Question

Proof verification: Does $\sup_{n\in\mathbb N} \int_{\mathbb R}|F(x, u_n)|^2 dx \le M_1$ hold?

score 149 · Answer 1 · 2023-12-29 10:42:19

149

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Proof verification: Does $\sup_{n\in\mathbb N} \int_{\mathbb R}|F(x, u_n)|^2 dx \le M_1$ hold?

Published on 29 Dec 2023 - 10:42

score 78 · Answer 2 · 2026-03-26 18:40:53

78

Views

Proving that $\frac{x}{x^2+\varepsilon^2}\to \mathsf{p.v.}\left(\frac1x\right)$ in $\mathscr{S}'$

Published on 26 Mar 2026 - 18:40

score 43 · Answer 3 · 2026-03-28 10:03:54

43

Views

Pointwise convergence up to a subsequence and Sobolev spaces

Published on 28 Mar 2026 - 10:03

score 45 · Answer 4 · 2026-04-11 20:27:06

45

Views

Interchanging limits and integrals for Cauchy sequences in $L^p$

Published on 11 Apr 2026 - 20:27

score 84 · Answer 5 · 2026-04-11 22:46:33

84

Views

Limit of a Fourier Trasformation

Published on 11 Apr 2026 - 22:46

score 27 · Answer 6 · 2026-04-11 21:28:34

27

Views

A restricted product of lp spaces

Published on 11 Apr 2026 - 21:28

score 86 · Answer 7 · 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 36 · Answer 8 · 2024-01-11 14:01:32

36

Views

When will Hardy's inequality for integral become equality?

Published on 11 Jan 2024 - 14:01

score 387 · Answer 9 · 2024-01-11 20:20:46

387

Views

Check that $T: \ell^2 \to \ell^2$ is not surjective where $T(x_1, x_2, x_3, \ldots) = (x_1 + x_2, x_2 + x_3, x_3 + x_4, \ldots)$

Published on 11 Jan 2024 - 20:20

score 45 · Answer 10 · 2024-01-12 00:31:14

45

Views

Understanding that $l^p(d)=\mathbb{R}^d$ and that $l^\infty(\mathbb{N})$ is not separable

Published on 12 Jan 2024 - 0:31

score 73 · Answer 11 · 2026-03-26 11:03:11

73

Views

Let $(X, \mathcal M, \mu)$ be a finite measure space. If $f_n, f$ are $L^1$ functions, $f_n$ unif. integrable, $f_n \to f$, then $f_n \to f$ in $L^1$

Published on 26 Mar 2026 - 11:03

score 51 · Answer 12 · 2024-01-12 06:46:36

51

Views

Prove if there exists $h\in L^1(\mathbb R)$ such that $|u(x)| +|u(x)|^{p-1}\le h(x)$

Published on 12 Jan 2024 - 6:46

score 49 · Answer 13 · 2026-04-12 11:10:47

49

Views

Showing that Sobolev Space $H^m$ is in $L^\infty$

Published on 12 Apr 2026 - 11:10

score 184 · Answer 14 · 2026-03-26 11:01:26

184

Views

If $u \notin L^p[0,1]$, can we find some $w \in L^{\frac{p}{p-1}}[0,1]$ such that $\int_0^1 u \cdot w = \infty$?

Published on 26 Mar 2026 - 11:01

score 70 · Answer 15 · 2024-01-14 20:03:34

70

Views

Is the function $x \mapsto \mu(A+x)$ continuous, where $\mu$ is a finite Borel measure on $\mathbb R^n$ and $A \in \mathcal B(\mathbb R^n)$

Published on 14 Jan 2024 - 20:03

List Question

Proof verification: Does $\sup_{n\in\mathbb N} \int_{\mathbb R}|F(x, u_n)|^2 dx \le M_1$ hold?

Proving that $\frac{x}{x^2+\varepsilon^2}\to \mathsf{p.v.}\left(\frac1x\right)$ in $\mathscr{S}'$

Pointwise convergence up to a subsequence and Sobolev spaces

Interchanging limits and integrals for Cauchy sequences in $L^p$

Limit of a Fourier Trasformation

A restricted product of lp spaces

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 $A \subset [0,1]$?

When will Hardy's inequality for integral become equality?

Check that $T: \ell^2 \to \ell^2$ is not surjective where $T(x_1, x_2, x_3, \ldots) = (x_1 + x_2, x_2 + x_3, x_3 + x_4, \ldots)$

Understanding that $l^p(d)=\mathbb{R}^d$ and that $l^\infty(\mathbb{N})$ is not separable

Let $(X, \mathcal M, \mu)$ be a finite measure space. If $f_n, f$ are $L^1$ functions, $f_n$ unif. integrable, $f_n \to f$, then $f_n \to f$ in $L^1$

Prove if there exists $h\in L^1(\mathbb R)$ such that $|u(x)| +|u(x)|^{p-1}\le h(x)$

Showing that Sobolev Space $H^m$ is in $L^\infty$

If $u \notin L^p[0,1]$, can we find some $w \in L^{\frac{p}{p-1}}[0,1]$ such that $\int_0^1 u \cdot w = \infty$?

Is the function $x \mapsto \mu(A+x)$ continuous, where $\mu$ is a finite Borel measure on $\mathbb R^n$ and $A \in \mathcal B(\mathbb R^n)$

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