An analogue of Hartogs lemma for separately harmonic functions with variable radius of harmonicity

In this note we prove that if a function u(x,y) is separately harmonic in a domain D × Vr = D × {y∈ℝ2:|y|<r, r>1} ⊂ ℝn × ℝ2 and for each fixed point x0 ∈ D the function u(x0,y) of variable y continues harmonically into the great circle {y∈ℝ2:|y|<R(x0), R(x0)>r}, then it continues harmonically into a domain {(x,y)∈ℝn×ℝ2:|y|<R*(x), x∈D} over a set of variables.

Hali tarjima qilinmagan