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Multiparameter extensions of the Christ-Kiselev maximal theorem: strong variational bounds

Himali Dabhi

TL;DR

This work extends the Christ–Kiselev maximal theorem to multiparameter and multilinear contexts by proving strong $r$-variational bounds for truncations over products of filtrations, valid whenever $1\le p<r\le q$ (and similarly for multilinear operators). The approach adapts the CK argument to a product setting, employing a dyadic, divisibility-based decomposition to control the $r$-variation $V_r$ of truncated operators, and then uses the linear (or multilinear) operator bounds to sum over dyadic pieces. The endpoint analysis shows sharpness of the $r>p$ range via a Fourier truncation example, with the $r=p$ case remaining an open question in the general framework. Overall, the results deliver strong variational bounds in a multiparameter/harmonic-analysis setting with implications for ergodic theory and generalized eigenfunction arguments.

Abstract

For a linear operator $T$ bounded from $L^p(Y)$ to $L^q(X)$, the Christ-Kiselev theorem gives $L^p \to L^q$ bounds for the maximal function $T^{*}$ associated to filtrations on $Y$. This result has been extended by establishing bounds for the maximal function associated to a product of filtrations, also known as the multiparameter extension of the Christ-Kiselev theorem. In this note, we strengthen the multiparameter theorem by proving the $r$-variational bounds for the multiparameter trunctations when $r>p$. Furthermore, we replace $T$ by a multilinear operator to obtain a strong variational, multilinear, multiparameter extension of the Christ-Kiselev theorem.

Multiparameter extensions of the Christ-Kiselev maximal theorem: strong variational bounds

TL;DR

This work extends the Christ–Kiselev maximal theorem to multiparameter and multilinear contexts by proving strong -variational bounds for truncations over products of filtrations, valid whenever (and similarly for multilinear operators). The approach adapts the CK argument to a product setting, employing a dyadic, divisibility-based decomposition to control the -variation of truncated operators, and then uses the linear (or multilinear) operator bounds to sum over dyadic pieces. The endpoint analysis shows sharpness of the range via a Fourier truncation example, with the case remaining an open question in the general framework. Overall, the results deliver strong variational bounds in a multiparameter/harmonic-analysis setting with implications for ergodic theory and generalized eigenfunction arguments.

Abstract

For a linear operator bounded from to , the Christ-Kiselev theorem gives bounds for the maximal function associated to filtrations on . This result has been extended by establishing bounds for the maximal function associated to a product of filtrations, also known as the multiparameter extension of the Christ-Kiselev theorem. In this note, we strengthen the multiparameter theorem by proving the -variational bounds for the multiparameter trunctations when . Furthermore, we replace by a multilinear operator to obtain a strong variational, multilinear, multiparameter extension of the Christ-Kiselev theorem.

Paper Structure

This paper contains 10 sections, 5 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Statement of the theorems
  3. Proof of the main theorems
  4. Preliminaries for the proof of Theorem \ref{['multi']}
  5. Classical dyadic intervals
  6. Remark
  7. Proof of Lemma \ref{['decomposition']}
  8. Proof of Theorem \ref{['multi']}
  9. Proof of Theorem \ref{['multilinpar']}
  10. The endpoint $r=p$

Key Result

Theorem 1.1

(Christ-Kiselev CK) If $1 \leq p < q \leq \infty$, then $T^{*}$ is bounded from $L^p(Y)$ to $L^q(X)$ with the operator norm

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.3
  • Remark 2.6
  • Lemma 3.2
  • Lemma 3.5
  • proof