On the equivalence of the modulus of smoothness and the K-functional over convex domains

It is well known that for any bounded Lipschitz graph domain Ω⊂RdΩ⊂Rd, r≥1r≥1 and 1≤p≤∞1≤p≤∞ there exist constants C1(d,r),C2(Ω,d,r,p)>0C1(d,r),C2(Ω,d,r,p)>0 such that for any function f∈Lp(Ω)f∈Lp(Ω) and t>0t>0C1(d,r)ωr(f,t)p≤Kr(f,tr)p≤C2(Ω,d,r,p)ωr(f,t)p,C1(d,r)ωr(f,t)p≤Kr(f,tr)p≤C2(Ω,d,r,p)ωr(f,t)p, where ωr(f,⋅)pωr(f,⋅)p is the modulus of smoothness and Kr(f,⋅)pKr(f,⋅)p is the KK-functional, both of order rr. As can be seen, the right hand side inequality depends on the geometry of the domain. One of our main results is that there exists an absolute constant C3(d,r,p)C3(d,r,p) such that for any convex domain Ω⊂RdΩ⊂Rd and all functions f∈Lp(Ω)f∈Lp(Ω), 1≤p≤∞1≤p≤∞, Kr(f,tr)p≤C3(d,r,p)μ(Ω,t)−(r−1+1/p)ωr(f,t)p,Kr(f,tr)p≤C3(d,r,p)μ(Ω,t)−(r−1+1/p)ωr(f,t)p, where μ(Ω,t)≔minx∈Ω|B(x,t)∩Ω||B(0,t)|,B(x,r)≔{y∈Rd:|x−y|≤r}. For bounded convex domains, the above estimate can be improved for ‘large’ values of ttKr(f,tr)p≤C4(d,r,p)((1−trdiam(Ω)r)μ(Ω,t)−(r−1+1/p)+1)ωr(f,t)p,0