Correction to the article "Global well-posedness and scattering in weighted space for nonlinear Schrödinger equations below the Strauss exponent without gauge-invariance"

Technical Summary: Correction to "Global Well-Posedness and Scattering in Weighted Space for Nonlinear Schrödinger Equations Below the Strauss Exponent Without Gauge-Invariance"

Problem Statement
This article serves as a correction to the authors' previous work [3], which addressed the global well-posedness and scattering of nonlinear Schrödinger equations (NLS) in weighted spaces for dimensions $d=1, 2, 3$ below the Strauss exponent. The original paper contained a critical error in the proof of the contraction mapping property for dimensions $d=2$ and $d=3$.

Specifically, the error occurred in the estimation of the difference term $\|A_{4,n}(\cdot, v_1) - A_{4,n}(\cdot, v_2)\|_X$ within the contraction argument. The original proof relied on an estimate (1.1) that implicitly required the nonlinearity exponent $\alpha \geq 3$ to hold for general non-smooth nonlinearities. However, the admissible ranges for $\alpha$ in the original theorem were $\alpha \in (2, 3)$ for $d=2$ and $\alpha = 2$ for $d=3$. Consequently, the proof for Lemma 3.4 in [3] was incomplete for these dimensions under the general assumptions of Assumption 1.1, rendering the main result (Theorem 1.4) unjustified for non-smooth nonlinearities in $d=2$ and $d=3$. While the result held for $d=1$ (where $\alpha > 3$) and for smooth nonlinearities in $d=3$, the general case required correction.

Methodology and Reformulation
To resolve the singularity issues arising from the time derivative of the nonlinearity and the lack of spatial integrability, the authors reformulate the integral equation and the associated solution space.

1. Reformulation of the Integral Equation:
   Instead of utilizing the identity involving the time derivative of $F_n(v)$ as in [3, (1.8)], the authors employ a new identity that isolates the term $i\partial_t v - \frac{p^2}{2}v$. This leads to a modified integral equation (2.1) where the term $A_{5,n}(t)$ is replaced by a new term $\tilde{A}_{5,n}(t)$. This new term incorporates the operator $L = i\partial_t - \frac{p^2}{2}$ acting on the solution, allowing for a more direct handling of the nonlinearity's time derivative without requiring the auxiliary term $E(v)$ used in the original proof.

2. Modification of the Solution Space:
   The authors introduce a new complete metric space $\tilde{Z}_M$ equipped with a modified norm $\|\cdot\|_{\tilde{Y}}$. A key innovation is the introduction of a truncated weight function $\chi_\gamma(x)$, defined as $|x|^{-\gamma}$ for $|x| \leq 1$ and $1$ for $|x| > 1$.

   • Purpose of $\chi_\gamma$: This function is designed to control spatial singularities at the origin ($x=0$) that arise when using resolvent estimates to avoid time singularities at $t=0$.
   • Dimensional Specifics:
     • For $d=2$, the weight handles the strong time singularity by imposing a stronger spatial singularity, utilizing resolvent estimates and the weight to localize the singularity near $x=0$.
     • For $d=3$, the weight addresses a lack of spatial integrability. By imposing spatial regularity and using Sobolev embeddings (specifically $\|f\|_{L^2} \lesssim \|\nabla^{1/2}f\|_{L^{3/2}}$), the authors manage the spatial singularity via $\chi_{1/2}$.

3. Contraction Mapping Argument:
   The map $\Phi$ from [3] is replaced by $\tilde{\Phi}$. The authors prove that $\tilde{\Phi}$ is a contraction on $(\tilde{Z}_M, d_{\tilde{Z}})$ for sufficiently small $M$. The proof involves:

   • Establishing boundedness estimates for $\tilde{\Phi}(v)$ in the $X$ and $\tilde{Y}$ norms, utilizing Strichartz estimates, Gagliardo-Nirenberg inequalities, and the properties of $\chi_\gamma$.
   • Proving the contraction property $d_{\tilde{Z}}(\tilde{\Phi}(v_1), \tilde{\Phi}(v_2)) \leq \frac{1}{2} d_{\tilde{Z}}(v_1, v_2)$ by estimating the difference of the nonlinear terms. Crucially, the metric $d_{\tilde{Z}}$ does not involve the truncated weight $\chi_\gamma$, allowing the estimates to close without the weight appearing in the difference term.

Key Contributions and Results

• Correction of Lemma 3.4 and Theorem 1.4: The paper provides a rigorous proof that the main result of [3] holds for the full class of nonlinearities specified in Assumption 1.1, including non-smooth cases in dimensions $d=2$ and $d=3$.
• New Analytical Framework: The introduction of the truncated weight $\chi_\gamma$ and the reformulation of the integral equation to include the operator $L$ directly in the auxiliary term $\tilde{A}_{5,n}$ allows the authors to bypass the regularity requirements ($\alpha \geq 3$) that previously limited the proof.
• Proposition 2.1: The authors establish that under the assumptions of the original Theorem 1.4, the map $\tilde{\Phi}$ is a contraction on the new space $\tilde{Z}_M$, and any fixed point is a solution to the original equation [3, (1.8)].

Significance
The paper claims that while a compactness argument could potentially recover the result, it is desirable to establish the theorem within a direct contraction mapping framework. By correcting the gap in the proof, the authors ensure that the method introduced in [3] for treating scattering solutions with fractional power nonlinearities remains valid and robust. The correction streamlines the argument by removing the need for the auxiliary term $E(v)$ and clarifies the handling of singularities in both time and space, thereby securing the global well-posedness and scattering results for the intended range of parameters in dimensions 2 and 3.