Questions tagged [lp-spaces]

For questions about $L^p$ spaces. That is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence classes of measurable functions such that $|f|^p$ is $\mu$-integrable. Questions can be about properties of functions in these spaces, or when the ambient space in a problem is an $L^p$ space.

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Rudin functional analysis: theorem 13.25 (involving functional calculus for unbounded operators)

Let $(\Omega, \mathfrak{R})$ be a measurable space, $H$ be a Hilbert space and let $E: \mathfrak{R}\to B(H)$ be a resolution of the identity. Let $f: \Omega \to \mathbb{C}$ be a measurable function ...
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3 votes
1 answer
21 views

Convergence of integrals without the dominated convergence theorem

Let $t>0$ and $\{f_n\}_{n \geq 1} $ be a sequence of functions that do not converge pointwise to any integrable function, but where: \begin{equation} \int_0^t f_n(s) ds \rightarrow F(t) < \...
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  • 31
2 votes
1 answer
44 views

Find a lower bound of the Maximal function of $f(x) = \frac{1}{x (\log x )^2}$ in $x \in (0,1/2)$.

Define $f(x) = \dfrac{1}{x (\log x )^2}$ in $x \in (0,1/2)$. Maximal function of $f$ is $$(Mf)(x) := \sup_{r>0} \frac{1}{m(B_r)} \int_{B(x,r)} |f(y)|\,dy,$$ where $B(x,r)$ denotes a ball of radius $...
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  • 541
1 vote
1 answer
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Limit of $L_p$ norm.

This is a well-known problem that has been asked in this website, but I just wanted to get some hints and see if I'm even doing the right thing. The problem is to show that $\lVert f \rVert_\infty = \...
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  • 1,352
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0 answers
19 views

$L^{\infty}([a,b])$ and the Lebesgue-Stieltjes integral

Let $[a,b]$ be a compact interval in $\mathbb{R}$, and $L^{\infty}([a,b])$ the space of all lebesgue measurable functions on $[a,b]$ essentially bounded; my question is whether these functions are ...
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  • 455
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1 answer
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How can you proove that every bounded function in $L^1[0;1]$ can be approximated by continuous function in $C[0;1]$?

Here my question, is this true that: Every bounded function in $L^1[0;1]$ can be approximated by continuous functions in $C[0;1]$ It seems to me true as we know that $C[0;1]$ is dense in $L^1[0;1]$, ...
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4 votes
1 answer
112 views

Does a bounded function like this exist?

is it possible to find a function $f$ such that $f \in L^{\infty}(\mathbb{T}^1)$ and $$\sum_{j=0}^{+\infty} \left(\sum_{n \in \mathbb{Z}: 2^j \le |n|<2^{j+1}}|\widehat{f}_{n}|^{2} \right)^{1/2}=+\...
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  • 224
4 votes
1 answer
52 views

The functionals $\|\cdot\|_{L^{p,q}}$ do not satisfy the triangle inequality.

Given a measurable function $f$ on a measure space $(X,\mu)$ and $0<p,q\leq \infty$, define $$\|f\|_{L^{p,q}}=\left\{ \begin{array}{ll} \displaystyle{\left(\int_{0}^\infty\left(t^{1/p}f^*(t)\right)...
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1 answer
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Exercise 4.19 (1) of Brezis

I am trying to solve the following exercise of Brezis' book on Functional Analysis. Let $(f_n)_{n \in \mathbb{N}}$ be a sequence in $L^p(\Omega)$ with $1 < p < \infty$ and let $f \in L^p(\Omega)$...
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Bound for $L^p$ Norm of a partial sum of a stationary sequence

I could need some help on stationary sequences. Assume that $X_1,X_2,\dots$ is a stationary sequence of real-valued random variables on $(\Omega, \mathcal{A},\mathbb{P})$. ($X_1,\dots,X_{t+s}$ and $...
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3 votes
0 answers
48 views

Is there a equivalent norm on $L^p$ induced by inner product?

Suppose $L^p[a,b]$ is the normed space with the usual norm $\|f\|_p=(\int_a^b|f(x)|^p\mathrm{d}x)^{1/p}$. By the parallelogram equality, we know the norm is induced by an inner product iff $p=2$. ...
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Proof: $C^\infty[0,1]$ is dense in $L^1[0,1]$

Intro: I would like to know if my demonstration of $C^\infty[0;1]$ is dense in $L^1[0,1]$ is correct because I didn't find any complete demonstration of that statement. -(i) As we know from here all ...
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Is $d_p$ with $p \in (0,1)$ good define?

He I try to see if $d_p(x,y)= \sum_{k=1}^\infty |x_k-y_k|^p $ with $x,y\in \ell^p$ is good define My idea I know if $a,b>0$ and 0<p<1 the inequality $(a+b)^p<a^p + b^p$ then if I use the ...
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4 votes
1 answer
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Proof: All functions in $L^2[0,1]$ are in $L^1[0,1]$

I would like to know if my demonstration of all functions in $L^2[0,1]$ are in $L^1[0,1]$? $\forall f\in L^2[0,1] $ we can split $f$ in two differents spaces: $A=\left \{0\leq x\leq 1:|f(x)|>1) \...
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1 vote
1 answer
35 views

Proof of the embedding of time dependent Sobolev spaces

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. $H^{-1}(\Omega)$ is the dual of $H_0^1(\Omega)$. For shorthand I write $\mathcal{H} = H^1(0,T,H_0^1(\Omega),H^{-1}(\Omega))$. I want to ...
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4 votes
1 answer
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problem on $L^2$ (pointwise) convergence and Carleson's Theorem

I am having some problem on my arguments and I want to see where it fails. It deals with pointwise convergence in $L^2[-\pi,\pi]$: pointwise convergence is given by Carleson's Theorem (so it is a hard ...
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2 votes
0 answers
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Proving $||f||^2_{L^2(\mathbb R^2)}\le 10||f||_{L^1(\mathbb R^2)}||\nabla f||_{L^2(\mathbb R^2)}$ with $f\in C^1,L^1\cap L^2$ and $\nabla f\in L^2$.

Prove that there exists a universal constant $K<10$, for all $C^1$ function $f : \mathbb R^2 \rightarrow\mathbb R$, if $f \in L^1 (\mathbb R^2)\cap L^2(\mathbb R^2)$ and $|\nabla f| \in L^2(\mathbb ...
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Is this shorter attempt on the existence of the maximizer in Kantorovich duality is correct?

I'm reading section 3.4 Existence of Maximisers to the Dual Problem in this lecture notes. The proof is quite involved and requires a complicated approximation argument. Below is my straightforward ...
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4 votes
1 answer
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Prove the uniqueness of $u\in H_0^1(\Omega)$ with $\Delta u=\vert u\vert^{q-1}u+f$ in $\Omega$ with $\Omega$ as a bounded domain with smooth boundary.

Problem: Let $\Omega\subset\mathbb R^2$ be a bounded domain with smooth boundary. Prove that, for all $p>1$ and $1\le q<\infty$, for all $f \in L^p(\Omega)$, there exists a unique $u\in H_0^1(\...
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  • 802
1 vote
0 answers
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Extending weak-type linear operator from $L^{p}\cap L^{q}$ to $L^{r}$ where $1\leq p<r<q<\infty$

Let $T$ be a linear mapping from $L^{p}\cap L^{q}$ into itself and $1\leq p<q<\infty$. Suppose $T$ is of weak-type $(p,p)$ and weak-type $(q,q)$ on functions $f$ in $L^{p}\cap L^{q}$, that is, $\...
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  • 195
1 vote
1 answer
24 views

Evaluate integral wrt Lebesgue measure and find the L^p space if it exists

I have been given the following the question: Consider the following function $f : \mathbb R → \mathbb R$, $$f=2·1_{(-3,1]}-3·1_{[5,+\infty)}$$ Here $1_A$ denotes the indicator function of set A. ...
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1 vote
0 answers
27 views

If $T$ is a compact operator and $S$ is a bounded operator then $TS-ST$ cannot be the identity operator.

Let $T:L_2[0,1] \to L_2[0,1]$ be a compact linear operator and $S:L_2[0,1] \to L_2[0,1]$ be a bounded linear operator. Prove $$TS-ST \neq e$$ where $e$ is the identity operator. My solution. I am ...
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2 votes
1 answer
23 views

Let $X$ be a locally compact Polish space. Is the space of continuous functions with compact support dense in that of $\mu$-integrable functions?

I'm reading this question for which I would like to clarify the theorem mentioned there. We have (S1) Let $X$ be a locally compact Hausdorff space. Then the space of continuous functions with compact ...
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3 votes
1 answer
101 views

Show that $C^\infty(0,T; L^2(\Omega(t)))$ is dense in $L^2(0,T; L^2(\Omega(t)))$.

Let $T > 0$, $\Omega \subset \mathbb R^2$ and $f:[0,T] \to \mathbb R^2$ a continuous and bounded function. We define $$\Omega(t) = \Omega + f(t),$$ and $\widetilde \Omega \subset \mathbb R^2$ such ...
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1 vote
1 answer
47 views

Show existence of bounded linear functional

To solve problem about a bounded linear functional, I am having a problem with the Hahn Banach Theorem. Problem is: For $n \in \mathbb{N}$ and $1 \leq p < \infty$, let $X_n \subset L^p([0,1]$ be a ...
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  • 541
0 votes
0 answers
13 views

Does the integral exist and belong to the L^p space

hey i have been given the folowing problem Consider the following function f:R→R $$f = 2·1_{(-3,1]} -3·1_{[5,+∞)}$$ Here $1_{A}$ denotes the indicator function of set A But im not sure how to check if ...
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  • 1
1 vote
0 answers
44 views

Equivalent definitions of Sobolev space on manifold and references

It is well-known that there are two equivalent definitions of Sobolev space on open subset $\Omega\subset\mathbb{R}^n$: D1. The completion of $C^\infty(\Omega)$ under $H^p_k$ norm. D2. All functions ...
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2 votes
0 answers
37 views

How to prove that normed space is complete?

$I=[0,1]$. For $k \in \mathbb{N}$, denote by $C^k(I)$ the space of real-valued functions on $I$ possessing continuous derivatives up to order $k$ on $I$, including one-sided derivative at the end ...
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  • 541
1 vote
1 answer
31 views

Completeness of weighted $L^p$ spaces

For $\infty >p>1$ consider the weighted $L^p$ spaces $(L^p(\mathbb{R}^n),\omega dx)$ where $\omega$ is some nonnegative weight. Is it true that $(L^p(\mathbb{R}^n),\omega dx)$ is complete iff $\...
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1 vote
1 answer
24 views

Is convolution integrable in $L_1(\mathbb{R}^n)$?

Let $|h(y)|\in L_1(\mathbb{R}^n)$, i.e. $\int\limits_{\mathbb{R}^n}|h(y)|\,dy<+\infty$. Consider the function $F(x)=\int\limits_{\mathbb{R}^n}|h(x-y)|\,dy$. It is known that $F(x)$ be bounded and ...
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1 vote
0 answers
27 views

Intersection of compact sets in different spaces

Let $A$ be a compact set in $L^1$ and $B$ a compact set in $L^2$. Determine if $A \cap B$ is compact in $L^1$ or $L^2$ or both. My idea: Since $L^2 \subset L^1$ and $\Vert \cdot \Vert_2$ is stronger ...
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1 vote
1 answer
64 views

The limit of a Lebesgue integral

I'm trying to prove the following exercise: Suppose that $f,f_n,g\in\mathcal{L}^2(\mathbb{R})$, $n=1,2,...$, $f_n$ converges to $f$ $\mu-$almost everywhere, and $$ \int_{\mathbb{R}}|f_n(x)|^2 dx\leq 1 ...
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-1 votes
3 answers
72 views

ways to calculate norm of the Bounded Linear Functional.

$\Lambda$ be a linear functional on $C([0,1])$ defined by $$ \Lambda(f) = \int_0^1 xf(x)dx \;\;\; \text{ for } f \in C([0,1]). $$ and use $\|f\|_{sup} = \sup_{x \in [0,1]} |f(x)| $ for $f \in C([0,1])$...
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  • 541
0 votes
0 answers
37 views

Find the norm of linear functional

Let $p \in [1,\infty)$ and let $\Lambda$ be a linear functional on $L^P(I)$ denoted by $$\Lambda(f) = \int_0^1 e^{2x} f(x) dx \;\;\; \text{for } f\in L^P(I).$$ where $I = [0,1] \subset \mathbb{R}$. ...
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  • 541
0 votes
0 answers
45 views

Fourier series expansion of a $L^2$ function.

For a function $f \in L^2(\mathbb{T})$ (where $\mathbb{T}$ denotes the unit circle) I know that it can be expressed as $f(z) = \sum_{j = -\infty}^{\infty}f_j z^j$. The Fourier coefficients are given ...
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1 vote
0 answers
16 views

Let $G$ be locally compact abelian, $T_2$ and $f\in L^1(G)$. Define $\mu(A)=\int_A f(x)\ dx$. Prove $\lVert \mu\rVert=\lVert f\rVert_1$

Here $\mu$ becomes a complex measure and $\lVert \mu\rVert =|\mu|(G)$ is the total variation norm of $\mu$. We have to show $|\mu|(G)=\int\limits_G |f(x)|\ dx$ Let $\{A_n\}$ be a partition of $G$. ...
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0 votes
0 answers
25 views

Complexity of Lebesgue measurable spaces

Consider a discrete finite set $\Omega=X\times Y \in \mathbb{R}^{m\times n}$ for finite $m,n$. Let $(\Omega,\Sigma,\mu)$ be the measure space. ($\Sigma$ is the power set and $\mu$ is $\sigma$-finite ...
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  • 1,666
2 votes
1 answer
40 views

How to prove that weakly convergence in $L^{p_2}$ implies weakly convergence in $L^{p_1}$ where $1\leq p_1\leq p_2<\infty$?

As the title suggested, suppose that $1\leq p_1\leq p_2<\infty$. Let $\{f_n\}$ be a sequence of functions in $L^{p_2}([0,1])$ and $f\in L^{p_2}([0,1])$. Show that $f_n \rightharpoonup f$ in $L^{p_2}...
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  • 89
1 vote
1 answer
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Finding the orthogonal complement to all linear functions on $(0, 1)\subset\mathbb{R}$.

So, this is really a two part question. It states the following. Let $H = L_2(0, 1)$ be a Hilbert space. Let $u_1, u_2\in L_2(0, 1)$ be given by $u_1(t) = 1$, and $u_2(t) = t$, $\forall t\in(0, 1)$. ...
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0 votes
0 answers
23 views

Is there an easy way to determine the orthogonal complement to the following set?

Let $M\subset L_2(0, 1)$ be the set of $L_2$-functions such that $$ \int_0^1u(x)~dx = a $$ $\forall u\in M$, and a constant $a\in\mathbb{C}$ (or $a\in\mathbb{R}$, although to ensure that all bases are ...
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1 vote
1 answer
48 views

How to solve this estimate in Grafakos' book?

Prove that for all $1<p<\infty $ there exist a constant $A_p>0$ such that for every $C^2_0(\mathbb{R}^2)$(twice continuously differentiable with compact support complex value function) such ...
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1 vote
0 answers
13 views

Stummel functions are uniformly locally $L^2$

In review articles of B. Simon about Schrödinger operators, he mentions that for $n \leq 3$ the condition $$ \lim_{\epsilon \to 0} \sup_x \int_{B(x,\epsilon)} |x-y|^{4-n}|V(y)|^2 dy =0 $$ on a ...
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  • 2,703
1 vote
1 answer
30 views

Function goes to 0 if all derivatives are $L^2$

Suppose that all derivatives of some function $f$ are $L^2$, so that $\left\lVert\frac{\partial^kf}{\partial x^k}\right\rVert<\infty$ for all $k\ge0$. Then is it true that $f\to0$ as $x\to\infty$? ...
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  • 1,232
0 votes
0 answers
24 views

Reference for chain rule for continuously Frechet differentiable maps

Let $f\colon L^p(\Omega;X) \to L^q(\Omega;Y)$ and $g\colon L^q(\Omega;Y) \to L^r(\Omega;Z)$ where $X,Y,Z$ are separable Hilbert spaces and $\Omega$ is a smooth and bounded open set (eg. $\Omega = [0,T]...
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  • 63
0 votes
1 answer
44 views

Confusion on the solution for Brezis Exercise 4.10

This is Exercise 4.10 on Brezis and I am having trouble understanding Brezis' solution for part 3 of the problem: This is the part of the solution where I am lost: My question is two-folds: Why do ...
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1 vote
0 answers
32 views

Proving the uniform convexity of $L^p$ for $1 < p \le 2$

We are asked to show the uniform convexity of $L^p$ for $1 < p \le 2$ using the following inequality: For all $1 < p < \infty$, there is a constant $C$ such that $|a - b|^p \leq C(|a|^p + |b|...
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1 vote
1 answer
25 views

The space $\operatorname{Lip}_{b}(X)$ is dense in $L_1(X, \mu)$ w.r.t. $\|\cdot\|_{L_1}$

I'm trying to prove this well-known property. Could you verify if my attempt is fine? Let $(X, d)$ be a metric space, $\operatorname{Lip}_{b}(X)$ the space of Lipschitz continuous bounded real-valued ...
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  • 1,153
2 votes
0 answers
61 views

Clarification on Proof from Folland's Analysis - Sobolev Embedding Theorem

This is Theorem 8.54, the Sobolev Embedding Theorem, in Folland's Analysis: If $s>k + (n/d)$, then $H_s \subset C_0^k$. (Here $H_s$ is defined to be the Sobolev space and $C_0^k = \big\{f \in C^k(\...
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1 vote
1 answer
63 views

Is $\ell^{p-1}$ a subspace of $\ell^p$?

I'm studying Introductory Functional Analysis with Applications by Erwin Kreyszig. In Chapter 2, section 2.4, while solving exercise, this question came to my mind, but I can't figure it out, please ...
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0 answers
25 views

Why does the Laplace transform of a function in L2 belong in the Hardy space?

Does anyone have a proof / mathematical explanation to why the Laplace transform of a function of $\mathbb{L}^2(0,\infty )$ belongs to the Hardy space? Any guidance would be fantastic! I cant find ...
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