Nonlinear Schrödinger equation and the twisted Laplacian-Global well posedness

Nonlinear Schrödinger equation for the twisted Laplacian in the critical case

“…Furthermore H m u m = u m and u m ∈ L q loc (R,W 1,p L (C n )) for every admissible pair (q, p). We deduce from Lemma 2.4 and Strichartz estimates (Theorem 3.3 in [7]) that…”

“…Proof. (of Theorem 1.1): (Local existence): Since G m (z, ·) : C → C is globally lipschitz for each m ≥ 1, see (1.8), therefore from [7] it follows that there exists a unique global solution u m ∈ C(R,W 1,2 L (C n )) of the initial value problem…”

“…In this paper, we consider the critical case α = 2 n−1 . In subcritical case 0 ≤ α < 2 n−1 for each α, we have some q > 2 such that (q, 2 + α) be an admissible pair (see Definition 3.1 in [7]), which is not the case when α = 2 n−1 . We overcome this difficulty by considering admissible pair (γ, ρ) and by using embedding theorem (Lemma 2.1), where ρ = 2n 2 n 2 − n + 1 , γ = 2n n − 1 .…”

Hardy-Sobolev Spaces Associated with Twisted Convolution

“…They also gave the atomic decomposition and Riesz transform characterizations for the Hardy spaces. In this paper, we first define HardySobolev spaces associated with twisted Laplacian based on [17,18] and then give the atomic decomposition of them. Finally, we give an application of the Hardy-Sobolev spaces associated with twisted Laplacian.…”

“…Recently, functional spaces associated with operators are considered by more and more mathematicians. In [17], the authors studied the Sobolev spaces associated with the twisted Laplacian and the Global well posedness of nonlinear Schrödinger equation. In [18], the authors defined the Hardy spaces associated with twisted 2…”