Counterexamples for High-Degree Generalizations of the Schrödinger Maximal Operator

Abstract

In 1980 Carleson posed a question on the minimal regularity of an initial data function in a Sobolev space $H^s({\mathbb {R}}^n)$ that implies pointwise convergence for the solution of the linear Schrödinger equation. After progress by many authors, this was recently resolved (up to the endpoint) by Bourgain, whose counterexample construction for the Schrödinger maximal operator proved a necessary condition on the regularity, and Du and Zhang, who proved a sufficient condition. Analogues of Carleson’s question remain open for many other dispersive partial differential equations. We develop a flexible new method to approach such problems and prove that for any integer $k\geq 2$, if a degree $k$ generalization of the Schrödinger maximal operator is bounded from $H^s({\mathbb {R}}^n)$ to $L^1(B_n(0,1))$, then $s \geq \frac {1}{4} + \frac {n-1}{4((k-1)n+1)}.$ In dimensions $n \geq 2$, for every degree $k \geq 3$, this is the first result that exceeds a long-standing barrier at $1/4$. Our methods are number-theoretic, and in particular apply the Weil bound, a consequence of the truth of the Riemann Hypothesis over finite fields.

On uniqueness properties of solutions of the generalized fourth-order Schrödinger equations

“…x k (u φ R ) and combine all the conclusions to see that sup t∈ [0,1] ˆ|x|⩾R |u| 2 + |∇u| 2 + |∇ 2 u| 2 + |∇ 3 u| 2 e λ|x| α /(10d) α dx ⩽ c 0 + c 0 exp (λR α ) . (7.1)…”

“…Applying lemma 4.2 to the above equation, we find that sup t∈ [0,1] ˆ|x|⩾R We repeat this application of corollary up to the equations of (i∂…”

“…We also define φ R (x) = 1 − θ R (x). • Let φ ∈ C ∞ satisfy φ(t) ∈ [0, M] on [0, 1] and φ (t) = M, on 1 2 − 1 8 , 1 2 + 1 8 , 0, on 0, 1 4 ∪ 3 4 , 1 .…”

“…To the best of the authors' knowledge, we believe that the current paper is the first result towards obtaining the unique continuation property of higher-order Schrödinger equations in higher dimensions. It is worth mentioning that this type of higher degree generalizations of the Schrödinger equation is not uncommon, see for example [1] for the same generalization in the context of the study of pointwise convergence of Schrödinger operators.…”

“…Before starting the proof, let us present the main idea of the calculation.Let us forget the perturbation H and potential V for a moment. By considering a change of variables: v = e βx1 u, we reduce (3.3) into the following inequality sup t∈[0,1]…”