An Equivalence Theorem for Integral Conditions Related to Hardy's Inequality

Редукционные Теоремы Для Весовых Интегральных Неравенств На Конусе Монотонных Функций

“…2) Недавно обнаружилось [21], [77], что наравне с указанными выше кри-териями существуют семейства (шкалы) равносильных условий, зависящих…”

Some new scales of weight characterizations of the class B p

“…In [6] the equivalence of four scales of integral conditions was proved. These conditions characterize the Hardy inequality and contain the usual Muckenhoupt condition as a special case.…”

“…These conditions characterize the Hardy inequality and contain the usual Muckenhoupt condition as a special case. The proof was carried out by rst proving the following equivalence theorem of independent interest that also will be applied in this paper (see Theorem 2.1 in [6]): …”

“…In fact, our Theorem 3.1 shows that there are six scales of WEIGHT CHARACTERIZATIONS OF THE CLASS Bp !$% weight characterizations of the class B p . The proof is elementary and mostly based on our result from [6]. In Section 4 we give some new characterizations of the classes B * p , B * ∞ and RB p , generalize a result of Y. Sagher [11] (Proposition 4.4) and apply Lemma 2.2 to give a new proof of a result of Andersen [1] (Proposition 4.6).…”

“…In Section 2, we formulate a theorem from [6], on which our results are based, and prove a gluing lemma (Lemma 2.2) of independent interest. In Section 3 we prove an equivalence theorem, which includes all results mentioned above.…”