Grand Variable Exponent Function Spaces

In this chapter we derive weighted inequalities with power-type weights for operators of harmonic analysis, such as maximal and singular integral operators and commutators of singular integrals, in grand variable exponent Lebesgue spaces (GV ELSs briefly) defined on spaces of homogeneous type (SHT briefly). Apart from obtaining some structural properties of GV ELSs and grand variable exponent Morrey spaces (GV EMSs briefly) over an SHT (e.g., duality and preduality of GV ELSs, the coincidence of the closure of \(L^{\infty }\) with the closure of \(L^{p(\cdot ), \lambda (\cdot )}\) in GV EMS denoted by \(L^{p(\cdot ), \lambda (\cdot ), \theta }\) ), the boundedness of the Hardy–Littlewood maximal operator and the weighted extrapolation in GV ELSs are also established provided that the Hardy–Littlewood maximal operator is bounded in appropriate V ELS. Moreover, we give some bounds of the norm of the Hardy–Littlewood maximal operator in these spaces. As corollaries, we have appropriate norm inequalities and the boundedness of operators of harmonic analysis such as maximal and sharp maximal functions: Calderón–Zygmund singular integrals and commutators of singular integrals in GV ELSs. To obtain most of the results regarding the boundedness of operators in GV ELSs, we establish appropriate weighted extrapolation. Finally, applying the boundedness results of integral operators of harmonic analysis, we have the direct and inverse theorems on the approximation of \(2\pi \) -periodic functions by trigonometric polynomials in the framework of GV ELSs.