REVISTA DE LA UNI ´ ON MATEM ´ ATICA ARGENTINA Vol. 57, No. 2, 2016, Pages 85–94 Published online: June 14, 2016 AN EXTENSION OF SOME PROPERTIES FOR THE FOURIER TRANSFORM OPERATOR ON L p (R) SPACES M. GUADALUPE MORALES, JUAN H. ARREDONDO, AND FRANCISCO J. MENDOZA Abstract. In this paper the Fourier transform is studied using the Henstock– Kurzweil integral on R. We obtain that the classical Fourier transform Fp : L p (R) → L q (R), 1/p +1/q = 1 and 1 <p ≤ 2, is represented by the integral on a subspace of L p (R), which strictly contains L 1 (R) ∩ L p (R). Moreover, for any function f in that subspace, Fp(f ) obeys a generalized Riemann–Lebesgue lemma. 1. Introduction If f belongs to the space of real valued Lebesgue integrable functions, L 1 (R), its classical Fourier transform is defined for every real number s as F 1 (f )(s) :=  R e −ist f (t) dt. (1.1) Here the integral is taken in the Lebesgue sense. When f is in L 2 (R), the Fourier transform of f is defined by F 2 (f )(s) := lim n→∞  R e −ist f n (t) dt, (1.2) where the limit is taken in the topology of the norm on L 2 (R) and (f n ) is a sequence in L 1 (R) ∩ L 2 (R) obeying ‖f n − f ‖ 2 → 0, as n →∞. For any unbounded set X ⊆ R, C ∞ (X) denotes the complex valued continuous functions on X vanishing at infinity (see [14]). In [18] the Henstock–Kurzweil integral was employed to study the Fourier trans- form. In [11, 12] it was proved that (1.1) makes sense as a Henstock–Kurzweil integral over BV 0 (R), and defines a function in C ∞ (R \{0}). In this paper we prove that the Fourier transform operator on L p (R), for 1 < p ≤ 2, is represented by a Henstock–Kurzweil integral on a subspace of L p (R), implying an extension of some properties for the operator. 2010 Mathematics Subject Classification. Primary 26A39, 43A32; Secondary 26A42. Key words and phrases. Fourier transform, L p (R) space, Henstock–Kurzweil integral. This work is partially supported by CONACyT-SNI and VIEP-BUAP (M´ exico). M.G.M. acknowledges support from a CONACyT postdoctoral fellowship. 85