Extrapolation and the Boundedness in Grand Variable Exponent Lebesgue Spaces Without Assuming the Log-Hölder Continuity Condition, and Applications

The boundedness of the Hardy–Littlewood maximal operator, and the weighted extrapolation in grand variable exponent Lebesgue spaces are established provided that Hardy–Littlewood maximal operator is bounded in appropriate variable exponent Lebesgue space. 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, commutators of singular integrals in grand variable exponent Lebesgue spaces. Finally, applying the boundedness results of integral operators of Harmonic Analysis, we have the direct and inverse theorems on the approximation of 2π-periodic functions by trigonometric polynomials in the framework of grand variable exponent Lebesgue spaces.