Question

Can we find $f: [0,1] \to [0, \infty) \notin L^{\frac{q}{q-1}}[0,1]$ such that $\int_A f \le |A|^{1/q}$ for some $q \ge 2$ and all $A \subset [0,1]$?

score 85 · Answer 1 · 2024-01-08 03:55:46

85

Views

Can we find $f: [0,1] \to [0, \infty) \notin L^{\frac{q}{q-1}}[0,1]$ such that $\int_A f \le |A|^{1/q}$ for some $q \ge 2$ and all $A \subset [0,1]$?

Published on 08 Jan 2024 - 3:55

score 38 · Answer 2 · 2024-01-11 22:56:14

38

Views

Proving an Integral Identity for the Laplacian in $W^{2,2}(\Omega)$ for Functions in $C_c^{\infty}(\Omega)$

Published on 11 Jan 2024 - 22:56

score 63 · Answer 3 · 2026-03-19 14:52:57

63

Views

A proof of Pólya-Szegő inequality

Published on 19 Mar 2026 - 14:52

score 147 · Answer 4 · 2015-02-16 07:30:53

147

Views

A proposition to prove the real interpolation of positive exponent

Published on 16 Feb 2015 - 7:30

score 230 · Answer 5 · 2015-03-16 23:47:44

230

Views

Condition on $f$ in $L^{p, \infty} $ implies $f \in L^q$

Published on 16 Mar 2015 - 23:47

score 367 · Answer 6 · 2015-04-24 03:46:45

367

Views

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Published on 24 Apr 2015 - 3:46

score 1.1k · Answer 7 · 2015-08-28 01:34:36

1.1k

Views

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Published on 28 Aug 2015 - 1:34

score 214 · Answer 8 · 2015-12-02 16:03:15

214

Views

Why $L^p$ is strictly contained in $L^{p,\infty}$

Published on 02 Dec 2015 - 16:03

score 414 · Answer 9 · 2026-03-04 19:22:46

414

Views

Inclusion of $L^p$ and weak $L^p$ spaces

Published on 04 Mar 2026 - 19:22

score 92 · Answer 10 · 2016-03-08 13:49:02

92

Views

Problem with $L^p(\Bbb R^n)$ spaces (Marcinkiewicz Theorem)

Published on 08 Mar 2016 - 13:49

score 481 · Answer 11 · 2026-03-11 08:45:32

481

Views

hint on exercise about weak $L^p$ space

Published on 11 Mar 2026 - 8:45

score 98 · Answer 12 · 2016-04-03 19:36:28

98

Views

If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$

Published on 03 Apr 2016 - 19:36

score 85 · Answer 13 · 2016-04-04 06:18:39

85

Views

If $||g_j-g||_{1,\infty}\to0$, then $||g_j||_{1,\infty}\to||g||_{1,\infty}$

Published on 04 Apr 2016 - 6:18

score 588 · Answer 14 · 2016-04-19 18:09:51

588

Views

Triangle inequality in weak $L^1$ space

Published on 19 Apr 2016 - 18:09

score 1.4k · Answer 15 · 2026-03-15 03:42:44

1.4k

Views

Triangle inequality fails in $L^{1,\infty}$

Published on 15 Mar 2026 - 3:42

List Question

Can we find $f: [0,1] \to [0, \infty) \notin L^{\frac{q}{q-1}}[0,1]$ such that $\int_A f \le |A|^{1/q}$ for some $q \ge 2$ and all $A \subset [0,1]$?

Proving an Integral Identity for the Laplacian in $W^{2,2}(\Omega)$ for Functions in $C_c^{\infty}(\Omega)$

A proof of Pólya-Szegő inequality

A proposition to prove the real interpolation of positive exponent

Condition on $f$ in $L^{p, \infty} $ implies $f \in L^q$

Proof of Hunt's Interpolation

Equivalent Definition of Weak $L^{p}$ (Quasi-) Norm

Why $L^p$ is strictly contained in $L^{p,\infty}$

Inclusion of $L^p$ and weak $L^p$ spaces

Problem with $L^p(\Bbb R^n)$ spaces (Marcinkiewicz Theorem)

hint on exercise about weak $L^p$ space

If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$

If $||g_j-g||_{1,\infty}\to0$, then $||g_j||_{1,\infty}\to||g||_{1,\infty}$

Triangle inequality in weak $L^1$ space

Triangle inequality fails in $L^{1,\infty}$

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