What is the motivation for Besov spaces?

Question

I am trying to understand the definition of Besov spaces. With such a complicated definition I wonder what is the motivation behind them and why are they so often used in PDE? What advantage do they give over Sobolev spaces?

Are there any nice (hopefully short) references that introduce them?

One way they 'naturally' arise is as (real) interpolation spaces between Sobolev spaces of different smoothness, so they're one way to quantify fractional order smoothness (see for example trace theorems for Sobolev spaces). I've also seen them used as substitutes in certain 'endpoint' cases where the result for more classical spaces is either too hard, or outright false. — Jose27, Sep 20 '23 at 01:15
@Jose27 Thank you, that explains why people describe them as generalizing Sobolev spaces. We may define Sobolev spaces $H^s = W^{s,2}$ of fractional order using Fourier analysis, are Besov spaces a sort of extension of this idea to $W^{s,p}$ where $p$ is no longer restricted to 2? — CBBAM, Sep 20 '23 at 02:01
Not only do they characterize Sobolev spaces, they also represent Holder spaces and (local) Hardy and BMO spaces. In a sense, they unify a multitude of spaces in one theme of characterizing smoothness by decay of Fourier modes. — Nick, Sep 20 '23 at 03:36
@Nick Thank you, I am starting to see the motivation behind them. Do you have any references one can learn more about these spaces from at an introductory level? — CBBAM, Sep 20 '23 at 03:45
@CBBAM I’m actually a fan of Hans Triebel’s A Theory of Function Spaces (the one published in the 90’s is the edition I think is called he best); he has a nice introduction to motivations and themes of these spaces before moving on to the (very) technical details. Grafakos’ Classical and Modern Forurier Analysis is very good too. — Nick, Sep 20 '23 at 03:50
If you're referring to $H^{s,p}$ as the space of distributions $f$ for which $(1-\Delta)^{s/2}f\in L^p$ (also called Bessel potential spaces), then these are a (special case of a) 'dual' notion to Besov spaces called Triebel-Lizorkin spaces (though they are also interpolation spaces between Sobolev, this time with the complex method). On the other hand, the spaces $W^{s,p}$, defined through the seminorm $$\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\dfrac{|f(x)-f(y)|^p}{|x-y|^{n+sp}}dxdy$$ is a Besov space (obtained by interpolation with the real method). See the wiki page for Sobolev spaces. — Jose27, Sep 20 '23 at 06:56
In the case $p=2$, both of these notions of 'fractional Sobolev space' coincide. This is explained in full detail in this paper dealing with the fractional laplacian. — Jose27, Sep 20 '23 at 06:58
It's worth knowing that there are lots of different (equivalent) definitions of Besov spaces and which one you see will depend on the context. I like the textbook "Fourier Analysis and Nonlinear Partial Differential Equations" by Bahouri, Chemin and Danchin for the Littlewood-Paley definition. — Rhys Steele, Sep 21 '23 at 09:26
@RhysSteele Thank you for the suggestion. Also I apologize if this is off-topic, but I have similar research interests to what you have listed on your profile (stochastic quantization in Euclidean QFT). I have read books on functional analysis/PDE but not so much on stochastic calculus or even probability. Do you have any suggestions on what to read to understand the stochastic approach to QFT? It seems I will need to first read a book/take a course on measure-theoretic probability and then another on stochastic calculus. Is there a shorter path? — CBBAM, Sep 21 '23 at 18:07
Since this is off-topic for this question, I made a math stackexchange chat room for it here — Rhys Steele, Sep 21 '23 at 18:36

score 4 · Accepted Answer · answered Mar 21 '24 at 16:18

Aside from the pointwise characterization (i.e. the Sobolev–Slobodeckij spaces) mentioned in the comment. I believe the best motivation whose norm requires the complicated Littlewood-Paley decomposition is the theory of paraproduct.

Think about the Leibniz's rule on Sobolev spaces $$\|fg\|_{W^{k,r}}\le\|f\|_{W^{k,p}}\|g\|_{L^q}+\|f\|_{L^p}\|g\|_{W^{k,q}},\quad\frac1p+\frac1q=\frac1r.$$ In order to make it more useful in PDEs we want to consider the concrete decomposition $fg=T_1(f,g)+T_2(f,g)$ such that $T_1:W^{k,p}\times L^q\to L^r$ and $T_2:L^p\times W^{k,q}\to L^r$ are bounded bilinear operators.

Roughly speaking $D^k T_1(f,g)\approx T_1(D^kf,g)$, and $T_1$ captures the "high frequency" of $f$. Similarly $D^k T_2(f,g)\approx T_2(f,D^kg)$.

To capture this idea one may want to use the Littlewood-Paley decomposition $f=\sum_jf$ and $g=\sum_kg_k$ (where $j,k\in\mathbb Z$ or $j,k\ge0$ depending on the context). In this decomposition $T_i(f,g)=\sum_{(j,k)\in\Lambda_i}f_jg_k$ where $\Lambda_1$ and $\Lambda_2$ are partition of the index space for $(j,k)$.

In general this also work for Sobolev spaces, which are special case of Triebel-Lizorkin spaces. But for the paraproduct decomposition along with their estimates, the Besov spaces should the best starting point.

I would recommend the book Fourier Analysis and Nonlinear Partial Differential Equations by Bahouri, Chemin and Danchin.

I have been going back to this question after reading parts of the book you suggested, and one thing I haven't figured out is why we include the weight $2^{js}$ when defining Besov norms using the Littlewood-Paley decomposition $|u|{B{p,r}^s} := \Big|\big(2^{js}|\Delta_ju|{L^p}\big){j \in \mathbb{Z}}\Big|_{\ell^r(\mathbb{Z})}$. Is this to preserve some scaling properties of the norm? — CBBAM, Oct 14 '24 at 01:22
@CBBAM You can think about it as the scaling in the Fourier side. Taking $s$ derivative (fractional Laplacian, precisely) of $u$ is $(|\xi|^s\hat u(\xi))^\vee$ and $\Delta_j$ is the smooth cutoff on $|\xi|\approx 2^j$. — Liding Yao, Nov 05 '24 at 17:09
Thank you very much! This answers a question I have been thinking about for a long time! — CBBAM, Nov 05 '24 at 17:59

What is the motivation for Besov spaces?

1 Answers1