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Avoid linear outer functions!
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F. H.
F. H.
  • 63
  • 5

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a Lipschitz nonlinearity with $f(0) = 0$ and suppose $u \in \textrm{H}^s(\mathbb{R}) \cap \textrm{L}^\infty(\mathbb{R})$ for some $s \in [0, \tfrac{1}{2}]$. Given some threshold frequency $\xi_0$, define the operators $$ u_{\textrm{low}} := \mathscr{F}^{-1}(1_{\{ |\xi| < \xi_0 \}} \widehat{u}) \qquad \text{and} \qquad u_{\textrm{high}} := \mathscr{F}^{-1}(1_{\{ |\xi| \geq \xi_0 \}} \widehat{u}). $$ Do any of the estimates $$ \| f(u)_{\textrm{low}} \|_\infty \lesssim \| v \|_\infty \qquad \text{or} \qquad \| f(u)_{\textrm{high}} \|_\infty \lesssim \| v \|_\infty $$ hold, where $v$ is one of $u_\mathrm{low}$, $u_\mathrm{high}$ or $u$?

Let $f \colon \mathbb{R} \to \mathbb{R}$ be Lipschitz and suppose $u \in \textrm{H}^s(\mathbb{R}) \cap \textrm{L}^\infty(\mathbb{R})$ for some $s \in [0, \tfrac{1}{2}]$. Given some threshold frequency $\xi_0$, define the operators $$ u_{\textrm{low}} := \mathscr{F}^{-1}(1_{\{ |\xi| < \xi_0 \}} \widehat{u}) \qquad \text{and} \qquad u_{\textrm{high}} := \mathscr{F}^{-1}(1_{\{ |\xi| \geq \xi_0 \}} \widehat{u}). $$ Do any of the estimates $$ \| f(u)_{\textrm{low}} \|_\infty \lesssim \| v \|_\infty \qquad \text{or} \qquad \| f(u)_{\textrm{high}} \|_\infty \lesssim \| v \|_\infty $$ hold, where $v$ is one of $u_\mathrm{low}$, $u_\mathrm{high}$ or $u$?

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a Lipschitz nonlinearity with $f(0) = 0$ and suppose $u \in \textrm{H}^s(\mathbb{R}) \cap \textrm{L}^\infty(\mathbb{R})$ for some $s \in [0, \tfrac{1}{2}]$. Given some threshold frequency $\xi_0$, define the operators $$ u_{\textrm{low}} := \mathscr{F}^{-1}(1_{\{ |\xi| < \xi_0 \}} \widehat{u}) \qquad \text{and} \qquad u_{\textrm{high}} := \mathscr{F}^{-1}(1_{\{ |\xi| \geq \xi_0 \}} \widehat{u}). $$ Do any of the estimates $$ \| f(u)_{\textrm{low}} \|_\infty \lesssim \| v \|_\infty \qquad \text{or} \qquad \| f(u)_{\textrm{high}} \|_\infty \lesssim \| v \|_\infty $$ hold, where $v$ is one of $u_\mathrm{low}$, $u_\mathrm{high}$ or $u$?

Unclear question
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F. H.
F. H.
  • 63
  • 5

Let $f \colon \mathbb{R} \to \mathbb{R}$ be Lipschitz and suppose $u \in \textrm{H}^s(\mathbb{R}) \cap \textrm{L}^\infty(\mathbb{R})$ for some $s \in [0, \tfrac{1}{2}]$. Given some threshold frequency $\xi_0$, define the operators $$ u_{\textrm{low}} := \mathscr{F}^{-1}(1_{\{ |\xi| < \xi_0 \}} \widehat{u}) \qquad \text{and} \qquad u_{\textrm{high}} := \mathscr{F}^{-1}(1_{\{ |\xi| \geq \xi_0 \}} \widehat{u}). $$ Do any of the estimates $$ \| f(u)_{\textrm{low}} \|_\infty \lesssim \| v \|_\infty \qquad \text{or} \qquad \| f(u)_{\textrm{high}} \|_\infty \lesssim \| v \|_\infty $$ hold, where $v$ is eitherone of $u_\mathrm{low}$, $u_\mathrm{high}$ or $u$?

Let $f \colon \mathbb{R} \to \mathbb{R}$ be Lipschitz and suppose $u \in \textrm{H}^s(\mathbb{R}) \cap \textrm{L}^\infty(\mathbb{R})$ for some $s \in [0, \tfrac{1}{2}]$. Given some threshold frequency $\xi_0$, define the operators $$ u_{\textrm{low}} := \mathscr{F}^{-1}(1_{\{ |\xi| < \xi_0 \}} \widehat{u}) \qquad \text{and} \qquad u_{\textrm{high}} := \mathscr{F}^{-1}(1_{\{ |\xi| \geq \xi_0 \}} \widehat{u}). $$ Do any of the estimates $$ \| f(u)_{\textrm{low}} \|_\infty \lesssim \| v \|_\infty \qquad \text{or} \qquad \| f(u)_{\textrm{high}} \|_\infty \lesssim \| v \|_\infty $$ hold, where $v$ is either $u_\mathrm{low}$, $u_\mathrm{high}$ or $u$?

Let $f \colon \mathbb{R} \to \mathbb{R}$ be Lipschitz and suppose $u \in \textrm{H}^s(\mathbb{R}) \cap \textrm{L}^\infty(\mathbb{R})$ for some $s \in [0, \tfrac{1}{2}]$. Given some threshold frequency $\xi_0$, define the operators $$ u_{\textrm{low}} := \mathscr{F}^{-1}(1_{\{ |\xi| < \xi_0 \}} \widehat{u}) \qquad \text{and} \qquad u_{\textrm{high}} := \mathscr{F}^{-1}(1_{\{ |\xi| \geq \xi_0 \}} \widehat{u}). $$ Do any of the estimates $$ \| f(u)_{\textrm{low}} \|_\infty \lesssim \| v \|_\infty \qquad \text{or} \qquad \| f(u)_{\textrm{high}} \|_\infty \lesssim \| v \|_\infty $$ hold, where $v$ is one of $u_\mathrm{low}$, $u_\mathrm{high}$ or $u$?

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F. H.
F. H.
  • 63
  • 5

Low/high-frequency estimates in $\mathrm{L}^\infty$ for Lipschitz nonlinearities

Let $f \colon \mathbb{R} \to \mathbb{R}$ be Lipschitz and suppose $u \in \textrm{H}^s(\mathbb{R}) \cap \textrm{L}^\infty(\mathbb{R})$ for some $s \in [0, \tfrac{1}{2}]$. Given some threshold frequency $\xi_0$, define the operators $$ u_{\textrm{low}} := \mathscr{F}^{-1}(1_{\{ |\xi| < \xi_0 \}} \widehat{u}) \qquad \text{and} \qquad u_{\textrm{high}} := \mathscr{F}^{-1}(1_{\{ |\xi| \geq \xi_0 \}} \widehat{u}). $$ Do any of the estimates $$ \| f(u)_{\textrm{low}} \|_\infty \lesssim \| v \|_\infty \qquad \text{or} \qquad \| f(u)_{\textrm{high}} \|_\infty \lesssim \| v \|_\infty $$ hold, where $v$ is either $u_\mathrm{low}$, $u_\mathrm{high}$ or $u$?