JFSA Journal of Function Spaces and Applications 1758-4965 0972-6802 Hindawi Publishing Corporation 825240 10.1155/2012/825240 825240 Research Article Nontangential Limits for Modified Poisson Integrals of Boundary Functions in a Cone Qiao Lei Yang Dachun Department of Mathematics and Information Science Henan University of Economics and Law Zhengzhou 450002 China 2012 1 8 2012 2012 17 05 2012 08 07 2012 2012 Copyright © 2012 Lei Qiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Our aim in this paper is to deal with non-tangential limits for modified Poisson integrals of boundary functions in a cone, which generalized results obtained by Brundin and Mizuta-Shimomura.

1. Introduction and Main Results

Let R and R+ be the set of all real numbers and the set of all positive real numbers, respectively. We denote by Rn(n2) the n-dimensional Euclidean space. A point in Rn is denoted by P=(X,xn), where X=(x1,x2,,xn-1). The Euclidean distance of two points P and Q in Rn is denoted by |P-Q|. Also |P-O| with the origin O of Rn is simply denoted by |P|. The boundary, the closure, and the complement of a set S in Rn are denoted by S, S¯, and Sc, respectively.

We introduce a system of spherical coordinates (r,Θ),  Θ=(θ1,θ2,,θn-1), in Rn which are related to cartesian coordinates (x1,x2,,xn-1,xn) by xn=rcosθ1.

For positive functions h1 and h2, we say that h1h2 if h1Mh2 for some constant M>0. If h1h2 and h2h1, we say that h1h2.

For PRn and R>0, let B(P,R) denote the open ball with center at P and radius R in Rn. The unit sphere and the upper half unit sphere are denoted by Sn-1 and S+n-1, respectively. For simplicity, a point (1,Θ) on Sn-1 and the set {Θ;(1,Θ)Ω} for a set Ω, ΩSn-1 are often identified with Θ and Ω, respectively. For two sets ΞR+ and ΩSn-1, the set {(r,Θ)Rn;rΞ,(1,Θ)Ω} in Rn is simply denoted by Ξ×Ω. In particular, the half space R+×S+n-1={(X,xn)Rn;xn>0} will be denoted by Tn.

By Cn(Ω), we denote the set R+×Ω in Rn with the domain Ω on Sn-1. We call it a cone. Then Tn is a special cone obtained by putting Ω=S+n-1. We denote the sets I×Ω and I×Ω with an interval on R by Cn(Ω;I) and Sn(Ω;I). By Sn(Ω) we denote Sn(Ω;(0,+)) which is Cn(Ω)-{O}.

Let Ω be a domain on Sn-1 with smooth boundary. Consider the Dirichlet problem: (1.1)(Λn+λ)φ=0onΩ,φ=0onΩ, where Λn is the spherical part of the Laplace operator Δn:(1.2)Δn=n-1rr+2r2+Λnr2. We denote the least positive eigenvalue of this boundary value problem by λΩ and the normalized positive eigenfunction corresponding to λΩ by φΩ(Θ), (1.3)ΩφΩ2(Θ)dσΘ=1, where dσΘ is the surface area on Sn-1. We denote the solutions of the equation t2+(n-2)t-λΩ=0 by αΩ,-βΩ (αΩ,βΩ>0). If Ω=S+n-1, then αΩ=1,βΩ=n-1, and φ1(Θ)=(2nsn-1)1/2cosθ1, where sn is the surface area 2πn/2(Γ(n/2))-1 of S1.

To simplify our consideration in the following, we will assume that if n3, then Ω is a C2,α-domain (0<α<1) on Sn-1 surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see (, pages 88-89) for the definition of C2,α-domain). Then by modifying Miranda’s method (, pages 7-8), we can prove the following inequality: (1.4)φΩ(Θ)dist(Θ,Ω)(ΘΩ).

For any (1,Θ)Ω, we have (see ) (1.5)φΩ(Θ)dist((1,Θ),Cn(Ω)), which yields that (1.6)δ(P)rφΩ(Θ), where δ(P)=dist(P,Cn(Ω)) and P=(r,Θ)Cn(Ω).

Let GΩ(P,Q)(P=(r,Θ),Q=(t,Φ)Cn(Ω)) be the Green function of Cn(Ω). We define the Poisson kernel KΩ(P,Q) by (1.7)KΩ(P,Q)=1cnnQGΩ(P,Q), where (1.8)cn={2πn=2,(n-2)snn3,QSn(Ω) and /nQ denotes the differentiation at Q along the inward normal into Cn(Ω).

In this paper, we consider functions fLp(Cn(Ω)), where 1p<. Then the Poisson integral WΩf(P)(PCn(Ω)) is defined by (1.9)WΩf(P)=Sn(Ω)KΩ(P,Q)f(Q)dσQ, where dσQ is the surface area element on Sn(Ω).

Remark 1.1.

Let Ω=S+n-1. Then (1.10)GS+n-1(P,Q)={log|P-Q*|-log|P-Q|n=2,|P-Q|2-n-|P-Q*|2-nn3, where Q*=(Y,-yn), that is, Q* is the mirror image of Q=(Y,yn) with respect to Tn. Hence, for the two points P=(X,xn)Tn and Q=(Y,yn)Tn, we have (1.11)cnKS+n-1(P,Q)=nQGS+n-1(P,Q)={2|P-Q|-2xnn=2,2(n-2)|P-Q|-nxnn3.

We fix an open, nonempty, and bounded set G(Ω)Cn(Ω). In Cn(Ω), we normalise the extension, with respect to G(Ω), by (1.12)PIΩf(P)=WΩf(P)WΩχG(Ω)(P), where χG(Ω) denotes the characteristic function of G(Ω).

Let (1.13)Γ(Ω,ζ)={P=(r,Θ)Cn(Ω):|(r,Θ)-ζ|δ(P)} be a nontangential cone in Cn(Ω) with vertex ζCn(Ω).

We define (1.14)p(f,l,P)=(1ln-1B(P,l)|f(Q)|pdσQ)1/p,Efp(G(Ω))={PG(Ω):p(f-f(P),l,P)0asl0}.

Note that, if fLp(Cn(Ω)), then |G(Ω)𝔼fp(G(Ω))|=0 (a.e. point is a Lebesgue point).

In Tn, the following conclusion was proved by Brundin (see (, pages 11–16)) and Mizuta and Shimomura (see (, Theorem 3)), respectively. In the unit disc, about related results, we refer the readers to the papers by Sjögren (see [6, 7]), Rönning (see ), and Brundin (see ).

Theorem A.

For a.e. ζG(S+n-1), 𝒫S+n-1f(P)f(ζ) (see Remark 1.1 for the definition of 𝒫S+n-1f(P)) as Pζ along Γ(S+n-1,ζ).

Our aim is to generalize Theorem A to the conical case.

Theorem 1.2.

For any ζ𝔼fp(G(Ω)) (in particular, for a.e. ζG(Ω)) one has that 𝒫Ωf(P)f(ζ) as Pζ along Γ(Ω,ζ).

2. Some Lemmas Lemma 2.1.

One has (2.1)KΩ(P,Q)r-βΩtαΩ-1φΩ(Θ),(resp.KΩ(P,Q)rαΩt-βΩ-1φΩ(Θ)), for any P=(r,Θ)Cn(Ω) and any Q=(t,Φ)Sn(Ω) satisfying 0<t/r4/5(resp.0<r/t4/5); (2.2)KΩ(P,Q)rφΩ(Θ)|P-Q|n, for any P=(r,Θ)Cn(Ω) and any Q=(t,Φ)Sn(Ω;(4r/5,5r/4)).

Proof.

These immediately follow from (, Lemma 2), (, Lemma 4 and Remark), and (1.4).

Lemma 2.2.

One has (2.3)WΩ1(P)=O(1) as PζG.

Proof.

Write (2.4)WΩ1(P)=E1+E2+E3=U1(P)+U2(P)+U3(P), where (2.5)E1=Sn(Ω;(0,45r]),E2=Sn(Ω;[54r,)),E3=Sn(Ω;(45r,54r)).

By (2.1), we have the following estimates (2.6)U1(P)r-βΩφΩ(Θ)E1tαΩ-1dσQsnβΩ(45)βΩφΩ(Θ),(2.7)U2(P)snαΩ(45)αΩφΩ(Θ).

Next, we will estimate U3(P). Take a sufficiently small positive number k such that (2.8)E3P=(r,Θ)Λ(k)B(P,12r), where (2.9)Λ(k)={P=(r,Θ)Cn(Ω);infzΩ|(1,Θ)-(1,z)|<k,0<r<}.

Since PζG, we only consider the case PΛ(k). Now put (2.10)Hi(P)={QE3;2i-1δ(P)|P-Q|<2iδ(P)}.

Since Sn(Ω){QRn:|P-Q|<δ(P)}=, we have (2.11)U3(P)i=0i(P)Hi(P)rφΩ(Θ)|P-Q|ndσQ, where i(P) is a positive integer satisfying 2i(P)-1δ(P)r/2<2i(P)δ(P).

By (1.6) we have (2.12)Hi(P)rφΩ(Θ)|P-Q|ndσQrφΩ(Θ)Hi(P)1δ(P)dσQ=rφΩ(Θ)δ(P)sn2i(P)sn2i(P) for i=0,1,2,,i(P).

So (2.13)U3(P)O(1).

Combining (2.6)–(2.13), Lemma 2.2 is proved.

Lemma 2.3.

One has (2.14)WΩχG(Ω)(P)=WΩ1(P)+O(1)asPζG(Ω).

Proof.

In fact, we only need to prove (2.15)U4(P)=Sn(Ω)-G(Ω)KΩ(P,Q)dσQO(1).

Write (2.16)U4(P)=(Sn(Ω)-G(Ω))E1+(Sn(Ω)-G(Ω))E2+(Sn(Ω)-G(Ω))E3=U5(P)+U6(P)+U7(P), where E1, E2, and E3 are sets on Sn(Ω) used in Lemma 2.2.

Obviously, (2.17)U5(P)U1(P)O(1),(2.18)U6(P)U2(P)O(1).

Further, we have by (2.2) (2.19)U7(P)rφΩ(Θ)(Sn(Ω)-G(Ω))E31|P-Q|ndσQsnd|ζ|φΩ(Θ)(PζG(Ω)), where (2.20)d=infQCn(Ω)-G(Ω)|Q-ζ|.

Combining (2.17)–(2.19), (2.15) holds which gives the conclusion.

3. Proof of the Theorem <xref ref-type="statement" rid="thm1">1.2</xref>

As PζG(Ω), WΩχG(Ω)(P)=O(1) from Lemmas 2.2 and 2.3.

Now, let fLp(Cn(Ω)) and ζ𝔼fp(G(Ω)) be given. We may, without loss of generality, assume that f(ζ)=0. Furthermore, we assume that P=(r,Θ)Γ(Ω,ζ). For short, let s=|(r,Θ)-ζ|. We write (3.1)WΩf(P)=E1+E2+E3B(ζ,2s)+E3Bc(ζ,2s)=V1f(P)+V2f(P)+V3f(P)+V4f(P), where E1, E2, and E3 are sets on Sn(Ω) used in Lemma 2.2.

By using Hölder’s inequality, (2.1), we have the following estimates (3.2)|V1f(P)|r-βΩφΩ(Θ)E1tαΩ-1f(Q)dσQr(1-n)/pfp,|V2f(P)|r(1-n)/pfp.

Similar to the estimate of U3(P) in Lemma 2.2, we only consider the following inequality by (1.6) (3.3)Hi(P)rφΩ(Θ)|P-Q|ndσQrφΩ(Θ)Hi(P)1{2i-1δ(P)}ndσQrαΩφΩ(Θ)E2t-βΩ-1|f(Q)|dσQr(1-n)/pfp for i=0,1,2,,i(P), which is similar to the estimate of V2f(P).

So (3.4)|V3f(P)|r(1-n)/pfp.

Notice that |P-Q|>(1/2)|ζ-Q| in the case QE3Bc(ζ,2s). By (1.6) and (2.2), we have (3.5)|V4f(P)|δ(P)E3Bc(ζ,2s)|f(Q)||P-Q|ndσQδ(P)i=1E3(B(ζ,2i+1s)B(ζ,2is))|f(Q)||ζ-Q|ndσQδ(P)i=1(12is)nE3B(ζ,2i+1s)|f(Q)|dσQδ(P)i=11(f,2i+1s,ζ)δ(P)i=12i+1s2i+2s1(f,l,ζ)ldlδ(P)s1(f,l,ζ)ldlδ(P)δ(P)1(f,l,ζ)ldl.

Thus, it follows that (3.6)|PIΩf(P)|1O(1)[|V1f(P)|+|V2f(P)|+|V3f(P)|+|V4f(P)|]r(1-n)/pfp+δ(P)    δ(P)1(f,l,ζ)ldl.

Using the fact that sδ(P)rφΩ(Θ), we get (3.7)|PIΩf(P)|1(f,2s,ζ)+δ(P)δ(P)1(f,l,ζ)ldl.

It is clear that (3.8)δ(P)1(f,l,ζ)ldl is a convergent integral, since (3.9)1(f,l,ζ)ls-1-nsn/qfps-1-(n/p)fp from the Hölder’s inequality.

Now, as δ(P)0, we also have s0. Since f(ζ)=0 and since we have assumed that ζ𝔼fp(G(Ω)) (and thus that ζ𝔼f1(G(Ω))), it follows that 𝒫Ωf(P)0=f(ζ) as P=(r,Θ)ζ along Γ(Ω,ζ). This concludes the proof.

Acknowledgments

This paper is supported by SRFDP (no. 20100003110004) and NSF of China (no. 11071020).

Gilbarg D. Trudinger N. S. Elliptic Partial Differential Equations of Second Order 1977 Berlin, Germany Springer x+401 0473443 Miranda C. Partial Differential Equations of Elliptic Type 1970 Berlin, Germany Springer xii+370 0284700 Courant R. Hilbert D. Methods of Mathematical Physics. Vol. I 1953 New York, NY, USA Interscience Publishers. xv+561 0065391 ZBL0053.02805 Brundin M. Boundary behavior of eigenfunctions for the hyperbolic Laplacian [Ph.D. thesis] 2002 Gothenburg, Sweden Göteborg University and Chalmers University of Technology Mizuta Y. Shimomura T. Growth properties for modified Poisson integrals in a half space Pacific Journal of Mathematics 2003 212 2 333 346 10.2140/pjm.2003.212.333 2038052 Sjögren P. Une remarque sur la convergence des fonctions propres du laplacien à valeur propre critique Théorie du Potentiel (Orsay, 1983) 1984 1096 Berlin, Germany Springer 544 548 Lecture Notes in Mathematics 10.1007/BFb0100130 890377 ZBL0563.31003 Sjögren P. Approach regions for the square root of the Poisson kernel and bounded functions Bulletin of the Australian Mathematical Society 1997 55 3 521 527 10.1017/S0004972700034183 1456282 ZBL0903.31002 Rönning J.-O. Convergence results for the square root of the Poisson kernel Mathematica Scandinavica 1997 81 2 219 235 1613784 ZBL0902.42009 Brundin M. Approach regions for the square root of the Poisson kernel and weak Lp boundary functions [thesis] 1999 Göteborg University and Chalmers University of Technology Essén M. Lewis J. L. The generalized Ahlfors-Heins theorem in certain d-dimensional cones Mathematica Scandinavica 1973 33 111 129 0348131 Azarin V. S. Generalization of a theorem of Hayman on subharmonic functions in an m-dimensional cone American Mathematical Society Translations 1969 2 80 119 138