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Is there a generalization of the Cauchy-Schwarz Inequality for multiple integrals?

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You mean like $|\iint f\overline g|\leq\sqrt{\iint|f|^2}\sqrt{\iint|g|^2}$ on subsets of $\mathbb{R}^2$, and higher dimensional analogues? If so, then yes, these are all special cases of the Cauchy-Schwarz inequality: en.wikipedia.org/wiki/Cauchy-schwarz –  Jonas Meyer Apr 4 '11 at 0:12

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According to Michael Steele, one generalization is the following for double integrals: $S \subset \mathbb{R}^2, f: S \to \mathbb{R}$ and $g: S \to \mathbb{R}$, then $$A = \iint_{S} f^{2} \ dx \ dy, \ B = \iint_{S} fg \ dx \ dy, \ C = \iint_{S} g^{2} \ dx \ dy$$ satisfy $|B| \leq \sqrt{A} \cdot \sqrt{C}$.

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