Bochner–Riesz regularity conjecture for ${}_{\mathrm{S}^{d-1}}L_p^k$

Let ${}_{\mathrm{S}^{d-1}}L_p^k$ be the Hardy-type space associated with the unit sphere, and let the Bochner–Riesz operator of order $\lambda>0$ be given as in the source. Bochner–Riesz regularity conjecture. There exists $l\in\mathbb{N}$, depending on $\lambda$, such that the Bochner–Riesz operator of order $\lambda$ acts continuously from ${}_{\mathrm{S}^{d-1}}L_p^k$ to $L_p$ for every $p$ and every $k>l$. The conjecture extends known improvements in continuity properties obtained by replacing $L_p$ with ${}_{\Sigma}L_p$; its resolution is not given in the source.