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On existence of interference between the components of biharmonic functions. (English. Russian original) Zbl 0984.31003
Mosc. Univ. Math. Bull. 55, No. 1, 29-32 (2000); translation from Vestn. Mosk. Univ., Ser. I 2000, No. 1, 45-48 (2000).
Let \(D^m:=\{x\in \mathbb R^m\mid \|x\|<1\}\) be the unit sphere in \(\mathbb R^m\), \(H_n = H_n(D^m)\) be the space of polyharmonic functions \(f\) of \(n\)th order in \(D^m\) \((\Delta^nf = 0)\), \(M^p(D^m)\) be the class of bounded functions of \(p\)th order. The main result of the paper is as follows. Let \( m\geq 3\), \( p\in \mathbb N\). Then there exists a biharmonic function \(f\in H_2(D^m)\cap M^p(D^m)\), \(f= P+f_0+f_1\), \(f_k= (1-|x|^2)^k\Phi_k\), \(\Phi_k\) polyanalytic, \(P\) a polynomial, such that \(f_0,f_1\notin M^p(D^m)\).
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
32A50 Harmonic analysis of several complex variables