Sharp estimates of the constants of equivalence between integral moduli of smoothness and \(K\)-functionals in the multivariate case (Q1959701)

The homogeneous Sobolev space is defined with the following seminorm: \[ \parallel f \parallel_{\dot{W}_{p}^{k}}=\Big(\sum_{i=1}^{n}||\partial_{i}^{k}f||_{p}^{p}\Big)^{1/p}. \] Here \(1\leq p< +\infty\), \(||f||_{p}=||f||_{L_{p}(\Omega)}\), where \(\Omega=\mathbb{R}^{n},\,\mathbb{T}^{n}\), or a Lie group \(G\) over \(\mathbb{R}\). The Peetre \(K_{p}\)-functional, \(0<p<\infty\), between two quasi-normed vector spaces can be defined in the following way. Let \(f\in A+B,\,t>0\). Then \(K_{p}(t,\,f;A,\,B)=\inf_{f=a+b}\big(||a||_{A}^{p}+t^{p}||b||_{B}^{p}\big)^{1/p}\). Let \(\{e_{i}\}_{i=1}^{n}\) be the standard basis in \(\mathbb{R}^{n}\). The integral modulus of smoothness is defined by the expression \[ \omega_{k}(f;t)_{p}=\sup_{0<h\leq t}\Big(\sum_{i=1}^{n}||\Delta_{he_{i}}^{k}f||_{p}^{p}\Big)^{1/p},\quad t>0, \] where \((\Delta_{y}^{k}f)(x)=\sum_{i=0}^{k}(-1)^{k+i}\Big({k \atop i}\Big)f(x+ky)\). Let \[ C_{k,p}(\mathbb{R}^{n})=\sup_{f\in L_{p}\cap \dot{W}_{p}^{k},\,f\neq 0}\frac{K_{p}(t^{k},f;L_{p},\, \dot{W}_{p}^{k})}{\omega_{k}(f;t)_{p}}, \] \[ D_{k,p}(\mathbb{R}^{n})=\sup_{f\in L_{p}\cap \dot{W}_{p}^{k},\,f\neq 0}\frac{\omega_{k}(f;t)_{p}}{K_{p}(t^{k},f;L_{p},\, \dot{W}_{p}^{k})}. \] The authors prove the following result: For any \(n\in \mathbb{N}\) and \(k\in \mathbb{N}\), \[ 2^{-1/2}\big(2\sin(1/2)\big)^{-k}\leq C_{k,2}(\mathbb{R}^{n})\leq 2^{1/2}\big(2\sin(1/2)\big)^{-k}, \] \[ 2^{k}\,\sqrt{n}\leq D_{k,2}(\mathbb{R}^{n})\leq 2^{k+1}\,\sqrt{n}. \]