On a necessary condition for the convergence of averages at Lebesgue \(p\)-points of functions from \(L_p\) (Q1381660)

In the space \(L_p(\mathbb{R}^n)\), \(1\leq p\leq \infty\), a necessary condition for the convergence of averages \(f_\varepsilon(x)= (\varphi_\varepsilon* f)(x)\), \(\varepsilon>0\), with \(\varepsilon\to\infty\) is given. Here \(\varphi(x)\) is the averaging kernel with normed zero moment, \(\varphi_\varepsilon(x)= \varepsilon^{-n}\varphi(x\varepsilon^{-1})\). The necessary condition consists of the convergence of a series with terms \(2^{kn/r}\|\varphi\|_{L_q(E_k)}\). Here \(1/p+ 1/q= 1\) and \(E_k\) are layers in \(\mathbb{R}^n\), \(E_k= \{y: 2^k<| y|< 2^{k+1}\}\), \(k= 0,\pm 1,\dots\)\ .