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We consider a system of two-dimensional electrons strongly localized bydisorder. Interactions create a gap in the average tunneling density of states$\nu(E)$ at energies, E, close to the Fermi level. We derive a system ofself-consistent equations for the correlators of local densities of occupiedand empty states. When either the interactions are screened by the gate or thetemperature, T, is high enough, so that the Coulomb gap is shallow, theperturbative solution of the system yields analytical expressions for theinteraction-induced correction $\delta\nu(E)$ and the level-level correlator.We show that even with short-range interactions, $\delta\nu (E)$ exhibits asingular energy dependence at T=0. We also demonstrate that at high T thisenergy dependence is a universal function of the the ratio $E/T$. Regarding thelevel correlations, we trace how the correlator falls off as a function ofspatial and energy separation between the levels. We also trace how thecorrelations vanish with increasing T. Our most noticeable observation is thatfor two close energies the correlator changes sign from positive (attraction)at small distances to negative (repulsion) at large distances.Comment: 14 pages; 3 figures include

Shallow Coulomb gap and weak level-level correlations in a deeply insulating electron system with interactions