  In this note, for $s \in \mathbb R$ and $1 \leq p, r \leq \infty$, we
introduce and study Sobolev-Fourier-Lorentz spaces $\dot{H}^s_{\mathcal{L}^{p,
r}}(\mathbb{R}^d)$. In the family spaces $\dot{H}^s_{\mathcal{L}^{p,
r}}(\mathbb{R}^d)$, the critical invariant spaces for the Navier-Stokes
equations correspond to the value $s = \frac{d}{p} - 1$. When the initial datum
belongs to the critical spaces $\dot{H}^{\frac{d}{p} -
1}_{\mathcal{L}^{p,r}}(\mathbb{R}^d)$ with $d \geq 2, 1 \leq p <\infty$, and $1
\leq r < \infty$, we establish the existence of local mild solutions to the
Cauchy problem for the Navier-Stokes equations in spaces $L^\infty([0, T];
\dot{H}^{\frac{d}{p} - 1}_{\mathcal{L}^{p, r}}(\mathbb{R}^d))$ with arbitrary
initial value, and existence of global mild solutions in spaces $L^\infty([0,
\infty); \dot{H}^{\frac{d}{p} - 1}_{\mathcal{L}^{p, r}}(\mathbb{R}^d))$ when
the norm of the initial value in the Besov spaces $\dot{B}^{\frac{d}{\tilde p}
- 1, \infty}_{\mathcal{L} ^{\tilde p,\infty}}(\mathbb{R}^d)$ is small enough,
where $\tilde p$ may take some suitable values.
