Garling sequence spaces

By generalizing a construction of Garling, for each $1⩽p<\mathrm{\infty }$ and each normalized, nonincreasing sequence of positive numbers $w\in c_0\setminus\ell_1$weexhibitan$\ell_p$-saturated, complementably homogeneous Banach space $g(w,p)$ related to the Lorentz sequence space $d(w,p)$. Using methods originally developed for studying $d(w,p)$, we show that $g(w,p)$ admits a unique (up to equivalence) subsymmetric basis, although when $w=(n^{-\theta }{)}_{n=1}^{\mathrm{\infty }}$ for some $0<\theta <1$, it does not admit a symmetric basis. We then discuss some additional properties of $g(w,p)$ related to uniform convexity and superreflexivity.