On p.399 of Stein's book "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals", one considers the operator

$f\mapsto \nabla (f * \text{d}\sigma)$,

where $\sigma$ denotes the standard surface measure on the unit sphere $S^2\subseteq \mathbb{R}^3$. Now, Stein asserts that, if $f$ is compactly supported with $f(x)=|x|^{-\alpha}$ for "small" $x$, then we have

$|\nabla (f*\text{d}\sigma)(x)|\gtrsim |1-|x||^{1-\alpha}$, provided $1<\alpha<3$.

However, I am not able to reproduce this estimate. My considerations so far: If $f$ has support in $B_{\varepsilon}$ (the ball with small radius $\varepsilon$ and center in the origin), then it follows immediately that the support of $f*\text{d}\sigma$ lies in $A_{1-\varepsilon, 1+\varepsilon}$ (the annulus with smaller radius $1-\varepsilon$ and larger radius $1+\varepsilon$, and center in the origin). If we assume $f(x)=|x|^{-\alpha} \chi_{B_\varepsilon}(x)$, then $\nabla (f*\text{d}\sigma)(x)=-\alpha\int_{S^2\cap \{|x+y|\leq \varepsilon\}} \frac{x+y}{|x+y|^{\alpha+2}} \text{d}\sigma(y)\quad $ for $x\in A_{1-\varepsilon, 1+\varepsilon}$. However, I am not able to estimate the latter integral in a clever way.

Any help would be greatly appreciated!



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