Uniform boundness of Steklov's operator in variable exponent Morrey space

Let $p(\,\cdot\,)$ be a continuous function on $I_0=[0,1]$ with values in $[1,\infty)$. We suppose that
\begin{equation} \label{N352:gag21} 1\le p_{-} \leq p(x)\leq p_{+}<\infty, \end{equation}
where $p_{-}:=\operatorname{ess\,\inf}_{x \in I_0}p(x) \ge 1$, $p_{+}:=\operatorname{ess\,\sup}_{x \in I_0}p(x)<\infty$, and also suppose the $p(\,\cdot\,)$ satisfy the log-condition i.e.
\begin{equation} \label{N352:gag22} |p(x)-p(y)|\leq \frac{A}{-\ln|x-y|}\mspace{2mu}, \qquad |x-y|\leq \frac{1}{2}\mspace{2mu}, \quad x,y\in I_{0}. \end{equation}

Let $\lambda(\,\cdot\,)$ be a measurable function on $I_0$ with values in $[0,1]$. We define the variable exponent Morrey space $M^{p(\,\cdot\,),\lambda(\,\cdot\,)}(I_0)$ as the set of integrable functions $f$ on $I_0$ such that
$$ I_{p(\,\cdot\,),\lambda(\,\cdot\,)}(f):= \sup_{\substack{x \in I_0 \\ 0< r <2 \pi}} r^{-\lambda(x)} \int_{\widetilde{I}(x,r)}|f|^{p(y)}\,dy < \infty. $$

The norm of space $M^{p(\,\cdot\,),\lambda(\,\cdot\,)}(I_0)$ may be defined in two forms,
$$ \|f\|_{1}:= \inf \biggl\{\eta>0: I_{p(\,\cdot\,),\lambda(\,\cdot\,)}\biggl(\frac{f}{\eta}\biggr)<1 \biggr\} , $$
and
$$ \|f\|_{2}:= \sup_{\substack{x \in I_0 \\ 0< r <2 \pi}} r^{-\frac{\lambda(x)}{p(x)}}\|f \chi_{\widetilde{I}(x,r)}\|_{L^{p(\,\cdot\,)}(I_0)} . $$

Since two norms coincide, we put
\begin{equation*} \|f\|_{M^{p(\,\cdot\,),\lambda(\,\cdot\,)}(I_0)} :=\|f\|_{1} = \|f\|_{2}. \end{equation*}

The Steklov operator is defined as
$$ s_{h}(f)(x) :=\frac{1}{h} \int_{0}^{h} f(x+t)\,dt. $$
Our main result is following.
Theorem. Let $f\in M^{p(\,\cdot\,),\lambda(\,\cdot\,)}(I_0)$, $\lambda_{+}:=\operatorname{ess\,\sup}_{x \in I_0} \lambda(x)$, $0 \leq \lambda(x) \leq \lambda_{+} < 1$, and $p(\,\cdot\,)$ satisfy conditions \eqref{N352:gag21} and \eqref{N352:gag22}, then the family of operators $s_{h}(f)$, $0 < h \le 1$, is uniformly bounded in $M^{p(\,\cdot\,),\lambda(\,\cdot\,)}(I_0)$.
This contribution is based on recent joint work with Professor Vagif Guliyev.