Open problems in Sobolev spaces

Question

What are the open problems in the theory of Sobolev spaces?

I would like to see problems that are yes or no only. Also I would like to see problems with the statements that are short and easy to understand for someone who has a basic knowledge in the theory, say at the level of the book by Evans and Gariepy.

The problems do not have to be a well know ones. Just the problems you think are interesting.

Please, list each problem as a separate answer.

That will allow people to leave comments related exclusively to this particular problem.

I have been working with Sobolev spaces for most of my adult live and I have some of my favorite problems that I will list below. But I will do it later, because first I would like to see your problems.

Answer 1

Let $H^{s,p}(\mathbb{R}, \mathbb{C})$ be the fractional order Sobolev space of scalar valued functions (distributions) over the real line, where $s\in \mathbb R$ and $1<p<\infty$.

It is a theorem by E. Shamir and R. Strichartz that the indicator function of the half line $1_{\mathbb{R}_+}$ (equal to $1$ for $x\geq 0$ and equal to $0$ for $x<0$) is a pointwise multiplier on $H^{s,p}(\mathbb{R}, \mathbb{C})$ if and only if ($p'$ dual exponent) $$- \frac{1}{p'} < s < \frac{1}{p}.$$ This means that $$\|1_{\mathbb{R}_+} \cdot f \|_{H^{s,p}} \leq C \|f\|_{H^{s,p}}$$ for all Schwartz functions $f$, with a constant $C > 0$ independent of $f$. This result is trivial for $s = 0$ (reducing to an $L^p$-space) but non-trivial for $s\neq 0$. Strictly outside this range, because of trace considerations, the inequality cannot hold.

My question regards the case of vector-valued functions. Let $X$ be a Banach space and let $H^{s,p}(\mathbb{R}, X)$ be the Sobolev space of $X$-valued functions (distributions), defined in the same way as in the scalar valued case. We could show the multiplier property of $1_{\mathbb{R}_+}$ in the same range as in the scalar-valued case provided the Banach space $X$ has the UMD property. See here or here, and here, Section 4 for an elementary proof of this fact. As a rule of thumb, all reflexive standard Banach spaces have UMD. Moreover, alle UMD spaces are reflexive. Space without UMD are thus $L^1$ and $L^\infty$.

My question is as follows:

Let $X$ be a Banach space. Suppose that the inequality $$\|1_{\mathbb{R}_+} \cdot f \|_{H^{s,p}(\mathbb{R}, X)} \leq C \|f\|_{H^{s,p}(\mathbb{R}, X)}$$ holds true for some $s\neq 0$ and some $1<p<\infty$, for all $X$-valued Schwartz functions $f$. Does this imply that $X$ has the UMD property?

I find this interesting because $X$ has the UMD property if and only if the Hilbert transform is a bounded operator on $L^p(\mathbb{R}, X)$, i.e. the signum function is a Fourier multiplier on this space. In other words, $F^{-1} sgn F$ is a bounded operator on $L^p(\mathbb{R}, X)$ ($F$ denoting the Fourier transform).

The pointwise multiplier property is equivalent to the boundedness of $$1_{\mathbb{R}_+} F^{-1}(1+|\cdot|^2)^{s/2} F$$ on $L^p(\mathbb{R}, X)$. So, given a positive answer the question, this would imply a new characterization of the boundedness of Hilbert transform in terms of a jump function in the time variable - and not in the frequency variable as in the usual definition.

Answer 2

Let $E \subset \mathbb R^n$. For $f : E \to \mathbb R$, let $$\|f\|_{L^{m,p}(E)} = \inf\{\|F\|_{L^{m,p}(\mathbb R^n)} : F|_E = f\}.$$ Here $\| \cdot \|_{L^{m,p}}$ is the homogeneous Sobolev seminorm $$\|F\|_{L^{m,p}(\mathbb R^n)} = \max\limits_{|\alpha| = m} \|\partial^\alpha F\|_{L^p(\mathbb R^n)}$$ Fefferman, Israel, and Luli have shown that in the case $p>n$ there is a linear extension operator $T : L^{m,p}(E) \to L^{m,p}(\mathbb R^n)$ such that $Tf|_E = f$ and $\|Tf\|_{L^{m,p}(\mathbb R^n)} \leq C \|f\|_{L^{m,p}(E)}$, where $C$ depends on $m,n,p$ only. To emphasize the point, $C$ does not depend at all on $E$, which can be completely arbitrary.

In principle, a result of this kind makes sense whenever $p > n/m$, but as far as I know nothing is known about the case $p \leq n$. Fefferman, Israel, and Luli have shown quite a bit more about these operators as well, but even the question of whether linear extension operators of uniformly bounded norm exist is open in the case $p \leq n$.