# Young Women in Harmonic Analysis and PDE

## December 2-4, 2016

#### An extension of some properties for the Fourier Transform operator on $L^{p}(\mathbb{R})$ spaces

In this work the Fourier Transform is studied using the Henstock-Kurzweil integral on $\mathbb{R}$. We obtain that the classical Fourier Transform $\mathcal{F}_{p}: L^{p}(\mathbb{R})\rightarrow L^{q}(\mathbb{R})$, $1/p+1/q=1$ and $1 < p \leq 2$, is represented by the integral on a subspace of $L^{p}(\mathbb{R})$, which strictly contains $L^{1}(\mathbb{R})\cap L^{p}(\mathbb{R})$. Moreover, for any function $f$ in that subspace, $\mathcal{F}_{ p} (f)$ obeys a generalized Riemann-Lebesgue Lemma.