ABSTRACT
Weighted function spaces arise naturally in many contexts of mathematics. Roughly speaking, weighted measure spaces, which generate weighted function spaces, are useful although they break some symmetries. One way to break the symmetry is to consider weighted measures. For the time being by a weight we mean a measurable function in a measure space which has a positive value almost everywhere. This chapter is interested in the weights for which the Hardy–Littlewood maximal operator is bounded by investigating the weighted estimates for Lebesgue spaces, including some sharp estimates on constants. It focuses on on the one-weight (maximal) inequality and deals with the two-weight norm inequality of Hardy operators, (fractional) maximal operators and singular integral operators. The chapter provides a tool to create some inequalities based on the boundedness property of the weighted Lebesgue spaces.
