Title: $\ell^p(\mathbb Z^d)$ boundedness for discrete operators of Radon type: maximal and variational estimates
Abstract: We will be concerned with estimates of $r$-variations for discrete operators of Radon type and their applications in pointwise ergodic theory. Let $\mathcal P=(\mathcal P_1,\ldots, \mathcal P_{d}): \mathbb Z^{k}\rightarrow \mathbb Z^{d}$ be a polynomial mapping where for each $j \in \{1, \ldots, d\}$ the function $\mathcal P_j:\mathbb Z^{k} \rightarrow \mathbb Z$ is an integer-valued polynomial of $k$ variables. We define, for a finitely supported function $f: \mathbb Z^{d} \rightarrow \mathbb C$, the Radon averages
$$M_N^{\mathcal P} f(x)=|\mathbb B_N|^{-1} \sum_{y\in\mathbb B_N} f\big(x-\mathcal P(y)\big)$$
where $\mathbb B_N = B_N\cap\mathbb Z^k$ for $N\in \mathbb N$ and $B_t=\{x\in\mathbb R^k: |x|\le t\}$ for any $t>0$. We will be also interested in discrete truncated singular integrals. Assume that $K \in\mathcal C^1\big(\mathbb R^k \setminus \{0\}\big)$ is a Calderón-Zygmund kernel satisfying
(i) the differential inequality $|y|^k |K(y)| + |y|^{k+1} |\nabla K(y)| \leq 1$ for all $y\in\mathbb R^k\setminus\{0\}$;
(ii) the cancellation condition $\int_{B_t\setminus B_s}K(y){\rm d}y=0$ for all $t>s>0$.
We define, for a finitely supported function $f: \mathbb Z^{d} \rightarrow \mathbb C$, the truncated singular Radon transforms
$$T_N^{\mathcal P} f(x)=\sum_{y\in\mathbb B_N\setminus\{0\}} f\big(x - \mathcal P(y)\big)K(y).$$
Recall that for any $r\ge1$ the $r$-variational seminorm $V_r$ of a sequence $\big(a_n(x): n\in \mathbb N\big)$ of complex-valued functions is defined by
$$V_r\big(a_n(x): n\in \mathbb N\big)=\sup_{J\in\mathbb N}\sup_{k_0<\ldots<k_J}\bigg(\sum_{j=0}^J|a_{k_{j+1}}(x)-a_{k_j}(x)|^r\bigg)^{1/r}.$$
The main result will be the following.
Main Theorem.
For every $p>1$ and $r>2$ there is $C_{p, r} > 0$ such that for all $f \in \ell^p\big(\mathbb Z^{d}\big)$
\[
\big\|
V_r\big( M_N^\mathcal P f: N\in\mathbb N\big)
\big\|_{\ell^p}+\big\|
V_r\big( T_N^\mathcal P f: N\in\mathbb N\big)
\big\|_{\ell^p}\le
C_{p, r}\|f\|_{\ell^p}.
\]
Moreover, $C_{p, r}\le C_p\frac{r}{r-2}$ and the constant $C_p>0$ is independent of the coefficients of the polynomial mapping $\mathcal P$.
This is a joint work with Elias M. Stein and Bartosz Trojan.