Another View on the Hoelder Inequality

TL;DR

This paper extends the Hölder inequality to negative exponents and measurable functions, introduces homogeneous weighted spaces, and provides a new perspective on operator norms of diagonal matrices.

Contribution

It generalizes Hölder norms for negative values and measurable functions, offering a broader framework for inequalities and operator norms.

Findings

Extended Hölder inequality for negative exponents

Introduction of homogeneous weighted spaces

Operator norm computation for diagonal matrices

Abstract

Every diagonalmatrix D yields an endomorphism on the n-dimensional complex vectorspace. If one provides this space with Hoelder norms, we can compute the operator norm of D. We define homogeneous weighted spaces as a generalization of normed spaces. We generalize the Hoelder norms for negative values, this leads to a proof of an extented version of the Hoelder inequality. Finally, we formulate this version also for measurable functions.