For questions about $L^p$ spaces, that is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence class of measurable functions with $p$-th power of the absolute value integrable. Question can be about properties of elements of these spaces, or when the ambient space on a ...

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5 views

Density of $L^\infty(\Omega)h$ in $L^p(\Omega)$ where $h \in L^p(\Omega)$

Let $(\Omega,\mu)$ be a finite measure space. Suppose $1\leq p <\infty$. Let $h$ be an element of $L^p(\Omega)$ with $h >0$ a.e.. How show that the subspace $L^\infty(\Omega)h=\{ f h\ :\ f\in ...
3
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1answer
17 views

Showing that the map that takes $u_0$ to solution $u(t)$ is self-adjoint

Let $u$ and $v$ be the solution of the heat equation $$w'(t) - \Delta w(t) =0$$ with initial data $u_0$ and $v_0$ respectively, and with either homogeneous Dirichlet or Neumann BCs on a bounded domain ...
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1answer
6 views

What is the parametrization of the set of points in $\mathbb{R}^2$ with $L^p$-(semi)norm $1$ for any $p$?

I'm looking for a curve $t_p: [0,L] \rightarrow \mathbb{R}^2$ that describes the set $T_p = \{ (x,y) \in \mathbb{R}^2 : |x|^p + |y|^p = 1\}.$
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0answers
27 views

Does this inequality imply a uniform $L^\infty$ bound?

Suppose I have the estimate for $t > 0$ $$\lVert u(t) \rVert_{L^\infty(\Omega)} \leq Ct^{-1}\lVert u_0 \rVert_{L^1(\Omega)}$$ for the solution $u$ of a parabolic PDE with initial data $u_0$ on a ...
1
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1answer
25 views

Example of a non square-integrable martingale?

Are there (simple) examples of martingales which aren't square integrable?
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0answers
19 views

Does every $\mathcal{L}_{2}$ signal have bounded $\mathcal{L}_{2}$ derivative?

Let a real signal $f(t) \in \mathcal{L}_{2}$. Does it always imply that $\dot{f}(t) \in \mathcal{L}_{2}$? It is assumed that $\dot{f}(t)$ exists for all $t \in \mathbb{R}^{+}$.
3
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1answer
41 views

Prove a vector in $\ell^2(\mathbb{Z})$ is zero

Suupose we take a vector $\vec{c}\in\ell^2(\mathbb{Z})$ where $$c(i)=\sum_{k=1}^\infty\frac{c(-k+i)+c(k+i)}{k+1}$$ That is, every elements of the vector is a series with the other terms in $\vec{c}$. ...
1
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1answer
34 views

$L^2$ convergence by the sequence of domain

Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Assume $u\in L^\infty(\Omega)$. We know a sequence $u_n\in L^\infty(\Omega)$ such that $$ \sup_{n}\|u_n\|_{L^\infty}<+\infty $$ ...
1
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1answer
31 views

Proof $\mathcal{C}^1(-1,1)$ is not a closed subspace of Sobolev space $H_0^1 \left[-1,1\right]$

Give a sequence of functions $\varphi_n\in \mathcal{C}^\infty(-1,1)$, Cauchy with respect to the Sobolev space $H^1_0$ norm $$|| \varphi||_1=\sqrt{\int_{-1}^1 (\varphi')^2+\int_{-1}^1 \varphi^2}$$ ...
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1answer
140 views

Convergence of Riemann sums of a periodic function

Short version for people who don't like reading: Let $f\colon\mathbb{R}\to\mathbb{R}$ be $1$-periodic, measurable and bounded. Is it true that, for almost all $x$, the average of $f(x)$, ...
2
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0answers
26 views

Closed subset for $L^2$ strong and weak convergence

I was trying to solve the following exercise. Let $K$ a closed subset of $\mathbb{R}$. $$X:=\{f\in L^2[0,1]:f(x)\in K \:a.e.\:x\in [0,1] \}$$ Then: 1)X is closed under strong convergence in $L^2$. ...
1
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1answer
20 views

Inequality for $u \in L^r(\Omega)$: $\int_{\Omega} (a_1+a_2|u|^{r/s})^s dx \leq a_3 \int_{\Omega} (1+|u|^r)dx$

This question is from one of the steps of the Proof of Proposition B.1 in Appendix B of P. H. Rabinowitz's "Minimax Methods in Critical Point Theory with Applications to Differential Equations." Let ...
0
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1answer
42 views

If $\sum_{n=1}^{\infty}x_n^2<\infty$ and $\sum_{m=1}^{\infty}x_n^2<\infty$, is $\sum_{k=1}^{\infty}(x_n)_k^2(x_m)_k^2<\infty$? [duplicate]

Let $$l^2=\left\{(x_n):\sum_{n=1}^{\infty}x_n^2<\infty\right\}$$ equipped with the norm $$\|(x_n)\|=\left(\sum_{n=1}^{\infty}x_n^2\right)^{1/2}.$$ Prove that $l^2$ is complete with ...
0
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1answer
44 views

Suppose $X_n \to_{p} X$, if $\limsup_n E|X_n|^r \leq E|X|^r$, how can I show that $X_n \to_r X$?

If I have that $X_n \to_p X$ (convergence in probability), and if $\limsup_n E|X_n|^r \leq E|X|^r$ for all $r \geq 1$, how can I show that $X_n \to_r X$ (this means $L^{r}$ convergence)? My goal is to ...
0
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0answers
20 views

$f_k\rightarrow g$ in $L^p(\mathbb{R}^n)$ and $f_k\rightarrow h$ in $L^r(\mathbb{R}^n)$, then $g=h$ $a.e.$ in $\mathbb{R}^n$

If $f_k\in L^p(\mathbb{R}^n)\cap L^r(\mathbb{R}^n) $ for some $p,r\in [1,\infty), f_k\rightarrow g$ in $L^p(\mathbb{R}^n)$ and $f_k\rightarrow h$ in $L^r(\mathbb{R}^n)$, then $g=h$ $a.e.$ in ...
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0answers
22 views

Integral inequality $L^p$ spaces

I'm trying to solve this problem: Let $1<p<\infty$. Then let $f:(0,\infty)\to [0,\infty]$ a measurable non negative function. It's true the following inequality: $$\int_0^\infty ( ...
1
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3answers
34 views

Justify or provide counterexamples [closed]

Justify or provide counterexamples $(i)$ If $f \in L^1[0,\infty)$ and $f\geq0$ then $\lim\limits_{x\rightarrow\infty}{f(x)}=0$ $(ii)$ If $\lim\limits_{x\rightarrow\infty} f(x)=0$ and $f\geq0$ on ...
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1answer
31 views

Showing that $(G' \circ u)u' \in L^p(I)$ where $G \in C^1(\mathbb{R})$ and $u \in W^{1,p}(I)$

I am trying to prove the rule of differentiation of a composition for weak derivatives in Sobolev spaces following the proof given in Corolary 8.11 in Functional Analysis, Sobolev Spaces and Partial ...
0
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1answer
24 views

Some insight about this integral limit

Given $u:\mathbb{R}^N \rightarrow \mathbb{R}$ is continuous and has compact support, we define the set $$K_u: = \{x\in \mathbb{R}^N : u(x) = \|u\|_\infty\}.$$ Looking at the following limit ...
1
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0answers
40 views

banach space bigger than $L^p$

we know that $L^p$ is banach space for any $p\geq 1$. My question: Is there any other banach space that is bigger than $L^p$?. In fact, I have an exercice that I don't have any idea: prove that ...
0
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1answer
33 views

Maple plot the $l^p$ balls: $\{(x,y) \mid |x|^p + |y|^p \le 1\}$ for $p = 1,…,10$

I have been asked to construct this plot in maple for an analysis assignment. I have been given no other instructions on how to do this. I am not familiar at all with defining or plotting these kinds ...
1
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2answers
39 views

Measure Theory and $L^{p}$ spaces

I have the two following very simple questions regarding measure theory that I want to show: If $f \in L^{p}(X, \mathcal{M}, \mu)$ for $1 \leq p < \infty$, then $f < \infty$ $\mu$-almost ...
2
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0answers
22 views

Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$, $\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$ where the scalar complex function ...
1
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1answer
31 views

Let $f \in L^{1}([0,1])$ be a real valued function. Prove the following

Let $f \in L^{1}([0,1])$ be a real valued function. Prove the following $1) x^k f(x) \in L^1([0,1])$ for all $k\in \mathbb{N}$ $2) \lim_{k\rightarrow\infty}\int_{0}^{1}x^k f(x) dx = 0$ $3)$ If ...
1
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1answer
22 views

Prove $F$ is in $L^1$

This is an old qualifier exam question at my school Let $f \in L^{1}([0,\infty))$ and for $x\geq 0$, define $F(x) = \int_{(x,\infty)} f(t) e^{x-t} dm(t) $ Show that $F \in L^{1}([0,\infty))$ The ...
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1answer
37 views

can you prove this theorem An introduction to wavelet Analysis? [closed]

Definition. The sequence $\{f_n(x)\}$, $n\in \mathbb{N}$ defined on an interval $I$ converges in mean-square to the function $f(x)$ on $I$ if $\lim_{n\to \infty} \int_I {|f_n(x)-f(x)|}^2\, dx =0$. We ...
1
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1answer
41 views

$\left\|f\right\|_{L^1(μ_1)}<∞$ $μ_2$-a.e.,$\left\|f\right\|_{L^1(μ_2)}<∞$ $μ_1$-a.e. $⇒$ $\left\|f\right\|_{L^1(μ_1\otimesμ_2)}<∞$

Let $(\Omega_i,\mathcal A_i,\mu_i)$ be a $\sigma$-finite measure space and $f:\Omega_1\times\Omega_2\to\mathbb R$ be measurable with respect to $\mathcal A_1\otimes\mathcal A_2$. Can we conclude, that ...
1
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0answers
35 views

Prove that this function is in $L^\infty$ with $\lVert g\rVert_\infty \le C$.

My professor used the following lemma in the proof that $L^1(X,\mu)^* = L^\infty(X,\mu)$ but left the proof as an exercise. Lemma. Assume that $(X,\mathcal A, \mu)$ is a measure space and $g \in ...
2
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1answer
51 views

Properties of mollification

We have this theorem For any $1\le p<\infty$ and $f\in L^p(\mathbb{R}^k)$, then $\|f*\phi_\delta - f\|_p\to 0$ as $\delta\to0$, where $\phi$ is any nonnegative measurable function on ...
2
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1answer
28 views

I do not understand a point in the proof of completness of $L^{\infty}$

do not understand a point in the proof of completness of $L^{\infty}$. I have this proof. We consider the sets $$A_{n,m}=\{x\in E:|f_{n}(x)-f_{m}(x)\|\leq\|f_{n}-f_{m}\|_\infty\}$$ for all ...
0
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1answer
39 views

Show that $(\int\left(\sum_k|f_k|\right)^p)^{1/p}\le\int(\sum_k|f_k|^p)^{1/p}$

How to show that $(\int\left(\sum_k|f_k|\right)^p)^{1/p}\le\int(\sum_k|f_k|^p)^{1/p}$, or is it not true in general ? $\{f_k\}\subset L^p, G_n=\sum_1^n|f_k|$, I think $|G_n|_p$ is the sum above ...
1
vote
2answers
35 views

Does the function $f(x)=\frac{1}{\sqrt x}$ belong to $L^p( \mathbb N , P(\mathbb N), \mu),p=1,2,\infty?$

Does the function $f(x)=\frac{1}{\sqrt x}$ belong to $L^p( \mathbb N , P(\mathbb N), \mu),p=1,2,\infty?$ $\mathbb N$- set of natural numbers, $P(\mathbb N)$- the partitive set of natural numbers. I ...
0
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2answers
38 views

Definition of $L^p$ norm of a vector-valued function

If $u$ is a vector the definition of the discrete norm will be $$\|u\|_{l^p}=(\sum |u_i|^p)^{1/p},$$ If $u$ is a function, $$\|u\|_{L^p}=\left(\int|u|^p\right)^{1/p}$$ But when $u$ is a vector-valued ...
2
votes
1answer
53 views

Given that $f_n \to f$ in $L^1(\Omega)$, $\mu(\Omega )=1$ and $ \|f_n\|_2^2 \leq M$, show $ \|f\|_2^2 \leq M$.

Given that $$\int_{\Omega} |f_n -f | \, d \mu \to 0,$$ $\mu(\Omega )=1$ and $ \|f_n\|_{L^2}^2 \leq M$, show $ \|f\|_{L^2}^2 \leq M$. Attempt: Note first that $f_n \in L^1(\Omega)$ since $$\|f_n ...
5
votes
1answer
50 views

uniform boundedness principle for $L^{1}$

i read this theorem from V.I.Bogachev vol 1 Measure Theory. A family $\mathcal{F}\subset L_{1}(\mu)$,where the measure $\mu$ takes values in $[0,+\infty]$, is norm bounded in $L_{1}(\mu)$ precisely ...
2
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1answer
44 views

about a product of random variables that converges weakly

Let $(\Omega,\mathcal{F},P)$ be a probability space. Suppose $f_n,g_n, n\in \mathbb{N}$ are sequences of functions on this space such that their product $f_ng_n$ converges weakly in $L^2$ to $h$, say. ...
2
votes
1answer
46 views

If a sequence of functionals converges weakly then it is bounded.

Let $f_k, f \in L^{\infty}(R)$ and $f_k \overset * \to f$ in $L^{\infty}(R)$. Is $f_k$ a bounded sequence in $L^{\infty}(R^n)$? (Definition: if $(v_n)$ is a sequence in $V = X^*$, we say that $v_n ...
0
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1answer
31 views

Does the weak divergence exist for each $\mathcal L^2(\Omega;\mathbb R^d)$-function?

Let $\Omega\subseteq\mathbb R^d$ be open. $v:\Omega\to\mathbb R$ is called weak divergence of $u:\Omega\to\mathbb R^d$ $:\Leftrightarrow$ $$\int_\Omega v\varphi\;{\rm d}\lambda=-\int_\Omega\langle ...
1
vote
1answer
16 views

If $u\in L^p(\Omega;\mathbb R^d)$, then $u_i\in L^p(\Omega)$ for all $i\in\left\{1,\ldots,d\right\}$. Does the reverse hold true?

Let $\Omega\subseteq\mathbb R^d$ and $u:\Omega\to\mathbb R^d$ be Borel measurable. Since $$|u_i|\le\left\|u\right\|_2\;\;\;\text{for all }i\in\left\{1,\ldots,d\right\}$$ we obtain ...
3
votes
1answer
33 views

Which assumptions on $Ω\subseteq\mathbb R^d$ do we need in order to show density of $C_c^∞(Ω)$ in $(L^p(Ω),\left\|\;\cdot\;\right\|_{L^p(Ω)})$?

Let $\Omega\subseteq\mathbb R^d$, $u\in\mathcal L^1(\Omega)$ and $$u_\varepsilon(x):=\frac 1{\varepsilon^d}\int_\Omega\rho\left(\frac{x-y}\varepsilon\right)u(y)\;{\rm d}\lambda(y)\;\;\;\text{for ...
0
votes
1answer
50 views

$x \cdot y \in L^1((a,b))$ for all $x \in L^1((a,b))$ implies $y \in L^{\infty}((a,b))$

If the integral of $x(t) \cdot y(t)$ on the interval $(a,b)$ converges for all $x \in L^1(a,b)$ then $y$ must be in $L^{\infty}(a,b)$, that is $\text{supess}|y|< inf$? Idea is to use Banach ...
2
votes
0answers
27 views

subset of L2 is precompact [closed]

I'm trying to prove that subset X of space $\mathcal L^2(a,b)$ is precompact if and only if $\forall \varepsilon>0$ $\exists \delta = \delta(\varepsilon)>0$ that $\forall$ $h, |h|<\delta$ ...
2
votes
1answer
41 views

Using Nash inequality to derive an inequality (from proof in paper)

We work on a domain $\Omega \subseteq \mathbb{R}^N$ with the Dirichlet Laplacian. Let $\lVert \cdot \rVert_p$ denote the $L^p$ norm. I am trying to understand why the following inequality is true: ...
1
vote
1answer
39 views

Measure Theory on integrals

Is it true that if $f\in L^1 (X,\mu)$ then $$\int_E |f| d\mu +\int_{E^c} |f| d\mu = \int_X |f| d\mu?$$
0
votes
2answers
119 views

Sequence in $l^p$ but not $l^q$ for all $q<p$

I need to find a sequence for real $p>1$ so it is in $l^p$ but not in any of the space $l^q$ with $1 \leq q <p$. I tried the sequence $(1/n)^{1/q}$ which is in $l^p\setminus l^q$. However, this ...
2
votes
0answers
29 views

A question about general Marcinkiewicz interpolation theorem

The general Marcinkiewicz interpolation theorem states as following: If $T$ is a linear operator of weak type $(p_0,q_0)$ and of weak type $(p_1,q_1)$ where $q_0\neq q_1$, then for each ...
4
votes
1answer
56 views

Prove that $X_n \to X$ in $L^1$ if and only if $E(X_n1_{A}) \to E(X1_{A})$ uniformly on $A \in \mathcal{F}$

This is a probability exercise from the Karr's book called "Probability". Prove that $X_n \overset{L^1}{\rightarrow} X$ if and only if $$\sup_{A \in \mathcal{F}} \left|E(X_n1_{A}) - ...
2
votes
1answer
49 views

What is the relation between $L^p([0,1])$ and the weak topology?

In my course of functional analysis, we always work with a Hausdorff locally convex space when we're creating the weak* and weak topology. My professor once stated that there is a reason for this ...
0
votes
1answer
14 views

Norm of pointwise product of Lp functions

Does the following inequality hold in $L_p$ spaces? $\|fg\|_p\leq\|f\|_p\|g\|_p$ How would I go about proving this? Do I need to apply Cauchy Schwarz?
8
votes
2answers
87 views

$(f_n)$ in $L^p(\Omega)$ satisfying $f_n(x) \to f(x)$ a.e. and $\|f_n\|_p \to \|f\|_p$, then $\|f_n - f\|_p \to 0$?

Let $1 < p < \infty$. If $(f_n)$ is a sequence in $L^p(\Omega)$ satisfying $f_n(x) \to f(x)$ a.e., $\|f_n\|_p \to \|f\|_p$, then does it follow that $\|f_n - f\|_p \to 0$? Edit. Here is my ...