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\dot H^{whatever} -> \smash{\dot H}^{whatever}, and <> -> \langle\rangle
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LSpice
LSpice
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Reference - measures in $\dot$\smash{\dot{H}}^{-1}$ have zero mean

The following fact seems to be well-known and easy to prove: Let $\sigma$ be a Borel measure on a Borel set $\Omega \subseteq \mathbb{R}^d$, then $$\|\sigma \|_{\dot{H}^{-1}}:\,=\sup\limits_{\|f\|_{\dot{H}^1}\leq 1} |<\sigma ,f>| < \infty \, \Rightarrow \sigma (\Omega )=0 \, ,$$$$\|\sigma \|_{\smash{\dot{H}}^{-1}}:\,=\sup\limits_{\|f\|_{\smash{\dot{H}}^1}\leq 1} |\langle\sigma ,f\rangle| < \infty \, \Rightarrow \sigma (\Omega )=0 \, ,$$ where $\|f\|_{\dot{H}^1}:\,=\|\nabla f\|_2$$\|f\|_{\smash{\dot{H}}^1}:\,=\|\nabla f\|_2$, and the inner product is just the integral of the product.

However, I couldn't find a proper reference in standard Sobolev Spaces textbooks (I tried Adams & Fournier and Leoni). Is this wrong? Does anyone know a solid reference?

(cross posted from MSE after a week with no answer).

Reference - measures in $\dot{H}^{-1}$ have zero mean

The following fact seems to be well-known and easy to prove: Let $\sigma$ be a Borel measure on a Borel set $\Omega \subseteq \mathbb{R}^d$, then $$\|\sigma \|_{\dot{H}^{-1}}:\,=\sup\limits_{\|f\|_{\dot{H}^1}\leq 1} |<\sigma ,f>| < \infty \, \Rightarrow \sigma (\Omega )=0 \, ,$$ where $\|f\|_{\dot{H}^1}:\,=\|\nabla f\|_2$, and the inner product is just the integral of the product.

However, I couldn't find a proper reference in standard Sobolev Spaces textbooks (I tried Adams & Fournier and Leoni). Is this wrong? Does anyone know a solid reference?

(cross posted from MSE after a week with no answer).

Reference - measures in $\smash{\dot{H}}^{-1}$ have zero mean

The following fact seems to be well-known and easy to prove: Let $\sigma$ be a Borel measure on a Borel set $\Omega \subseteq \mathbb{R}^d$, then $$\|\sigma \|_{\smash{\dot{H}}^{-1}}:\,=\sup\limits_{\|f\|_{\smash{\dot{H}}^1}\leq 1} |\langle\sigma ,f\rangle| < \infty \, \Rightarrow \sigma (\Omega )=0 \, ,$$ where $\|f\|_{\smash{\dot{H}}^1}:\,=\|\nabla f\|_2$, and the inner product is just the integral of the product.

However, I couldn't find a proper reference in standard Sobolev Spaces textbooks (I tried Adams & Fournier and Leoni). Is this wrong? Does anyone know a solid reference?

(cross posted from MSE after a week with no answer).

A typo.
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user64494
user64494
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reference Reference - measures in $\dot{H}^{-1}$ have zero mean

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Amir Sagiv
Amir Sagiv
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reference - measures in $\dot{H}^{-1}$ have zero mean

The following fact seems to be well-known and easy to prove: Let $\sigma$ be a Borel measure on a Borel set $\Omega \subseteq \mathbb{R}^d$, then $$\|\sigma \|_{\dot{H}^{-1}}:\,=\sup\limits_{\|f\|_{\dot{H}^1}\leq 1} |<\sigma ,f>| < \infty \, \Rightarrow \sigma (\Omega )=0 \, ,$$ where $\|f\|_{\dot{H}^1}:\,=\|\nabla f\|_2$, and the inner product is just the integral of the product.

However, I couldn't find a proper reference in standard Sobolev Spaces textbooks (I tried Adams & Fournier and Leoni). Is this wrong? Does anyone know a solid reference?

(cross posted from MSE after a week with no answer).