x x <br()}, x R n. A funcion u on R n+ + is said o have Ω R -limi L a a poin x R n if i is rue for ever <b<+ ha u(x, ) L as (x, ) (x, ) wihin Ω R,b (x ). In he case Φ(r) =r p wih <p< and K/ L p,wehave k(r, ) = B(,r) P K eφ,b(,r) = B(,r) /p P K L p (B(,r)) This implies ( β σ n /p P K p ) p/n τ β (). B(,r) /p P K p. Hence, for R() = P K p p/n,iff L p K and u = P [f], hen, for almos all x R n, he Ω R -limi of u exiss a x and equals f(x ). This is a resul of Nagel, Rudin and Shapiro [4]. The [4] also showed ha, if (3.6) holds, hen ω() C P K p p/n. We exend his resul o he following.

7 TANGENTIAL BOUNDARY BEHAVIOR OF THE POISSON INTEGRALS 93 Theorem 3.5. Le Φ be an N-funcion, Φ 2 and K / L eφ (R n ). Le τ β, <β<+, be as in (3.). Then here exiss a coninuous increasing funcion R :(, + ) (, + ) wih R() as such ha b > β > >:br() τ β (). In his case he Ω Φ K -limi is also he Ω R-limi. Moreover, if Φ 2 2, hen he funcion R also have he proper β > b > >:τ β () br(). In his case he Ω Φ K -limi and he Ω R-limi are he same. Remark 3.2. We can choose R τ β for an fixed β > (see Proof of Theorem 3.5). In he following, we calculae he funcion τ = τ R. For a funcion ρ :(, + ) (, + ), le K ρ (x) = ρ( x ) x n. We assume ha ρ(r)/r n is decreasing and ha + ρ(r) dr < +. r Then K ρ is a kernel. From he decreasingness of ρ(r)/r n i follows ha ρ(2r) 2 n ρ(r) for r>. Le r ρ() ρ(r) = d. Then ρ(r) C ρ(r) for all r>. If ρ(r)/r α is almos increasing for small r> wih α>, hen ρ ρ for small r>. If ρ(r)/r β is almos decreasing for small r> wih β>, hen ρ(r)/r β is also almos decreasing for small r> (see Lemma 6.2 (iii), (iv)). Proposiion 3.6. Le τ = τ be as in (3.) for a kernel K ρ and an N-funcion Φ wih K ρ / L eφ (R n ). (i) If Φ 2, hen τ() ( τ() (3.7) C τ() n n ) ρ() n + τ()n ρ() n+ n d C for small >. (ii) If Φ 2 2, hen ( τ() (3.8) C τ() n n ) ρ() n + τ()n ρ() n+ n d C for small >. (iii) If Φ 2, Φ(r)/r p is almos decreasing wih <p<, and,ρ(r)/r β is almos (3.9) decreasing for small r> wih < β < n/p, hen ( ) n ( ) C τ()n ρ() τ() n C for small >. Theorem 3.7. Le Φ(r) =r p wih <p<. Le ρ(r)/r α be almos increasing and ρ(r)/r β be almos decreasing for small r>, wih α n/p and α β<n.

8 94 EIICHI NAKAI AND SHIGEO OKAMOTO (i) In he case ha α>, le/p +/p =and (3.) R() = ( ( ) p ) p/(np ) ρ() d n/p for small >. (ii) In he case ha < β < n/p, le /ρ() p/n (α>) (3.) R() = / ρ() p/n (α =) for small >. If f L p K ρ and u = P [f], hen, for almos all x R n, he Ω R -limi of u exiss a x and equals f(x ). Moreover, his is opimal in he sense of Proposiion 3.4. Remark 3.3. If he inegral in (3.) is finie as, hen K ρ L p (R n ). Le SV be he se of all coninuous funcions l :(, + ) (, + ) saisfing, for some consan C>, C l(s) l(r) C for 2 log r s 2, r,s. Then, for ever ɛ>, l(r)r ɛ is almos increasing and l(r)/r ɛ is almos decreasing. Theorem 3.8. Le N-funcion Φ(r) be of he form r p l(r) wih <p< and l SV, and ρ(r) be of he form r α m(r) wih α n/p and m SV. (i) In he case ha α = n/p, le/p +/p =and (3.2) ( ( ) ) p /p p/(np ) R() = m() p l d for small >. (ii) In he case ha <α<n/p,le αp/n l(/)/n (3.3) R() = m() p/n (iii) In he case ha α =,le for small >. l(/ m())/n (3.4) R() = for small >. m() p/n If f L Φ K ρ and u = P [f], hen, for almos all x R n, he Ω R -limi of u exiss a x and equals f(x ). Moreover, his is opimal in he sense of Proposiion 3.4. Remark 3.4. Le ( ) /p φ(r) =m(r)l. r If φ(r) is almos increasing for small r>, hen L Φ K ρ (R n ) L Φ (R n ) BMO φ (R n ). If φ()p / d < + as, hen K ρ L eφ (R n ). If φ()p / d + as, hen (2.6) fails. The following resul is no necessaril opimal.

9 TANGENTIAL BOUNDARY BEHAVIOR OF THE POISSON INTEGRALS 95 Theorem 3.9. Le Φ 2 and R() = ρ(). /n If f MK Φ ρ and u = P [f], hen, for almos all x R n, he Ω R -limi of u exiss a x and equals f(x ). A he end of his secion, we sae examples. Examples 3. and 3.2 follow immediael from Theorems 3.7 and 3.8, respecivel. Example 3.3 is for he case of Φ 2 \ 2. A proof of Example 3.3 is in Secion 6. Example 3.3 is no necessaril opimal. Example 3.. Le <p<, α n/p, <β<+, <γ<+, Φ(r) =r p, and, ρ(r) =r α (log(/r)) β (log log(/r)) γ for small r>. Le ( log log ) (γ )p/n when α =,β =,γ >, ( log ) (β )p/n ( log log ) γp/n when α =,β >, ( αp/n log ) βp/n ( log log ) γp/n when <α<n/p, R() = ( log ) ( /p β)p/n ( log log ) γp/n when α = n/p, β < /p, ( log log ) ( /p γ)p/n when α = n/p, β = /p, γ < /p, ( log log log ) (p )/n when α = n/p, β = /p, γ = /p. If f L p K ρ and u = P [f], hen, for almos all x R n, he Ω R -limi of u exiss a x and equals f(x ). Moreover, his is opimal in he sense of Proposiion 3.4. (If α > n/p, orif α = n/p and β> /p, orifα = n/p, β = /p and γ> /p, hen K ρ L eφ (R n ).) Example 3.2. Le <p<, <θ<+ and r p (log r) θp for large r>, Φ(r) = r p (log(/r)) θp for small r>. For consans α and β wih α n/p and <β<+, leρ(r) =r α (log(/r)) β for small r>. Le ( log ) (β )p/n ( log log ) θp/n when α =,β>, ( αp/n log ) (β+θ)p/n when < α < n/p, R() = ( log ) ( /p β θ)p/n when α = n/p, /p > β + θ, ( log log ) (p )/n when α = n/p, /p = β + θ. If f L Φ K ρ and u = P [f], hen, for almos all x R n, he Ω R -limi of u exiss a x and equals f(x ). Moreover, his is opimal in he sense of Proposiion 3.4. (If α > n/p, orif α = n/p and /p < β + θ, hen K ρ L eφ (R n ).)

10 96 EIICHI NAKAI AND SHIGEO OKAMOTO Example 3.3. Le exp r for large r>, Φ(r) = / exp(/r) for small r>, and ρ(r) = (log(/r)) 2 for small r>. Le ( R() = ɛ log ) /n for small >, where ɛ> is small enough. If f MK Φ ρ and u = P [f], hen, for almos all x R n,he Ω R -limi of u exiss a x and equals f(x ). 4. Proofs of he properies (3.2) and (3.3) To show he properies (3.2) and (3.3), we invesigae he properies of k(r, ). Firs we sae wo lemmas. Lemma 4.. Le Φ be an N-funcion and g. Then,forz R n and r>, Φ( f g(x) ) dx g sup Φ( f(x) ) dx. B(z,r) z R n B(z,r) Proof. Le α = g, β =sup Φ( f(x) ) dx. z R n B(z,r) If α =, hen his inequaion is clear. We assume α. Leµ(A) = g(x) dx/α for A A R n. Then µ is a probabili measure. We noe b χ E he characerisic funcion of E R n. Then we have ( ) Φ( f g(x) ) dx Φ f(x x ) g(x ) dx χ B(z,r)(x) dx B(z,r) R n R ( n ) = Φ α f(x x ) dµ(x ) χ B(z,r)(x) dx R n R ( n ) α Φ f(x x ) dµ(x ) χ B(z,r)(x) dx R n R ( n ) α Φ( f(x x ) ) dµ(x ) χ B(z,r)(x) dx R n R n α βdµ(x )=αβ. R n Lemma 4. shows ha, if f is radial and decreasing and g, hen (4.) f g Φ,B(,r) g f Φ,B(,r). Lemma 4.2. Le K be a kernel and Φ be an N-funcion. Then, for all r>, c n (4.2) P K Φ,B(,r) P Φ,B(,r) Φ () n for >. (4.3) P 2 K Φ,B(,r) P K Φ,B(,r) for < 2. If K/ L Φ (R n ), hen, for ever r>, (4.4) P K Φ,B(,r) + as. If Φ 2, hen, for r>, > and >, (4.5) P K Φ,B(,r) P K Φ,B(,r) < n P K Φ,B(,r).

11 TANGENTIAL BOUNDARY BEHAVIOR OF THE POISSON INTEGRALS 97 Proof. P and K are radial and decreasing. Hence P K is also radial and decreasing. B P = K =,P + = P P and P (x) c n / n, using (4.), we have (4.2) and (4.3). If K L (R n ) \ L Φ (R n ), hen ( ) K(x) Φ dx =+ for all r>,λ>. B(,r) λ Since P K K a.e. as, we have (4.4). B he inequali ( ) ( ) P K(x) P K(x) Φ dx Φ dx, B(,r) B(,r) λ B(,r) B(,r) λ we have he firs inequali in (4.5). If Φ 2, hen, for λ = P K Φ,B(,r), ( ) ( ) P K(x) P K(x) Φ B(,r) B(,r) λ/ n dx Φ dx B(,r) B(,r) λ ( ) P K(x) > Φ dx =. B(,r) B(,r) λ Hence P K Φ,B(,r) >λ/ n, his is he second inequali in (4.5). We assume ha Φ 2 and K/ L eφ. Then 2. We appl Lemma 4.2 o. Then (4.3) and (4.4) impl (4.6) k(r, ) k(r, 2 )for < 2 and k(r, ) + as. Le l(λ, ) = (P B K(x)/λ) dx. Then l(λ, ) is coninuous wih respec o λ and, sricr decreasing wih respec o λ, l(λ, ) + as λ, and, l(λ, ) asλ +. Hence P K eφ,b(,r) is coninuous wih respec o, andsoisk(r, ). B (4.5) and (4.2) we have ha k(r, ) is coninuous wih respec o r and (4.7) k(r,) <k(r 2,)forr <r 2 and k(r, ) asr. Le m = P K(x) for x =. For all λ>, here exiss λ > such ha λ () ( m ) λ for all >. Then, for B = B(,r), ( ) B (P K)(x) B λ B dx B B(,) ( ) B m dx λ σ n ( B ) λ > B for large r>. Hence (4.8) k(r, ) = B (P K) eφ,b + as r +. These properies (4.6), (4.7), (4.8) and he coninui of k(r, ) ield (3.2). Nex we show (3.3). We noe ha ( ) n τβ () k(τ β (),) β = σ n n = P K eφ,b(,τβ ()) σ n n, P K eφ,b(,τβ ()) and we show (4.9) n P K eφ,b(,τβ ()) c < + for all >, (4.) n P K eφ,b(,τβ ()) as.

12 98 EIICHI NAKAI AND SHIGEO OKAMOTO B (4.2) we have (4.9). For all ε>, le K = G ε + H ε, G ε L, G ε <, H ε <ε. For C ε = G ε / (), using Lemma 4., we have ( ) ( ) P G ε (x) Gε (x) dx sup dx B(,r). B(,r) C ε z R n B(z,r) C ε Then P G ε eφ,b(,r) C ε for all r>. B (4.) and (4.2) we have εc n P H ε eφ,b(,r) () n for all r>. Hence, for n ε/c ε and r = τ β (), ( ) n P K eφ,b(,τβ ()) εc n () + C ε n c n ε () +. Therefore we have (3.3). 5. Proofs of Theorem Le Mf(x) =sup f(z) dz and M Φ f(x) =sup f Φ,B, B x B B B x where he supremum is aken over all balls B conaining x. The nex wo lemmas are [, Lemma 5.2] and [4, Lemma 2.2]. Lemma 5.. There exiss C> such ha, for all F L Φ (R n ) and for all >, ( ) F (x) x R n : M Φ F (x) >} C Φ dx. R n Lemma 5.2. If F L,andg is radial and decreasing, hen F (x) g(x) dx (MF)() g(x) dx. R n R n We have he following: Theorem 5.3. Le Φ and be a complemenar pair of N-funcions. Then here exiss a consan C> such ha ) (K F )(x) C ((M Φ F )(x ) x x n K eφ,b(, x x ) +(MF)(x ) K whenever F L Φ, K is a nonnegaive, radial and decreasing funcion on R n, x R n, x R n. Proof. Take x =, wihou loss of generali. Fix x. Then (K F )(x) K(x z) F (z) dz = I + I 2, R n where I and I 2 are inegrals over B = B(, 2 x ) andb C respecivel. Hölder s inequali shows ha I = K(x z) F (z) dz 2 B K(x ) eφ,b F Φ,B B 2 B K(x ) eφ,b (M Φ F )().

13 TANGENTIAL BOUNDARY BEHAVIOR OF THE POISSON INTEGRALS 99 Since B(, x ) B(x,2 x ) andk is radial and decreasing, ( ) K(z x) dz B(, 2 x ) B(,2 x ) λ ( ) K(z) = dz B(x, 2 x ) B(x,2 x ) λ ( ) K(z) dz. B(, x ) B(, x ) λ Hence K(x ) eφ,b(,2 x ) K eφ,b(, x ). and I 2 n+ σ n x n K eφ,b(, x ) (M Φ F )(). If z B C, hen x z z /2, hence K(x z) K(z/2), so ha I 2 = K(x z) F (z) dz K(z/2) F (z) dz B C B C MF() K( /2) =2 n MF() K, b Lemma 5.2. These wo esimaes prove he heorem. Since P and K are nonnegaive, radial and decreasing, so is P K. Hence Theorem 5.3 holds wih P K in place of K. B Hölder s inequali, MF CM Φ F. B Fubini s heorem, P K = P K = K. Hence Theorem 5.3 implies he following: Theorem 5.4. Le Φ and be a complemenar pair of N-funcions. If F L Φ,anduis defined in R n+ + b u(x, ) =(P K F )(x), hen u(x, ) C(M Φ F )(x )( K + k( x x,)), where k(r, ) = B(,r) P K eφ,b(,r) and x R n. Proof of Theorem 3.. If u = P [f] andf = K F, Theorem 5.4 shows ha (5.) u(x, ) C(M Φ F )(x )(β + K ) in Ω Φ K,β (x ). Thus M[Ω Φ K,β]f(x ) C(β + K )M Φ F (x ) for all x R n. Combining Lemma 5., we have Theorem 3.. Proof of Theorem 3.2. Le } E = E(β,ɛ) = x R n : lim sup u(x, ) f(x ) >ɛ. (x,) Ω Φ K,β,(x,) (x,) We shall prove ha E = for all β and for all ɛ. Foreachj N, here exiss G j C comp (R n ) such ha F G j Φ /j. Le g j = K G j and v j (x, ) =P g j (x). Then f g j Φ K F G j Φ as j +.

14 2 EIICHI NAKAI AND SHIGEO OKAMOTO We can use (2.5) for 3C(β + )(F G j )/ɛ and 3(f g j )/ɛ, where C is he consan in Theorem 3.. Le } E,j = x R n : lim sup u(x, ) v j (x, ) >ɛ/3, (x,) Ω Φ K,β,(x,) (x,) } E 2,j = x R n : lim sup v j (x, ) g j (x ) >ɛ/3, (x,) Ω Φ K,β,(x,) (x,) E 3,j = x R n : g j (x ) f(x ) >ɛ/3}. Then E E,j E 2,j E 3,j for j =, 2,. B Theorem 3. we have ( ) C(β +) F (x) Gj (x) E,j Φ dx R ɛ/3 n as j +. Since v j is coninuous on he closure of R n+ +, lim sup v j (x, ) g j (x ) =. (x,) Ω Φ K,β,(x,) (x,) Hence we have E 2,j =. And we have E 3,j = dx ( ) gj (x) f(x) Φ dx E 3,j Φ() R ɛ/3 n as j +. Proof of Proposiion 3.4. Le B = B(,τ β ()). Since P K eφ,b equals he norm of P K in he Orlicz space L eφ (B,dx/ B ), using (2.3), we have P K eφ,b = P K L eφ (B,dx/ B ) sup (P K)(x)F (x) dx/ B : B B } Φ( F (x) )dx/ B. Then here exiss F L Φ (R n ) such ha F (x), F (x) =forx/ B, Φ( F (x) )dx/ B and P K eφ,b 2 (P K)(x)F (x) dx/ B. B Le u = P [f], f = K F. Then u(,)=p K F () = (P K)( x)f (x) dx = B B (P K)(x)F (x) dx. Hence u(,) B P K eφ,b /2=k(τ β (),)/2 =β/2. If x B(,ω()), hen (,) Ω(x), so ha u(,) M[Ω]f(x). Hence B(,ω()) M[Ω]f(x) u(,)} M[Ω]f(x) > β/4}. We have, for β 4c, ( ) (5.2) σ n ω() n c F (x) M[Ω]f(x) >β/4} Φ dx β/4 Φ(F (x)) dx B σ n τ β () n, b (3.6). Then we have, for β 4c, ω() τ β ().

15 TANGENTIAL BOUNDARY BEHAVIOR OF THE POISSON INTEGRALS 2 In (5.2), if Φ 2, hen, for all β>, ( ) (5.3) σ n ω() n c F (x) M[Ω]f(x) >β/4} Φ dx β/4 C c,β Φ(F (x)) dx C c,β B C c,βσ n τ β () n, where C c,β depends on Φ, c and β. Proof of Theorem 3.5. Fix β > and le R = τ β. Since τ β () is increasing wih respec o β, and P K eφ,b(,r) is decreasing wih respec o r, wehave P K eφ,b(,τβ ()) P K eφ,b(,τβ ()) for β β. Hence ( ) n τβ () = β/ P K eφ,b(,τβ ()) β. τ β () β / P K eφ,b(,τβ ()) β For β b n β,wehave ( ) /n β bτ β () b τ β () τ β (). β Le Φ 2 2. B Theorem 3. x R n : M[Ω Φ K,β]f(x) >} ( ) C(β +) F (x) Φ dx. R n In Proof of Proposiion 3.4 and (5.3), using τ β, C(β +) and β insead of ω, c and β, respecivel, we have τ β () C β,β τ β (), where C β,β > depends on Φ, β, β. 6. Proofs of Proposiion 3.6 and Theorems To prove Proposiion 3.6, we sae a proposiion and wo lemmas. Proposiion 6.. Assume ha Φ 2, ha K and H are kernels, no in L eφ, and ha here are consans a, b, ɛ> such ha (6.) <a K(x) b<+ if < x <ɛ. H(x) Then here are consans a, b, such ha <a P K eφ,b(,r) (6.2) b < + P H eφ,b(,r) for <r<+ and <<. Proof. Le K = K + K, H = H + H, where K and H are he resricions of K and H o x <ɛ}. Then K L L. Hence ( P K P K eφ,b(,r) =inf λ>: ) } (x) dx B(,r) B(,r) λ ( K inf λ>: B(,r) sup ) } (x) dx K z R n λ () < +. B(z,r)

16 22 EIICHI NAKAI AND SHIGEO OKAMOTO The same is rue for P H eφ,b(,r). Since P K eφ,b(,r) and P H eφ,b(,r) end o + when (see (4.4) in Lemma 4.2), i follows ha he upper and lower limis of he raio in (6.2) are unchanged if K and H are replaced b K and H. Since ah K bh and 2, (6.2) holds. Lemma 6.2. (i) If ρ(r) is almos increasing for small r>, hen r ρ() d rρ(r) for small r>. (ii) If ρ(r) is almos increasing and ρ(r)/r β is almos decreasing for small r>wih β<n, hen ρ() ρ(r) d r n+ r n for small r>. (iii) If ρ(r)/r α is almos increasing for small r> wih <α n, hen r ρ() ρ(r) = d ρ(r) for small r>. (iv) If ρ(r)/r β is almos decreasing for small r> wih β>, hen ρ(r)/r β is also almos decreasing for small r>. Proof. (i) We noe ha ρ(r)/r n is decreasing. r r rρ(r) =ρ(r) n + r n n d ρ() d Crρ(r). (ii) If <r</2, hen ρ(r) 2n r n ρ(r) ρ() ρ(r) d C d C r n+ r n+ r β d C ρ(r) r n β+ n β r n. (iii) Using he decreasingness of ρ(r)/r n,wehave r ρ(r) =ρ(r) n r n n d (iv) For r<s,le =(r/s)u. Then r ρ() d = s r ρ() ρ((r/s)u) u d C ρ(r) r α r ( r β s du s) α+ d C α ρ(r). Lemma 6.3. Suppose ρ is almos increasing and ρ(r)/r β is almos decreasing for small r> wih <β<n.ifφ 2, hen r ( ρ() P K ρ eφ,b(,r) inf λ >:r n λ n + ρ() ) } (6.3) λ n+ n d, for < r/2. Proof. Le Using H ρ (x) = x x 2 2 ρ() P (x) d. ( x ), ( x <), ρ(u) u du.

17 TANGENTIAL BOUNDARY BEHAVIOR OF THE POISSON INTEGRALS 23 we have H ρ (x) x n+ B Lemma 6.2, we have B P P = P +,wehave For x + </2, le Then I 2 = If x, hen If x >, hen I Hence x + Therefore B(,) I = x + I ρ() x + ρ() x + x ρ() ρ() d + d for x < /2. x n+ K ρ (x) H ρ (x) for x < /2. (H ρ P )(x) = ρ() P + (x) d. ρ() P + (x) d, I 2 = x + c n ( + ) d. ( x 2 +( + ) 2 )(n+)/2 ρ() + d ( x + + ) n+ ρ() P H ρ (x) n ρ( x ) x n ρ() P + (x) d. x + ρ() ρ( x + ) d n+ ( x + ) n. + ρ() d ρ( x + ) ( x + + ) n+ n n. x n+ x + ρ() ( + ) d = ρ( x ) (( x + )ρ( x + )+ ρ( x + )) x n+ x n + ρ( x ) x n+. + ρ( x ) x n+ ( ) ( P H ρ (x) dx λ ρ() λ n 2 ) n ( x ), ( x ), when x + </2. ( ) ρ() 2 ( ) ρ() λ n n d λ n+ n d, r ( ρ() dx λ n + ρ() ) λ n+ n d. and ( ) P H ρ (x) B(,r)\B(,) λ This shows (6.3). Proof of Proposiion 3.6. Le τ = τ.from k(τ(),)= B(,τ()) P K ρ eφ,b(,τ()) =, i follows ha P K ρ eφ,b(,τ()) = B(,τ()) τ() n.

18 24 EIICHI NAKAI AND SHIGEO OKAMOTO B Lemma 6.3, we have τ() P K ρ eφ,b(,τ()) inf λ>:τ() n B (2.4) we have (3.7). If 2, hen Hence τ() n τ() Therefore we have (3.8). If, hen + ( ) s n s n ds < +. ( ρ() λ n + ρ() ) } λ n+ n d. ( τ() n ) ρ() n + τ()n ρ() n+ n d ( ) τ() Cτ() n n /τ() ( ) τ() n n d C s n s n ds C. τ() n ρ() n + τ()n ρ() n+ 2 τ()n ρ() n. p From he almos increasingness of (r)/r and he almos decreasingness of ρ(r)/r β,i follows ha ( ) ( τ() n ) ( ) ( ) p ρ() n + τ()n ρ() n+ C τ()n ρ() τ()n ρ() τ()n ρ() n C n n ( τ()n ρ() Hence τ() n τ() Cτ() n ( τ()n ρ() β ( ) ( ) p τ()n ρ() τ()n ρ() C n p β ( ) p (n β) τ()n ρ() n ( τ() n ) ρ() n + τ()n ρ() n+ n d ( ) ) p τ()n ρ() ( ) n n p ( ) p (n/p β) = C τ()n ρ() τ() n On he oher hand τ() ( τ() τ() n n ) ρ() n + τ()n ρ() n+ n d 2 ( ) τ() n τ()n ρ() n d 2 ( τ() n τ()n ρ() n n+ ) n d = C ( ) n τ() n ) p for. ( τ()n ρ() n ( τ()n ρ() Proof of Theorem 3.7. We show ha R is equivalen o τ in Proposiion 3.6. Then we have he conclusion b Theorem 3.5. n ). ).

19 TANGENTIAL BOUNDARY BEHAVIOR OF THE POISSON INTEGRALS 25 In he case (i), using ρ ρ, wehave τ() n ρ() n B (r) r p,wehave ( τ() τ() n n ρ() n + τ()n ρ() n+ τ()n ρ() n for. ) + τ()n ρ() n+ n τ() np /p ( ) p ρ(). n/p Using (3.8) in Proposiion 3.6, we have ha τ is equivalen o R in (3.). In he case (ii), i follows from (3.9) in Proposiion 3.6. Proof of Theorem 3.8. We have ha (r) r p l(r) p /p. Acuall, Φ(r) =r p l(r) implies Φ (r) r /p l(r) /p. From (2.) i follows ha (r) r /p l(r) /p. This implies (r) r p l(r) p /p. In he case (i), using ρ ρ, wehave and ( τ() n ρ() Le n τ() n ρ() n ) + τ()n ρ() n+ + τ()n ρ() n+ ( τ() n ) ρ() n ( τ() n ρ() τ() ( τ() E() =τ() n n ρ() n τ()n ρ() n for, n ) p ) p ( τ() n ρ() l n n ) p /p ( τ() n ) p ρ() /p l n d. for. Then, b (3.7) in Proposiion 3.6, we have C E() C. Choose δ>andν>so ha p δp < δp <ν<, /n 2δ and le B () =τ() n B 2 () =τ() n τ() ν τ() ν ( τ() n ρ() n ( τ() n ρ() n ) p ) p ( τ() n ) p ρ() /p l n d, l n ( ) p /p n d. If we show ha (6.4) <τ() ν <τ() < for small >, ( τ() n ) ( ) ρ() (6.5) l n l for τ() ν, (6.6) hen we have B (), B 2 () as, ( τ() E() τ() n ) p ( ) p ρ() /p n l n d ( ) p = τ() np /p /p m() p l d for small >,

20 26 EIICHI NAKAI AND SHIGEO OKAMOTO and hen ( ( ) ) p /p p/(np ) τ() R() = m() p l d. In he following we show (6.4) (6.6). From he almos increasingness of r δp l(r) p /p and i follows ha τ() n ρ() n C n, ( τ() n ρ() (6.7) l n ) p /p ( τ() n ) δp ( ρ() τ() n ρ() = n n ( τ() n ) δp ( ) δp ( ρ() C n n l ( = Cτ() δnp δnp /p m() δp l ) δp ( τ() n ρ() l ) p /p ) p /p Hence, using he almos increasingness of δnp /p m() p δp l ( ) p /p,wehave E() =τ() np /p τ() ( τ() n ) p ρ() /p m() p l d n τ() ( ) p Cτ() np /p δnp δnp /p m() p δp /p l d From C E() i follows ha n ) p /p for <, τ() <. τ() Cτ() np /p δnp 2δnp /p d Cτ() np /p δnp 2δnp /p. Cτ() ( δp)/(2δ) <τ() ν for small >. Then we have (6.4). For τ() ν <, we have C n τ()n ρ() n n/ν ρ() n = m() δ C n n/p δ n/ν, n/p δ n/ν We noe ha n/p δ n/ν >. Hence we have (6.5). B (6.7) we have B () =τ() np /p m() p l τ() ν Cτ() np /p δnp ( τ() n ρ() n τ() ν δnp Cτ() np /p δnp ) p /p d ( ) p /p m() p δp /p l d τ() ν 2δnp /p d Cτ() np /p δnp τ() 2νδnp /p = Cτ() (np /p)( δp 2νδ) as.

21 TANGENTIAL BOUNDARY BEHAVIOR OF THE POISSON INTEGRALS 27 Using he almos increasingness of δnp /p m() p l ( ) p /p,wehave ( ) p B 2 () =τ() np /p m() p /p l d τ() ν = τ() np /p τ() ν δnp Cτ() np /p ( /p+δnp /p m() p l τ() ν δnp /p d Cτ() np /p τ() νδnp /p ) p /p d = Cτ() (np /p)( νδ) as. In he cases (ii) and (iii), le W (r) = (r)/r r p l(r) p /p. Then W (r) r /(p ) l(r). Using (3.9) in Proposiion 3.6, we have ( ) τ()n ρ() W n ρ(), and hen τ() n ( ) ( ) ρ() n W ρ() ρ() l. /(p ) ρ() Hence τ() l(/ ρ())/n. ρ() p/n We noe ha ρ() α m() in he case (ii) and ha ρ() m() in he case (iii). Therefore we have ha τ is equivalen o R in (3.3) or in (3.4). Proof of Theorem 3.9. Le (6.8) w(, ) = τ()n ρ() n + τ()n ρ() n+. Then w(, ) w(, ) for>. B he increasingness of (r)/r, wehave τ() τ() τ() n (w(, )) n d τ() n w(, ) (w(, )) n d w(, ) = (w(, )) τ() ( ρ() + ρ() ) w(, ) 2 d C (w(, )) ρ(τ()). w(, ) B Proposiion 3.6 (i), we have (w(, )) (6.9) ρ(τ()) C. w(, ) From ρ(τ()) as, i follows ha w(, ) + as. Therefore we have R() =/ ρ() /n τ() for small >. Proof of Example 3.3. Le l(r) = log r for large r>, / log(/r) for small r>.

22 28 EIICHI NAKAI AND SHIGEO OKAMOTO Then (r) rl(r). Le w(, ) be as in (6.8). For small >, w(, ) is large. Then l(w(, )) = log w(, ). B (6.9), we have ( ) C τ()n ρ() ρ(τ())l(w(, )) ρ(τ()) log This shows ha C ρ(τ()) nlog τ() + n log + log ρ(). B ρ() (log(/)) for small >, we have τ() (log(/)) /n ɛ = R(), ɛ = n n + C. References [] J. R. Dorronsoro, Poisson inegrals of regular funcions, Trans. Amer. Mah. Soc. 297 (986), [2] S. Janson, On funcions wih condiions on he mean oscillaion, Ark. Ma. 4 (976), [3] Y. Mizua, Inegral represenaions, differeniabili properies and limis a infini for Beppo Levi funcions, Poenial Anal. 6 (997), [4] A. Nagel, W. Rudin and J. H. Shapiro, Tangenial boundar behavior of funcion in Dirichle-pe spaces, Ann. of Mah. 6 (982), [5] A. Nagel and E. M. Sein, On cerain maximal funcions and approach regions, Adv. in Mah. 54 (984), [6] E. Nakai, On he resricion of funcions of bounded mean oscillaion o he lower dimensional space, Arch. Mah. 43 (984), [7], On generalized fracional inegrals, Taiwanese J. Mah. 5 (2), [8], On generalized fracional inegrals in he Orlicz spaces on spaces of homogeneous pe, Sci. Mah. Jpn. 54 (2), [9], On generalized fracional inegrals on he weak Orlicz spaces, BMO φ, he Morre spaces and he Campanao spaces, Funcion spaces, inerpolaion heor and relaed opics (Lund, 2), 389 4, Waler de Gruer, Berlin, New York, 22. [] C. Pérez and R. L. Wheeden, Uncerain Principle esimaes for vecor fields, J. Func. Anal. 8 (2), [] M. M. Rao and Z. D. Ren, Theor of Orlicz Spaces, Marcel Dekker, Inc., New York, Basel and Hong Kong, 99. Eiichi Nakai, Deparmen of Mahemaics, Osaka Koiku Universi, Kashiwara, Osaka , Japan address: enakai@cc.osaka-koiku.ac.jp Shigeo Okamoo, Ikeda Senior High School Aached To Osaka Koiku Universi, -5-, Midorigaoka Ikeda-ci, Osaka , Japan address: sgo@mb9.seikou.ne.jp n