Continuity of infinitely degenerate weak solutions via the trace method

In 1971 Fediĭ proved in [3] the remarkable theorem that the linear second order partial differential operator L f u ( x , y ) ≡ { ∂ ∂ x 2 + f ( x ) 2 ∂ ∂ y 2 } u ( x , y ) is hypoelliptic provided that f ∈ C ∞ ( R ) , f ( 0 ) = 0 and f is positive on ( − ∞ , 0 ) ∪ ( 0 , ∞ ) . Variants of this result, with hypoellipticity replaced by continuity of weak solutions, were recently given by the authors, together with Cristian Rios and Ruipeng Shen, …