Solvability of elliptic functional-differential equations with compressions of the arguments in weighted spaces

Abstract

In the paper, the equation $$\sum\limits_{k = 0}^l {\sum\limits_{\left| \alpha \right| = 2m} {a_{k\alpha } D^\alpha (u(q^{ - k} x)) = f(x)(x \in \mathbb{R}^n )} } $$ is considered in the scale of the weighted spaces H β (ℝ n ) (q > 1, a kα ∈ ℂ). We prove that if the expression $$\sum\limits_{k = 0}^l {\sum\limits_{\left| \alpha \right| = 2m} {a_{k\alpha } \xi ^\alpha z^k } } $$ does not vanish on the set {ξ ∈ ℝ n ∖ 0, |z| ≤ q β−s+n/2−2m}, then this equation has a unique solution u ∈ H +2 (ℝ n ) for every function f ∈ H (ℝ n ) provided that β, s ≠ ∈ ℝ, β − s ≠ n/2 + p, and β − s − 2m ≠ − n/2 − p (p = 0, 1, ...).