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

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].

Higher order Adams’ inequality with the exact growth condition

Adams’ inequality is the complete generalization of the Trudinger–Moser inequality to the case of Sobolev spaces involving higher order derivatives. The failure of the original form of the sharp inequality when the problem is considered on the whole space [Formula: see text] served as a motivation to investigate in the direction of a refined sharp inequality, the so-called Adams’ inequality with the exact growth condition. Due to the difficulties arising in the higher order case from the lack of direct symmetrization techniques, this refined result is known to hold on first- and second-order Sobolev spaces only. We extend the validity of Adams’ inequality with the exact growth to higher order Sobolev spaces.

Adams inequality with exact growth in the hyperbolic space ℍ4 and Lions lemma

In this paper, we prove Adams inequality with exact growth condition in the four-dimensional hyperbolic space [Formula: see text] [Formula: see text] [Formula: see text]. We will also establish an Adachi–Tanaka-type inequality in this setting. Another aspect of this paper is the Lions lemma in the hyperbolic space. We prove Lions lemma for the Moser functional and for a few cases of the Adams functional on the whole hyperbolic space.

Best Constants for Adams’ Inequalities with the Exact Growth Condition in ℝn

In this paper, we establish the following sharp Adams inequality with exact growth condition in the entire space ℝ,where