On Garling sequence spaces

The aim of this paper is to introduce and investigate a new class of separable Banach spaces modeled after an example of Garling from 1968. For each $1⩽p<\mathrm{\infty }$ and each nonincreasing weight $$\mathbf{\text{w}}$\in c_0\setminus\ell_1$weexhibitan$\ell_p$-saturated, complementably homogeneous, and uniformly subprojective Banach space $g(\mathbf{\text{w}},p)$. We also show that $g(\mathbf{\text{w}},p)$ admits a unique subsymmetric basis despite the fact that for a wide class of weights it does not admit a symmetric basis. This provides the first known examples of Banach spaces where those two properties coexist.