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Blowup of Sobolev norms in approximating a nowhere differentiablenon-absolutely continuous function

Let $f: [0, 1] \to \mathbb R$ be a continuous function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly.

Questions:

i) Is it true that if $f$ is nowhere differentiable, then $\|u_n\|_{W^{1, 1}} \to \infty?$

ii) Is it true that if $f$ fails to be absolutely continuous, then $\|u_n\|_{W^{1, p}} \to \infty$ for $p > 1$?

Blowup of Sobolev norms in approximating a nowhere differentiable function

Let $f: [0, 1] \to \mathbb R$ be a continuous function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly.

Questions:

i) Is it true that if $f$ is nowhere differentiable, then $\|u_n\|_{W^{1, 1}} \to \infty?$

ii) Is it true that if $f$ fails to be absolutely continuous, then $\|u_n\|_{W^{1, p}} \to \infty$ for $p > 1$?

Blowup of Sobolev norms in approximating a non-absolutely continuous function

Let $f: [0, 1] \to \mathbb R$ be a continuous function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly. Is it true that if $f$ fails to be absolutely continuous, then $\|u_n\|_{W^{1, p}} \to \infty$ for $p > 1$?

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Nate River
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Let $f: [0, 1] \to \mathbb R$ be a continuous, nowhere differentiable function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly.

Questions:

i) Is it true that if $\|u_n\|_{W^{1, p}} \to \infty?$$f$ is nowhere differentiable, then $\|u_n\|_{W^{1, 1}} \to \infty?$

ii) Is it true that if $f$ fails to be absolutely continuous, then $\|u_n\|_{W^{1, p}} \to \infty$ for $p > 1$?

Let $f: [0, 1] \to \mathbb R$ be a continuous, nowhere differentiable function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly. Is it true that $\|u_n\|_{W^{1, p}} \to \infty?$

Let $f: [0, 1] \to \mathbb R$ be a continuous function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly.

Questions:

i) Is it true that if $f$ is nowhere differentiable, then $\|u_n\|_{W^{1, 1}} \to \infty?$

ii) Is it true that if $f$ fails to be absolutely continuous, then $\|u_n\|_{W^{1, p}} \to \infty$ for $p > 1$?

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Nate River
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Question: Let Let $f: [0, 1] \to \mathbb R$ be a continuous, nowhere differentiable function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly. Is it true that $\|u_n\|_{W^{1, p}} \to \infty?$

Question: Let $f: [0, 1] \to \mathbb R$ be a continuous, nowhere differentiable function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly. Is it true that $\|u_n\|_{W^{1, p}} \to \infty?$

Let $f: [0, 1] \to \mathbb R$ be a continuous, nowhere differentiable function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly. Is it true that $\|u_n\|_{W^{1, p}} \to \infty?$

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Nate River
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Nate River
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Nate River
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