Kolmogorov type inequalities for norms of Riesz derivatives of multivariate functions and some applications (Q742268)

Let \(L_{\infty,p}^1\) be the space of essentially bounded functions \((f\in L_{\infty}(\mathbb R^m))\) such that \(\frac{\partial f}{\partial x_i}\in L_p(\mathbb R^m),\) \(i=1,\dots,m,\) \(D^{\alpha}f\) be the Riesz derivative of order \(\alpha,0<\alpha<1,\) \[ D^{\alpha}f=\frac{1}{d_{m,1}(\alpha)}\int_{\mathbb R^m}\frac{f(x)-f(x+t)}{|t|^{m+\alpha}}dt, \] where \(d_{m,1}(\alpha)\) is the related normalizing factor. The main result: If \(p>m,\) \(0<\alpha<1-m/p\) then for any \(f\in L_{\infty,p}^1\) \[ \| D^{\alpha}f\|_{\infty}\leq C(\alpha,p,m)\| f\|_{\infty}^{1-\frac{\alpha}{1-m/p}}\;\|\nabla f\|_p^{\frac{\alpha}{1-m/p}}, \] where \(C(\alpha,p,m)\) is given explicitly and the inequality is sharp. Applications for the best approximation of the operator \(D^{\alpha}\) by bounded operators and its optimal recovery from inaccurately given information are given.