The Stechkin problem for partial derivation operators on classes of finitely smooth functions (Q1581461)

Let \(D\) be a domain of \(\mathbb{R}^d\), \(h\in\mathbb{R}^d\), \(D_h= \{x\in D: x+th\in D\) for all \(t\in [0,1]\}\), \(\Delta_hf(x)= f(x+ h)- f(x)\) for \(x\in D_h\) and \(f: D\to\mathbb{R}\), \(\Delta^0_hf=f\), \(\Delta^m_hf(x)= \Delta_h(\Delta^{m-1}_h f)(x)\) for \(m\in\mathbb{Z}_+\) and \(x\in D_{mh}\), \(\Omega^m(f, t)_p= \text{ess sup}\{|\Delta^m_h f|_p\), \(h\in\mathbb{R}^d\), \(|h|\leq t\}\) for \(p\in ]1,\infty[\), \(t> 0\) and \(f\in L^p(D)\), \([\alpha]= \max\{n\in\mathbb{N}, n> \alpha\}\) for \(\alpha> 0\), \(q\geq 1\), \(s\leq\infty\), \(\lambda\in\mathbb{N}^d\), \(|\lambda|_1< \alpha\) such that \(\tau\equiv |\lambda|+ (d/s- d/q)\vee 0> 0\), \(\alpha- d/p+ d/s> 0\) for \(s> p\), \(\alpha-|\lambda|- d/p+ d/q> 0\) for \(q> p\), \[ H^{\alpha,p}(D)= \Biggl\{f\in W^{[\alpha],p}(D): \sum_{\lambda\in \mathbb{N}^d,|\lambda|_1= [\alpha]} {\sup\{\Omega^m(\partial^\lambda f,t)_p,t> 0\}\over t^{\alpha-[\alpha]}}\leq 1\Biggr\}, \] \[ B^{\alpha,p,\theta}(D)= \Biggl\{f\in W^{[\alpha],p}(D): \sum_{\lambda\in\mathbb{N}^d, |\lambda|_1= [\alpha]} \Biggl(\int^\infty_0 t^{-1-\theta(\alpha- [\alpha])}(\Omega^m(\partial^\lambda f,t)_p)^\theta dt\Biggr)^{1/\theta}\leq 1\Biggr\} \] for \(\theta\in [1,\infty[\), where \(m= 1\) if \(\alpha\not\in \mathbb{N}\), \(m=2\) if \(\alpha\in\mathbb{N}\). Then there exist \(\rho_1(\alpha, s,q,\lambda)>0\), \(\rho_2(s,q,\lambda)> 0\), \(c_1(\alpha, s,p,q,\lambda)> 0\), \(c_2(\alpha, s,p,q,\theta,\lambda)> 0\) such that \(\inf\{\sup\{|\partial^\lambda f-Vf|_q\), \(f\in H^{\alpha,p}(]0,1[^d)\}\), \(V\in B(L^s(]0,1[^d)\), \(L^q(]0,1[^d))\), \(|V|\leq \rho\}\leq c_1(\alpha, s,p,q,\lambda)\rho^{(|\lambda|+ (d/p-d/q)\vee 0-\alpha)/\tau}\) for \(\rho> \rho_1(\alpha, s,q,\lambda)\), \(\inf\{\sup\{|\partial^\lambda f- Vf|_q\), \(f\in B^{\alpha,p,\theta}(]0,1[^d)\}\), \(V\in B(L^s(]0,1[^d)\), \(L^q(]0,1[^d))\), \(|V|\leq \rho\}\geq c_2(\alpha, s,p,q,\theta,\lambda)\rho^{(|\lambda|+ (d/p-d/q)\vee 0-\alpha)/\tau}\) for \(\rho>\rho_2(s,q,\lambda)\).