Notes on the Sobolev (Semi)Norms of Quadratic Functions

TL;DR

This paper derives explicit formulas for the Sobolev norms and seminorms of quadratic functions over $l_p$-balls, providing insights into their behavior in various functional spaces and domains.

Contribution

It provides explicit expressions for Sobolev norms of quadratic functions on $l_p$-balls, enhancing understanding of their analytical properties.

Findings

Explicit formulas for $H^0$ norm of quadratic functions.

Explicit formulas for $H^1$ seminorm of quadratic functions.

Applicable to $l_p$-balls in $\,R^n$ for 1 ≤ p ≤ ∞.

Abstract

This paper studies the $H^0$ norm and $H^1$ seminorm of quadratic functions. The (semi)norms are expressed explicitly in terms of the coefficients of the quadratic function under consideration when the underlying domain is an $l_p$-ball (1 \leq p \leq \infty) in $R^n$ .