nonlinear water waves, one is led to study the operator associated to the symbol (see [@Lannes]) $$\label{symbDN}  
    \sigma(x,\xi):=\sqrt{(1+\vert\nabla a\vert^2)\vert\xi\vert^2-
    (\nabla a\cdot\xi)^2},$$ which is of the form described above, with $\Sigma(v,\xi)=\sqrt{(1+\vert v\vert^2)\vert\xi\vert^2-(v\cdot\xi)^2}$ and $v(\cdot)=\nabla a$. Such symbols $\sigma(x,\xi)$ are not infinitely smooth with respect to $x$, since their regularity is limited by the regularity of the function $v$. One must therefore be able to handle symbols of limited smoothness to deal with such situations; moreover, one must be able to say which norms of $v(\cdot)$ are involved in the constant $\underline{C}(\sigma^1)$ of (\[intro1\]), (\[intro2\]) and (\[intro3\]).\
But even knowing precisely the way the constants $\underline{C}(\sigma^1)$ depend on $\sigma^1$, the estimates (\[intro1\]) and (\[intro2\]) may not be precise enough in some situations. Indeed, when one has to use, say, a Nash-Moser iterative scheme, *tame* estimates are needed. For instance, in such situations, the product estimate $\vert uv\vert_{H^s}
\lesssim \vert u\vert_{H^s}\vert v\vert_{H^s}$ ($s>d/2$) is inappropriate and must be replaced by Moser’s tame product estimate $\vert uv\vert_{H^s}\lesssim 
\vert u\vert_\infty\vert v\vert_{H^s}+\vert u\vert_{H^s}\vert v\vert_\infty$ ($s\geq 0$). Obviously, (\[intro1\]) is not precise enough to contain this latter estimate. Part of this paper is therefore devoted to the derivation of sharper versions of (\[intro1\]).

In the works dealing 