New Contributions to Hardy-type Inequalities and Boundary-Domain Integral Equations
No Thumbnail Available
Date
2025-06
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Addis Ababa University
Abstract
For a large class of operators acting between weighted `1 spaces, exact
formulas are obtained for their norms and the norms of their restrictions to the
cones of nonnegative sequences and nonnegative monotone sequences. The
weights involved are arbitrary nonnegative sequences and may differ in the
domain and codomain spaces. The results are applied to the Ces`aro and Copson
operators, giving their norms and their distances to the identity operator on the
whole space and on the cones. Simplifications of these formulas are derived in
the case of these operators acting on power-weighted `1. As an application, best
constants are given for inequalities relating the weighted `1 norms of the Ces`aro
and Copson operators both for general weights and for power weights. Moreover,
we characterize the optimal non-absolute domain for the Hardy operator (and its
dual) minus the identity, in the Lebesgue space Lp(0;1), 1 _ p _ 1.
For variable coefficient Helmholtz equation, using appropriate parametrix,
we formulate boundary-domain integral equations (BDIEs) for the Dirichlet
and mixed (Dirichlet-Neumann) boundary value problems (BVPs) in a twodimensional
bounded domain. The Dirichlet BVP is reduced to two different
BDIE systems, depending on whether the trace or co-normal derivative of the
third Green identity is employed on the boundary. On the other hand, the mixed
BVP is reduced to four different BDIE systems, depending on whether the trace
or co-normal derivative of the third Green identity is employed on the Dirichlet
and Neumann boundaries. It is not clear in advance which of them will be
more suitable for particular applications and for numerical implementation, and
hence we analyzed all the BDIE systems. The equivalence between the BVPs
and the formulated BDIE systems are shown. Fredholm properties, invertibility
and unique solvability of BDIE systems are investigated in appropriate Sobolev
spaces.
Description
Keywords
New Contributions, Hardy-type Inequalities, Boundary-Domain, Integral Equations