PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 3, Pages 745–754 S 0002-9939(02)06870-3 Article electronically published on October 15, 2002 EQUIVALENT QUASI-NORMS ON LORENTZ SPACES DAVID E. EDMUNDS AND BOHUM ´ IR OPIC (Communicated by Andreas Seeger) Abstract. We give new characterizations of Lorentz spaces by means of cer- tain quasi-norms which are shown to be equivalent to the classical ones. 1. Introduction and results Lorentz spaces play an important role in many branches of mathematical anal- ysis. The aim of this paper is to derive new formulae which provide equivalent quasi-norms on such spaces. These results are motivated by mapping properties of fractional maximal operators, Riesz potentials and the Hilbert transform. Our proofs are based on weighted norm inequalities for integral operators. To explain our results in more detail, we first need some notation. Given two quasi-Banach spaces X and Y , we write X = Y if X and Y are equal in the algebraic and the topological sense (their quasi-norms are equivalent). The symbol X → Y means that X ⊂ Y and the natural embedding of X in Y is continuous. We write A  B (or A  B) if A ≤ cB (or cA ≥ B) for some positive constant c independent of appropriate quantities involved in the expressions A and B, and A ≈ B if A  B and B  A. Throughout the paper we use the abbreviation LHS(∗) (RHS(∗)) for the left (right) hand side of the relation (∗). Moreover, we adopt the convention that 1/∞ = 0. Let Ω ⊂ R n be a measurable subset (with respect to n-dimensional Lebesgue measure), |Ω| its measure and χ Ω its characteristic function. The symbol M(Ω) is used to denote the family of all scalar-valued (real or complex) measurable functions on the set Ω. By M + (Ω) we mean the subset of M(Ω) consisting of all non-negative functions on Ω. If Ω = (a, b) ⊆ R, we simply write M(a, b) and M + (a, b) instead of M((a, b)) and M + ((a, b)). Given p, r ∈ (0, ∞], the Lorentz space L p,r (Ω) is defined by (cf. [Lo1], [Lo2], [BS]) (1.1) L p,r (Ω) = {f ∈M(Ω); ‖f ‖ p,r = ‖f ‖ p,r;Ω < ∞}, Received by the editors July 1, 2001. 2000 Mathematics Subject Classification. Primary 46E30, 26D10, 47B38, 47G10. Key words and phrases. Lorentz spaces, equivalent quasi-norms, weighted norm inequalities, fractional maximal operators, Riesz potentials, Hilbert transform. This research was supported by NATO Collaborative Research Grant PST.CLG 970071 and by grant no. 201/01/0333 of the Grant Agency of the Czech Republic. c 2002 American Mathematical Society 745 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use