  This paper is devoted to the $L^p(\mathbb R)$ theory of the fractional
Fourier transform (FRFT) for $1\le p < 2$. In view of the special structure of
the FRFT, we study FRFT properties of $L^1$ functions, via the introduction of
a suitable chirp operator. However, in the $L^1(\mathbb{R})$ setting, problems
of convergence arise even when basic manipulations of functions are performed.
We overcome such issues and study the FRFT inversion problem via approximation
by suitable means, such as the fractional Gauss and Abel means. We also obtain
the regularity of fractional convolution and results on pointwise convergence
of FRFT means. Finally we discuss $L^p$ multiplier results and a
Littlewood-Paley theorem associated with FRFT.
