Weighted Boundedness of the Maximal, Singular and Potential Operators in Variable Exponent Spaces

We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces $L^{p(\cdot )}(\varrho )$ with variable exponent $p(x)$, mainly in the Euclidean setting and dwell on a new result of the boundedness of the Hardy-Littlewood maximal operator in the space $L^{p(\cdot )}(X,\varrho )$ over a metric measure space $X$ satisfying the doubling condition. In the case where $X$ is bounded, the weight function satisfies a certain version of a general Muckenhoupt-type condition For a bounded or unbounded $X$ we also consider a class of weights of the form $\varrho (x)=[1+d(x_0,x){]}^{{\text{bt}}_{\mathrm{\infty }}}{\prod }_{k=1}^m{w}_k(d(x,x_k))$, $x_k\in X$, where the functions $w_k(r)$ have finite upper and lower indices $m(w_k)$ and $M(w_k)$. Some of the results are new even in the case of constant $p$.