Potential-type operators in \(L^{p(x)}\) spaces
Statements in the introduction of this paper refer to papers of \textit{O. Kováčik} and \textit{J. Rakosnik} [Czech. Math. J. 41 (116), 592-618 (1991; Zbl 0784.46029)] and \textit{S. G. Samko} [Integral Transforms Spec. Funct. 7, 123-144 (1998; Zbl 0934.46032)] in which there are considerations of \(L^p\)-spaces with function parameters. In particular, if \(\Omega\) is a domain of Euclidean space \(R^n\), where \(n>1\), and if \(p:\Omega\to (1,\infty)\), then \(L^{p(.)}\) denotes the set of measurable functions \(f\) on \(\Omega\) such that \(I_p(f)= \int_\Omega|f(x)|^{p(x)} dx<\infty\). The norm \(\|\cdot \|_{p(.)}\) on \(L^{p(.)}\) is stated as defined by \[ \|f\|_{p(.)}= \inf\bigl\{ \lambda >0: I_p(f/ \lambda)\leq 1\}. \] In the results introduced in this paper, the authors consider the Riemann-Liouville and Weyl fractional integral operators with function parameters in the form \[ R_{\alpha(x)} (f)(x)=\int_{[0,x]} (x-y)^{\alpha(x)- 1}f(y) dy; \quad W_{\alpha(x)} (f)(x)= \int_{[x,\infty]} (y-x)^{ \alpha(x) -1}f(y) dy. \] By applying estimates involving maximal functions, results are derived which include estimates of the form \[ \bigl\|(2^{\alpha (.)} -1)R^0_{\alpha(.)} (f)\bigr\|_{p(.)} \leq c\|f\|_{p(.)} \text{ in }L^{p(.)} (0,1);\;R^0_{\alpha(x)} (f)(x)=x^{-\alpha (x)}R_{\alpha(x)} (f)(x); \] \[ \bigl\|vR_{\alpha(.)} (fw)\bigr\|_{p(.)} \leq c\|f\|_{p(.)};\;\bigl\|wW_{\alpha(.)} (gv)\bigr\|_{p'(.)} \leq c\|g\|_{p'(.)}; \] \[ w(x)= (1+x)^{(2/p(x)) -a(x)-1};\;v(x)= (1+x)^{(2/p'(x)) +1}x^{-\alpha(x)} 2^{x/(1+x)}-1). \]
scientific article; zbMATH DE number 1827605
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