Integral Operators on Weighted Grand Lebesgue Spaces (WGLS)

In this chapter, we deal with the boundedness problems for fundamental integral operators of harmonic analysis in generalized weighted grand Lebesgue spaces \(L^{p),\varphi }_w\) defined on domains G in \({\mathbb {R}}^n\) without assuming that the underlying measure \(\mu \) on G is doubling. Operators under consideration involve maximal and Calderón–Zygmund operators, also commutators of singular integrals with BMO functions. The weighted Sobolev-type inequalities in these spaces for fractional integrals with non-doubling measures satisfying the growth condition are also given. We discuss the case when a weight function defines the absolute continuous measure of integration, or plays the role of multiplier in the definition of the norm. The results are obtained under the Muckenhoupt condition on weights. These results are obtained for weight functions satisfying the Muckenhoupt \(A_p\) condition defined with respect to nonhomogeneous spaces.