BEHAVIOR OF THE SOLUTIONS OF HYPOELLIPTIC EQUATIONS IN R~2

摘要

A polynomial P(ξ) = P(ξ_1, ξ_2, … , ξ_n) is said to prevail over a polynomial Q(ξ) if R_(Q,P) (ξ) = |Q((ξ))|/(1 + |P (ξ)|) → 0 as |ξ| → ∞. The paper suggests a new description for the set m(P,Q) = {β : β ∈ R_+~2| |ξ~β|R_(Q,P) (ξ) < ∞, any ξ ∈ R~2} by means of the prevalence measure of P over Q, leading to the concept of hypoellipticity measure of a differential operator P(D). For n = 2 the hypoellipticity measure is used for description of the best Gevrey classes, which contain the solutions of the hypoelliptic equation P(D)u = 0.