Double Ore extensions versus graded skew PBW extensions

In this paper we present necessary and sufficient conditions for a graded (trimmed) double Ore extension to be a graded (quasi-commutative) skew PBW extension. Using this fact, we prove that a graded skew PBW extension $A = \sigma(R)\langle x_1,x_2 \rangle$ of an Artin-Schelter regular algebra $R$ is Artin-Schelter regular. As a consequence, every graded skew PBW extension $A = \sigma(R)\langle x_1,x_2 \rangle$ of a connected skew Calabi-Yau algebra $R$ of dimension $d$ is skew Calabi-Yau of dimension $d+2$.