Euler type partial differential operators on real analytic functions

We describe all Euler partial differential operators which act on the space of real analytic functions and we identify them among the Taylor multipliers on these spaces. Partial differential operators of the form f par right arrow Sigma a(alpha)D(alpha)f, D-alpha := D-1(alpha 1) ... D-d(alpha d), D-j(f)(x) := q(j,1)(xj)partial derivative f/partial derivative x(j)(x + q(j,0)(xj)f(x), where qj,i, qi,0 : (aj, bj) -> C, are called generalized Euler differential operators whenever all Di are conjugate to the classical Euler differential theta, theta(f) (t) = tf'(t). We find criteria when a linear differential operator with analytic coefficients on the space of real analytic functions is a generalized Euler differential operator. It turns out that this happens for a wide variety of linear operators with variable coefficients. Using our earlier results on solvability of classical Euler operators of finite order we then study the question of surjectivity or "big image" for generalized Euler partial differential operators with analytic coefficients, i.e., global solvability of the considered equations in spaces of real analytic functions. (C) 2016 Published by Elsevier Inc.