One-dimensional and Multidimensional Hardy Operators in Grand Lebesgue Spaces

Grand Lebesgue spaces over sets of innite measure are dened with using an additional characteristic a() called a grandizer. Conditions on the grandizer a(x) for the Hardy operators to be bounded in the grand Lebesgue spaces Lp) a (Rn) are found, and the lower and upper estimates for a sharp constant in the one-dimensional and multidimensional Hardy inequalities are given in dependence on the grandizer. For some special choice of the grandizer it is proved that this sharp constant is equal to the sharp constant for the classical Lebesgue spaces.