positive definite and $n=3$ was solved by Duke and Schulze-Pillot [@DSP] for $K = {\mathbb{Q}}$. For arbitrary totally real $K$, the result was established by Cogdell, Piatetski-Shapiro and Sarnak [@CPSS]; an account of the key ideas appeared in [@Co]. In fact, the systematic study of subconvexity over number fields was initiated by [@CPSS] about a decade ago motivated by this striking application. The relevant subconvex bound was subsequently generalized over arbitrary number fields by Venkatesh [@Ve], while our Corollary \[cor5\] allows a better approximation for the number of representations.

Let $K$ be a totally real number field and let $Q$ be a positive integral ternary quadratic form over $K$. Then there is an ineffective constant $c>0$ such that every totally positive squarefree integer $r \in {\mathfrak{o}}$ with ${\mathcal{N}}r {\geqslant}c$ is represented integrally by $Q$ if and only if it is integrally represented over every completion of $K$. More precisely, the number of representations for such $r$ equals $({\mathcal{N}}r)^{\frac{1}{2}+o(1)}+O(({\mathcal{N}}r)^{\frac{7}{16}+\frac{\theta}{8}+o(1)})$.

This result, with a slightly weaker error term, was originally proved in [@CPSS]. The representation of non-squarefree integers is quite subtle, but in principle can again be characterized by more involved local considerations, cf.[@SP].

Another application of Theorem \[theorem1\] can be found 