On point to point reflection of harmonic functions across real-analytic hypersurfaces in R(n)

Let Gamma be a non-singular real-analytic hypersurface in some domain U subset of R(n) and let Har(0)(U, Gamma) denote the linear space of harmonic functions in U that vanish on Gamma. We seek a condition on X(0),x(1) is an element of U \ Gamma such that the reflection law (RL) u(X(0)) + Ku(X(1)) = 0, For All u is an element of Har(0)(U, Gamma) holds for some constant K. This is equivalent to the class Har(0)(U, Gamma) not separating the points X(0), X(1). We find that in odd-dimensional spaces (RL) never holds unless Gamma is a sphere or a hyperplane, in which case there is a well known reflection generalizing the celebrated Schwarz reflection principle in two variables. In even-dimensional spaces the situation is different. We find a necessary and sufficient condition (denoted the SSR-strong Study reflection-condition), which we describe both analytically and geometrically, for (RL) to hold. This extends and complements previous work by e.g. P. R. Garabedian, H. Lewy, D. Khavinson and H. S. Shapiro.