Product of Sobolev functions

Question

Suppose that $\Omega$ is 2-dimensional bounded open set with smooth boundary and $f\in W^{2,2}( \Omega) $ and $ g,h\in W^{1,2} ( \Omega) $. What can we say about the regularity of the product of these three functions $ f\cdot g \cdot h $. I don't believe that it is even a $W^{1,2} $ function.

Answer 1

The product is not in $H^1$:

(1) $fgh\in L^p(\Omega)$ for all $p<\infty$ due to the Sobolev embeddings into the $L^q$ spaces

(2) the limiting terms are the products of type $fg\nabla h$, $f\nabla gh$, which are in $L^p(\Omega)$ for all $p<2$ ($f\in L^\infty(\Omega)$, $\nabla g\in L^2(\Omega)$, $h\in L^q(\Omega)$ $\forall q<\infty$ implies that the product $f\nabla gh$ is in $L^p(\Omega)$ for all $p<2$).

Answer 2

Due to the embeddings $W^{2,2}(\Omega)\hookrightarrow W^{1,p}(\Omega)$ and $W^{1,2}(\Omega)\hookrightarrow L^p(\Omega)$ with any $p\in (2,\infty)$, by Hölder's inequality it readily follows that the product $f\cdot g\cdot h\in W^{1,q}(\Omega)$ with any $q\in [1,2)$. In terms of the Sobolev space $W^{1,q}(\Omega)$, the regularity of the product cannot be stronger, i.e., $f\cdot g\cdot h\not\in W^{1,2}(\Omega)$, which is corroborated by routine example: $$ \Omega=\{x\in\mathbb{R}^2\,\colon |x|<e^{-1}\}, \quad f\equiv 1, \,\, g(x)=h(x)=\Bigl(\ln{\frac{1}{|x|}}\Bigr)^{1/4} \;\forall\,x\in \Omega, $$ where $\,g,h\in W^{1,2}(\Omega)\,$ while $\,g\cdot h\not\in W^{1,2}(\Omega)$.

The best known regularity result for the product in question can be formulated in terms of the Orlicz space due to the Trudinger's theorem 8.27 in Adams–Fournier's textbook of Sobolev Spaces http://bookza.org/book/492535/11bd42 (see page 277 therein), which implies the embedding $W^{1,2}(\Omega)\hookrightarrow L_A(\Omega)$ into the Orlicz space $L_A(\Omega)$ with an $N$-function $A(t)=e^{t^2}-1$. Hence, there is some constant $c>0$ such that the product $\varphi\overset{\rm def}{=}f\cdot g\cdot h$ does satisfy the condition $$ \int\limits_{\Omega}e^{c|\varphi(x)|}dx<\infty $$ due to the embedding $\, W^{2,2}(\Omega)\hookrightarrow C(\overline{\Omega})\,$ along with an obvious inequality $\,2|g\cdot h|\leqslant g^2+h^2$ followed by applying Hölder's inequality to a product of exponential functions.