Convexity of Generators of L ^p -like Paranorms

Let

be a measure space with at least two disjoint sets of finite and positive measure, and

S_{+}=S_{+}(\Omega,\Sigma,\mu ) S + = S + ( Ω , Σ , μ )

denote the set of all

\mu μ

-integrable simple functions

\mathbf{x}:\Omega \rightarrow \mathbb{R}_{+} x : Ω → R +

having support

\Omega \left( \mathbf{x}\right) Ω ( x )

of positive measure. Then, for an arbitrary bijection

\varphi:\left(0,\infty \right) \rightarrow \left( 0,\infty \right) φ : ( 0 , ∞ ) → ( 0 , ∞ )

, the functional

\mathbf{P}_{\varphi }:S_{+}\rightarrow \mathbb{R}_{+} P φ : S + → R +

given by

\mathbf{P}_{\varphi }\left( \mathbf{x}\right):=\varphi ^{-1}\big( \int_{\Omega (\mathbf{x})}\varphi \circ xd\mu \big) P φ ( x ) : = φ − 1 ( ∫ Ω ( x ) φ ∘ x d μ )

is well defined. The results presented support the conjecture that subadditivity of

\mathbf{P}_{\varphi } P φ

implies the convexity of

\varphi φ

. The case of superadditivity of

\mathbf{P}_{\varphi} P φ

is also discussed.