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$L^p$-estimates for singular integral operators along codimension one subspaces

Mikel Flórez-Amatriain

TL;DR

This work studies maximal directional singular integrals along codimension-one subspaces in $\mathbb{R}^n$ with the subspace allowed to depend measurably on the first $n-1$ variables. The authors develop a time-frequency discretization and a refined tree-based decomposition to prove $L^p$ bounds for $\frac{3}{2}<p<\infty$, and extend to $1<p<\infty$ under a single-band frequency restriction, while showing non-degeneracy is essential in general. The method hinges on discretizing the operator into a model sum over tiles, establishing orthogonality for lacunary trees, and proving a Kakeya-type maximal bound to control top collections, enabling a full range of $p$-boundedness. These results extend the Bateman–Thiele framework to higher dimensions and codimension one, shedding light on differentiation along Lipschitz vector fields and the necessity of a non-degeneracy condition for boundedness.

Abstract

In this paper we study maximal directional singular integral operators in $ \mathbb{R}^n $ given by a Hörmander--Mihlin multiplier on an $ (n-1)$-dimensional subspace and acting trivially in the perpendicular direction. The subspace is allowed to depend measurably on the first $ n-1 $ variables of $ \mathbb{R}^n $. Assuming the subspace to be non degenerate in the sense that it is away from a cone around $e_n$ and the function $ f $ to be frequency supported in a cone away from $ \mathbb{R}^{n-1} $, we prove $ L^p $-bounds for these operators for $ p > 3/2 $. If we assume, additionally, that $ \widehat{f} $ is supported in a single frequency band, we are able to extend the boundedness range to $ p >1 $. The non-degeneracy assumption cannot in general be removed, even in the band-limited case.

$L^p$-estimates for singular integral operators along codimension one subspaces

TL;DR

This work studies maximal directional singular integrals along codimension-one subspaces in with the subspace allowed to depend measurably on the first variables. The authors develop a time-frequency discretization and a refined tree-based decomposition to prove bounds for , and extend to under a single-band frequency restriction, while showing non-degeneracy is essential in general. The method hinges on discretizing the operator into a model sum over tiles, establishing orthogonality for lacunary trees, and proving a Kakeya-type maximal bound to control top collections, enabling a full range of -boundedness. These results extend the Bateman–Thiele framework to higher dimensions and codimension one, shedding light on differentiation along Lipschitz vector fields and the necessity of a non-degeneracy condition for boundedness.

Abstract

In this paper we study maximal directional singular integral operators in given by a Hörmander--Mihlin multiplier on an -dimensional subspace and acting trivially in the perpendicular direction. The subspace is allowed to depend measurably on the first variables of . Assuming the subspace to be non degenerate in the sense that it is away from a cone around and the function to be frequency supported in a cone away from , we prove -bounds for these operators for . If we assume, additionally, that is supported in a single frequency band, we are able to extend the boundedness range to . The non-degeneracy assumption cannot in general be removed, even in the band-limited case.

Paper Structure

This paper contains 40 sections, 28 theorems, 284 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic definitions and main results
  3. On the novelties, techniques and difficulties
  4. Past literature and related work
  5. Structure of the paper
  6. Reduction to a model operator
  7. Some basic reductions
  8. Grids and tiles
  9. Trees
  10. Adapted classes
  11. Discretization of the single annulus operator
  12. Orthogonality estimates for lacunary trees
  13. Decompositions of tiles and proof of the estimate for the single annulus model operator
  14. Density and size of a collection of tiles
  15. A general decomposition algorithm
  16. ...and 25 more sections

Key Result

Theorem 1

Let $0 < \varepsilon < 1$ and $\mathscr{M} = \{ m_{\sigma} \in L^\infty( \mathbb{R}^{n-1}) : \, \sigma \in \mathrm{\bf{Gr}}(n-1,n) \}$ be a family of multipliers satisfying Eq:MihlinFamDef and $\sigma \colon \mathbb{R}^n \longrightarrow \Sigma_{1-\varepsilon}$ be a measurable function depending

Figures (4)

  • Figure 1: Graphical explanation of the action of the multiplier of the singular integral operator $T_{m}$.
  • Figure 2: The Fourier support of $g \mapsto T_{ m_{\sigma} } ( P_0 P_{\mathop{\mathrm{cn}}\nolimits,\gamma} g )$.
  • Figure 3: Example of the frequency and space component of a tile.
  • Figure 4: The vector field for $\sigma$.

Theorems & Definitions (56)

  • Theorem 1
  • Theorem 2
  • Corollary 2.1
  • Definition 2.1: Triadic grid
  • Definition 2.2: Projected grids
  • Definition 2.3: Parallelepipeds in $\mathbb{R}^n$
  • Definition 2.4: Tiles
  • Remark 2.5
  • Lemma 2.6: Geometric Lemma
  • Lemma 2.7
  • ...and 46 more