Introduction

• Akihito Uchiyama
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

First, we prepare notations. R = {the set of real numbers}, C = {the set of complex numbers}, N = {1, 2, 3,…}, N0 = {1, 2, 3,…}, Z = {0, ±1, ±2, ±3,…}, [t] = {the greatest integer not exceeding a real number t} x = (x 1,…,x n), ξ = (ξ1,…ξn) ∈ R n, dx = dx 1dx n = {the Lebesgue measure on R n},
$$\left| x \right| = \left\{ {x_1^2 + \cdots + x_n^2 } \right\}^{1/2}$$
$$S^{n - 1} = \left\{ {\xi \in {\rm{R}}^n :\left| \xi \right| = 1} \right\},$$
$${\rm{R}}_ + ^{n + 1} = \left\{ {\left( {x,t} \right):x \in {\rm{R}}^n ,t \in \left( {0, + \infty } \right)} \right\},$$
$${\rm{B}}\left( {x,t} \right) = \left\{ {y \in {\rm{R}}^n :\left| {x - y} \right| < t} \right\}.$$
$$\begin{array}{l} {\rm{If }}B = B\left( {x_0 ,t_0 } \right){\rm{and }}\delta > 0,{\rm{ then}} \\ {\rm{ }}\delta B = B\left( {x_0 ,\delta t_0 } \right),{\rm{ }}x_B = x_0 ,{\rm{ }}\ell \left( B \right) = t_0 , \\ {\rm{ }}Q\left( B \right) = \left\{ {\left( {x,t} \right) \in {\rm{R}}_ + ^{n + 1} :x \in B,t \in \left( {0,t_0 } \right]} \right\}. \\ \end{array}$$
$$I\left( {x,t} \right) = \left\{ {y = \left( {y_1 , \cdots ,y_n } \right) \in {\rm{R}}^n {\rm{:}}\mathop {\max }\limits_{1 \le j \le n} {\rm{ }}\left| {x_j - y_j } \right| \le t/2} \right\}.$$
$$\begin{array}{l} {\rm{If }}I = I\left( {x_0 ,t_0 } \right),{\rm{then}} \\ {\rm{ }}\delta I = I\left( {x_0 ,\delta t_0 } \right),{\rm{ }}x_2 = x_0 ,{\rm{ }}\ell \left( I \right) = t_0 . \\ \end{array}$$

Keywords

Harmonic Function Hardy Space Dual Space Linear Functional Continuous Linear
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.