Suppose that $B\subset R^m$, $m\geq 2$, be a unit ball, and $W^{1,2}(B )$ be the Sobolev space. My quesion is whther $(W^{1,2}(B))^*\supset L^1(B)$ holds or not?
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$\begingroup$ Do you know the answer if $m=1$? (I'm thinking of Sobolev embeddings.) $\endgroup$– Oskar LimkaDec 8, 2015 at 9:39
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$\begingroup$ I would start with the following (maybe counter) example $$l(\phi)=\int_B\phi(x)\frac{dx}x,$$ $m=2$ and check whether this is a bounded-linear functional or not on $W^1_2(B)$. $\endgroup$– Oskar LimkaDec 8, 2015 at 9:45
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1$\begingroup$ This is essentially equivalent to $W^{1,2} \hookrightarrow L^\infty$ which fails for $m$ large enough, IIRC even for all $m \geq 2$. $\endgroup$– PhoemueXDec 8, 2015 at 18:31
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