Narrow Results

By applying Rohlin's result on the classification of homomorphisms of Lebesgue space, the random inertial manifold of a stochastic damped nonlinear wave equations with singular perturbation is proved to be approximated almost surely by that of a stochastic nonlinear heat equation which is driven by a new Wiener process depending on the singular perturbation parameter. This approximation can be seen as the Smolukowski–Kramers approximation as time goes to infinity. However, as time goes infinity, the approximation changes with the small parameter, which is different from the approximation on a finite time interval.

The paper introduces the $k$th-order slant Toeplitz operator on the Lebesgue space of $n$-torus, where $k=(k_1,k_2,\dots ,k_n)$ such that $k_t\ge 2$ for all $t$. It investigates certain properties of $k$th-order slant Toeplitz operators on the Lebesgue space $L^2({𝕋}^n)$. The paper deals with a system of operator equations, characterizing the $k$th-order slant Toeplitz operators. At the end, we discuss certain spectral properties of the considered operator.

Recently, the trace space of Sobolev functions with variable exponents has been characterized by the authors [L. Diening and P. Hästö: Variable exponent trace spaces, Preprint (2005)]. In this note we relax the assumptions on the exponent need for some basic results on trace spaces, like a characterization of zero boundary value spaces in terms of traces.

In this survey we summarize recent results from the variable exponent, metric measure space setting, and some other closely related material. We show that the variable exponent arises very naturally in the metric measure space setting. We also give an example which shows that the maximal operator can be bounded for piecewise constant, non-constant, exponents.