# A note on the boundary behavior for a modiﬁed Green function in the upper-half space

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3 Zhag ad Pisarev Boudary Value Problems (015) 015:114 Page 3 of 7 the there exists a Borel set E H with properties: x α β+δ+1 (1) lim x 0,x H E (1 + x ) G α,m(x, μ)=0; +m α+δ+ () i( α+β+δ) C α,β,δ (E i )<, i=1 where E i = x E : i x < i+1 }. Remar By usig Lemma 4 below, coditio () i Theorem with α =,β =0,δ =0 meas that E is -thi at H i the sese of [10]. Some lemmas Throughout this paper, let M deote various costats idepedet of the variables i questios, which may be differet from lie to lie. Lemma 1 There exists a positive costat M such that G α (x, y) M, where 0< x y α+ α, x =(x 1, x,...,x ) ad y =(y 1, y,...,y ) i H. This ca be proved by a simple calculatio. Lemma Gegebauer polyomials have the followig properties: (1) C ω(t) Cω Ɣ(ω+) (1) = Ɣ(ω)Ɣ(+1), t 1; d () dt Cω (t)=ωcω+1 1 (t), 1; (3) =0 Cω (1)r =(1 r) ω ; (4) C α (t) C α (t ) ( α)c α+ 1 (1) t t, t 1, t 1. Proof (1) ad () ca be derived from [1], p.3. Equality (3) follows from expressio (1.1) by taig t = 1; property (4) is a easy cosequece of the mea value theorem, (1) ad also (). Lemma 3 For x, y R (α = =),we have the followig properties: (1) I m =0 x m 1 x x y +1 =0 ; y + () I =0 x+m+1 m+1 x y x m ; (3) G,m (x, y) G (x, y) M m x y x 1 =1 ; y +1 (4) G,m (x, y) M x y x 1 =m+1. y +1 The followig lemma ca be proved by usig Fuglede (see [11], Théorèm 7.8). Lemma 4 For ay Borel set E i H, we have C α (E)=Ĉ α (E), where Ĉ α (E)=if λ(h), α = α,0,0, the ifimum beig tae over all o-egative measures λ o H such that α (λ, x) 1 for every x E. Followig [10], we say that a set E H is α-thi at the boudary H if i( α) C α (E i )<, i=1 where E i = x E : i x < i+1 }. x y

4 Zhag ad Pisarev Boudary Value Problems (015) 015:114 Page 4 of 7 3 Proof of Theorem We write [ G α,m (x, μ)= G α (x, y) + G α (x, y) + Gα,m (x, y) G α (x, y) ] G 1 G G 3 + G α,m (x, y) + G α,m (x, y) G 4 G 5 = U 1 (x)+u (x)+u 3 (x)+u 4 (x)+u 5 (x), where G 1 = y H : x y x }, G = y H : y 1, x } < x y 3 x, G 3 = y H : y 1, x y 3 x }, G 4 = y H : y 1, x y >3 x }, G 5 = y H : y <1, x y > x }. We distiguish the followig two cases. Case 1. 0 < α <. By assumptio (1.)wecafidasequecea i } of positive umbers such that lim i a i = ad i=1 a ib i <,where b i = y H: i 1 <y < i+ } Cosider the sets +m α+δ+. E i = x H : i x < i+1 x α β+δ+1 }, (1 + x ) U 1(x) a 1 +m α+δ+ i (i 1)β for i =1,,...Set G = ( B x, x ). x E i The G y H : i 1 < y < i+ }.Letν be a o-egative measure o H such that S ν E i,wheres ν is the support of ν.thewehave α,β,δ (y, ν) 1fory H ad H dν a i ( i+1)β x α β+δ+1 H (1 + x ) U 1(x) dν(x) +m α+δ+ Ma i ( i+1)β ( i+1)( α+δ+1) α,β,δ (y, ν) G Ma i ( i+1)β ( i+1)( α+δ+1) i+1 M α+β+δ+ i( α+β+δ) a i b i. y H: i 1 <y < i+ } y δ +m α+δ+ +m α+δ+

5 Zhag ad Pisarev Boudary Value Problems (015) 015:114 Page 5 of 7 So that C α,β,δ (E i ) M i( α+β+δ) a i b i, which yields i( α+β+δ) C α,β,δ (E i )<. i=1 Settig E = i=1 E i, we see that () i Theorem is satisfied ad lim x 0,x H E x α β+δ+1 (1 + x ) +m α+δ+ U 1(x)=0. (3.1) For U (x), by Lemma 1 we have U (x) y Mx G x y α+ Mx α 1 x +m α+δ+ 1 G y δ Mx α 1 x +m α+ G +m α+δ+ +m α+δ+. (3.) Note that C0 ω x y (t) 1. By (3) ad (4) i Lemma,wetaet = x y, t = x y x y i Lemma (4) ad obtai U 3 (x) G 3 m =1 Mx x m x α+ ( α)c y α+ 1 (1) x y x y m =1 α+ y C 1 1 (1) G3 δ y +m α+δ+ +m α+δ+ +m α+δ+ Mx x m. (3.3) Similarly, we have by (3) ad (4) i Lemma U4 (x) G 4 =m+1 Mx x m x y α+ ( α)c α+ 1 (1) x y x y =m+1 α+ y C 1 1 (1) G4 δ+1 1 y +m α+δ+ +m α+δ+ +m α+δ+ Mx x m. (3.4) Fially, by Lemma 1,wehave U 5 (x) Mx α 1 G5. (3.5) +m α+δ+

7 Zhag ad Pisarev Boudary Value Problems (015) 015:114 Page 7 of 7 Refereces 1. Szegö, G: Orthogoal Polyomials. America Mathematical Society Colloquium Publicatios, vol. 3. Am. Math. Soc., Providece (1975). Re, YD, Yag, P: Growth estimates for modified Neuma itegrals i a half space. J. Iequal. Appl. 013, 57 (013) 3. Xu, G, Yag, P, Zhao, T: Dirichlet problems of harmoic fuctios. Boud. Value Probl. 013, 6 (013) 4. Yag, DW, Re, YD: Dirichlet problem o the upper half space. Proc. Idia Acad. Sci. Math. Sci. 14(), (014) 5. Qiao, L: Itegral represetatios for harmoic fuctios of ifiite order i a coe. Results Math. 61, 6-74 (01) 6. Qiao, L, Pa, GS: Geeralizatio of the Phragmé-Lidelöf theorems for subfuctios. It. J. Math. 4(8), (013) 7. Liao, Y, Su, BY: Solutios of the Dirichlet problem i a tube domai. Acta Math. Si. 57(6), (014) 8. Su, BY: Dirichlet problem for the Schrödiger operator i a half space. Abstr. Appl. Aal. 01, Article ID (01) 9. Polidoro, S, Ragusa, MA: Harac iequality for hypoelliptic ultraparabolic equatios with a sigular lower order term. Rev. Mat. Iberoam. 4(3), (008) 10. Armitage, H: Tagetial behavior of Gree potetials ad cotractive properties of L p -potetials. Toyo J. Math. 9, 3-45 (1986) 11. Fuglede, B: Le théorèm du miimax et la théorie fie du potetiel. A. Ist. Fourier 15, (1965)

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