One-Sided BMO Spaces

Some one-sided estimates for oscillatory singular integrals

“…The study of one-sided spaces emerged naturally alongside the study of one-sided operators. In one previous study, the authors studied one-sided BMO spaces associated with one-sided sharp functions and their relationship to good weights for the one-sided Hardy-Littlewood maximal functions [26]. It is well known that the classical Hardy spaces were the dual spaces of BMO spaces and were the natural alternative for Lebesgue spaces when p < 1.…”

One-sided extrapolation at infinity and singular integrals

“…Rubio de Francia in 1984, see [14], roughly speaking, if an operator preserves L p 0 (w) for all w ∈ A p 0 , then it necessarily preserves the L p (w) space for every w ∈ A p , and every 1 < p < ∞. Extensions of this result were obtained by several authors, for instance in [5] and [6], E. Harboure, R.A. Macías and C. Segovia study A(p, q) classes, pairs of weights and extreme cases, p 0 = ∞; more recently F. Martín-Reyes, P. Ortega and A. de la Torre, see [9], considered this problem for one-sided classes of weights for 1 ≤ p 0 < ∞. (The one-sided classes of weights were first introduced by Sawyer in [13].)…”

“…In [9], an one-sided version of the BM O space, BM O + , is defined as the set of those functions such that ||f # + || ∞ < ∞. We consider the quantity given by…”

“…These allow us to show several applications which are contained in Paragraph 3, including new proofs of the results appearing in [1], [9], and [10].…”

Boundedness of the Weyl fractional integral on one-sided weighted Lebesgue and Lipschitz spaces

“…For its proof we shall use the method employed in Theorem 3 in [6] and the techniques of Lemma 1 in [5]. …”