Quasilinearization with regularizing tensor paraproducts
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915695658270720 |
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| author | Fasina, Oluwadamilola |
| author_facet | Fasina, Oluwadamilola |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_12629 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasilinearization with regularizing tensor paraproducts Fasina, Oluwadamilola Analysis of PDEs Discrete Mathematics 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. |
| title | Quasilinearization with regularizing tensor paraproducts |
| topic | Analysis of PDEs Discrete Mathematics |
| url | https://arxiv.org/abs/2503.12629 |