Surely yes, and in more generality, but can it be proved?
It seems that most, if not all, statements about quadratic forms representing primes fall back on algebraic number theory (i.e. splitting of primes in $\mathbb{Q}(\sqrt{7})$) for their proofs, and so are incompatible with the condition that $0 < y < x/10$.
Some related references which didn't lead to a proof: First of all there is this previous MO post, which suggests a negative answer.
There is also this paper of Iwaniec, which uses sieve methods but which also uses the multiplicative structure of solutions to the quadratic form.
There is also the interesting Theorem 5.36 of Iwaniec and Kowalski, which states that the arguments of prime elements of $\mathbb{Z}[i]$ are equidistributed in $(0, 2\pi)$. This is proved using the Hecke $L$-function $\sum_{\alpha \in \mathbb{Z}[i]} \big( \frac{\alpha}{|\alpha|} \big)^{ik} |\alpha|^{-s}$, for all $k$ divisible by 4. This generalizes further, but presumably not to real quadratic fields, where the infinite unit group would foul the construction up.
Finally, using a straight-up sieve (with only the additive structure of solutions to $x^2 – 7 y^2$) seems hopeless, as sieves tend to be bad at finding primes. There is the recent work of Friedlander-Iwaniec on $x^2 + y^4$ and Heath-Brown on $x^3 + 2y^3$, but these use algebraic number theory in $\mathbb{Q}(i)$ and $\mathbb{Q}(\sqrt{-3})$, and seem unlikely to generalize here.
I wonder if there is a promising approach out there which I have overlooked? Thank you!
Best Answer
The units of $k=\mathbf{Q}(\sqrt{7})$ have the form $\pm (8+3 \sqrt{7})^n$ with $n \in \mathbf{Z}$. If $\pi = x+y\sqrt{7}$ is a prime element of $k$, then $\lambda(\pi):= \log |x+y\sqrt{7}|$ is well-defined in $\mathbf{R}/\alpha \mathbf{Z}$ where $\alpha = \log(8+3\sqrt{7})$. Note that $\lambda$ factors as $\lambda = f \circ \sigma$ where $\sigma : k^{\times} \to \mathbf{R}^{\times}$ is a given embedding of $k$ and $f : \mathbf{R}^{\times} \to \mathbf{R}/\alpha \mathbf{Z}$ is a continuous group homomorphism. We can apply Hecke's theory of equidistribution (see Lang, Algebraic number theory, Chap. XV, especially Example 3 at the end of the chapter) to show that the sequence $\lambda(\pi)$ is equidistributed in $\mathbf{R}/\alpha \mathbf{Z}$ where $\pi$ runs through the primes of $k$ (with respect to the usual ordering on the norm of $\pi$).
You want $0 < y < x/10$ which translates into the inequality
\begin{equation*} \sqrt{p} \leq x+y\sqrt{7} \leq C \sqrt{p} \end{equation*} where $C=\frac{10+\sqrt{7}}{\sqrt{93}}>1$ and $p=N_{k/\mathbf{Q}}(x+y\sqrt{7})$. This in turn is equivalent to $\lambda(\pi) \in [\frac12 \log p , \frac12 \log p + \log C]$ inside $\mathbf{R}/\alpha \mathbf{Z}$.
Using the equidistribution result above, the set $X=\{\pi : \lambda(\pi) \in [0,\frac12 \log C]\}$ has a positive natural density (here we consider only primes of $k$ which don't belong to $\mathbf{Q}$, but this is ok because the norm of a rational prime $p$ is equal to $p^2$, so these rational primes are negligible). Moreover, the set $Y=\{\frac12 \log p : \pi \in X\}$ is dense in $\mathbf{R}/\alpha \mathbf{Z}$ because of the prime number theorem. So we can find infinitely many primes $\pi \in X$ with $\frac12 \log p \in [-\frac12 \log C,0]$ inside $\mathbf{R}/\alpha \mathbf{Z}$, which implies what you want using the above discussion.