  The fractional Hilbert transform was introduced by Zayed [30, Zayed, 1998]
and has been widely used in signal processing. In view of is connection with
the fractional Fourier transform, Chen, the first, second and fourth authors of
this paper in [6, Chen et al., 2021] studied the fractional Hilbert transform
and other fractional multiplier operators on the real line. The present paper
is concerned with a natural extension of the fractional Hilbert transform to
higher dimensions: this extension is the fractional Riesz transform which is
defined by multiplication which a suitable chirp function on the fractional
Fourier transform side. In addition to a thorough study of the fractional Riesz
transforms, in this work we also investigate the boundedness of singular
integral operators with chirp functions on rotation invariant spaces, chirp
Hardy spaces and their relation to chirp BMO spaces, as well as applications of
the theory of fractional multipliers in partial differential equations. Through
numerical simulation, we provide physical and geometric interpretations of
high-dimensional fractional multipliers. Finally, we present an application of
the fractional Riesz transforms in edge detection which verifies a hypothesis
insinuated in [26, Xu et al., 2016]. In fact our numerical implementation
confirms that amplitude, phase, and direction information can be simultaneously
extracted by controlling the order of the fractional Riesz transform.
