Abstract:
The Riesz potentials $I^\alpha f$, $0<\alpha<\infty$, are considered in the framework of a grand Lebesgue space $L^{p),\theta}_a$, $1<p<\infty$, $\theta>0$, on $\mathbb{R}^n$ with grandizers $a\in L^1(\mathbb{R}^n)$, which are understood in the case $\alpha\ge n/p$ in terms of distributions on test functions in the Lizorkin space. The images under $I^\alpha$ 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 $\alpha$ in this subspace.
Keywords:Riesz potential, space of Riesz potentials, hypersingular integral, grand Lebesgue space, grandizer, Lizorkin space of test functions, identity approximation.
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