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In this thesis we prove sharp Adams inequality with exact growth condition for the Riesz potential as well as the more general strictly Riesz-like potentials on R[superscript n]. Then we derive the Moser-Trudinger type inequality with exact growth condition for fractional Laplacians with arbitrary 0 [less than] [alpha] [less than] n, higher order gradients and homogeneous elliptic differential operators. Next we give an application to a quasilinear elliptic equation, and prove the existence of ground state solution of this equation. Lastly, we extend our result to the Heisenberg group. By applying the same technique used in R[superscript n], we derive a sharp Adams inequality with critical growth condition on H[superscript n] for integral operators whose kernels are strictly Riesz-like on H[superscript n]. As a consequence we then derive the corresponding sharp Moser-Trudinger inequalities with exact growth condition for the powers of sublaplacian -L[subscript 0] [superscript alpha/2] when [alpha] is an even integer, and for the subgradient [del] H[subscript n].

Adams inequalities with exact growth condition : on Rn and the Heisenberg group