On the theory of harmonic functions of several variables
On the theory of harmonic functions of several variables
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DOI：
10.1007/bf02546524
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发表时间：
1960-07
期刊：
影响因子：
3.7
通讯作者：
E. Stein;G. Weiss
中科院分区：
文献类型：
--
作者：
E. Stein;G. Weiss
This paper is the first of series concerned with certain aspects of the theory of harmonic functions of several variables. Our particular interest will be to extend to n variables some of the deeper properties known to hold in the case of two variables.The study of the more fundamental properties of harmonic functions of two real variables is' linked, by the notion of the conjugate harmonic function, to the study of analytic functions of one complex variable. Therefore, the investigation of the deeper properties of harmonic functions of several variables appears, at first sight, to be connected with either the theory of analytic functions of several complex variables or with an appropriate extension of the notion of conjugate harmonic function. The theory of analytic functions of several complex variables, though widely studied, does not seem to have direct applications to the theory of harmonic functions of several real variables. On the other hand, there are known notions of" conjugacy" of harmonic functions which seem to us to be both more natural and more fruitful for the development of the latter theory. It is these notions that form the starting point for our investigation. We begin by sketching their background. Let us first consider a function u= u (re i~ which is harmonic in the interior of the unit circle 0~ r< l. Suppose that, for 0~< r< l and~ 0>~ 1,
This paper is the first of series concerned with certain aspects of the theory of harmonic functions of several variables. Our particular interest will be to extend to n variables some of the deeper properties known to hold in the case of two variables.The study of the more fundamental properties of harmonic functions of two real variables is' linked, by the notion of the conjugate harmonic function, to the study of analytic functions of one complex variable. Therefore, the investigation of the deeper properties of harmonic functions of several variables appears, at first sight, to be connected with either the theory of analytic functions of several complex variables or with an appropriate extension of the notion of conjugate harmonic function. The theory of analytic functions of several complex variables, though widely studied, does not seem to have direct applications to the theory of harmonic functions of several real variables. On the other hand, there are known notions of" conjugacy" of harmonic functions which seem to us to be both more natural and more fruitful for the development of the latter theory. It is these notions that form the starting point for our investigation. We begin by sketching their background. Let us first consider a function u= u (re i~ which is harmonic in the interior of the unit circle 0~ r< l. Suppose that, for 0~< r< l and~ 0>~ 1,