Boundedness of classical operators on classical Lorentz spaces

Weighted norm inequalities for averaging operators of monotone functions

“…We point out that the double weight inequality (*) has been characterized in a recent paper by E. Sawyer [7] for non-increasing functions with the q-norm of the averaging operator on the left and the p-norm on the right. It is also possible to prove some of our results by the methods developed in the paper by D.W. Boyd [2] .…”

On weighted iterated Hardy-type inequalities

“…The first results on the problem Λ p (v) ֒→ Γ p (v), 1 < p < ∞, which is equivalent to inequality (1.1) restricted to the cones of non-increasing functions, were obtained by Boyd [5] and in an explicit form by Ariño and Muckenhoupt [3]. The problem with w v and p q, 1 < p, q < ∞ was first successfully solved by Sawyer [40]. Many articles on this topic followed, providing the results for a wider range of parameters.…”

“…[24][25][26][27]42]. At the initial stage the main tool was the Sawyer duality principle [40], which allowed one to reduce an L p − L q inequality for monotone functions with 1 < p, q < ∞ to a more manageable inequality for arbitrary non-negative functions. This principle was extended by Stepanov in [46] to the case 0 < p < 1 < q < ∞.…”

Best approximation properties in spaces of measurable functions

“…The opposite inclusion Λ p,w ⊂ Γ p,w is satisfied if and only if w ∈ B p (see [22]). It is worth mentioning that the spaces Γ p,w and Λ p,w are also connected by Sawyer's result (Theorem 1 in [28]; see also [30]), which states that the Köthe dual of Λ p,w , for 1 < p < ∞ and ∞ 0 w(t)dt = ∞, coincides with the space Γ p ′ , w , where 1/p + 1/p ′ = 1 and w(t) = t/ t 0 w(s)ds p ′ w(t).…”