Negative-order Bochner–Riesz conjecture

Let $d\ge2$, let $0<\alpha<\frac{d+1}{2}$, and define the negative-order Bochner–Riesz operator by analytic continuation,

$$\mathcal R^{-\alpha}f=\mathcal F^{-1}\left(\frac{(1-|\xi|^2)_+^{-\alpha}}{\Gamma(1-\alpha)}\widehat f(\xi)\right).$$

For $d\ge2$ and $0<\alpha\le\frac{d+1}{2}$, set

$$\mathcal P_\alpha(d)=\left\{(x,y)\in I^2:x-y\ge\frac{2\alpha}{d+1},\ x>\frac{d-1}{2d}+\frac\alpha d,\ y<\frac{d+1}{2d}-\frac\alpha d\right\}.$$

Negative-order Bochner–Riesz conjecture. The operator $\mathcal R^{-\alpha}$ is bounded from $L^p(\mathbb R^d)$ to $L^q(\mathbb R^d)$ if and only if $(1/p,1/q)\in\mathcal P_\alpha(d)$. This conjecture gives the proposed exact $L^p$–$L^q$ boundedness range for negative-order Bochner–Riesz operators. The surrounding discussion relates the endpoint case to restriction-extension estimates, but no resolution of the stated range is supplied here.