Lq null space property (${\sum }_{i\in S}|v_i{|}^q<{\sum }_{i\notin S}|v_i{|}^q$)

For $0<q\le 1$, this strict inequality for every nonzero $v\in \ker A$ and every $|S|\le s$ is equivalent to unique constrained ${\ell }^q$ recovery of every sparse vector of order $s$. For sufficiency, use $|x_i+v_i{|}^q\ge |x_i{|}^q-|v_i{|}^q$ on the support of a vector. For necessity compare $x=-v_S$ with $v_{S^c}$, which have the same measurements. The statement concerns uniform recovery over the whole sparse class, rather than a single signed vector.

If the Lq null space property holds at $q\le 1$, it holds at every $0<p<q$. Order a nonzero null vector's magnitudes $a_1\ge \cdots \ge a_N$ and put $t=a_s>0$. Since $p-q<0$, the top-$s$ $p$-power sum is at most $t^{p-q}$ times its $q$-power sum, while the tail $p$-power sum is at least that multiple of the tail $q$-power sum. The strict $q$ inequality therefore gives the strict $p$ inequality. The largest $s$ magnitudes are the worst support of a vector, so all other supports satisfy it too.