Poisson-type inequalities for growth properties of positive superharmonic functions

Journal of Inequalities and Applications, Jan 2017

In this paper, we present new Poisson-type inequalities for Poisson integrals with continuous data on the boundary. The obtained inequalities are used to obtain growth properties at infinity of positive superharmonic functions in a smooth cone.

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Poisson-type inequalities for growth properties of positive superharmonic functions

Luan and Vieira Journal of Inequalities and Applications Poisson-type inequalities for growth properties of positive superharmonic functions Kuan Luan 0 John Vieira 0 0 1 Introduction Cartesian coordinates of a point G of R In this paper, we present new Poisson-type inequalities for Poisson integrals with continuous data on the boundary. The obtained inequalities are used to obtain growth properties at infinity of positive superharmonic functions in a smooth cone. where  ≤ r < +∞, -  π ≤ θn- <  π and  ≤ θj ≤ π for  ≤ j ≤ n -  (n ≥ ). We denote the unit sphere and the upper half unit sphere by Sn- and Sn+-, respectively. Let ⊂ Sn-. The point (, ) and the set { ; (, ) ∈ } are identified with and , respectively. Let × denote the set {(r, ) ∈ Rn; r ∈ , (, ) ∈ }, where ⊂ R+. The set R+ × is denoted by n( ), which is called a cone. Especially, the set R+ × Sn+- is called the upper-half space, which is denoted by Tn. Let I ⊂ R. Two sets I × and I × ∂ are denoted by n( ; I) and n( ; I), respectively. We denote n( ; R+) by n( ), which is ∂ n( ) - {O}. Let B(G, l) denote the open ball, where G ∈ Rn is the center and l >  is the radius. Poisson-type inequality; continuous data; growth property - xn = r cos θ, x = r( jn=– sin θj), n = , xn–m+ = r( jm=– sin θj) cos θm, n ≥ , E ⊂ In spherical coordinate the Laplace operator is n = r– n + r–(n – ) h =  on ∂ . where n is the Beltrami operator. Now we consider the boundary value problem If the least positive eigenvalue of it is denoted by τ , then we can denote by h ( ) the normalized positive eigenfunction corresponding to it. We denote by ι (> ) and –κ (< ) two solutions of the problem t + (n – )t – τ = , Then ι + κ is denoted by for the sake of simplicity. Remark  In the case = Sn+–, it follows that (I) ι =  and κ = n – . (II) h ( ) = wnn cos θ, where wn is the surface area of Sn–. It is easy to see that the set ∂ n( ) ∪ {∞} is the Martin boundary of n( ). For any G ∈ n( ) and any H ∈ ∂ n( ) ∪ {∞}, if the Martin kernel is denoted by MK(G, H), where a reference point is chosen in advance, then we see that (see []) MK(G, ∞) = rι h ( ) and where G = (r, ) ∈ n( ) and c is a positive real number. We shall say that two positive real valued functions f and g are comparable and write f ≈ g if there exist two positive constants c ≤ c such that cg ≤ f ≤ cg. Remark  Let ∈ . Then h ( ) and dist( , ∂ ) are comparable. Remark  Let (G) = dist(G, ∂ n( )). Then h ( ) and (G) are comparable for any (, ) ∈ (see []). Remark  Let  ≤ α ≤ n. Then h ( ) ≤ c( , n){h ( )}–α , where c( , n) is a constant depending on and n (e.g. see [], pp.-). Definition  For any G ∈ n( ) and any H ∈ n( ). If the Green function in n( ) is defined by GF (G, H), then: (I) The Poisson kernel can be defined by POI (G, H) = ∂nH where G ∈ n( ) and ν is a positive measure in n( ). (II) The Poisson integral with g can be defined by POI [g](G) = on Rn – n( ; (, +∞)). h ( )t–κ dν on n( ; (, +∞)),  on Rn – n( ; (, +∞)). Definition  Let μ and ν be defined in Definitions  and , respectively. Then the positive measure ξ is defined by Remark  Let = Sn+–. Then GF Sn+– (G, H) = log |G – H∗| – log |G – H| if n = , |G – H|–n – |G – H∗|–n if n ≥ , where G = (X, xn), H∗ = (Y , –yn), that is, H∗ is the mirror image of H = (Y , yn) on ∂Tn. Hence, for the two points G = (X, xn) ∈ Tn and H = (Y , yn) ∈ ∂Tn, we have d = Remark  Let = Sn+–. Then we define d (y) = Definition  Let λ be any positive measure on Rn having finite total mass. Then the maximal function M(G; λ, β) is defined by EX( ; λ, β) = G = (r, ) ∈ Rn – {O}; M(G; λ, β)rβ > where is a sufficiently small positive number. Remark  Let β >  and λ({P}) >  for any P = O. Then (I) Then M(G; λ, β) = +∞. (II) {G ∈ Rn – {O}; λ({P}) > } ⊂ EX( ; λ, β). Recently, Qiao and Wang (see [], Corollary . with m = ) proved classical Poissontype inequalities for Poisson integrals in a half space. Applications of them were also developed by Pang and Ychussie (see []) and Xue and Wang (see []). In particular, Huang (see []) further obtained Schrödinger-Poisson-type inequalities for Poisson-Schrödinger integrals and gave their related applications. ∂Tn g(y)  + |y| –n dy < ∞. as |x| → ∞ in Tn. 2 Results Our first aim in this paper is to prove the following result, which is a generalization of Theorem A. For similar results with respect to Schrödinger operator, we refer the reader to the literature (see [, ]). Theorem  Let POI μ(G) ≡ +∞ for any G = (r, ) ∈ n( ), where μ is a positive measure on n( ). Then < ∞. dt < ∞. Then the Poisson integral POI [g](G) is harmonic in n( ) and Remark  If = Sn+–, then it is easy to see that () is equivalent to () and () is a finite sum, then the set EX( ; μ , ) is a bounded set and () reduces to () in the case α = n from Remark . Corollary  If μ is a positive measure on ∂Tn satisfying POISn+– μ(x) ≡ +∞ for any x = (X, xn) ∈ Tn, then < ∞. Let dμ = |g| dσH for any H = (t, ) ∈ n( ). Then we have the following result, which generalizes Theorem A to the conical case. Corollary  If g is a measurable function on n( ) satisfying < ∞. The following result is very well known. We quote it from []. Theorem B (see []) Let  < w(G) be a superharmonic function in Tn. Then there exist a positive measure μ on ∂Tn and a positive measure ν on Tn such that w(x) can be uniquely decomposed as w(x) = cxn + POISn+– μ(x) + GF Sn+– ν(x), Theorem C (see [], Theorem ) Let  < w(G) be a superharmonic function in n( ). Then there exist a positive measure μ on n( ) and a positive measure ν in n( ) such that w(G) can be uniquely decomposed as c(w) = and c(w) = As an application of Theorem  and Lemma  in Section , we give the growth properties of positive superharmonic functions at infinity in a cone. w(G) – c(w)MK(G, ∞) – c(w)MK(G, O) = o rι Theorem  immediately gives the following corollary. Corollary  Let w(x) (≡ +∞) (x = (X, xn) ∈ Tn) be defined by (). Then w(x) – cxn = o(|x|) for any x ∈ Tn – EX( ; , n – ) as |x| → ∞, where EX( ; , n – ) is a subset of n( ) and has a covering satisfying (). 3 Lemmas In order to prove our main results we need following lemmas. In this paper let M denote various constants independent of the variables in questions, which may be different from line to line. Lemma  (see [], Lemma ) Let any G = (r, ) ∈ n( ) and any H = (t, ) ∈ n( ), we have the following estimates: POI (G, H) ≤ Mr–κ tι –h ( ) POI (G, H) ≤ Mrι t–κ –h ( ) for  < rt ≤  , and for r < t ≤ r . < ∞. Lemma  (see [], Lemma ) If β ≥  and λ is positive measure on Rn having finite total mass, then exceptional set EX( ; λ, β) has a covering {rk, Rk} (k = , , . . .) satisfying The estimation of the Green potential at infinity is the following, which is due to []. 4 Proof of Theorem 1 Let G = (r, ) be any point in the set n( ; (L, +∞)) – EX( ; μ , n – α), where r is a sufficiently large number satisfying r ≥ l . Put POI μ(G) = POI (G) + POI (G) + POI (G), POI (G) = n( ;(,r]) POI (G, H) dμ(H), POI (G) = n( ;(r,r)) POI (G, H) dμ(H), POI (G) = n( ;[r,∞)) POI (G, H) dμ(H). We have the following estimates: POI (G) ≤ Mrι h ( )  r – from (), (), and [], Lemma . By (), we write POI (G) ≤ POI(G) + POI(G), POI(G) = M POI(G) = M We first have from [], Lemma . Next, we shall estimate POI(G). We can find a number k satisfying k ≥  and for any G = (r, ) ∈ (k), where Then the set n( ) can be split into two sets (k) and n( ) – (k). Let G = (r, ) ∈ n( ) – (k). Then |G – H| ≥ kr, where H ∈ n( ) and k is a positive number. So n( ;(  r,∞)) from [], Lemma . If G ∈ (k), we put Fl(G) = H ∈ n ; l– (G) ≤ |G – H| < l (G) . i= Fl(G) POI(G) = M where l(G) is a positive integer satisfying l(G)– (G) ≤ r < l(G) (G). By Remark  we have rh ( ) ≤ M (G) (G = (r, ) ∈ n( )), and hence dμ (H) ≤ Mrκ –α+ h ( ) –αμ Fl(G) l (G) α–n for l = , , , . . . , l(G). Since G = (r, ) ∈/ EX( ; μ , n – α), we have μ Fl(G) l (G) α–n ≤ μ B G, l (G) l (G) α–n ≤ M G; μ , n – α ≤ rα–n for l = , , , . . . , l(G) –  and μ Fl(G)(G) l (G) α–n ≤ μ B G, r From (), (), (), (), (), and Remark , we obtain POI μ(G) = o(rι {h ( )}–α) for any G = (r, ) ∈ n( ; (L, +∞)) – EX( ; μ , n – α) as r → ∞, where L is a sufficiently large real number. With Lemma  we have the conclusion of Theorem . Let G = (r, ) be a fixed point in n( ). Then there exists a number R satisfying max{ r , } < R. There exists a positive constant M such that P OI (G, H) ≤ M rι t–κ –h ( ) from Remark  and (), where H = (t, ) ∈ Let M = M cn–rι h ( ). Then we have from () and () n( ) satisfying  < rt ≤  . P OI [g](G) is a harmonic function of G ∈ proof of Corollary  is completed. Competing interests The authors declare that they have no competing interests. n( ;(R,+∞)) dt < ∞. For any G ∈ n( ), it is easy to see that P OI [g](G) is finite, which means that n( ). 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Kuan Luan, John Vieira. Poisson-type inequalities for growth properties of positive superharmonic functions, Journal of Inequalities and Applications, 2017, 12, DOI: 10.1186/s13660-016-1278-7