On some diophantine inequalities involving primes

We present the following result obtained by the speaker:
Theorem 1. Suppose that $H>\sqrt{N}\exp(-\ln^{0.1}N)$. Then the inequality
$$ |p_1^2+p_2^2-N|\leqslant H $$
is solvable in primes $p_1$ and $p_2$.
Theorem 2. Let $H>N^{\,49/144}\exp(\ln^{0.8}N)$. Then the inequality
$$ |p_1^2+p_2^2+p_3^2-N|\leqslant H $$
is solvable in primes $p_1$, $p_2$ and $p_3$.
Theorem 3. Suppose that $H>N^{\,7/72}\exp(\ln^{0.8}N)$. Then the inequality
$$ |p_1+p_2-N|\leqslant H $$
is solvable in primes $p_1$ and $p_2$.
The proof of Theorem 1 would be also presented.