Grand Lebesgue space for p = ∞ and its application to Sobolev–Adams embedding theorems in borderline cases

We define the grand Lebesgue space corresponding to the case p=∞$ p = \infty$ and similar grand spaces for Morrey and Morrey type spaces, also for p=∞$ p = \infty$, on open sets in Rn$ \mathbb {R}^n$. We show that such spaces are useful in the study of mapping properties of the Riesz potential operator in the borderline cases αp=n$ \alpha p = n$ for Lebesgue spaces and αp=n−λ$ \alpha p = n-\lambda$ for Morrey and Morrey type spaces, providing the target space more narrow than BMO. While for Lebesgue spaces there are known results on the description of the target space in terms better than BMO, the results obtained for Morrey and Morrey type spaces are entirely new. We also show that the obtained results are sharp in a certain sense.