# Small Lipschitz space

• May 31st 2009, 09:13 AM
Carl140
Small Lipschitz space
Let Lip_m[a,b] be the set of all Lipschitz functions of order m in [a,b] that is all functions satisfying |f(x)-f(y)|<= K * |x-y|^m for a certain constant K and for all x ,y in [a,b].

Now define the following function:

A_m (f) = sup { |f(x) - f(y)|/|x-y|^m : x,y are in [a,b] and x is not equal to y}. sup represents supremum.

Now for z >0 and f in Lip_m[a,b] and t>0 define:

R_z (f,t) = sup { |f(x)- f(y)| / |x-y|^z : x,y are in [a,b], 0< |x- y| <= t}.

Now another definition:

The "small Lipschitz" space denoted by lip_m[a,b] is the following set:

lip_m[a,b] = { f in Lip_m[a,b] : limit ( R_m (f,t) when t-> +0) = 0 } where t->+0 means from the right.

If m = 1 show that lip_m [a,b] contains only the constant functions.

Choose $\displaystyle \epsilon>0$. Since the limit of $\displaystyle R_1(f,t)$ is zero for $\displaystyle t\rightarrow 0+$, we get some $\displaystyle \delta>0$ such that $\displaystyle |f(x)-f(y)|<\epsilon|x-y|$ for $\displaystyle |x-y|<\delta$. Consider a covering of $\displaystyle [a,b]$ consisting of intervals of length $\displaystyle \delta$. Choose $\displaystyle x\in [a,b]$ and show that $\displaystyle |f(a)-f(x)|<\epsilon$ by adding and substracting inside the absolute value.