Showing a function belongs to a Besov space.

Question

In this paper, just before Theorem 3 it is stated that the kernels given in equations 4a-f are in $\mathbb{B}_{pq}^s$ for all $p,q\geq 1$ and some $s>0$. This is stated without proof/reference and I cannot find any literature which explicitly shows this. How would one go about doing this? For example if we take $$g(x)=\exp\left(-\frac{1}{2}x^2\right)$$ and want to show that $g\in \mathbb{B}_{pq}^s$ and we want to work in the one-dimensional case, then we want to show that $$|g|_{pq}^s=\left(2\int_0^1h^{-1-sq}\|\Delta_h^mg\|_p^qdh\right)^q<\infty.$$ In Theorem 3 it is mentioned that $\|\Delta_h^mg\|_p\leq 2^m\|g\|_p$, which I tried to use to prove the above inequality, but it does not make sense to use since $\int_0^1h^{-1-sq}dh<\infty$ only for $sq<0$, which is not the case. How can I go about this? Any reference with exercises/examples to show functions belong in a Besov space? Working with the finite difference operator $\|\Delta_h^mg\|_p$ seems cumbersome.

Answer 1

The idea is that you belong in the Besov space $B^{s}_{pq}$ if $D^ku$ satisfies an $q$ version of Dini continuity in $L^p$, where $k$ is the largest integer below $s$. Also, this is why you take $m>s$ (the integer endpoints are always special). In fact, if $s=0$, then this definition is exactly that $u$ satisfies a $q$-Dini condition in $L^p$, which further agrees with the usual one for $p=\infty$.

In other words, the $q$ exponent is telling you how good your $L^p$ modulus of continuity is. $s$ is telling you which derivatives this modulus is being applied to.

So for instance, if $m=1$, then you have $$ \Delta_h u(x)= u(x+h)-u(x), $$ and you can show $$ \| \Delta_h u(x)\|_{L^p}\lesssim |h| \|\nabla u\|_{L^p}. $$ This means $$ h^{-1-sq}\| \Delta_h u\|_{L^p}^q \lesssim h^{-1+(1-s)q}\| \nabla u\|_{L^p}^q. $$ We conclude $$ |u|_{B_{pq}^s} \lesssim_{s,q,p} \| \nabla u\|_{L^p}, \qquad s\leq 1. $$ You should compare the previous argument to the proof of the following: If $u$ is bounded and Lipschitz, then it's Hölder continuous of any order $\alpha\leq 1$.

For higher $m$ you use $$ \| \Delta^m_h u\|_{L^p} =\| \Delta_h (\Delta^{m-1}_hu)\lesssim |h| \| \nabla( \Delta^{m-1}_h u) \|_{L^p} \lesssim |h|^m\| D^m u\|_{L^p}. $$ In the case of your Gaussian example this is enough to give that $g\in B_{pq}^s$ for any $s>0$ and $p,q \geq 1$.

In general of course, you won't have this excess regularity of being in every Sobolev space. In those cases you have to look at the difference quotients, but the idea is the same: look at the $k$ derivative ($k$ as before) and try to get an $L^p$ Hölder estimate directly, i.e. something like $v=D^k u$ $$ \| v(\cdot +h)-v\|_{L^p}\lesssim \omega(|h|). $$ Once you have this, your function will be in $B_{pq}^s$ for $p,q,s$ determined by $k, \omega$.