Research Article Nontangential Limits for Modified Poisson Integrals of Boundary Functions in a Cone

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1 Function Spaces and Applications Volume 2012, Article ID , 9 pages doi: /2012/ Research Article Nontangential Limits for Modified Poisson Integrals of Boundary Functions in a Cone Lei Qiao Department of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou , China Correspondence should be addressed to Lei Qiao, qiaocqu@163.com Received 17 May 2012; Accepted 8 July 2012 Academic Editor: Dachun Yang Copyright q 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 R n n 2 the n-dimensional Euclidean space. A point in R n is denoted by P X, x n, where X x 1,x 2,...,x n 1. The Euclidean distance of two points P and Q in R n is denoted by P Q. Also P O with the origin O of R n is simply denoted by P. The boundary, the closure, and the complement of a set S in R n are denoted by S, S,and S c, respectively. We introduce a system of spherical coordinates r, Θ, Θ θ 1,θ 2,...,θ n 1,inR n which are related to cartesian coordinates x 1,x 2,...,x n 1,x n by x n r cos θ 1. For positive functions h 1 and h 2, we say that h 1 h 2 if h 1 Mh 2 for some constant M>0. If h 1 h 2 and h 2 h 1, we say that h 1 h 2. For P R n and R>0, let BP, R denote the open ball with center at P and radius R in R n. The unit sphere and the upper half unit sphere are denoted by S n 1 and S n 1, respectively. For simplicity, a point 1, Θ on S n 1 and the set {Θ; 1, Θ Ω} for a set Ω, Ω S n 1 are often identified with Θ and Ω, respectively. For two sets Ξ R and Ω S n 1,theset{r, Θ R n ; r Ξ, 1, Θ Ω} in R n is simply denoted by Ξ Ω. In particular, the half space R S n 1 {X, x n R n ; x n > 0} will be denoted by T n. By C n Ω, we denote the set R Ω in R n with the domain Ω on S n 1. We call it a cone. Then T n is a special cone obtained by putting ΩS n 1. We denote the sets I Ω and I Ω

2 2 Function Spaces and Applications with an interval on R by C n Ω; I and S n Ω; I. ByS n Ω we denote S n Ω; 0, which is C n Ω {O}. Let Ω be a domain on S n 1 with smooth boundary. Consider the Dirichlet problem: Λ n λϕ 0onΩ, ϕ 0on Ω, 1.1 where Λ n is the spherical part of the Laplace operator Δ n : Δ n n 1 r r 2 r 2 Λ n r We denote the least positive eigenvalue of this boundary value problem by λ Ω and the normalized positive eigenfunction corresponding to λ Ω by ϕ Ω Θ, Ω ϕ 2 Ω Θdσ Θ 1, 1.3 where dσ Θ is the surface area on S n 1. We denote the solutions of the equation t 2 n 2t λ Ω 0byα Ω, β Ω α Ω,β Ω > 0. IfΩS n 1, then α Ω 1, β Ω n 1, and ϕ 1 Θ 2ns 1 n 1/2 cos θ 1, where s n is the surface area 2π n/2 Γn/2 1 of S 1. To simplify our consideration in the following, we will assume that if n 3, then Ω is a C 2,α -domain 0 <α<1on S n 1 surrounded by a finite number of mutually disjoint closed hypersurfaces e.g., see 1, pages for the definition of C 2,α -domain. Then by modifying Miranda s method 2, pages 7-8, we can prove the following inequality: ϕ Ω Θ distθ, Ω Θ Ω. 1.4 For any 1, Θ Ω, we have see 3 ϕ Ω Θ dist1, Θ, C n Ω, 1.5 which yields that δp rϕ Ω Θ, 1.6 where δp distp, C n Ω and P r, Θ C n Ω. Let G Ω P, QP r, Θ,Q t, Φ C n Ω be the Green function of C n Ω. We define the Poisson kernel K Ω P, Q by K Ω P, Q 1 c n n Q G Ω P, Q, 1.7

3 Function Spaces and Applications 3 where 2π n 2, c n n 2s n n 3, 1.8 Q S n Ω and / n Q denotes the differentiation at Q along the inward normal into C n Ω. In this paper, we consider functions f L p C n Ω, where 1 p <. Then the Poisson integral W Ω fpp C n Ω is defined by W Ω fp S n Ω where dσ Q is the surface area element on S n Ω. K Ω P, QfQdσ Q, 1.9 Remark 1.1. Let ΩS n 1. Then G S n 1P, Q log P Q log P Q n 2, P Q 2 n P Q 2 n n 3, 1.10 where Q Y, y n,thatis,q is the mirror image of Q Y, y n with respect to T n. Hence, for the two points P X, x n T n and Q Y, y n T n, we have c n K S n 1P, Q 2 P Q 2 x n n 2, G n S n 1P, Q Q 2n 2 P Q n x n n We fix an open, nonempty, and bounded set GΩ C n Ω. InC n Ω, we normalise the extension, with respect to GΩ,by PI Ω fp W Ω fp W Ω χ GΩ P, 1.12 where χ GΩ denotes the characteristic function of GΩ. Let ΓΩ,ζ {P r, Θ C n Ω : r, Θ ζ δp} 1.13 be a nontangential cone in C n Ω with vertex ζ C n Ω. We define ( ) 1/p ( ) 1 ℵ p f, l, P fq p dσq, l n 1 BP,l E p f GΩ { P GΩ : ℵ p ( f fp,l,p ) 0asl 0 }. 1.14

4 4 Function Spaces and Applications Note that, if f L p C n Ω, then GΩ \ E p GΩ 0 a.e. point is a Lebesgue f point. In T n, the following conclusion was proved by Brundin see 4, pages and Mizuta and Shimomura see 5, Theorem 3, respectively. In the unit disc, about related results, we refer the readers to the papers by Sjögren see 6, 7, Rönning see 8, and Brundin see 9. Theorem A. For a.e. ζ GS n 1 PI S n 1 fp)asp ζ along ΓS n 1, PI S n 1fP fζ (see Remark 1.1 for the definition of,ζ. Our aim is to generalize Theorem A to the conical case. Theorem 1.2. For any ζ E p f GΩ (in particular, for a.e. ζ GΩ) one has that PI ΩfP fζ as P ζ along ΓΩ,ζ. 2. Some Lemmas Lemma 2.1. One has K Ω P, Q r β Ω t αω 1 ϕ Ω Θ, ( ) resp.k Ω P, Q r α Ω t βω 1 ϕ Ω Θ, 2.1 for any P r, Θ C n Ω and any Q t, Φ S n Ω satisfying 0 <t/r 4/5 resp. 0 <r/t 4/5; K Ω P, Q rϕ ΩΘ P Q n, 2.2 for any P r, Θ C n Ω and any Q t, Φ S n Ω; 4r/5, 5r/4. Proof. These immediately follow from 10, Lemma 2, 11, Lemma 4 and Remark, and 1.4. Lemma 2.2. One has W Ω 1P O1 as P ζ G. 2.3 Proof. Write W Ω 1P E 1 E 2 E 3 U 1 P U 2 P U 3 P, 2.4 where E 1 S n (Ω; (0, 45 ( [ ( ( 5 4 ]), r E 2 S n Ω; )), 4 r, E 3 S n Ω; 5 r, 5 )) 4 r. 2.5

5 Function Spaces and Applications 5 By 2.1, we have the following estimates U 1 P r β Ω ϕ Ω Θ t αω 1 dσ Q E 1 s ( ) 2.6 n 4 βω ϕ Ω Θ, β Ω 5 U 2 P s ( ) n 4 αω ϕ Ω Θ. 2.7 α Ω 5 Next, we will estimate U 3 P. Take a sufficiently small positive number k such that E 3 Pr,Θ Λk B (P, 12 ) r, 2.8 where Λk { } P r, Θ C n Ω; inf 1, Θ 1,z <k,0 <r< z Ω. 2.9 Since P ζ G, we only consider the case P Λk. Nowput H i P { } Q E 3 ;2 i 1 δp P Q < 2 i δp Since S n Ω {Q R n : P Q <δp}, we have ip U 3 P i0 H i P rϕ Ω Θ P Q n dσ Q, 2.11 where ip is a positive integer satisfying 2 ip 1 δp r/2 < 2 ip δp. By 1.6 we have H i P rϕ Ω Θ P Q n dσ 1 Q rϕ Ω Θ H i P δp dσ Q rϕ ΩΘ δp s n 2 ip s n 2 ip 2.12 for i 0, 1, 2,...,iP. So U 3 P O Combining , Lemma 2.2 is proved. Lemma 2.3. One has W Ω χ GΩ P W Ω 1P O1 as P ζ GΩ. 2.14

6 6 Function Spaces and Applications Proof. In fact, we only need to prove U 4 P K Ω P, Qdσ Q O1. S n Ω GΩ 2.15 Write U 4 P S n Ω GΩ E 1 S n Ω GΩ E 2 S n Ω GΩ E 3 U 5 P U 6 P U 7 P, 2.16 where E 1, E 2,andE 3 are sets on S n Ω used in Lemma 2.2. Obviously, U 5 P U 1 P O1, U 6 P U 2 P O Further, we have by U 7 P rϕ Ω Θ S n Ω GΩ E 3 P Q n dσ Q s n d ζ ϕ ΩΘ P ζ GΩ, 2.19 where d inf Q ζ Q C nω GΩ Combining , 2.15 holds which gives the conclusion. 3. Proof of the Theorem 1.2 As P ζ GΩ, W Ω χ GΩ P O1from Lemmas 2.2 and 2.3. Now, let f L p C n Ω and ζ E p GΩ be given. We may, without loss of f generality, assume that fζ 0. Furthermore, we assume that P r, Θ ΓΩ,ζ. For short, let s r, Θ ζ. We write W Ω fp E 1 E 2 E 3 Bζ,2s E 3 B c ζ,2s V 1 fp V 2 fp V 3 fp V 4 fp, 3.1 where E 1, E 2,andE 3 are sets on S n Ω used in Lemma 2.2.

7 Function Spaces and Applications 7 By using Hölder s inequality, 2.1, we have the following estimates V1 fp r β Ω ϕ Ω Θ t αω 1 fqdσ Q E 1 r 1 n/p f p, V2 fp r 1 n/p f p. 3.2 Similar to the estimate of U 3 P in Lemma 2.2, we only consider the following inequality by 1.6 H i P rϕ Ω Θ P Q n dσ 1 Q rϕ Ω Θ H i P {2 i 1 δp} n dσ Q r α Ω ϕ Ω Θ t β Ω 1 fq dσq E r 1 n/p f p for i 0, 1, 2,...,iP, which is similar to the estimate of V 2 fp. So V3 fp r 1 n/p f p. 3.4 have Notice that P Q > 1/2 ζ Q in the case Q E 3 B c ζ, 2s. By1.6 and 2.2, we V4 fp δp δp i1 E 3 B c ζ,2s ( ) 1 n δp 2 i s i1 fq P Q n dσ Q E 3 Bζ,2 i1 s\bζ,2 i s ) δp ℵ 1 (f, 2 i1 s, ζ i1 2 i2 s E 3 Bζ,2 i1 s ( ) ℵ 1 f, l, ζ δp dl i1 2 i1 s l ( ) ℵ 1 f, l, ζ δp dl s l ( ) ℵ 1 f, l, ζ dl. l δp δp fq ζ Q n dσ Q fq dσq 3.5

8 8 Function Spaces and Applications Thus, it follows that PI Ω fp 1 [ V 1 fp V 2 fp V 3 fp V 4 fp ] O1 r 1 n/p ( ) ℵ 1 f, l, ζ f p δp dl. l Using the fact that s δp rϕ Ω Θ, weget δp PI Ω fp ℵ 1 ( f, 2s, ζ ) δp δp ℵ 1 ( f, l, ζ ) l dl It is clear that δp ℵ 1 ( f, l, ζ ) l dl 3.8 is a convergent integral, since ( ) ℵ 1 f, l,ζ s 1 n s n/q f p l s 1 n/p f p 3.9 from the Hölder s inequality. Now, as δp 0, we also have s 0. Since fζ 0 and since we have assumed that ζ E p f GΩ andthusthatζ E1 f GΩ, it follows that PI ΩfP 0 fζ as P r, Θ ζ along ΓΩ,ζ. This concludes the proof. Acknowledgments This paper is supported by SRFDP no and NSF of China no References 1 D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, Germany, C. Miranda, Partial Differential Equations of Elliptic Type, Springer, Berlin, Germany, R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I, Interscience Publishers., New York, NY, USA, M. Brundin, Boundary behavior of eigenfunctions for the hyperbolic Laplacian [Ph.D. thesis], Göteborg University and Chalmers University of Technology, Gothenburg, Sweden, Y. Mizuta and T. Shimomura, Growth properties for modified Poisson integrals in a half space, Pacific Mathematics, vol. 212, no. 2, pp , P. Sjögren, Une remarque sur la convergence des fonctions propres du laplacien à valeur propre critique, in Théorie du Potentiel (Orsay, 1983), vol of Lecture Notes in Mathematics, pp , Springer, Berlin, Germany, 1984.

9 Function Spaces and Applications 9 7 P. Sjögren, Approach regions for the square root of the Poisson kernel and bounded functions, Bulletin of the Australian Mathematical Society, vol. 55, no. 3, pp , J.-O. Rönning, Convergence results for the square root of the Poisson kernel, Mathematica Scandinavica, vol. 81, no. 2, pp , M. Brundin, Approach regions for the square root of the Poisson kernel and weak Lp boundary functions [thesis],göteborg University and Chalmers University of Technology, M. Essén and J. L. Lewis, The generalized Ahlfors-Heins theorem in certain d-dimensional cones, Mathematica Scandinavica, vol. 33, pp , V. S. Azarin, Generalization of a theorem of Hayman on subharmonic functions in an m-dimensional cone, American Mathematical Society Translations, vol. 2, no. 80, pp , 1969.

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