Question

How to show that $f \in L^p[0,1]$ for all $1 \leq p <2 $ and $\lVert f \rVert_p$ uniformly bounded implies $f \in L^2[0,1]$?

score 49 · Answer 1 · 2026-03-29 12:39:18

49

Views

How to show that $f \in L^p[0,1]$ for all $1 \leq p <2 $ and $\lVert f \rVert_p$ uniformly bounded implies $f \in L^2[0,1]$?

Published on 29 Mar 2026 - 12:39

score 27 · Answer 2 · 2026-03-28 13:28:05

27

Views

Prove monotonicity of Lp-norm optima in 1D

Published on 28 Mar 2026 - 13:28

score 52 · Answer 3 · 2023-07-30 02:27:16

52

Views

A sequence $f_k$ defined on the unit ball converging in measure to zero and satisfying $\|f_k\|_{L^2(B(0,1))}\le M$ for all $k\ge 1$.

Published on 30 Jul 2023 - 2:27

score 81 · Answer 4 · 2023-07-30 20:17:25

81

Views

Bound on $L_p$ norm by the $L_1$norm

Published on 30 Jul 2023 - 20:17

score 135 · Answer 5 · 2023-07-30 23:30:56

135

Views

If $f_n$ is a sequence weakly convergent to $f$ in $L^1[0,1]$, is the convolution $f_ng$ weakly convergent to $fg$?

Published on 30 Jul 2023 - 23:30

score 51 · Answer 6 · 2026-04-01 16:23:55

51

Views

Brezis' exercise 6.23.7: prove that $\|T^n\| \le \frac{1}{n!}$

Published on 01 Apr 2026 - 16:23

score 96 · Answer 7 · 2026-04-09 00:51:07

96

Views

How does $f$ being uniformly continuous imply $f_\epsilon \rightarrow f$ uniformly?

Published on 09 Apr 2026 - 0:51

score 86 · Answer 8 · 2026-04-07 00:24:45

86

Views

Measure of $\Omega_k = \{x\in \Omega\, : \, \operatorname{dist}(x,\partial \Omega)>\frac{1}{k}\}$, $\Omega$ is a bounded domain with smooth boundary

Published on 07 Apr 2026 - 0:24

score 54 · Answer 9 · 2026-04-10 18:18:33

54

Views

Is it true that an open ball $U\subseteq L^p(X, \sum, \mu)$ centered at $0$ contains an indicator function of a positive measure?

Published on 10 Apr 2026 - 18:18

score 36 · Answer 10 · 2023-08-06 01:24:32

36

Views

Is there a bound on $\frac{1}{\lvert Q \rvert } \int_Q \lvert f(x) \rvert dx$ for periodic $L^1$ functions and hypercubes $Q$?

Published on 06 Aug 2023 - 1:24

score 46 · Answer 11 · 2026-04-10 23:05:52

46

Views

A question on maximal operators on $\mathbb{R}^2$

Published on 10 Apr 2026 - 23:05

score 56 · Answer 12 · 2023-08-08 04:17:47

56

Views

Finiteness of the functional $\int_0^1 \frac{\phi(x)}{\sqrt{x}}dx$ in $L^2$

Published on 08 Aug 2023 - 4:17

score 60 · Answer 13 · 2026-04-06 11:13:51

60

Views

Convergence in $L^p$ preserves density

Published on 06 Apr 2026 - 11:13

score 75 · Answer 14 · 2026-03-27 14:52:52

75

Views

The space $L^p_{\text{loc}} (\mathbb R^d)$ is complete w.r.t. the norm $\|f\|_{\tilde L^p} := \sup_{x \in \mathbb R^d} \|1_{B(x, 1)} f\|_{L^p}$

Published on 27 Mar 2026 - 14:52

score 53 · Answer 15 · 2023-08-13 19:01:00

53

Views

Must the domain of a Hilbert space operator be limited to square-summable elements? If so, what type of space does not restrict, thus?

Published on 13 Aug 2023 - 19:01

List Question

How to show that $f \in L^p[0,1]$ for all $1 \leq p <2 $ and $\lVert f \rVert_p$ uniformly bounded implies $f \in L^2[0,1]$?

Prove monotonicity of Lp-norm optima in 1D

A sequence $f_k$ defined on the unit ball converging in measure to zero and satisfying $\|f_k\|_{L^2(B(0,1))}\le M$ for all $k\ge 1$.

Bound on $L_p$ norm by the $L_1$norm

If $f_n$ is a sequence weakly convergent to $f$ in $L^1[0,1]$, is the convolution $f_ng$ weakly convergent to $fg$?

Brezis' exercise 6.23.7: prove that $\|T^n\| \le \frac{1}{n!}$

How does $f$ being uniformly continuous imply $f_\epsilon \rightarrow f$ uniformly?

Measure of $\Omega_k = \{x\in \Omega\, : \, \operatorname{dist}(x,\partial \Omega)>\frac{1}{k}\}$, $\Omega$ is a bounded domain with smooth boundary

Is it true that an open ball $U\subseteq L^p(X, \sum, \mu)$ centered at $0$ contains an indicator function of a positive measure?

Is there a bound on $\frac{1}{\lvert Q \rvert } \int_Q \lvert f(x) \rvert dx$ for periodic $L^1$ functions and hypercubes $Q$?

A question on maximal operators on $\mathbb{R}^2$

Finiteness of the functional $\int_0^1 \frac{\phi(x)}{\sqrt{x}}dx$ in $L^2$

Convergence in $L^p$ preserves density

The space $L^p_{\text{loc}} (\mathbb R^d)$ is complete w.r.t. the norm $\|f\|_{\tilde L^p} := \sup_{x \in \mathbb R^d} \|1_{B(x, 1)} f\|_{L^p}$

Must the domain of a Hilbert space operator be limited to square-summable elements? If so, what type of space does not restrict, thus?

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