Dual of homogeneous Triebel-Lizorkin

Question

Let $ p, q \in (1,\infty)$ and consider the homogeneous Triebel- Lizorkin space $\dot{F}^{s}_{p,q}$ to be the space of all tempered distributions (modulo polynomials) with
$$ [f]^{p}_{\dot{F}^{s}_{p,q}} = \int_{\mathbb{R}^{n}} \left( \sum_{j\in \mathbb{Z}} \vert \Delta_{j} f \vert^{q} \right)^{p/q} dx < \infty $$ Is there a duality characterization of this semi-norm in terms of $ g \in \dot{F}^{-s}_{p',q'} $. I know this is the case for the inhomogeonous Triebel space. In other words can we have something like $$ [f]_{\dot{F}^{s}_{p,q}} = \sup_{g \in F^{\sigma}_{p',q'}, |g|_{F^{\sigma}_{p',q'}} \leq 1} \mbox{ } \int_{\mathbb{R}^{n}} \sum_{j} \Delta_{j} f \Delta_{j} g $$ For some $ \sigma $. I suspect if such statement is true then $ \sigma = -s$. Remark: I am interested in such equality for $ f \in C^{\infty}_{0}$. Not sure if this helps.