Degenerate self-similar measures, spectral asymptotics and small deviations of Gaussian processes

We find the logarithmic small ball asymptotics for the $L_2$-norm with respect to a degenerate self-similar measures of a certain class of Gaussian processes including Brownian motion, Ornstein - Uhlenbeck process and their integrated counterparts.Comment: 15 pages, 1 figur

Exact constants in Poincare type inequalities for functions with zero mean boundary traces

In the paper, we investigate Poincare type inequalities for the functions having zero mean value on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. We derive exact and easily computable constants for some basic domains (rectangles, cubes, and right triangles). In the last section, we derive an a estimate of the difference between the exact solutions of two boundary value problems. Constants in Poincare type inequalities enter these estimates, which provide guaranteed a posteriori error control.Comment: A gap in the proof of Theorem 3.2 is fixed; 19 pages, 3 figure

A note on truncations in fractional Sobolev spaces

We study the Nemytskii operators $u\mapsto |u|$ and $u\mapsto u^{\pm}$ in fractional Sobolev spaces $H^s(\mathbb R^n)$, $s>1$.Comment: 9 page