Author

D. Bhimani

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Preprint Aug 2026

Inhomogeneous nonlinear Schr\"odinger equation in Fourier-Lebesgue and modulation spaces

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 $\widehat{L}^{p}$ spaces for $4/3<p<8$. The analysis is carried out differently for the cases $p<2$ and $p>2$. Primarily, in both cases, we prove global well-posedness for arbitrarily large initial data via the data decomposition method adapted for the Fourier-Lebesgue spaces. Furthermore, we obtain analogous results for INLS in modulation spaces $M^{p,p'}$ for $4/3<p<2$.

D. Bhimani, Diksha Dhingra, Vijay Kumar Sohani · 0 citations

Preprint Jul 2026

Fefferman--Stein-type estimates and fractional NLS in Fourier Sobolev spaces

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.

D. Bhimani, Ryosuke Hyakuna · 0 citations