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We count the number$S$($x$) of quadruples $ {\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right)} \in \mathbb{Z}^{4} $ for which $$ p = x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4} \leqslant x $$ is a prime number and satisfying the determinant condition:$x$_{1}$x$_{4}−$x$_{2}$x$_{3}= 1. By means of the sieve, one shows easily the upper bound$S$($x$) ≪$x$/log$x$. Under a hypothesis about prime numbers, which is stronger than the Bombieri–Vinogradov theorem but is weaker than the Elliott–Halberstam conjecture, we prove that this order is correct, that is$S$($x$) ≫$x$/log$x$.

Suppose that$X$is a vector field on a manifold$M$whose flow, exp$tX$, exists for all time. If μ is a measure on$M$for which the induced measures$μ$_{$t$}≡(exp$tX$)_{*}$μ$are absolutely continuous with respect to μ, it is of interest to establish bounds on the$L$^{$p$}(μ) norm of the Radon-Nikodym derivative$dμ$_{$t$}/$dμ$. We establish such bounds in terms of the divergence of the vector field$X$. We then specilize$M$to be a complex manifold and derive reverse hypercontractivity bounds and reverse logarithmic Sololev inequalities in some holomorphic function spaces. We give examples on$C$^{m}and on the Riemann surface for$z$^{1/$n$}.

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