Best approach regions for potential spaces

Abstract

Abstract. We characterize the approach regions so that the non-tangential maximal function is of weak-type on potential spaces, for which we use a simple argument involving Carleson measure estimates.

A counterexample to a weak-type estimate for potential spaces and tangential approach regions

“…The main result of this paper is to show that, contrary to the case of [NS84], the assumptions on the region assumed in [RS97], which is natural as we mentioned before, from the point of view of convergence, turns out to give different boundedness results for the corresponding maximal operators. In order to clarify this statement, let us introduce some notations:…”

“…and this is only possible if N(t) is bounded, since r(t k ) ∼ r(2t k ) (this follows from the related relation for the Poisson kernel, P t (x) ∼ P 2t (x)). Therefore, Ω K cannot satisfy the necessary condition (2.5), and hence, M Ω K cannot be of weak type (1,1) (by Theorem 2.6 in [RS97]). Now that the region Ω is defined (and we have dealt with (i), (ii), (iv), and (v) as soon as we show the existence of N), we need to prove the weak type of the maximal operator (i.e., (iii)).…”

“…In [NRS82] Fatou's theorem was extended to some tangential approach regions, when the functions were assumed to have some a priori smoothness (they belonged to a potential space). This result was later on generalized in [RS97] to characterize all the approach regions (under a completion hypothesis similar to the one in [NS84]) for which the convergence holds for the potential spaces.…”