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Pointwise convergence of solutions of the Schr\"odinger equation along general curves on Damek-Ricci spaces by Dewan, Utsav is a scholarly article available to read on EtoBox.

What is Pointwise convergence of solutions of the Schr\"odinger equation along general curves on Damek-Ricci spaces about?

One of the most celebrated problems in Euclidean Harmonic analysis is the Carleson's problem: determining the optimal regularity of the initial condition $f$ of the Schr\"odinger equation given by \begin{equation*} \begin{cases} i\frac{\partial u}{\partial t} =\Delta u\:,\: (x,t) \in \mathbb{R}^n \times \mathbb{R} \newline u(0,\cdot)=f\:, \text{ on } \mathbb{R}^n \:, \end{cases} \end{equation*} in terms of the index $\beta$ such that $f$ belongs to the inhomogeneous Sobolev space $H^\beta(\mathbb{R}^n)$, so that the solution of the Schr\"odinger operator $u$ converges pointwise to $f$, \begin{equation*} \displaystyle\lim_{t \to 0+} u(x,t)=f(x), \text{ almost everywhere}. \end{equation*} Recently, the author considered the Carleson's problem for the Schr\"odinger equation with radial initial data on Damek-Ricci spaces and obtained the sharp bound up to the endpoint $\beta \ge 1/4$. Interpreting the above as convergence along vertical lines, in this article, we consider the problem of pointwise convergence via more general approach paths. By constructing a counter-example on the $3$-dimensional Real Hyperbolic space, we show that the solutions of the Schr\"odinger equation, unlike Ha

Author

Dewan, Utsav

Published

2024

Language

EN