Vertical versus horizontal Sobolev spaces

摘要: Abstract Let α ⩾ 0 , 1 p ∞ and let H n be the Heisenberg group. Folland in 1975 showed that if f : → R is a function horizontal Sobolev space S 2 ( ) then φf belongs to Euclidean + for any test φ. In short, ⊂ loc . We show localisation can omitted one only cares regularity vertical direction: continuously contained V Our search sharper result was motivated by following two applications. First, combined with short additional argument, it implies bounded Lipschitz functions on have -order derivative BMO Second, yields fractional order generalisation of (non-endpoint) versus Poincare inequalities V. Lafforgue A. Naor.