A question which belongs to a class of Zygmund functions

Question

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 \begin{equation} \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}} \end{equation} is true? If it is how can I show?

Answer 1

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.