The well-posedness of nonlinear Schrödinger equations in Triebel-type spaces (Q2346678)

In this paper, via the frequency-uniform decomposition techniques, a new class of Triebel-Lizorkin type space is introduced and applied in the Cauchy problem for generalized nonlinear Schrödinger equations. To be precise, let \(\rho: \mathbb{R}^n\to [0,1]\) be a smooth radial bump function adapted to the cube \([-1,1]^n\) so that \(\rho\) equals to 1 on \([-1/2,1/2]^n\) and vanishes outside \([-1,1]^n\). Let \(\rho_k=\rho(\cdot-k)\) for all \(k\in\mathbb{Z}^n\) and \(\sigma_k=\rho_k(\sum_{i\in\mathbb{Z}^n}\rho_i)^{-1}\). Then \(\square_k=\mathcal{F}^{-1}\sigma_k\mathcal{F}\) denotes the frequency-uniform decomposition operator, where \(\mathcal{F}\) is the Fourier transform. The space \(N^s_{p,q}(\mathbb{R}^n)\), where \(s\in\mathbb{R}\), \(p\in(0,\infty)\) and \(q\in (0,\infty]\), is then defined as the set of all tempered distributions \(f\) so that \[ \|f\|_{N^s_{p,q}(\mathbb{R}^n)}=\bigg\|\bigg(\sum_{k\in\mathbb{Z}^n} (1+|k|)^{sq}|\square_k f|^q\bigg)^{1/p}\bigg\|_p \] is finite. In the case \(p=\infty\), the space \(N^s_{\infty,q}(\mathbb{R}^n)\) consists of all tempered distributions \(f\) so that there exists a sequence \(\{f_j\}_{j\in\mathbb{Z}}\in L^\infty(\mathbb{R}^n,\ell^q)\) such that \(f=\sum_{j\in\mathbb{Z}}\square_k f_k\). In this paper, the authors first obtain sufficient and necessary conditions for embeddings between the spaces \(N^s_{p,q}(\mathbb{R}^n)\) and the Triebel-Lizorkin spaces \(F^s_{p,q}(\mathbb{R}^n)\). As an application, the Cauchy problem for generalized nonlinear Schrödinger equations in \(L^r(0,T,N^s_{p,q}(\mathbb{R}^n))\) is studied.