Question

If $μ$ is reversible with respect to a Markov kernel $κ$, is $L^2(μ)∋g↦\frac{\|(x,y)↦g(x)-g(y)\|_{L^2(μ⊗κ)}^2}{\|g\|_{L^2(μ)}^2}$ convex?

score 38 · Answer 1 · 2019-12-10 15:14:13

38

Views

If $μ$ is reversible with respect to a Markov kernel $κ$, is $L^2(μ)∋g↦\frac{\|(x,y)↦g(x)-g(y)\|_{L^2(μ⊗κ)}^2}{\|g\|_{L^2(μ)}^2}$ convex?

Published on 10 Dec 2019 - 15:14

score 92 · Answer 2 · 2026-03-29 14:59:24

92

Views

$f_n\rightharpoonup f$ in $L^P(E)$ iff $\lim_{n\to \infty}\int_A f_n d\mu= \int_A f d\mu$

Published on 29 Mar 2026 - 14:59

score 40 · Answer 3 · 2019-12-11 08:47:06

40

Views

Proving an inequality related to ratio of norms to measures

Published on 11 Dec 2019 - 8:47

score 71 · Answer 4 · 2019-12-12 04:22:26

71

Views

prove that $\lim_{p\to 0^+}\int_X |f|^p d\mu = \mu\Big(\{x\in X :f(x)\neq 0\} \Big)$

Published on 12 Dec 2019 - 4:22

score 100 · Answer 5 · 2026-03-26 16:03:45

100

Views

$L^p$ space inequality

Published on 26 Mar 2026 - 16:03

score 208 · Answer 6 · 2026-02-24 00:47:57

208

Views

Do pointwise convergence and norm boundedness imply Lp convergece?

Published on 24 Feb 2026 - 0:47

score 32 · Answer 7 · 2026-03-31 15:49:36

32

Views

Computing Fourier Transforms in $L^2$.

Published on 31 Mar 2026 - 15:49

score 48 · Answer 8 · 2026-04-03 22:07:05

48

Views

Prove that $\int_{1}^{\infty}{\frac{e^{-x}}{\sqrt{1+x^{2}}}\text{dx}}\leq \frac{\sqrt{\pi}}{2\sqrt{2}e}$

Published on 03 Apr 2026 - 22:07

score 37 · Answer 9 · 2026-03-25 19:06:37

37

Views

show that $\|f\|_p = \max_{g\in L^q(E)\\\|g\|_q\le 1} \int_E f.g~d\mu$

Published on 25 Mar 2026 - 19:06

score 15 · Answer 10 · 2026-03-06 01:40:48

15

Views

Let $\mathcal{G} \subseteq \mathcal{F}$ and show that $X \in L^{1}(P) \iff X \in L^{1}(P\vert_{\mathcal{G}})$

Published on 06 Mar 2026 - 1:40

score 59 · Answer 11 · 2026-03-25 12:51:01

59

Views

if $\{f_n\}\to f$ in $L^{p_2}(E)$ then $\{f_n\}\to f$ in $L^{p_1}(E)$.

Published on 25 Mar 2026 - 12:51

score 49 · Answer 12 · 2026-03-31 15:45:00

49

Views

How to compute the Fourier transform of this function?

Published on 31 Mar 2026 - 15:45

score 124 · Answer 13 · 2019-12-15 16:32:19

124

Views

Is the pointwise supremum of a family of $L^p$ functions also $L^p$?

Published on 15 Dec 2019 - 16:32

score 315 · Answer 14 · 2019-12-15 16:41:02

315

Views

Linear operator commutes with translations if and only if exits a Fourier multiplier. (Stein, Singular Integrals and differenciability properties)

Published on 15 Dec 2019 - 16:41

score 278 · Answer 15 · 2026-03-29 19:28:27

278

Views

if $\|f\|_p \le C$ for all $p \ge 1$, then $f \in L^\infty$.

Published on 29 Mar 2026 - 19:28

List Question

If $μ$ is reversible with respect to a Markov kernel $κ$, is $L^2(μ)∋g↦\frac{\|(x,y)↦g(x)-g(y)\|_{L^2(μ⊗κ)}^2}{\|g\|_{L^2(μ)}^2}$ convex?

$f_n\rightharpoonup f$ in $L^P(E)$ iff $\lim_{n\to \infty}\int_A f_n d\mu= \int_A f d\mu$

Proving an inequality related to ratio of norms to measures

prove that $\lim_{p\to 0^+}\int_X |f|^p d\mu = \mu\Big(\{x\in X :f(x)\neq 0\} \Big)$

$L^p$ space inequality

Do pointwise convergence and norm boundedness imply Lp convergece?

Computing Fourier Transforms in $L^2$.

Prove that $\int_{1}^{\infty}{\frac{e^{-x}}{\sqrt{1+x^{2}}}\text{dx}}\leq \frac{\sqrt{\pi}}{2\sqrt{2}e}$

show that $\|f\|_p = \max_{g\in L^q(E)\\\|g\|_q\le 1} \int_E f.g~d\mu$

Let $\mathcal{G} \subseteq \mathcal{F}$ and show that $X \in L^{1}(P) \iff X \in L^{1}(P\vert_{\mathcal{G}})$

if $\{f_n\}\to f$ in $L^{p_2}(E)$ then $\{f_n\}\to f$ in $L^{p_1}(E)$.

How to compute the Fourier transform of this function?

Is the pointwise supremum of a family of $L^p$ functions also $L^p$?

Linear operator commutes with translations if and only if exits a Fourier multiplier. (Stein, Singular Integrals and differenciability properties)

if $\|f\|_p \le C$ for all $p \ge 1$, then $f \in L^\infty$.

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