ON THE SMALL BALLS PROBLEM FOR EQUIVALENT GAUSSIAN MEASURES

Let mu be a centred Gaussian measure in a linear space X with Cameron- Martin space H, let q be a mu-measurable seminorm, and let Q be a mu-m easurable second-order polynomial. We show that it is sufficient for t he existence of the limit lim(epsilon-->0) E(expQ\q less than or equal to epsilon), where E is the expectation with respect to mu, that the second derivative D(H)(2)Q of the function Q be a nuclear operator on H. This condition is also necessary for the existence of the above-men tioned limit for all seminorms q. The problem under discussion can be reformulated as follows: study lim(epsilon-->0) upsilon(q less than or equal to epsilon)/mu(q less than or equal to epsilon) for Gaussian me asures upsilon equivalent to mu.