Weighted dispersive estimates for point interactions

Let $H_{\alpha,y}$ be the Schrödinger operator with a point interaction of strength $\alpha$ at $y\in\mathbb{R}^3$, let $P_{ac}$ denote the projection onto its absolutely continuous spectrum, and let $G_i$ be the kernel appearing in the preceding weighted estimates. Fix $q\in[3,+\infty]$ and let $w_q$ be a weight satisfying $w_q(x)\equiv1$ outside a ball centered at the origin and $w_q^{-1}G_i\in L^q(\mathbb{R}^3)$. Weighted dispersive estimates conjecture. For every $\alpha\ne0$ and $y\in\mathbb{R}^3$, one has

\left\\|w_q(\cdot-y)^{-1}e^{itH_{\alpha,y}}P_{ac}f\right\\|_{L^q(\mathbb{R}^3)}\lesssim t^{-\frac32\left(\frac1p-\frac1q\right)}\left\\|w_q(\cdot-y)f\right\\|_{L^p(\mathbb{R}^3)}.

When $\alpha=0$, a similar estimate should hold with slower decay:

\left\\|w_q(\cdot-y)^{-1}e^{itH_{\alpha,y}}f\right\\|_{L^q(\mathbb{R}^3)}\lesssim t^{-\frac12}\left\\|w_q(\cdot-y)f\right\\|_{L^p(\mathbb{R}^3)}.

These estimates are proposed for the regime $q\ge3$ because more general weighted Fourier inequalities could extend the paper's method beyond the range covered by Pitt's inequality. The conjecture is motivated by the principle that removing the local singularity should suffice for dispersive estimates; it would improve the cited polynomially weighted estimates, with a logarithmic weight expected when $q=3$.