Fractional Adams–Moser–Trudinger type inequality on Heisenberg group. (June 2020) - Gupta, Madhu; Tyagi, Jagmohan
Abstract: The aim of this paper is to establish singular fractional Adams–Moser–Trudinger inequality for both bounded and unbounded domains in the Heisenberg group H n . We first establish fractional Adams–Moser–Trudinger type inequality on domain Ω ⊂ R n with finite measure (Theorem 1.12) and then using this inequality and Hardy–Littlewood–Sobolev inequality adapted to the result of R. O'Neil (1963), we establish singular fractional Adams–Moser–Trudinger type inequality on domain Ω ⊂ R n with finite measure (Theorem 1.13). We also establish singular fractional Adams–Moser–Trudinger type inequality in H n, using the approach of N. Lam and G. Lu (2012, 2013). The main idea of our approach is that any function in higher order fractional Sobolev space in Heisenberg group can be represented in terms of Riesz potential and then using techniques from harmonic analysis and kernel properties of the associated operator, we establish fractional Adams–Moser–Trudinger type inequality in H n . In this paper, our approach is quite simple and free from symmetrization arguments. As an applications of our theorems, we establish the existence of solution to the following class of problems T α u = f ( ξ, u ) | ξ | a + b ( ξ ) | u | γ − 1 u in Ω, u = 0 in H n ∖ Ω, where Ω is a bounded subset of H n of class C 0, 1 with bounded boundary and 0 ≤ a < Q, f satisfies either the subcritical exponential growth or critical exponential growth condition and b is a small L 2 -perturbation, that is, there exists a small η > 0 with 0 < ‖ b ‖ L 2 ( Ω ) < η, 0 ≤ γ < 1 and α = Q 2 .