On the behaviour of harmonic functions at the boundary
On the behaviour of harmonic functions at the boundary
复制标题
调和函数在边界处的行为
DOI：
10.1090/s0002-9947-1950-0032863-9
复制
发表时间：
1950
影响因子：
1.3
通讯作者：
A. Calderón
中科院分区：
文献类型：
--
作者：
A. Calderón
1. Let F(x, y) be a function harmonic for y >0, and suppose that for every point (x, 0) of a set E of positive measure of the x-axis there exist a triangular region with vertex at that point, where the function is bounded; then it is well known that: (A) Almost everywhere in E, F(x, y) has a limit as (x, y) tends nontangentially to (x, 0) G£. This result, first proved by Priwaloff [l](2), when applied to analytic functions leads, as shown by Plessner [2], to the following stronger result: (B) Let F(z), z = x+iy, be a function analytic for y>0. Then, except for a set of points of measure zero, at every point (x, 0) of the x-axis, either the function has a finite limit as z tends nontangentially to (x, 0), or the range of F(z) in every triangular region with vertex at that point is dense in the whole complex plane. Actually these results were proved not only for functions harmonic or analytic in a half-plane, but also in domains limited by rectifiable curves. However, even the special cases mentioned above were obtained by methods of conformai mapping [l], [2] which cannot be applied to harmonic or analytic functions of more variables. A purpose of the present paper is to give a different proof of (A) which leads to its generalization to functions of any number of variables: (a) Let F(P), P = (xx, x2, · · · , x„), be a function harmonic for xn>0 such that for every point Q of a set E of positive measure of the hyperplane x„ = 0 there exists a region Yq limited by a cone with vertex at Q and a hyperplane xn = const, where F(P) is bounded. Then almost everywhere in E the function has a limit as P tends to QÇLE nontangentially to xn = 0. A further generalization of (A), which will enable us to extend (B) to functions of several complex variables, deals with functions which are harmonic in sets of variables, and may be stated as follows: (b) Let E = ExXE2X · · -X-Et» be the Cartesian product of the spaces Ek of points Pk = (x?\ xf, ■ ■■ , xf ), and F(P), P=(Px, P», · · · , Pm)EE, be defined and continuous in asJfX), k = 1, 2, · · · , m, and harmonic in Pk, that is, such that
1. Let F(x, y) be a function harmonic for y >0, and suppose that for every point (x, 0) of a set E of positive measure of the x-axis there exist a triangular region with vertex at that point, where the function is bounded; then it is well known that: (A) Almost everywhere in E, F(x, y) has a limit as (x, y) tends nontangentially to (x, 0) G£. This result, first proved by Priwaloff [l](2), when applied to analytic functions leads, as shown by Plessner [2], to the following stronger result: (B) Let F(z), z = x+iy, be a function analytic for y>0. Then, except for a set of points of measure zero, at every point (x, 0) of the x-axis, either the function has a finite limit as z tends nontangentially to (x, 0), or the range of F(z) in every triangular region with vertex at that point is dense in the whole complex plane. Actually these results were proved not only for functions harmonic or analytic in a half-plane, but also in domains limited by rectifiable curves. However, even the special cases mentioned above were obtained by methods of conformai mapping [l], [2] which cannot be applied to harmonic or analytic functions of more variables. A purpose of the present paper is to give a different proof of (A) which leads to its generalization to functions of any number of variables: (a) Let F(P), P = (xx, x2, • • • , x„), be a function harmonic for xn>0 such that for every point Q of a set E of positive measure of the hyperplane x„ = 0 there exists a region Yq limited by a cone with vertex at Q and a hyperplane xn = const, where F(P) is bounded. Then almost everywhere in E the function has a limit as P tends to QÇLE nontangentially to xn = 0. A further generalization of (A), which will enable us to extend (B) to functions of several complex variables, deals with functions which are harmonic in sets of variables, and may be stated as follows: (b) Let E = ExXE2X • • -X-Et» be the Cartesian product of the spaces Ek of points Pk = (x?\ xf, ■ ■■ , xf ), and F(P), P=(Px, P», • • • , Pm)EE, be defined and continuous in asJfX), k = 1, 2, • • • , m, and harmonic in Pk, that is, such that