Weighted Lorentz 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 fix $q\in[3,+\infty]$. Define

$$w_q(x)=1+|x|^{\frac3q-1}.$$

Weighted Lorentz 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,\infty}(\mathbb{R}^3)}\lesssim t^{-\frac32\left(\frac1p-\frac1q\right)}\left\\|w_q(\cdot-y)f\right\\|_{L^{p,1}(\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,\infty}(\mathbb{R}^3)}\lesssim t^{-\frac12}\left\\|w_q(\cdot-y)f\right\\|_{L^{p,1}(\mathbb{R}^3)}.

This alternative formulation uses weighted Lorentz spaces and is motivated by Lorentz-space generalizations of Pitt's inequality. It gives a concrete candidate weight in the range $q\ge3$, where the preceding argument suggests that optimal dispersive estimates may be accessible through more general weighted Fourier inequalities.