In this paper, we prove a sharp Fefferman--Stein-type estimate for the fractional Schr\"odinger equation, which can be regarded as a generalized Strichartz estimate for data in the Fourier Lebesgue space $\widehat{L^p}$. Then, as an application of the Fefferman--Stein inequality and its off-diagonal generalization, we prove large data local well-posedness and small data global well-posedness results for the one dimensional fractional nonlinear Schr\"odinger equation with pure power nonlinearities in the homonegeneous and inhomogeneous Fourier--Sobolev spaces $\widehat{\dot{H}^s_p},\widehat{H^s_p}$. Solutions are established in $L_x^r(\mathbb{R} ;L^q_t(I))$ spaces in order to overcome the difficulty of a loss of derivatives in the standard Strichartz estimates.
Inspired by the sharp coefficient estimates established by Cho \emph{et al.}\cite{CKKLS2018} for starlike functions of order $\alpha$ in the unit disk, we investigate the corresponding problems for starlike mappings of order $\alpha$ defined on the unit ball of a complex Banach space. Employing Fr\'echet derivatives to...
The purpose of this work is to provide a broader framework for analyzing the inhomogeneous nonlinear Schr\"odinger equation (INLS) \[iu_t + u_{xx} \pm |x|^{-b}|u|^{\alpha-1}u=0, \quad 1<\alpha<5-2b\; \text{and}\; 0<b\leq 1/4.\] Specifically, we establish low-regularity well-posedness in the Fourier-Lebesgue $\widehat{L...
D. Bhimani, Diksha Dhingra, Vijay Kumar Sohani· 0 citations
In this paper, we establish a Fefferman--Stein inequality in terms of area function and non-tangential maximal function associated with the Schr\"odinger operator $\mathcal{L} = -\Delta + V$ on stratified Lie groups $\mathcal G$, where $\Delta$ denotes the sub-Laplacian on $\mathcal G$ and $V$ is a nonnegative locally...
We establish two pointwise estimates for fractional difference operators, tracking explicitly the dependence of the constants on the smoothness index $s\in(0,1)$. Using these, within the framework of ball Banach function spaces we obtain two fractional Gagliardo--Nirenberg inequalities, including the BMO endpoint case....
Pingxu Hu, Yinqin Li, Da-Chun Yang et al.· 0 citations
We study fractional-Sobolev Tikhonov regularization for linear inverse problems on a bounded Lipschitz domain. The regularization penalty is generated by the restricted Dirichlet fractional Laplacian, and the associated variational problem is shown to admit a unique minimizer that depends Lipschitz continuously on the...
We construct divergence-free, vector-valued approximation functions on the whole space $\mathbb{R}^d$, $d\geq 2$, as well as on general unbounded domains of uniform $\mathrm{C}^{1,1}$-type. These approximations converge simultaneously in both Sobolev and Lebesgue spaces. In the whole-space setting, we employ Bogovski\u...
Akram Khan, Sagar Gautam, Manil T. Mohan· 0 citations
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