1
$\begingroup$

Consider the classical Weierstrass function $$ W(x)=\sum_{n=1}^\infty \frac{e^{i2^nx}}{2^n}. $$ It is a well-known result that this function is nowhere differentiable (Hardy, TAMS 1916, Thm 1.31). In particular, by Rademacher theorem, it does not belong to the class $Lip(T)$. The question is whether there may exist some point $x_0$ such that $W\in Lip(x_0)$, that is, for some $\delta>0$ it holds $$ |W(x_0+h)-W(x_0)|\leq C |h| $$ when $|h|<\delta$.

$\endgroup$
2
$\begingroup$

The argument in Zygmund's paper is valid for any continuous function $f$ in the class $\Lambda^*$, that is with $$ \|f\|_*=\sup_{x\in{\mathbf T},\; h\not=0}\Big|\frac{f(x+h)+f(x-h)-2f(x)}h\Big|<\infty. $$ Assume first that $f$ has a local minimum at some point $x_0\in\mathbb T$. Then for some $\delta>0$ and all $h\in(0,\delta)$ one has $$ \frac{f(x_0+h)-f(x_0)}h + \frac{f(x_0-h)-f(x_0)}h =\frac{\Delta^2_hf(x_0)}h\leq \|f\|_*. $$ Since the first two summands are non-negative, one deduces that $$0\leq \frac{f(x_0\pm h)-f(x_0)}h \leq \|f\|_*,$$ and so $f\in Lip(x_0)$. The same applies if $f$ has a local maximum (replacing $f$ by $-f$).

In the case of $W$, this already implies the existence of at least 2 points with the local Lipschitz condition.

To obtain an everywhere dense set, for a general $f\in\Lambda^*$, pick any interval $(a,b)\subset\mathbb T$ and a line function $L(x)$ with $L(a)=f(a)$ and $L(b)=f(b)$. Then $F=f-L$ belongs to $\Lambda^*$, is continuous in $[a,b]$ and vanishes at the end-points. So it must have either a maximum or minimum at some $x_0\in(a,b)$. Then $F$, and hence $f$, must belong to $Lip(x_0)$.

$\endgroup$
1
$\begingroup$

Sorry, I apologize for posing this question. I just realized that the answer is yes, by an elementary argument in Zygmund's paper (Duke Math, 1945, proof of Theorem 1). The function $W$ is pointwise Lipschitz at an everywhere dense set of points.

$\endgroup$
  • $\begingroup$ will you give a quick sketch of Zygmund's argument? $\endgroup$ – Pietro Majer May 10 '18 at 16:22
  • 1
    $\begingroup$ Which paper of Zygmund? He had several papers in Duke in 1945. $\endgroup$ – Piotr Hajlasz May 10 '18 at 21:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.