The $L^p\to L^p$ local smoothing conjecture for spherical averages

Let $A_t$ denote the spherical averaging operator, let $\partial_t^\gamma$ be the fractional time derivative, and let $L^p_s(\mathbb{R}^d)$ denote the inhomogeneous Sobolev space. The spherical-average local smoothing conjecture. For $d\geq2$, $\gamma\geq0$, and all $p\geq\frac{2d}{d-1}$,

$$\left(\int_1^2\int_{\mathbb{R}^d}|\partial_t^\gamma A_t(f)(x)|^p\,dx\,dt\right)^{1/p}\lesssim_{\varepsilon}\|f\|_{L^p_{\gamma-\frac{d}{p}+\varepsilon}(\mathbb{R}^d)}.$$

The estimate is motivated by the wave-equation local smoothing conjecture and would yield scale-localized spherical-average bounds; its status is not resolved in the supplied text.