TY - JOUR
T1 - L → L norm estimates of Cauchy transforms on the Dirichlet problem and their applications
AU - Zhu, Jian-Feng
AU - Rasila, Antti
PY - 2021/10/1
Y1 - 2021/10/1
N2 - Denote by the space of the functions f on the unit disk which are Hölder continuous with the exponent α, and denote by the space which consists of differentiable functions f such that their derivatives are in the space . Let be the Cauchy transform of Dirichlet problem. In this paper, we obtain the norm estimates of , where and . Suppose and is the Green potential of g. By using Sobolev embedding theorem, we show that if , then , where . We also show that if , then , where . Finally, for the case , we show that f is not necessarily in , but its gradient, i.e., is Lipschitz continuous with respect to the pseudo-hyperbolic metric. This paper is inspired by [2, Chapter 4] and [9].
AB - Denote by the space of the functions f on the unit disk which are Hölder continuous with the exponent α, and denote by the space which consists of differentiable functions f such that their derivatives are in the space . Let be the Cauchy transform of Dirichlet problem. In this paper, we obtain the norm estimates of , where and . Suppose and is the Green potential of g. By using Sobolev embedding theorem, we show that if , then , where . We also show that if , then , where . Finally, for the case , we show that f is not necessarily in , but its gradient, i.e., is Lipschitz continuous with respect to the pseudo-hyperbolic metric. This paper is inspired by [2, Chapter 4] and [9].
U2 - 10.1016/j.jmaa.2021.125255
DO - 10.1016/j.jmaa.2021.125255
M3 - 文章
SN - 0022-247X
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
ER -