Theoretical analysis demonstrates sharp discrete Hardy inequalities and Polya-Szego rearrangement principles on lattice graphs, highlighting connections to continuous torus bounds.
Key Points
Sharp weighted discrete functional inequalities hold on one-dimensional integers with power weights, establishing precise bounds for discrete operators.
Theoretical analysis uses the Fourier method and super-solutions to determine the asymptotic behavior of the sharp discrete Hardy inequality as dimensions grow.
Framework connects the discrete Polya-Szego inequality directly to graph isoperimetric inequalities, extending functional analysis tools across lattice graphs.