Differential inequalities in -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 L p -spaces: We define the spectrum Σ ( f ) of f ∈ L p ( R ) ( 1 ⩽ p < ∞ ), as Σ ( f ) = ⋃ k ∈ L q ( R ) sp B { f ⁎ k } ¯ ( 1 p + 1 q = 1 ) , where sp B { ⋅ } is the Beurling spectrum. It is shown that if τ ∈ R satisfies the condition 0 ⩽ τ σ < π , then f ′ ∈ L p ( R ) and ‖ f ′ ‖ p ⩽ σ 2 sin τ σ ‖ f ( ⋅ + τ ) − f ( ⋅ − τ ) ‖ p , where σ : = sup { | λ | : λ ∈ Σ ( f ) } . Some related problems are also discussed.