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International conference on Function Spaces and Approximation Theory dedicated to the 110th anniversary of S. M. Nikol'skii
May 27, 2015 15:20–15:45, Ôóíêöèîíàëüíûå ïðîñòðàíñòâà, Moscow, Steklov Mathematical Institute of RAS

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

Institute for Advanced Studies in Basic Sciences (IASBS), Iran

Abstract: Let $p( \cdot )$ be a continuous function on $I_0=[0,1]$ with values in $[1,\infty)$. We suppose that
$$\label{N352:gag21} 1\le p_{-} \leq p(x)\leq p_{+}<\infty,$$
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.
$$\label{N352:gag22} |p(x)-p(y)|\leq \frac{A}{-\ln|x-y|}\mspace{2mu}, \qquad |x-y|\leq \frac{1}{2}\mspace{2mu}, \quad x,y\in I_{0}.$$

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,
$$\|f\|_{1}:= \inf \{\eta>0: I_{p( \cdot ),\lambda( \cdot )}(\frac{f}{\eta})<1 \} ,$$
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.

Materials: abstract.pdf (138.3 Kb)

Language: English

References
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2. P. L. Butzer, R. J. Nessel, Fourier Analysis and Approximation, Academic Press, New York, 1971
3. O. Kovavcik, J. Rakosnik, “On spaces $L^{p (x)}$ and $W^{k, p(x)}$”, Czechoslovak Math. J., 41 (1991), 592–618
4. I. I. Sharapudinov, “On direct and inverse theorems of approximation theory in variable Lebesgue and Sobolev spaces”, Azerbaijan J. Math., 4:1 (2014), 53–71
5. I. I. Sharapudinov, “Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by finite Fourier–Haar series”, Mat. Sb., 205:2 (2014), 145–160
6. I. I. Sharapudinov, “Approximation of functions in $L^{p(x)}_{2\pi}$ by trigonometric polynomials”, Izv. RAN. Ser. Mat., 77:2 (2013), 197–224

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