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

Abstract

For a measure space $(\Omega ,\Sigma ,\mu )$ and a bijective increasing function $\phi :\left[0,\infty \right)\to \left[0,\infty \right)$ the $L^p$-like paranormed ($F$-normed) function space with the paranorm of the form ${\mathbf{p}}_{\phi }(x)={\phi }^{-1}\left({\int }_{\Omega }\phi \circ \left|x\right|d\mu \right)$ 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 $\phi$ 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 $\phi$ 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 $\phi$ is given by $\phi (t)={\mathrm{e}}^t-1$.

arXiv comment30 pages