If $\delta>-\frac12$, show that $(1-x^2)_+^\delta\in W^{s,2}$, where $s\in (0,\delta+\frac12)$.
2026-04-24 19:46:26.1777059986
Show the example belong to the Bessel potentials space (fractional order sobolev space), where $p=2$
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Find (or look up) the Fourier transform. In 1 D, it's essentially $$ \xi \mapsto |\xi|^{-\delta - 1/2} J_{\delta + 1/2}(\xi) $$ up to scaling. Then use the known behavior of Bessel functions at $\infty$. This extends readily to the n-dimensional case.