[Paper Review] Variational Inequalities For The Differences Of Averages Over Lacunary Sequences

Sakin Demir|arXiv (Cornell University)|Mar 4, 2022

Approximation Theory and Sequence Spaces4 citations

TL;DR

This paper establishes strong and weak type weighted inequalities for the variation operator $\mathcal{V}_s f(x)$, defined as the $\ell^s$-norm of differences between averages over lacunary sequences. By leveraging convolution-type singular integral operators and weighted norm theory, it proves that $\mathcal{V}_s$ is bounded on $L^p(w)$ for $1 < p < ∞$ and weak-type $(1,1)$ when $w$ satisfies $A_p$ or $A_1$ conditions, extending known results to general lacunary sequences with $2 \leq s < \infty$. The key contribution is the extension of variational $L^p$ and BMO estimates to arbitrary lacunary sequences via a dyadic-like decomposition and kernel $D_r$-condition analysis.

ABSTRACT

Let $f$ be a locally integrable function defined on $\mathbb{R}$, and let $(n_k)$ be a lacunary sequence. Define the operator $A_{n_k}$ by $$A_{n_k}f(x)=\frac{1}{n_k}\int_0^{n_k}f(x-t)\, dt.$$ We prove various types of new inequalities for the variation operator $$\mathcal{V}_sf(x)=\left(\sum_{k=1}^\infty|A_{n_k}f(x)-A_{n_{k-1}}f(x)|^s ight)^{1/s}$$ when $2\leq s&lt;\infty$.

Motivation & Objective

• To extend known variational $L^p$ and BMO estimates from dyadic to general lacunary sequences.
• To establish strong and weak type weighted norm inequalities for the $\ell^s$-variation of averages over lacunary sequences.
• To prove boundedness of the variation operator $\mathcal{V}_s f$ on $H^1$, $L^p$, and BMO spaces with respect to $A_p$ weights.
• To generalize results from dyadic settings (e.g., $2^n$) to arbitrary lacunary sequences satisfying $n_{k+1}/n_k \geq \beta > 1$.

Proposed method

• Constructs a dyadic-like sequence $(m_j)$ from a given lacunary sequence $(n_k)$ such that $\beta \leq m_{j+1}/m_j \leq \beta^2$, preserving the variation structure.
• Uses the identity $A_{n_k}f(x) - A_{n_{k-1}}f(x) = \sum_{j \in J(k)} (A_{m_j}f(x) - A_{m_{j-1}}f(x))$ to majorize the variation over $n_k$ by that over $m_j$, reducing to a controlled dyadic setting.
• Represents the variation operator as $\|K * f\|_{\ell^s(\mathbb{Z}^+)}$ where $K(x) = \{\phi_k(x) - \phi_{k-1}(x)\}$, treating it as an $\ell^s$-valued convolution operator.
• Applies weighted norm theory from Rubio de Francia et al. [5] to $\ell^s$-valued singular integral operators with kernels satisfying the $D_r$ condition.
• Establishes that the kernel $K$ satisfies $D_r$ for $1 \leq r < \infty$, enabling application of weighted inequalities for $\ell^s$-valued operators.
• Uses the fact that $\|K * f(x)\|_{\ell^s} = \mathcal{V}_s f(x)$ to transfer weighted estimates from operator theory to the variation operator.

Experimental results

Research questions

• RQ1 Can strong and weak type weighted $L^p$ inequalities for the variation operator $\mathcal{V}_s f$ be extended from dyadic to general lacunary sequences?
• RQ2 What conditions on the weight $w$ ensure that $\mathcal{V}_s$ is bounded on $L^p(w)$ for $1 < p < \infty$?
• RQ3 Does the variation operator $\mathcal{V}_s f$ satisfy weak-type $(1,1)$ estimates with respect to $A_1$ weights when $2 \leq s < \infty$?
• RQ4 How does the $\ell^s$-variation of averages over lacunary sequences relate to classical function spaces like $H^1$, $BMO$, and $L^p$?
• RQ5 Can the boundedness of $\mathcal{V}_s$ on $H^1$ and $BMO$ be established via kernel $D_r$-condition and weighted operator theory?

Key findings

• The variation operator $\mathcal{V}_s f$ satisfies $\|\mathcal{V}_s f\|_{L^1(\mathbb{R})} \leq C \|f\|_{H^1(\mathbb{R})}$ for all $f \in H^1(\mathbb{R})$ and $2 \leq s < \infty$, extending $H^1$-boundedness to lacunary sequences.
• For $1 < p < \infty$, $\|\mathcal{V}_s f\|_{L^p(w)} \leq C_{p,\rho}(w) \|f\|_{L^p(w)}$ holds if $w \in A_{p/r'}$ and $r' \leq p < \infty$, or $w \in A_p^{r'}$ and $1 < p \leq r'$, with $r'$ the H"older conjugate of $r$.
• The weak-type $(1,1)$ inequality $w(\{x : \mathcal{V}_s f(x) > \lambda\}) \leq C_\rho(w) \lambda^{-1} \int |f(x)| w(x) dx$ holds if $w(x)^{r'} \in A_1$, extending weak-type estimates to general lacunary sequences.
• The variation operator $\mathcal{V}_s f$ is bounded on $BMO(\mathbb{R})$ with $\|\mathcal{V}_s f\|_{BMO} \leq C_2 \|f\|_{L^\infty}$ for compactly supported $f$, extending known $BMO$ estimates to lacunary averages.
• The kernel $K(x) = \{\phi_k(x) - \phi_{k-1}(x)\}$ associated with $\mathcal{V}_s f$ satisfies the $D_r$ condition for $1 \leq r < \infty$, enabling application of weighted norm theory.
• The operator $Tf = \{(\phi_k - \phi_{k-1}) * f\}_{k \in \mathbb{Z}^+}$ is an $\ell^s$-valued singular integral operator of convolution type, and its $\ell^s$-norm equals $\mathcal{V}_s f(x)$, providing a functional analytic framework for the results.