Let $0<\alpha and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$\left\|\int_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha}} \right\|_{L^q(\mathbb{R}^n)} \leq C\left\| f \right\|_{L^p(\mathbb{R}^n)}$$.
Sobolev-type inequality.
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analysis
functional-analysis
fourier-analysis
sobolev-spaces
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0Here are some helpful references http://math.stackexchange.com/questions/204322/step-in-proof-of-sobolev-imbedding , http://math.stackexchange.com/questions/181339/compactness-and-boundedness-of-integral-operator , http://math.stackexchange.com/questions/178938/a-marcinkiewicz-approach – 2012-10-19
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1Would you mind and formulate a question? – 2012-10-19