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.

本文致力于 ${\mathrm{大号}}^p（\mathbb{[R}）$ 分数傅里叶变换（FRFT）的理论 $\mathrm{1个}\le p<\mathrm{2个}$。鉴于FRFT的特殊结构，我们研究了FRFT的FRFT特性。${\mathrm{大号}}^{\mathrm{1个}}$通过引入合适的线性调频运算符来实现功能。但是，在${\mathrm{大号}}^{\mathrm{1个}}（\mathbb{[R}）$设置时，即使执行基本的功能操作，也会出现收敛问题。我们克服了这些问题，并通过分数高斯和Abel均值等合适的方法通过近似来研究FRFT反演问题。我们还获得了分数卷积的正则性，并得到了FRFT均值逐点收敛的结果。最后我们讨论${\mathrm{大号}}^p$ 乘数结果和与FRFT相关的Littlewood-Paley定理。