Differential inequalities in Lp-spaces

In this article, we present inequalities related to the continuous representations of one-parameter groups. As an application, we obtain some differential inequalities of Bernstein type in Lp-spaces: We define the spectrum Σ(f) of f∈Lp(R) (1⩽p<∞), asΣ(f)=⋃k∈Lq(R)spB{f⁎k}¯(1p+1q=1), where spB{⋅} is the Beurling spectrum. It is shown that if τ∈R satisfies the condition 0⩽τσ<π, then f′∈Lp(R) and‖f′‖p⩽σ2sinτσ‖f(⋅+τ)−f(⋅−τ)‖p, where σ:=sup{|λ|:λ∈Σ(f)}. Some related problems are also discussed.