= Angular quotient functions belong to Sobolev spaces below the dimension
{title2=$u(x)=x_1/|x|\in W^{1,p}(B^n)\quad(1\leq p<n,\ n>1)$}
The bounded angular <function> has ordinary derivatives $D_ju=\delta_{1j}/r-x_1x_j/r^3$ away from zero. Their magnitudes are at most $2/r$, whose $p$-th power is integrable on the unit ball exactly below the dimension threshold in this estimate. To verify the <weak derivative> across zero, integrate by parts outside a ball of radius $\varepsilon$ against a compactly supported smooth test. The boundary term has magnitude at most $C\varepsilon^{n-1}$, tending to zero for $n>1$, while the derivative volume term converges by integrability. This proves the stated Sobolev membership with those <weak derivatives>. The dimensional condition distinguishes it from the one-dimensional sign <function>, whose jump produces a Dirac derivative.
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