Embeddings of concave functions and duals of Lorentz spaces

Abstract

A simple expression is presented that is equivalent to the norm of the L p v → L q u embedding of the cone of quasi-concave functions in the case 0 < q < p < ∞. The result is extended to more general cones and the case q = 1 is used to prove a reduction principle which shows that questions of boundedness of operators on these cones may be reduced to the boundedness of related operators on whole spaces. An equivalent norm for the dual of the Lorentz space Γp(v) = f :

Characterization of a rearrangement-invariant hull of a Besov space via interpolation

“…Function spaces and more general structures involving f * * − f * are known to constitute natural indispensable function classes in various situations. They occur for example as optimal range function spaces in Sobolev-type embeddings [2,16,25], in the expression that determines the associate space of a classical Lorentz space [30], or in necessary and sufficient conditions for boundedness of maximal Calderón-Zygmund singular integral operators on classical Lorentz spaces [4]. Our point of departure is the following well-known pointwise inequality (let us recall that is a Lipschitz domain): there exists a positive constant C, depending only on n, such that, for every weakly-differentiable function f : → R and every t ∈ (0, ∞), one has…”

Some estimates in GΓ(p,m,w) spaces

“…All such considerations apply to the familiar Lorentz spaces Λ p,q and Γ p,q defined setting m = q and ν(x) = x q/p−1 . There is a wide literature on the study of Lorentz spaces and on the several questions about their normability, their equivalence to Banach spaces, their associate spaces, their generalizations and variants: besides the references already given, we quote here the papers [1,7,10,26,28] and references therein.…”

“…Motivated by the important role played by the associate norm of such spaces (see [17,Theorem 3]), and since such spaces turn out to be at the same time a generalization of some Lorentz spaces Γ (ν), of classical Lebesgue spaces and of the small Lebesgue spaces (originated in [11]; see also [4,6,13,15,16] and references therein), in this paper we discuss some of their properties and we prove some estimates of the norm of the associate space of GΓ (p, m, w) for certain values of p, m, w. A full explicit description of the associate norm remains as an open problem; however, this paper, together with the deep study (see e.g. [9,19,28]) on the duality of Lorentz spaces Γ (ν), make a progress in this direction. When the parameters p, m, w are such that GΓ (p, m, w) coincide with the small Lebesgue spaces, our estimates can be compared with the expression of the associate norm appearing in [13].…”

A Calderón couple of down spaces

“…Halperin's investigation of "D-type Hölder inequalities" used the norm to improve the usual Hölder inequality when one factor is monotone. Later, down spaces and the related level function construction were studied in [8,[16][17][18][19][20][21][22] and applied to prove weighted Hardy inequalities, to prove general versions of Sawyer's duality theorem, to study Banach envelopes of Orlicz-Lorentz spaces, to characterize the dual of the Lorentz spaces Γ p (w), and to give a weight characterization for the boundedness of the Fourier transform on weighted Lorentz spaces.…”