0.3 Embeddings

Sobolev, Besov and Bessel-potential spaces satisfy two obvious embedding relations:

• For fixed $p$ (and arbitrary $q$ in the case of Besov spaces), the spaces get larger as $s$ decreases.
• In the case where $\Omega$ a bounded domain, for fixed $s$ (and fixed $q$ in the case of Besov spaces), the spaces get larger as $p$ decrease, since ${\parallel f\parallel }_{L^{p_1}}\le C{\parallel f\parallel }_{L^{p_2}}$ if $p_1\le p_2$ .

A less trivial type of embedding is known as Sobolev embedding . In the caseof Sobolev spaces, it states that the continuous embedding

holds except in the case where $p_2=+\infty$ and $s_2$ is an integer, for which one needs to assume $s_1-s_2>d(1/p_1-1/p_2)$ . For example in the univariate case, any $H^1$ function has also ${\mathcal{C}}^{1/2}$ smoothness. In the case of Besov spaces the embedding relation are given by

with no other restrictions on the indices $s_1,s_2\ge 0$ . In the case where $\Omega$ is a bounded domain, these embedding are compact if and only if the strict inequality $s_1-s_2>d(1/p_1-1/p_2)$ holds. The proof of these embeddings can be found in [link] for Sobolev spaces and [link] for Besov spaces.

As an exercise, let us see how these embeddings can be used to derive the range of $r$ such that $B_{2,q}^r\left([0,1]\right)$ can contain discontinuous functions. If $r>1/2$ , then there exists $\epsilon >0$ such that $r-2\epsilon >1/2$ ; We remark that $B_{2,q}^r\subset B_{2,q}^{r-\epsilon }\subset B_{\infty ,\infty }^{\epsilon }={\mathcal{C}}^{\epsilon }$ , so all functions in $B_{2,q}^r$ are continuous.Therefore only $B_{2,q}^r$ with $r\le 1/2$ can contain discontinuous functions. In the limiting case $r=1/2$ , a closer inspection reveals that the functions in $B_{2,q}^{1/2}$ are continuous if $q<\infty$ , while $B_{2,\infty }^{1/2}$ includes discontinuous functions, such as the characteristic funciton of an interval $[0,a]$ for $0<a<1$ .