Weighted Lebesgue spaces

This chapter examines weighted norm inequalities. 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. The chapter notes that a group can be used to determine the symmetry of a space. In fact, a group action allows the symmetry of the spaces to be measured. However, when handling something in Rn, often the whole space Rn is neglected. Occasionally, some open sets in Rn are used instead of Rn itself. The Gaussian distribution is a natural one appearing in many branches of sciences. Weighted Lebesgue spaces arise naturally when considering the change of variables.