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Kernel Smoothing Operators on Thick Open Domains

Dimitrios Giannakis, Mohammad Javad Latifi Jebelli

Abstract

We define the notion of a thick open set $Ω$ in a Euclidean space and show that a local Hardy-Littlewood inequality holds in $L^p(Ω)$, $p \in (1, \infty]$. We then establish pointwise and $L^p(Ω)$ convergence for families of convolution operators with a Markov normalization on $Ω$. We demonstrate application of such smoothing operators to piecewise-continuous density, velocity, and stress fields from discrete element models of sea ice dynamics.

Kernel Smoothing Operators on Thick Open Domains

Abstract

We define the notion of a thick open set in a Euclidean space and show that a local Hardy-Littlewood inequality holds in , . We then establish pointwise and convergence for families of convolution operators with a Markov normalization on . We demonstrate application of such smoothing operators to piecewise-continuous density, velocity, and stress fields from discrete element models of sea ice dynamics.
Paper Structure (27 sections, 18 theorems, 70 equations, 5 figures)

This paper contains 27 sections, 18 theorems, 70 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Motivation: Sea Ice Dynamics
  4. Organization of the paper
  5. Kernel smoothing on thick domains
  6. Notation
  7. Radial kernels and associated integral operators
  8. Smoothing with integral operators
  9. Thick domains and Markov normalization
  10. Smoothing using Markovian kernels
  11. Convergence analysis
  12. Convolution and thick open sets
  13. Smoothing bounded measurable functions
  14. Proof of \ref{['thm:main']}
  15. Thick sets as metric measure spaces
  16. ...and 12 more sections

Key Result

Lemma 2.1

\newlabellem:conv_contin0 Set $p \in (1,\infty]$ and $q \in [1,\infty)$ with $\frac{1}{p} + \frac{1}{q} = 1$. Then, for every $h \in L^q(\mathbb R^n)$ and $f \in L^p(\mathbb R^n)$, the function $K_\epsilon f$ is continuous and bounded with $\lVert K_\epsilon f \rVert_\infty \leq \lVert h \rVert_{L

Figures (5)

  • Figure 1: Piecewise-continuous mass density function $m(x)$ associated with two ice floes (left) and the corresponding smooth mass density field obtained by smoothing $m$ by a Markovian integral operator (right).
  • Figure 1: Illustration of the center-of-mass velocity $\bm u_{t,\ell}$, rotational velocity $\omega_{t,\ell} \bm r^\perp$, and total velocity $\bm v_{t,\ell}$ at a point $x \in S_{t,\ell}$.
  • Figure 2: Schematic of a triangulation of a single polygonal domain $Y$ together with associated quadrature nodes $y_{j,N}\in Y$ (each coming with a corresponding weight $w_{j,N}$).
  • Figure 2: Nares Strait simulation. The top row shows snapshots of the ice floes from the DEM at simulation times $t_j =j \, \Delta t$ with $j= 50$ (left), 200 (center), and 350 (right). The second, third, and fourth rows from the top show snapshots of kernel-smoothed mass density $\hat{m}_t$, meridional velocity ($y$ component of $\hat{\bm v}_t$), and trace of the stress field $\mathop{\mathrm{tr}}\nolimits \hat{\bm \sigma_t}$, respectively, at the times $t_j$.
  • Figure 3: As in \ref{['fig:nares']}, but for the lateral compression experiment. The time instances $t_j = j \, \Delta t$ shown are for $j=20$ (left), 50 (center), and 92 (right).

Theorems & Definitions (40)

  • Lemma 2.1
  • Proof 1
  • Lemma 2.2
  • Proof 2
  • Theorem 2.3: Local Hardy-Littlewood maximal inequality
  • Theorem 2.4
  • Corollary 2.5
  • Proof 3
  • Definition 2.6: Thick set
  • Proposition 2.7
  • ...and 30 more