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Let $\mu $ be a positive Borel measure on $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{d}$ such that $\mu \left( B\left( a,r\right) \right) \leq Cr^{n}$ for some $n\in (0,d]$ and for any ball $B\left( a,r\right) $ in $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{d}$. Could you help me to prove that $\int_{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{d}}\frac{1}{\left\vert x\right\vert ^{n}}d\mu \left( x\right) =\infty $?

My effort: $\int_{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{d}}\frac{1}{\left\vert x\right\vert ^{n}}d\mu \left( x\right) \geq \int_{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{d}\backslash B\left( 0,1\right) }\frac{1}{\left\vert x\right\vert ^{n}}% d\mu \left( x\right) =\sum_{k=0}^{\infty }\int_{B\left( 0,2^{k+1}\right) \backslash B\left( 0,2^{k}\right) }\frac{1}{\left\vert x\right\vert ^{n}}% d\mu \left( x\right) \geq \sum_{k=0}^{\infty }\frac{1}{\left( 2^{k+1}\right) ^{n}}\mu \left( B\left( 0,2^{k+1}\right) \backslash B\left( 0,2^{k}\right) \right) $.

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    What about $n$-dimensional Hausdorff measure restricted to the $n$-sphere (embedded in the obvious way in $\mathbb{R}^d$)?2012-12-31
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    Which page of Stein's book is this from? as @Jose27 remarked, it can't be true as stated. Here's a one-dimensional counterexample ($d=n=1$): $d\mu(x) = \frac{|x|}{x^2+1}|dx|$.2012-12-31
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    @PavelM : I wanna learn that Hardy-Littlewood-Sobolev inequality (for p=1) is wrong just like lebesgue measure case in Stein book. But now,for a measure that satisfy the above condition.2012-12-31
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    @PavelM: How to check that that your measure satisfying $\mu (B(a,r))\leq Cr^n$? Since $d\mu(x)=\frac{|x|}{x^2+1} dx$, then $\mu(x)=\int \frac{|x|}{x^2+1} dx$, right?2012-12-31
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    The function $|x|/(x^2+1)$ is bounded (say, by $M$), which implies $\mu(B(a,r))\le 2Mr$. (Integral estimated by supremum * size of the region of integration).2012-12-31
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    @beginner: I think your inequality for the measure needs to be reversed. That is, I think you need $\mu(B(a,r))\geq Cr^n$. To see why we need this, if it were not the case we could take the trivial measure $\mu=0$.2013-01-01

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