I am aware of decay rates for the Fourier transform of $B_p(1) = \{x = (x_1, \dots, x_d) \in \mathbb{R}^d|~|x_1|^{p} + \dots + |x_d|^{p} \leq 1\}$, the $\ell^p(\mathbb{R}^d)$ unit ball when $p \geq 1$. For example, theorem $5.5$ of "Decay of the Fourier Transform", Alex Iosevich , Elijah Liflyand, states that $\int_{\mathbb{S}^{d-1}} |\widehat{\mathbf{1}_{C}}(R \omega)|^2 d\omega \lesssim R^{-d-1}$ for $C$ a convex bounded set of $\mathbb{R}^d$ ($\mathbb{S}^{d-1}$ is the $\ell^2(\mathbb{R}^d)$ unit sphere). Thus, this theorem applies to $B_p(1)$ when $p \geq 1$.
However, I didn't found any result concerning the Fourier transform of $B_p(1)$ for $p < 1$. My question is: is there any bound on $\int_{\mathbb{S}^{d-1}} |\widehat{\mathbf{1}_{B_p(\mathbb{R}^d)}}(R \omega)|^2 d\omega$ and directly on $k \mapsto |\widehat{\mathbf{1}_{B_p(\mathbb{R}^d)}}(k)|$ for $p < 1$ ?
I expect the decay to be slower, as $B_p(\mathbb{R}^d)$ has less smooth boundary when $p < 1$.