Euler type partial differential operators on real analytic functions

• Published in: Journal of mathematical analysis and applications 2016-11, Vol.443 (2), p.652-674
• Main Authors: Domański, Paweł, Langenbruch, Michael
• Format: Article
• Language: English
• Subjects: Euler differential operator; Hadamard multiplier; Linear partial differential operator with variable coefficients; Solvability of generalized Euler partial differential equation of finite order; Solvability of partial differential operators with variable coefficients; Spaces of real analytic functions
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2016.05.018

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 formf↦∑aαDαf,Dα:=D1α1⋯Ddαd,Dj(f)(x):=qj,1(xj)∂f∂xj(x)+qj,0(xj)f(x), where qj,1,qj,0:(aj,bj)→C, are called generalized Euler differential operators whenever all Dj are conjugate to the classical Euler differential θ, θ(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.