Sobolev spaces embeddings and conjugate exponents

Question

Let $N\geq 1$ and $1<p<N$ with conjugate exponenr $p^{\prime}$. I would like to understand if there exists some inclusion relation between $W^{1, p}(\mathbb{R}^N)$ and $W^{1, p^{\prime}}(\mathbb{R}^N)$, maybe something $$W^{1, p}(\mathbb{R}^N)\hookrightarrow W^{1, p^{\prime}}(\mathbb{R}^N).$$ If not, it makes sense to consider $$W^{1, p}(\mathbb{R}^N)\cap W^{1, p^{\prime}}(\mathbb{R}^N)?$$

Could someone please help?

Thank you in advance!

Answer 1

Sketch. In the case $p=2=p'$ there is nothing to prove. Otherwise, say $p<2$ so that $p'>2$. Let $\chi$ be a bump function equal to $1$ near the origin. Then it is a standard exercise to check that $f(x)=\frac1{|x|^\theta}\in W^{1,p}(B_1(0))$ and $f\notin W^{1,p'}(B_1(0))$ for a certain $\theta$ depending on $N$ and $p$. (The basic calculation needed is that $\frac1{|x|^r}\in L^1_{\text{loc}}$ if $r<n$). Then $\chi f$ is a counterexample on $\mathbb R^n$. The $p>2$ case is also false, and for this you instead consider $(1-\chi)f$. Since $p''=p$ this also covers the reverse inclusion.

I don't know what it means to "consider" it; you can certainly write down the set.

Answer 2

If $p=1$ fixed, (thanks Calvin in the comments) you cannot say much in general, hence it makes sense to consider the intersection. Some nice results are for a domain $\Omega$ but you need to increase the order of "differentiation".

If the domain $\Omega$ in $\mathbb{R^{n}}$ has the cone property, the following embeddings are continuous: $$ W^{j+m, p}(\Omega) \subset W^{j, q}(\Omega), \text { when } p \leq q \leq \frac{n p}{n-m p} . $$ If $\Omega$ is a Lipschitz domain, then: $$ W^{j+m, p}(\Omega) \subset C^{j, \lambda}(\bar{\Omega}), \text { for } 0<\lambda \leq m-\frac{n}{p} $$ For the proof of the above, see Adams, Sobolev Spaces, (1975), p. 97.

Note: $\Omega$ has the cone property if there exists a finite cone $C$ such that each point $x \in \Omega$ is the vertex of a finite cone $C_{x}$ contained in $\Omega$ and congruent to $C .$ (That means, $C_{x}$ is obtained from $C$ by a rigid motion). Think of this as a regularity assumption on the boundary of $\Omega$.

Let $\Omega$ be a bounded domain in $\mathbb{R^{n}}$ having the cone property. Then the following embedding is also compact: $$ W^{j+m, p}(\Omega) \subset W^{j, q}(\Omega) \text { if } 0<n-m p \text { and } j+m-\frac{n}{p} \geq j-\frac{n}{q} $$