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I would like to know whether multiplication defines a bounded map $$H^{1/2} \otimes H^{1/2} \to H^{-1/2}$$ dimension of the domain is $3$.

I have checked two different sources and one said that it works but the other that this map is bounded as a map $$H^{1/2} \otimes H^{1/2} \to H^{s}$$ where $s < -\frac{1}{2}$. That is why I am confused.

Notation: $H^{1/2} = W^{1/2, 2} = L^2_{1/2}$

I would be satissfied with a source i can rely on.

Maciej Starostka
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2 Answers2

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For $\Omega \subseteq \mathbb{R}^3$ there are Sobolev embeddings $$ H^{1/2} \subset L^3, \, L^{3/2} \subset H^{-1/2} $$ where the second one follows from the first one by duality. Thus $$ f, \, g \in H^{1/2} \subset L^3 \; \Rightarrow \; fg \in L^{3/2} \; (\text{by Cauchy-Schwarz}) \; \Rightarrow \; fg \in H^{-1/2} \, . $$

Hans Engler
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See this notes by Terry Tao. Also, Sobolev multiplication below the borderline.

Martín-Blas Pérez Pinilla
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  • Thanks for interest. Unfortunetelly notes by Terrence Tao doesnt give an answer. The second link i have checked earlier and it gives positive answer. However no refferences are given and the proof is for $m \in \mathbb{N}$ which is not my case. I need something i could cite in the paper. – Maciej Starostka Oct 31 '14 at 11:06
  • @BigBolzano, I will search more in my books. – Martín-Blas Pérez Pinilla Oct 31 '14 at 11:09
  • @BigBolzano, read the answer by paul garrett in http://math.stackexchange.com/users/52330/frank-zermelo. – Martín-Blas Pérez Pinilla Nov 03 '14 at 07:40
  • Thanks. The answer given there proves that $H^{s} \times H^{s}$ gives a well defined element in $H^{s-\frac{3}{2}}$ but the assumption is $s - \frac{3}{2} > 0$ which is not my case (cannot give the same argument). Also $\frac12-\frac{3}{2} = -1$ but this does not say that the element is not in $H^{-\frac{1}{2}} $ (which would be stronger). I am trying to find a counterexample maybe. – Maciej Starostka Nov 04 '14 at 12:16
  • @BigBolzano, I've found this: http://www.sciencedirect.com/science/article/pii/S0022247X00972093. – Martín-Blas Pérez Pinilla Nov 06 '14 at 10:07
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    (The above mentioned answer of p. garrett is presumably this one: https://math.stackexchange.com/a/315044/80734) – Calvin Khor Aug 14 '21 at 08:19