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 86 · Answer 1 · 2026-03-25 17:44:36

86

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 25 Mar 2026 - 17:44

score 39 · Answer 2 · 2026-03-25 19:09:58

39

Views

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

Published on 25 Mar 2026 - 19:09

score 69 · Answer 3 · 2026-05-04 20:33:14

69

Views

A proof of Pólya-Szegő inequality

Published on 04 May 2026 - 20:33

score 148 · Answer 4 · 2026-03-25 19:05:33

148

Views

A proposition to prove the real interpolation of positive exponent

Published on 25 Mar 2026 - 19:05

score 231 · Answer 5 · 2026-03-25 17:40:34

231

Views

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

Published on 25 Mar 2026 - 17:40

score 369 · Answer 6 · 2026-04-01 14:32:48

369

Views

Proof of Hunt's Interpolation

Published on 01 Apr 2026 - 14:32

score 1.1k · Answer 7 · 2026-03-25 17:40:38

1.1k

Views

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

Published on 25 Mar 2026 - 17:40

score 215 · Answer 8 · 2026-04-20 14:13:35

215

Views

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

Published on 20 Apr 2026 - 14:13

score 415 · Answer 9 · 2026-04-06 16:45:57

415

Views

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

Published on 06 Apr 2026 - 16:45

score 93 · Answer 10 · 2026-04-22 22:40:50

93

Views

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

Published on 22 Apr 2026 - 22:40

score 482 · Answer 11 · 2026-05-10 08:35:02

482

Views

hint on exercise about weak $L^p$ space

Published on 10 May 2026 - 8:35

score 99 · Answer 12 · 2026-04-07 00:25:14

99

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 07 Apr 2026 - 0:25

score 86 · Answer 13 · 2026-04-07 00:25:14

86

Views

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

Published on 07 Apr 2026 - 0:25

score 590 · Answer 14 · 2026-04-01 18:51:07

590

Views

Triangle inequality in weak $L^1$ space

Published on 01 Apr 2026 - 18:51

score 1.4k · Answer 15 · 2026-04-07 01:54:35

1.4k

Views

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

Published on 07 Apr 2026 - 1:54

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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