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

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>0 C1(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, Turn MathJax on 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,