Radial sine-power kernel integrability (${\Vert \frac{\sin (x^2+y^2)}{(x^2+y^2{)}^{\alpha }}\Vert }_1\le \pi \left(\frac{1}{2-\alpha }+\frac{1}{\alpha -1}\right),1<\alpha <2$)

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, $|\sin u|\le u$ gives $u^{1-\alpha }$; at infinity, $|\sin u|\le 1$ 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.