# A question which belongs to a class of Zygmund functions

Let $f$ be an absolutely continuous, periodic with period 1 and satisfies the condition $$|f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const}\frac{\delta}{(\log\frac{1}{\delta})^{\epsilon}}, \,\,\,\delta\in (0,1), \,\,\epsilon\in (0, 1].$$ My question is that for any $I=[a, b],$ which is $|I|<1$ the following inequality $$\int_{a}^{b}\Big|\frac{f(x)-f(a)}{x-a}-\frac{f(b)-f(x)}{b-x}\Big|dx\leq \text{const}\frac{|I|}{(\log\frac{1}{|I|})^{\epsilon}}$$ is true? If it is how can I show?

• The Article "A Note on Smooth Functions" from Weiss and Zygmund contains also information about your class of functions. goo.gl/ha7i4l – CPJ Sep 19 '14 at 15:55

The standard Zygmund class (with $\epsilon=0$) is the Besov space $B^1_{\infty,\infty},$ that is using a Littlewood-Paley decomposition $$1=\sum_{\nu\ge 0}\varphi_\nu(\xi),\quad \varphi_\nu(\xi) =\phi(\vert\xi\vert 2^{-\nu}) \text{ for \nu\ge 1, supp\phi=[r,R], r>0,\ \varphi_0\in C^\infty_c,}$$ $$f\in B^1_{\infty,\infty} \text{ means}\ \sup_\nu2^\nu\Vert \varphi_\nu(D)u\Vert_{L^\infty}<+\infty.$$ I guess that your class of functions is characterized by $\sup_\nu\nu^\epsilon2^\nu\Vert \varphi_\nu(D)u\Vert_{L^\infty}<+\infty$ and hope it can help for your problem by generalizing what is known for the true Zygmund class.