P-adic Derivative Fourier Multiplier Multiplicative System Ultrametric Space
Issue Date:
2010
Publisher:
Bulgarian Academy of Sciences - National Committee for Mathematics
Citation:
Mathematica Balkanica New Series, Vol. 24, Fasc 3-4 (2010), 275p-284p
Abstract:
We perform analysis of certain aspects of approximation in multiplicative systems
that appear as duals of ultrametric structures, e.g. in cases of local fields, totally disconnected
Abelian groups satisfying the second axiom of countability or more general ultrametric spaces
that do not necessarily possess a group structure. Using the fact that the unit sphere of a
local field is a Vilenkin group, we introduce a new concept of differentiation in the field of
p-adic numbers. Some well known convergence tests are generalized to unbounded Vilenkin
groups, i.e. to the setting where the standard boundedness assumption related to the sequence
of subgroups generating the underlying topology is absent. A new Fourier multiplier theorem
for Hardy spaces on such locally compact groups is obtained. The strong Lq, q > 1, and weak
L1 boundedness of Fourier partial sums operators in the system constructed on more general
ultrametric spaces is proved.
