# Open problems in Sobolev spaces

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.

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.

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.

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 $$1, for all $$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.

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 $$p \leq n$$.