Counterexamples for the fractal Schrödinger convergence problem with an intermediate space trick

TL;DR

The paper addresses fractal refinements of the Schrödinger convergence problem, establishing counterexamples that match the known lower bounds for weighted $L^2$ restriction estimates. It combines a fractal extension of Bourgain's counterexample with the Du–Kim–Wang–Zhang intermediate space trick, and uses the Mass Transference Principle to explicitly construct and quantify the divergence set via its Hausdorff dimension. By analyzing a family of counterexamples parameterized by the intermediate-dimension $m$, it derives detailed Sobolev regularity thresholds $s_m(\alpha)$ and determines the maximal regularity curve $s(\alpha)=\max_m s_m(\alpha)$, including several regimes and transition lines. The results yield sharp, highly structured limits for fractal convergence, sharpen understanding of fractal maximal estimates, and extend Bourgain-type phenomena to fractal measures with explicit dimension formulas.

Abstract

We construct counterexamples for the fractal Schrödinger convergence problem by combining a fractal extension of Bourgain's counterexample and the intermediate space trick of Du--Kim--Wang--Zhang. We confirm that the same regularity as Du's counterexamples for weighted $L^2$ restriction estimates is achieved for the convergence problem. To do so, we need to construct the set of divergence explicitly and compute its Hausdorff dimension, for which we use the Mass Transference Principle, a technique originated from Diophantine approximation.

Figures (8)

• Figure 1: Representation of Theorem \ref{['thm:Main_Theorem']} for $n = 15$, where we show the improvement with respect to the former lower bound \ref{['eq:BestLowerBound']}. The positive result refers to Theorem 2.3 in du_etal2018.
• Figure 2: Arrangement of the slabs of $F_k$.
• Figure 3: In blue, the unit cell $\widetilde{X}_R^{a_1,a_2}$. On the right, $\Omega_R^{a_1,a_2} = T(\widetilde{X}_R^{a_1,a_2})$. In black on the right, the image by $T$ of the original slabs, and in yellow, the image of the slabs dilated by $a_1,a_2$. To apply the Mass Transference Principle, we must prove that $\Omega_k^{\boldsymbol{a}}$ covers a positive portion of the unit cell.
• Figure 4: Restrictions on $\boldsymbol{u} = (u_1, u_2, u_3)$.
• Figure 5: For fixed $\alpha$, the maximum regularity is attained on the blurred, green line.

Theorems & Definitions (32)

• Theorem 1.1
• Proposition 2.1
• proof
• Lemma 2.2: Lemma 3.4 of eceizabarrena2021
• Proposition 2.3
• Corollary 2.4
• proof : Proof of Corollary \ref{['cor:Divergence']}
• proof : Proof of Proposition \ref{['prop:Initial_Datum']}
• Proposition 3.1
• proof