Refinements on the Restriction and Kakeya problems range in $\mathbb{R}^4$

Abstract: We obtain improved Fourier restriction and Kakeya maximal estimates in $\mathbb R<sup>4$. More precisely, by proving the two-ends incidence estimates $$\mathbf{TE}\left(\frac{133}{44},\frac{177}{88},\frac34\right)$$ and $$\mathbf{TE}\left(\frac{386453}{125988},\frac{386453}{125988},\frac{358951}{377964}\right)$$, we extend the restriction range to $p&gt;2+ \frac{176}{221}$ and the Kakeya maximal estimate up to dimension $3.06737$. The implications to the known lower-bounds for the Hausdorff dimension of Kakeya sets in $\mathbb{R}<sup>4$ and the Bochner--Riesz conjecture are also studied.