On two S. M. Nikol'skii problems

1. Equivalence of different spherical moduli of smoothness.
Let $S^{n-1}$ be a unit sphere in $\mathbb{R}^n$, $n\ge 3$, $f$ be a measurable real-valued function belonging to $L_p(S^{n-1})$, $1<p< \infty$, $\Delta_t f(\mu)$ stand for the spherical difference with the step $t\in (0, \pi/2)$ defined as
\begin{equation} \label{442:eq1} \Delta_t f(\mu):= (\sin t)^{2-n} \int_{(\sigma,\mu)=\cos t} (f(\sigma) - f(\mu))\, d\sigma \end{equation}
where the integration is taken over the $(n-2)$-dimensional parallels, the point $\mu$ being a northpole.
In 1981 M. Wehrens [1] considered the so-called “amplified” moduli of smoothness
\begin{equation} \label{442:eq2} \omega^{\star}_k (\delta; f; L_p):= \sup_{0<t_1, t_2 \dots t_k\le \delta} \|\Delta_{t_1} (\Delta_{t_2}(\,\dots (\Delta_{t_k} f(\mu) )\dots ) \|_p, \qquad k\in \mathbb{N},\quad \delta<\pi/2, \end{equation}
and proved that this quantity is equivalent to J. Peetre $K$-functional
\begin{equation} \label{442:eq3} K(\delta^{2k}, f; L_p, W_p^{2k}):= \inf \{ \delta^{2k}\|{\mathbb{D}^k}g\|_p + \|f-g\|_p: g \in C^{\infty}(S^{n-1}) \} \end{equation}
with $\mathbb{D}$ denoting the Laplace-Beltrami operator on $S^{n-1}$. It was also claimed in [1] that the ordinary moduli of smoothness $\omega_k (\delta; f; L_p):= \sup_{0<t\le \delta} \| \Delta_{t}^k f(\mu)\|_p$ do not yield such an equivalence. The latter assertion was refuted by P. I. Lizorkin and S. M. Nikol'skii [2] for $p=2$. The whole range of $p$ was covered in [3]:
Theorem 1. There is a constant $c:=c(k, n, p)>0$, $1<p<\infty$, such that:
\begin{equation} \label{442:eq4} c^{-1} \omega^{\star}_k (\delta; f; L_p) \le K(\delta^{2k}, f; L_p, W_p^{2k}) \le c \omega_k (\delta; f; L_p); \end{equation}
in particular, the ordinary and the amplified moduli of smoothness are equivalent to each other.
The proof uses the Strichartz theorem on multipliers with respect to spherical harmonics and the estimates of derivatives (over positive parameter $m$) for Mehler integral form presentation of Gegenbauer polynomials $C_{m}^{0.5n - 1}(\cos t)$, $m\in \mathbb{N}$.
2. On functions whose differences belong to $L_p(0, 1)$.
Let $\mathfrak{M}\equiv \mathfrak{M}(I)$, ${I}:=(0, 1) $ be the set of all measurable and almost everywhere finite functions $f\colon {I} \to \mathbb{C}$, $0<p\le \infty$, $L_p\equiv L_p(I)$ be the Lebesgue spaces of functions possessing a finite quasi-norm (a norm as $p\ge 1$):
\begin{equation} \label{442:eq5} \| f \|_{L_p({I})} := \left(\int_I \ |f(x)|^p \ dx \right)^{1/p},\qquad \| f \|_{L_{\infty}(I)} := \operatorname{ess\,sup}\limits_{x\in I}\ |f(x)|. \end{equation}

Introduce a difference of $f$ with the step $h\in I$ at the point $x\in (0, 1-h)$: $\Delta_hf(x):=f(x+h)-f(x)$, and the higher differences ($k\ge 2$, [4, § 4.2])
$$ \Delta^k_hf(x):= \Delta_h\left(\Delta^{k-1}_hf(x+h) -\Delta^{k-1}_h f(x)\right),\qquad h\in (0, 1/k),\quad x\in (0, 1-kh), $$
for which one has the estimates $\|\Delta^k_hf(\,\cdot\,)\|_{L_p(0, 1-kh)} \le c(p,k) \|f(\,\cdot\,)\|_{L_p(I)}$.
In the 1970s, Sergey Mikhailovich Nikol'skii raised the following question (according to O. V. Besov): if $\| \Delta_h^k f \|_{L_p(0, 1-kh)}$ is finite for all $h$ from sufficiently “massive” set $\mathcal{H} \subset \mathbb{R}_+$ is it true that $f$ belongs to $L_p(I)$? Our goal is to answer this question in the affirmative; moreover the answer is given for general families of linear combination of shifts.
Definition. Given two finite collections of numbers
\begin{equation} \label{442:eq6} \begin{gathered} \{\beta\}\equiv \{\beta_j\}_{j=0}^k,\qquad \{\tau\}\equiv \{\tau_j\}_{j=0}^k,\qquad \beta_j\in\mathbb{C}\setminus \{0\},\qquad \tau_j \in \mathbb{R}, \\ j \in \{0, 1, \dots k\},\qquad k\ge 1, \qquad \tau_0<\tau_1<\tau_2 < \dots <\tau_k. \end{gathered} \end{equation}

Denote $H=H(\{\tau\}):= (\tau_k - \tau_0)^{-1}$ and consider the family of linear operators
\begin{equation} \label{442:eq7} \begin{gathered} {\mathcal{D}}^k_h\equiv {\mathcal{D}}^k_{h, \{ \beta \} \{ \tau \}}\colon \mathfrak{M}(I) \to \mathfrak{M}(I_h), \qquad I_h:=(-\tau_0h, 1- \tau_kh); \\ {\mathcal{D}}^k_h f (x):= \sum_{j=0}^k \beta_j f(x+\tau_jh), \qquad 0<h<H, \quad x\in I_h. \end{gathered} \end{equation}

Theorem 2. Let $f \in \mathfrak{M}(I)$ and for some $p\in (0, \infty] $ the functions ${\mathcal{D}}^k_h f \in L_p(I_h)$ for all $h$ from a set ${\mathcal{H}}\subset (0, H/2)$ of positive Lebesgue measure. Then $f \in L_p(I)$.
Remark. In the proof only the classical results on the measurable functions [6] are involved.
Remark. Simple examples show that the number $H/2$ in Theorem 2 cannot be replaced by any larger one.
Remark. As for multidimensional case (say for “good” bounded domains in $\mathbb{R}^d$) the question posed by S. M. Nikol'skii remains still open except when it can be reduced to $d=1$.
The work was supported by grant of Russian Foundation of Basic Research, project No 14-01-00684.
The author is deeply grateful to O. V. Besov and V. I. Burenkov for their interest to this work and a number of valuable remarks.