Intermediate rates of growth of Lebesgue constants in the two-dimensional case

The behavior as $R\to\infty$ of the Lebesgue constants
$$ L(RW)=\dfrac{1}{4\pi^2}\int^\pi_{-\pi}\int^\pi_{-\pi}\biggl|\sum_{(n,m)\in RW\cap\mathbf Z^2}e^{i(nx+my)}\biggr|\,dx\,dy, $$
where $RW$ is homothetic to a compact, convex set $W$ is considered. that
a) for any $p>2$ there exists $W$ for which
$$ C_1(\ln R)^p\leqslant L(RW)\leqslant C_2(\ln R)^p,\quad R\geqslant2; $$

b) for any $p\in\biggl(0,\dfrac12\biggr)$ and $\alpha>1$ there exists $W$ for which
$$ C_1R^p(\ln R)^{-\alpha p}\leqslant L(RW)\leqslant C_2R^p(\ln R)^{2-2p},\quad R\geqslant2. $$