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Abstract  We prove the existence of a global fundamental solution    Γ  (  x  ;  y  )    (with pole   x  ) for any Hörmander operator    L  =   ∑   i  =  1   m    X  i  2on    R  n    which is    δ  λ   ‐homogeneous of degree 2. Here homogeneity is meant with respect to a family of non‐isotropic diagonal maps    δ  λ    of the formδ  λ    (  x  )   =   (   λ   σ  1x  1   ,  …  ,   λ   σ  nx  n   )    , with    1  =   σ  1   ⩽  ⋯  ⩽   σ  n    . Due to a global lifting method for homogeneous operators proved by Folland [ Comm. Partial Differential Equations  2 (1977) 161–207], there exists a Carnot group   G   and a polynomial surjective map    π  :  G  →   R  nsuch that   L   is   π  ‐related to a sub‐Laplacian    L  G    on   G  . We show that it is always possible to perform a (global) change of variable on   G   such that the lifting map   π   becomes the projection of    G  ≡   R  n   ×   R  ponto    R  n   . We prove that an integration argument over the (non‐compact) fibers of   π   provides a fundamental solution for   L  . Indeed, ifΓ  G    (  x  ,    x   ′   ;  y  ,    y   ′   )(   x  ,  y  ∈   R  n    ;x  ′   ,    y   ′   ∈   R  p    ) is the fundamental solution of    L  G   , we show thatΓ  G    (  x  ,  0  ;  y  ,    y   ′   )is always integrable with respect to  y   ′   ∈   R  p    , and its    y  ′   ‐integral is a fundamental solution for   L  .

The existence of a global fundamental solution for homogeneous Hörmander operators via a global lifting method