Uniform convexity of paranormed generalizations of $L^p$ spaces

TL;DR

This paper investigates the uniform convexity of generalized $L^p$ spaces defined via paranorms involving a bijective increasing function, extending classical results and providing explicit formulas for the modulus of convexity in specific cases.

Contribution

It generalizes Clarkson's theorem to a broader class of paranormed spaces and derives explicit convexity moduli under certain conditions.

Findings

Conditions for uniform convexity of generalized $L^p$ spaces are established.

The uniform convexity of finite-dimensional paranormed spaces is proven.

Explicit modulus of convexity formulas are provided for specific functions and spaces.

Abstract

For a measure space $(\Omega ,\Sigma ,\mu)$ and a bijective increasing function $\varphi :\left[ 0,\infty \right) \rightarrow \left[0,\infty \right)$ the $L^{p}$-like paranormed ($F$-normed) function space with the paranorm of the form $\mathbf{p}_{\varphi}(x)=\varphi ^{-1}\left(\int_{\Omega}\varphi \circ \left|x\right|d\mu \right)$ is considered. Main results give general conditions under which this space is uniformly convex. The Clarkson theorem on the uniform convexity of $L^{p}$-space is generalized. Under some specific assumptions imposed on $\varphi$ we give not only a proof of the uniform convexity but also show the formula of a modulus of convexity. We establish the uniform convexity of all finite-dimensional paranormed spaces, generated by a strictly convex bijection $\varphi$ of $[0, \infty)$. However, the {\it a contrario} proof of this fact provides no information on a modulus of convexity of these spaces. In some cases it can be done, even an exact formula of a modulus can be proved. We show how to make it in the case when $S={\mathbb R}^2$ and $\varphi$ is given by $\varphi(t)={\rm e}^t-1$.