An improved necessary condition for the Schrödinger maximal estimate

Abstract

We improve the necessary condition for Carleson's problem regarding convergence for the Schrödinger equation in dimensions $n\ge 3$. We prove that if the solution converges almost everywhere to its initial datum as time tends to zero, for all data in $H^s({\mathbb{R}}^n)$, then $s\ge \frac{n}{2(n+2)}$.

7 pages

Cited by in corpus (9)

• A sharp Schrodinger maximal estimate in ${\mathbb{R}}^2$
• $L^p$-estimates of maximal function related to Schrödinger Equation in ${\mathbb{R}}^2$
• $L^p$-estimate of Schrödinger maximal function in higher dimensions
• On convergence properties for generalized Schrödinger operators along tangential curves
• Pointwise convergence of solution to Schrodinger equation on manifolds
• A Note on Non-tangential Convergence for Schrödinger Operators
• Pointwise convergence problem of Ostrovsky equation with rough data and random data
• Pointwise Convergence for sequences of Schrödinger means in ${\mathbb{R}}^2$
• Sharp convergence for sequences of nonelliptic Schrödinger means