On the small balls problem for equivalent Gaussian measures

Let μ be a centred Gaussian measure in a linear space X with Cameron-Martin space H, let q be a μ-measurable seminorm, and let Q be a μ-measurable second-order polynomial. We show that it is sufficient for the existence of the limit limε→0E(exp Q|q≤ε), where E is the expectation with respect to μ, that the second derivative DH2Q of the function Q be a nuclear operator on H. This condition is also necessary for the existence of the above-mentioned limit for all seminorms q. The problem under discussion can be reformulated as follows: study lim ε→0ν(q≤ε)/μ(q≤ε) for Gaussian measures ν equivalent to μ