# Sharp Logarithmic Sobolev and related inequalities with monomial weights

## TL;DR

This paper establishes a sharp Logarithmic Sobolev inequality with monomial weights, extending classical results and providing new characterizations of equality cases, with implications for related inequalities like Shannon and Heisenberg's uncertainty.

## Contribution

It introduces a novel proof for the sharp Logarithmic Sobolev inequality with monomial weights, including new equality characterizations, even in the unweighted case.

## Key findings

- Derived a sharp Logarithmic Sobolev inequality with monomial weights
- Established related inequalities such as Shannon and Heisenberg's uncertainty
- Provided a new proof for the unweighted case with equality characterization

## Abstract

We derive a sharp Logarithmic Sobolev inequality with monomial weights starting from a sharp Sobolev inequality with monomial weights. Several related inequalities such as Shannon type and Heisenberg's uncertain type are also derived. A characterization of the equality case for the Logarithmic Sobolev inequality is given when the exponents of the monomial weights are all zero or integers. Such a proof is new even in the unweighted case.