Weighted Lorentz Spaces and the Hardy Operator

Discretization and anti-discretization of rearrangement-invariant norms

“…e.g. [1], [26], [8], [9], [10], [16], [30], [31], [32], [28]. A summary of the results on embeddings of classical Lorentz spaces known by the end of 1990's, as well as some more references, can be found in [7].…”

Bilinear weighted Hardy inequality for nonincreasing functions

“…Furthermore, if q > 1, or if q < 1 and the function s → v(s)s p V −p (s) is integrable near the origin, then A (8) A (9) , where Let us say more on the equivalence A (8) A (9) . If q > 1 and the function u, defined by u(s) := v(s)s p V −p (s) for s > 0, is not integrable near the origin (a simple example of such function u was given in [28, p. 93]), then both A (8) and A (9) are infinite. However, if q < 1 and u is not integrable near the origin, then A (8) = ∞ but A (9) = 0, since the exponent (q−1)p p−q is negative.…”

“…The reason is that the function involving w appears only once in there and the resulting expression may be understood as the (quasi-)norm in a certain space. Nevertheless, for the final conditions which we state in the lemmas or theorems, we prefer the "safe" form in the style of A (8) , i.e. avoiding the potentially negative exponents.…”

“…The problem with v = w and p = q, 1 < p, q < ∞ was first successfully solved by Sawyer [26]. Many articles on this topic followed, providing the results for a wider range of parameters, see [30,8,9,28,10,7,6]. In [7] the results available in 2000 were surveyed.…”

Interpolation restricted to decreasing functions and Lorentz spaces

“…[2], [19], [7], etc.) to characterize when a variety of classical operators are bounded on Lorentz spaces associated to pairs («, co) of weights, AS(a>) = {/ € L o ; ll/ll^, = ||/ril M<u) < oo},…”