Global Sobolev-Morrey estimates for hypoelliptic operators with drift on homogeneous groups

Abstract

Let <italic>G</italic> be a homogeneous group and <italic>X</italic>₀,<italic>X</italic>₁;<italic>X</italic>₂, ... ,<italic>X</italic>_{<italic>p</italic>0} be left invariant real vector fields on <italic>G</italic> satisfying the Hörmanders rank condition. Assume that <italic>X</italic>₁,<italic>X</italic>₂, ... ,<italic>X</italic>_{<italic>p</italic>0} are homogeneous of degree one and <italic>X</italic>₀ is homogeneous of degree two. In this paper, we study the following hypoelliptic operator with drift: where (<italic>a_ij</italic>) is a constant matric satisfying the uniform ellipticity condition and <italic>a</italic>₀ is a constant away from zero, and obtain the global Sobolev-Morrey estimates on <italic>G</italic> by establishing the Morrey boundedness of the singular integrals on homogeneous spaces and interpolation inequalities depending on vector fields.