Functional Analysis and Applications Seminar 

27/04/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Natasha G. Samko, Centro de Análise Funcional e Aplicações, Faro

Indices of almost monotonic functions depending on a parameter and their applications to Hölder spaces of variable order

Hölder spaces of variable order $\lambda(x)$ varying from point to point are well known. Meanwhile, it is also possible to consider generalized Hölder spaces with a characteristic $\omega(h)=\omega(x,h)$ which may also depend on the point $x$, similarly to the case of a Musielak-Orlicz space with the Young function $\Omega(x,u)$ depending on the point $x$. Since the Zygmund-Bary-Stechkin classes of characteristics $\omega$ are described in terms of the so called index numbers of $\omega$, in this generalization we arrive at indices depending on the parameter $x$ (this parameter in general may belong to an arbitrary abstract set, in applications this set may be a set in metric measure spaces). In this talk we consider properties of such parameter dependent index numbers, one of the main points being the study of conditions under which the Zygmund type inequality for $\omega(x,\cdot)$ is uniform with respect to the parameter. We shall discuss an application to measuring local dimensions of a metric measure space at a point $x$ and give an application to the study of generalized Hölder spaces on the unit sphere in the Euclidean space.