Description of the Space of Riesz Potentials of Functions in a Grand Lebesgue Space on ℝn

The Riesz potentials L a f , 0 < α < ∞, are considered in the framework of a grand Lebesgue space L a p ),θ, 1 < p < ∞, θ > 0, on Rn with grandizers a ∈ L 1 (ℝ n ), which are understood in the case α ≥ n/p in terms of distributions on test functions in the Lizorkin space. The images under I α of functions in a subspace of the grand space which satisfy the so-called vanishing condition is studied. Under certain assumptions on the grandizer, this image is described in terms of the convergence of truncated hypersingular integrals of order α in this subspace.