An inequality from Bessel potential space to Besov space

Question

I'm not sure this question is suitable for MathOverflow. Currently, I'm reading a paper "Inhomogeneous Dirichlet Problem in Lipschitz domain" by Jerison and Kenig.

I have a question on some inequality on Bessel potential space and Besov space.

For convenience, let us fix some notation. $L_s^q$ denote the Bessel potential space and $B^{p,q}_\alpha$ the Besov space. When $p=q$, we simply denote $B^p_\alpha$.

Let $\frac{1}{p}<\alpha<1+\frac{1}{p}$ and $1\leq p<\infty$. Fix values of $q$ and $s$ so that $qs>\alpha p$ and $s<1+\frac{1}{q}$.

While I'm reading a paper, I have a trouble to prove the following statement. (p.180)

Since the $L_s^q$ norm is larger than the $B_\alpha^p$ norm,...

I tried to establish this inequality via some embedding result, but I failed. How can I prove this inequality?

Thanks in advance.

Answer 1

I think that your idea is completely correct, but the choice of $s$ and $q$ is indeed somewhat curious in the paper. However, the case $q=p$ should be sufficient for the proof to work:

We need $s$ and $q$ such that

1. $L^q_s(\Omega)$ continuously embeds into $B^p_\alpha(\Omega)$, and
2. Corollary 3.11 is useable for $L^q_s(\Omega)$ there, so $\frac1q < s < 1+\frac1q$.

These conditions are fulfilled for $q=p$ and $s > \alpha$ with $s-\alpha$ so small that still $\frac1p < s < 1+\frac1p$, because standard embeddings for the Bessel/Besov scales then imply that $L^p_s \hookrightarrow B^p_\alpha$ (which transfers to the quotient spaces on $\Omega$), see e.g. the book of Triebel, Ch. 2.3.3, Remark 4.

(Sidenote: I also think that the sequence $(f_j)$ in the proof should in fact be taken from $C_c^\infty$ (instead of $C^\infty$) to have $f_j$ in all the function spaces one needs..)