Integral representation and asymptotic property of analytic functions with order less than two in the half-plane

Integral representation and asymptotic behavior of harmonic functions in half space

“…A similar result of harmonic functions whose order is less than 2 in the half plane is given in [14]. Analogous to the classical Hadamard factorization theorem of entire function with finite order, together with the inner and outer factorization theorem of analytic function in the Hardy space of half plane, Deng [15] uses Carleman's formula to obtain the analytic function of finite order in the right half plane that can be factorized into the product Ge Q e g of the three types of analytic functions, G is a weighted Blaschke product, Q is a polynomial of degree not greater than n and e g is a weighted outer function.…”

“…In [13], Zhang and Deng gave the following integral representation theorem of analytic function whose order is less than 2 on a half plane.…”

Asymptotic behavior of fractional Laplacians in the half space

“…One can find interesting properties and interpretations of fractional calculus in [14,15,17], which also give a useful mathematical tool for modeling many process in nature. Moreover, this paper provides the details of the asymptotic properties for the fractional Laplacian operator, as the applications of theory, i.e., Theorem 3.1 is the generation of asymptotic behaviors in [24][25][26], especially when a ¼ 2, Theorem 3.1 reduces to the first result of paper [25]; when a ¼ 2; m ¼ 1, Theorem 3.3 reduces to the second result of paper [27].…”

“…1 outside of an exceptional set explicitly defined. The growth properties of analytic functions with order less than 2 in the half plane was studied in [24]. Asymptotic behaviors for different kinds of potentials were also discussed in [1,12,13,16,18].…”

“…In the present article, motivated by those works in [1,18,24,25], we study the asymptotic behaviors of Poisson integral fractional Laplacians in the half space of R n . By the similar methods in studying Poisson integral fractional Laplacians, we estimate the asymptotic behavior of modified Poisson integrals of fractional Laplacians.…”