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Let $0 < \alpha < 1$. The space $\text{Lip}_{\alpha}(\mathbb{R}^n)$ of the Lipschitz continuous functions with order $\alpha$ is

$$ \text{Lip}_{\alpha}(\mathbb{R}^n)=\{f: |f(x)-f(y)|\le C|x-y|^{\alpha} \quad\text{ for a.e. }\,x, y \in \mathbb{R}^n\}. $$ The smallest such constant $C$ is called the $\text{Lip}_{\alpha}(\mathbb{R}^n)$ norm of $f$ and is denoted by $\|f\|_{\text{Lip}_{\alpha}(\mathbb{R}^n)}$.

This norm has the following integral representation:

$$ \|f\|_{\text{Lip}_{\alpha}(\mathbb{R}^n)}\thickapprox\sup_{B}\frac{1}{|B|^{1+\alpha/n}}\int_{B}|f(x)-f_{B}|dx, $$ where $f_{B}=\frac{1}{|B|}\int_{B}f(y)dy$. It is clear that for $\alpha=0$ the right hand side is the definition of $BMO$ norm of $f$.

According to these facts can we say that Lipschitz space includes $BMO$ space? What is the relation between Lipschitz and $BMO$ spaces?

Calvin Khor
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user315531
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    Let's replace $\Bbb R^n$ by $\Bbb T^n$ for now since the compactness leads to nice embeddings of the relevant spaces. I will check this more precisely later, but if I remember correctly then the inclusion goes something like $$B^{\alpha}{\infty,\infty} \subsetneq L^{\infty}(\Bbb T^n) \subsetneq BMO(\Bbb T^n) \subsetneq B^0{\infty,\infty}(\Bbb T^n).$$ Here $\alpha>0$ and $B^{\alpha}{\infty,\infty}(\Bbb T^n)$ is the same (when $\alpha >0$) as $Lip{\alpha}(\Bbb T^n)$ in your notation. – shalop Aug 17 '20 at 22:27
  • In general, there's a bunch of different equivalences of these spaces for $\alpha \notin \Bbb Z$ but these equivalences always fail for $\alpha \in \Bbb Z$. The reason why they fail at integer values always comes down to the simple fact that $\sum_n 2^{-\alpha n}$ converges for $\alpha \in (0,1)$ but diverges for $\alpha =0$. – shalop Aug 17 '20 at 22:32
  • I'll wait your precise answer. So as i understand we can not say "$BMO(\mathbb{R}^n)$ is a special case of $\text{Lip}_{\alpha}(\mathbb{R}^n)$." Is it true? – user315531 Aug 18 '20 at 10:05
  • Yes it's not a special case. Give me a day or two and I'll write an answer. – shalop Aug 18 '20 at 17:29

1 Answers1

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I'm going to define a bunch of spaces which all describe functions of "regularity $\alpha$" in some sense.

Hölder spaces: Here $\alpha$ will be in $[0,1]$. Define $Lip_{\alpha}(\Bbb T^n)$ to be the space of all functions $f:\Bbb T^n \to \Bbb R$ such that $|f(x)-f(y)| \leq C|x-y|^{\alpha}$ for some $C>0$ independent of $x,y \in \Bbb T^n$. The smallest such constant $C$ is called the Holder seminorm, denoted by $[f]_{\alpha}$. The Banach space norm on $Lip_{\alpha}(\Bbb T^n)$ is defined by $\|f\|_{L^{\infty}(\Bbb T^n)}+[f]_{\alpha}.$ Note that when $\alpha = 0$ we just get $L^{\infty}(\Bbb T^n)$. Equivalently one may describe $Lip_{\alpha}(\Bbb T^n)$ as the set of functions $f$ such that $\sup_{x\in Q}|f(x)-f_Q| \leq C|Q|^{\alpha/n},$ for all cubes $Q \subset \Bbb T^n$, where $f_{Q} := \frac1{|Q|} \int_Q f$, and $|Q|$ is the Lebesgue measure of $Q$. (Proving this equivalence is difficult.)

Besov spaces: Here $\alpha$ can be any real number. Any function $f:\Bbb T^n \to \Bbb R$ admits a canonical decomposition called the Littlewood-Paley decomposition $f = \sum_{j\ge 0} f_j$. The Besov space $B^{\alpha}_{\infty,\infty}(\Bbb T^n)$ consists of those functions $f$ such that $\|f_j\|_{L^{\infty}(\Bbb T^n)} \leq C2^{-\alpha j}$ for some $C$ which is independent of $j$. The smallest constant $C$ for which the inequality holds is called the Besov norm. This induces a Banach space structure on $B^{\alpha}_{\infty,\infty}$. The space $B^1_{\infty,\infty}$ is called the Zygmund class and is equivalently described as the set of all functions $f$ such that $$|f(x+h)+f(x-h)-2f(x)| \leq C|h|,$$ and $B^0_{\infty,\infty}$ consists of the distributional derivatives of functions from the Zygmund class.

BMO spaces: Here $\alpha$ will be in $[0,1]$. Let us define the space $BMO_{\alpha}(\Bbb T^n)$ to be the space of all functions $f:\Bbb T^n \to \Bbb R$ such that $\sup_Q \frac{1}{|Q|^{1+\alpha/n}}\int_{Q} |f-f_Q|dx <\infty$, where the sup is over all cubes $Q\subset \Bbb T^n$, and $f_{Q} := \frac1{|Q|} \int_Q f$, and $|Q|$ is the Lebesgue measure of $f$. The norm on $BMO_{\alpha}$ is defined to be that supremum, which makes it a Banach space.

Continuous function spaces: Here $\alpha=:k$ must take values in $\Bbb N$. Then $C^{k}(\Bbb T^n)$ is defined to be the set of all functions $f:\Bbb T^n \to \Bbb R$ such that all partial derivatives of order up to $k$ are continuous. The norm is defined to be sum of the uniform norms of all of the partial derivatives up to order $k$. Again, we get a Banach space.

So now the question is: how are all of these spaces related?

Theorem 1: If $\alpha \in (0,1)$ then $$ Lip_{\alpha}(\Bbb T^n) = B^{\alpha}_{\infty,\infty} (\Bbb T^n)= BMO_{\alpha}(\Bbb T^n).$$ All of the norms are equivalent.

Theorem 2: For $\alpha = 0$ we have the following inclusions: $$C^0(\Bbb T^n) \subsetneq L^{\infty}(\Bbb T^n) \subsetneq BMO_0(\Bbb T^n) \subsetneq B^0_{\infty,\infty}(\Bbb T^n).$$ So none of the norms are equivalent. For $\alpha=1$ we have the corresponding sequence of proper inclusions.

Basically the equivalences in Theorem 1 always boil down to a computation on dyadic blocks. They fail for $\alpha=0$ due to the fact that the series $\sum 2^{-\alpha n}$ diverges for $\alpha=0$.

Sorry if this was unclear. Will try to update with references.

shalop
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  • Sorry for my weak English. What do you mean for saying "For $\alpha=1$ we have the corresponding sequence of proper inclusions." Can you write more explicitly? Is there special names for the spaces $Lip_1$ and $BMO_1$? – user315531 Aug 21 '20 at 14:01
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    @user315531 For $\alpha=1$, I believe you will have $$C^1(\Bbb T^n) \subsetneq Lip_1(\Bbb T^n) \subsetneq BMO_1(\Bbb T^n) \subsetneq B^1_{\infty,\infty}(\Bbb T^n).$$ For $n=1$ the first inclusion fails because you can have functions with corners like $|x|$. The second inclusion fails because $BMO_1$ can have functions like $|x| \log |x|$ which is not Lipchitz in any neighborhood of zero. The third inclusion fails because you can have functions like $\sum_n 2^{-n} \sin(2^{n}x)$, which is nowhere differentiable if I remember correctly. – shalop Aug 23 '20 at 05:44
  • I think $Lip_1$ is just called the Lipchitz class. I don't think $BMO_1$ has any specific names. It might just be all functions whose first partial derivatives are in $BMO_0$, but I'm not 100% sure about that. – shalop Aug 23 '20 at 05:45