2020/01/13 by Van Hoang Nguyen, Nguyen, Van Hoang
Engineering · Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Numerical methods in engineering #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2001.04017
openalex publication_date 2020/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let 2≤ m < n and q ∈ (1,∞), we denote by WmL\frac nm,q(\mathbb Hn) the Lorentz-Sobolev space of order m in the hyperbolic space \mathbb Hn. In this paper, we establish the following Adams inequality in the Lorentz-Sobolev space Wm L\frac nm,q(\mathbb Hn) sup_u∈ WmL\frac nm,q(\mathbb Hn), ‖∇gm u‖\frac nm,q≤ 1 ∫\mathbb Hn Φ\frac nm,q(βn,m^\frac qq-1 |u|^\frac qq-1) dVg lt; ∞ for q ∈ (1,∞) if m is even, and q ∈ (1,n/m) if m is odd, where βn,mq/(q-1) is the sharp exponent in the Adams inequality under Lorentz-Sobolev norm in the Euclidean space. To our knowledge, much less is known about the Adams inequality under the Lorentz-Sobolev norm in the hyperbolic spaces. We also prove an improved Adams inequality under the Lorentz-Sobolev norm provided that q≥ 2n/(n-1) if m is even and 2n/(n-1) ≤ q ≤ \frac nm if m is odd, sup_u∈ WmL\frac nm,q(\mathbb Hn), ‖∇gm u‖\frac nm,qq -λ‖u‖\frac nm,qq ≤ 1 ∫\mathbb Hn Φ\frac nm,q(βn,m^\frac qq-1 |u|^\frac qq-1) dVg lt; ∞ for any 0< λ< C(n,m,n/m)q where C(n,m,n/m)q is the sharp constant in the Lorentz-Poincaré inequality. Finally, we establish a Hardy-Adams inequality in the unit ball when m≥ 3, n≥ 2m+1 and q ≥ 2n/(n-1) if m is even and 2n/(n-1) ≤ q ≤ n/m if m is odd sup_u∈ WmL\frac nm,q(\mathbb Hn), ‖∇gm u‖\frac nm,qq -C(n,m,\frac nm)q ‖u‖\frac nm,qq ≤ 1 ∫\mathbb Bn exp(βn,m^\frac qq-1 |u|^\frac qq-1) dx lt; ∞.