Question

Prove that $f\in L^p[0,1]$ for all $p\in[1,2)$:

score 72 · Answer 1 · 2014-07-07 20:40:22

72

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Prove that $f\in L^p[0,1]$ for all $p\in[1,2)$:

Published on 07 Jul 2014 - 20:40

score 1.7k · Answer 2 · 2026-03-29 12:38:43

1.7k

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Monotonically approximate $L^p$ function by step function

Published on 29 Mar 2026 - 12:38

score 71 · Answer 3 · 2014-07-09 11:35:42

71

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If $u_n^p \rightharpoonup v$ in $L^1$, then does it follow that $u_n \rightharpoonup v^{\frac 1p}$ in $L^p$?

Published on 09 Jul 2014 - 11:35

score 507 · Answer 4 · 2026-03-30 06:48:35

507

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Proving that weak limit in $L^p$ and strong limit in $H^{-1}$ are the same

Published on 30 Mar 2026 - 6:48

score 310 · Answer 5 · 2014-07-09 20:16:55

310

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An application of Holder's inequality to show one norm is smaller than another

Published on 09 Jul 2014 - 20:16

score 195 · Answer 6 · 2026-03-29 15:16:55

195

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Preliminaries of the Martingale Representation Theorem

Published on 29 Mar 2026 - 15:16

score 581 · Answer 7 · 2014-07-11 08:17:03

581

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p-norm of a function

Published on 11 Jul 2014 - 8:17

score 158 · Answer 8 · 2014-07-13 06:21:35

158

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$\|g\|_{L^{1}(\mathbb R)}=\sup \{ {|\int_{\mathbb R} fg|: f\in C_{c}^{\infty}(\mathbb R), \|f\|_{L^{\infty}(\mathbb R)}=1\}} ?$

Published on 13 Jul 2014 - 6:21

score 254 · Answer 9 · 2026-04-01 03:45:13

254

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Help with closed graph theorem

Published on 01 Apr 2026 - 3:45

score 88 · Answer 10 · 2014-07-13 18:23:27

88

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When does the convergence of the regularization of a function is decreasing?

Published on 13 Jul 2014 - 18:23

score 98 · Answer 11 · 2014-07-14 08:15:49

98

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Formula for $L^{q}$ norm using $C_{c}^{\infty}$ functions

Published on 14 Jul 2014 - 8:15

score 219 · Answer 12 · 2026-03-26 08:02:18

219

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How to obtain the inequality $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p} $ from Jensen's inequality?

Published on 26 Mar 2026 - 8:02

score 432 · Answer 13 · 2026-04-01 00:26:52

432

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$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

Published on 01 Apr 2026 - 0:26

score 234 · Answer 14 · 2026-03-30 06:46:40

234

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Pointwise (in time) convergence in $H^{-1}$ implies pointwise weak convergence in $L^q$, why?

Published on 30 Mar 2026 - 6:46

score 2k · Answer 15 · 2026-03-30 06:46:38

2k

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Does Strong Convergence in $L^1$ Imply Weak Convergence in $L^2$?

Published on 30 Mar 2026 - 6:46

List Question

Prove that $f\in L^p[0,1]$ for all $p\in[1,2)$:

Monotonically approximate $L^p$ function by step function

If $u_n^p \rightharpoonup v$ in $L^1$, then does it follow that $u_n \rightharpoonup v^{\frac 1p}$ in $L^p$?

Proving that weak limit in $L^p$ and strong limit in $H^{-1}$ are the same

An application of Holder's inequality to show one norm is smaller than another

Preliminaries of the Martingale Representation Theorem

p-norm of a function

$\|g\|_{L^{1}(\mathbb R)}=\sup \{ {|\int_{\mathbb R} fg|: f\in C_{c}^{\infty}(\mathbb R), \|f\|_{L^{\infty}(\mathbb R)}=1\}} ?$

Help with closed graph theorem

When does the convergence of the regularization of a function is decreasing?

Formula for $L^{q}$ norm using $C_{c}^{\infty}$ functions

How to obtain the inequality $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p} $ from Jensen's inequality?

$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

Pointwise (in time) convergence in $H^{-1}$ implies pointwise weak convergence in $L^q$, why?

Does Strong Convergence in $L^1$ Imply Weak Convergence in $L^2$?

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