Equivalence of some integrals and modulus of smoothness

Let f∈C(R)f∈C(R). We are interested in lower and upper bounds of the integrals∫hHΔt2f(x)t1+αdt, where 03α>3 there holdshα‖∫h∞Δt2f(⋅)t1+αdt‖≍ω2(f,h),∀f∈C(R),h>0, where ω2(f,h)ω2(f,h) is the modulus of smoothness defined by ω2(f,t)=sup0<δ⩽t‖Δδ2f‖. For α>2α>2 we will show that for some 1⩽A<∞1⩽A<∞ the following relations are valid:hαsupH>h‖∫hHΔt2f(⋅)t1+αdt‖≍hαmaxh⩽t⩽Ah‖∫t∞Δu2f(⋅)u1+αdu‖≍ω2(f,h),∀f∈C(R),h>0.