Equivalence of some integrals and modulus of smoothness

Let f ∈ C ( R ) . We are interested in lower and upper bounds of the integrals ∫ h H Δ t 2 f ( x ) t 1 + α d t , where 0 < h < H ⩽ ∞ and Δ t 2 f ( x ) = f ( x + t ) + f ( x − t ) − 2 f ( x ) . We will show that for fixed α > 3 there holds h α ‖ ∫ h ∞ Δ t 2 f ( ⋅ ) t 1 + α d t ‖ ≍ ω 2 ( f , h ) , ∀ f ∈ C ( R ) , h > 0 , where ω 2 ( f , h ) is the modulus of smoothness defined by ω 2 ( f , t ) = sup 0 < δ ⩽ t ‖ Δ δ 2 f ‖ . For α > 2 we will show that for some 1 ⩽ A < ∞ the following relations are valid: h α sup H > h ‖ ∫ h H Δ t 2 f ( ⋅ ) t 1 + α d t ‖ ≍ h α max h ⩽ t ⩽ A h ‖ ∫ t ∞ Δ u 2 f ( ⋅ ) u 1 + α d u ‖ ≍ ω 2 ( f , h ) , ∀ f ∈ C ( R ) , h > 0 .