No abstract

We show the boundedness of fractional integral operators by means of extrapolation. We also show that our result is sharp.

The classical Hardy inequality holds in Sobolev spaces W 1,p 0 when 1 p < N . In the limiting case where p = N , it is known that by introducing a logarithmic weight function in the Hardy potential, some inequality which is called the critical Hardy inequality holds in W 1,N 0 . In this note, in order to give an explanation of the appearance of the logarithmic function in the potential, we derive the logarithmic function from the classical Hardy inequality with best constant via some limiting procedure as p N . We show that our limiting procedure is also available for the classical Rellich inequality in second order Sobolev spaces W 2,p 0 with p ∈ (1, N2 ) and the Poincaré inequality.

We introduce B u w -function spaces which unify Lebesgue, Morrey-Campanato, Lipschitz, B p , CMO, local Morrey-type spaces, etc., and investigate the interpolation property of B u w -function spaces. We also apply it to the boundedness of linear and sublinear operators, for example, the Hardy-Littlewood maximal and fractional maximal operators, singular and fractional integral operators with rough kernel, the Littlewood-Paley operator, Marcinkiewicz operator, and so on.

The classical Hardy inequality holds in Sobolev spaces W 1,p 0 when 1 ≤ p < N. In the limiting case where p = N, it is known that by adding a logarithmic function to the Hardy potential, some inequality which is called the critical Hardy inequality holds in W 1,N 0 . In this note, in order to give an explanation of appearance of the logarithmic function at the potential, we derive the logarithmic function from the classical Hardy inequality with the best constant via some limiting procedure as p ր N. And we show that our limiting procedure is also available for the classical Rellich inequality in second order Sobolev spaces W 2,p 0 with p ∈ (1, N 2 ) and the Poincaré inequality.