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Quasilinearization with regularizing tensor paraproducts

Oluwadamilola Fasina

TL;DR

The paper addresses nonlinear actions $A(f)$ where $f$ has mixed dyadic Hölder regularity by developing continuous and multiscale tensor paraproducts to quasilinearize $A(f)$. The core approach replaces the operator $T: f \mapsto A(f)$ with a tensor paraproduct representation $\Pi^{(t,t')}_{(A',A'')}$, and its multiscale discretization $\Pi^{(N,N')}_{(A',A'')}$, yielding $A(f) = \tilde{A}_{(N,N')}(f) + \Delta_{(N,N')}(A,f)$, where the residual $\Delta_{(N,N')}(A,f)$ has twice the regularity of $f$, i.e., $\Delta_{(N,N')}(A,f) \in \Lambda_{2\alpha}([0,1]^2)$ with $\|\Delta_{(N,N')}(A,f)\|_{\Lambda_{2\alpha}} \le C_A \|f\|_{\Lambda_{\alpha}}$. The construction relies on a telescoping expansion across tensor scales, bilinear interpolation in scale parameters, and a careful bounding of residual terms; the residual regularity is validated via a key lemma linking mixed Hölder regularity to decay of tensor-wavelet coefficients. A computational example demonstrates the practical utility by illustrating how the multiscale paraproduct isolates singular structure while smoothing the nonlinear action. Overall, the framework provides a principled way to separate singular and smooth components in multiscale data and PDE-like settings, with provable regularity gains for the residual.

Abstract

We extend Bony's celebrated work on paraproducts to continous and multiscale \emph{tensor} paraproducts. For $A \in \mathcal{C}^2(\mathbb{R})$ and $f \in Λ_α([0,1]^2, d_d(x,y)^α \times d'_d(x',y')^α)$, we construct an approximation, $\tilde{A}_{(N,N')}(f)$ to $A(f)$, replacing the operator $T: f \to A(f)$ with the continous tensor paraproduct, $Π^{(t,t')}_{(A',A'')}$, and the multiscale tensor paraproduct $Π^{(N,N')}_{(A',A'')}:f \to \tilde{A}_{(N,N')}(f) + Δ_{ (N,N')}(A,f)$. In the multiscale case, we provide estimates on the residual, $Δ_{(N,N')}(A,f)$, and show it has twice the regularity of $f$ such that $Δ_{(N,N')}(A,f) \in Λ_{2 α}([0,1]^2)$ and $\lVert Δ_{(N,N')}(A,f) \rVert_{Λ_{2α}([0,1]^2)} \leq C_A \lVert f \rVert_{Λ_α([0,1]^2)} $. Our theoretical findings are supplemented with a computational example.

Quasilinearization with regularizing tensor paraproducts

TL;DR

The paper addresses nonlinear actions where has mixed dyadic Hölder regularity by developing continuous and multiscale tensor paraproducts to quasilinearize . The core approach replaces the operator with a tensor paraproduct representation , and its multiscale discretization , yielding , where the residual has twice the regularity of , i.e., with . The construction relies on a telescoping expansion across tensor scales, bilinear interpolation in scale parameters, and a careful bounding of residual terms; the residual regularity is validated via a key lemma linking mixed Hölder regularity to decay of tensor-wavelet coefficients. A computational example demonstrates the practical utility by illustrating how the multiscale paraproduct isolates singular structure while smoothing the nonlinear action. Overall, the framework provides a principled way to separate singular and smooth components in multiscale data and PDE-like settings, with provable regularity gains for the residual.

Abstract

We extend Bony's celebrated work on paraproducts to continous and multiscale \emph{tensor} paraproducts. For and , we construct an approximation, to , replacing the operator with the continous tensor paraproduct, , and the multiscale tensor paraproduct . In the multiscale case, we provide estimates on the residual, , and show it has twice the regularity of such that and . Our theoretical findings are supplemented with a computational example.

Paper Structure

This paper contains 6 sections, 12 theorems, 62 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries: Key Definitions and Lemmas
  4. Tensor Paraproducts
  5. Residual Estimates
  6. Computational Example

Key Result

Theorem 1.1

Suppose $A \in \mathcal{C}^2(\mathbb{R})$, $f \in \Lambda_\alpha([0,1]^2), 0 < \alpha < \frac{1}{2}$, then for the operator $T: f \to A(f)$ we can approximate $A(f)$ with such that the multiscale tensor paraproduct transforms $T : f \to A(f)$ to where $\Delta_{(N,N')}(A,f) = A(f) - \tilde{A}_{(N,N')}(f) \in \Lambda_{2\alpha}([0,1]^2)$ is the residual which has twice the regularity of $f$ and

Figures (1)

  • Figure 1: Visualization of the paraproduct decomposition of $A(f) = e^{- 0.2 f(z) }$ for fixed scales $N,N' = 4$ (third row) and $N,N' = 6$ (last row) for $\alpha = 4 \times 10^{-1} , 4 \times 10^{-2}, 4 \times 10^{-3}$. Each column is a different $\alpha$ level and row 1 is the original function, $f$, row 2 is the smooth nonlinear composition, $A(f)$, and rows 3 and 4 are the approximations, $\tilde{A}_{(N=4,N'=4)}(f)$ and $\tilde{A}_{(N=6,N'=6)}(f)$, respectively. $X$ and $iY$ are the labels for the real and complex axes, respectively.

Theorems & Definitions (39)

  • Theorem 1.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Definition 3.5
  • Definition 3.6
  • Remark 3.7
  • Definition 3.8
  • Definition 3.9
  • ...and 29 more