= Radial sine-power kernel integrability
{title2=$\left\|\dfrac{\sin(x^2+y^2)}{(x^2+y^2)^\alpha}\right\|_1\leq\pi\left(\dfrac1{2-\alpha}+\dfrac1{\alpha-1}\right),\quad1<\alpha<2$}
Assign an arbitrary finite value at the origin. <Polar integration> and the <substitution rule> $u=r^2$ reduce the joint <L1 norm> exactly to $\pi\int_0^\infty|\sin u|u^{-\alpha}\,du$. Near zero, $|\sin u|\leq u$ gives $u^{1-\alpha}$; at infinity, $|\sin u|\leq1$ gives $u^{-\alpha}$. These are integrable when $\alpha<2$ and $\alpha>1$, respectively. Their explicit <integrals> give the displayed bound. This is a planar version of <sine power kernel integrability>, with the radial area element determining the exponents.
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