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We introduce the concept of locally homogeneous space, and prove in this context L p and C α estimates for singular and fractional integrals, as well as L p estimates on the commutator of a singular or fractional integral with a B M O or V M O function. These results are motivated by local a priori estimates for subelliptic equations.

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Let H n = C n × R be the n-dimensional Heisenberg group, Q = 2n+ 2 be the homogeneous dimension of H n . We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of P. L. Lions to the setting of the Heisenberg group H n . Furthermore, we also obtain the corresponding concentrationcompactness principle for the Sobolev space HW 1,Q (H n ) on the entire Heisenberg group H n .Our results improve the sharp Trudinger-Moser inequality on domains of finite measure in H n by Cohn and the second author [8] and the corresponding one on the whole space H n by Lam and the second author [21]. All the proofs of the concentrationcompactness principles in the literature even in the Euclidean spaces use the rearrangement argument and the Polyá-Szegö inequality. Due to the absence of the Polyá-Szegö inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of Q-Laplacian subelliptic equations on H n :with nonlinear terms f of maximal exponential growth exp αt Q Q−1 as t → +∞. 1 q .

Let W 1,n (R n ) be the standard Sobolev space and · n be the L n norm on R n . We establish a sharp form of the following Trudinger-Moser inequality involving the L n norm sup u W 1,n (R n ) =1ˆR n