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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68 views

Real Analysis , Folland Problem 6.1.5

Problem 6.1.5 - Suppose $0 < p < q < \infty$. Then $L^p \not\subset L^q$ if and only if $X$ contains sets of arbitrary small positive measure, and $L^q\not\subset L^p$ if and only if $X$ ...
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0answers
11 views

Neumann Poincare operator maps $L^2$ in itself

How can I show that the Neumann-Poincare operator $$ K_{\partial \Omega}[\phi](x) = \int_{\partial \Omega} \dfrac{(x-y) \cdot \nu(y)}{|x-y|^d} \phi(y) \ dy $$ maps $L^2(\partial \Omega)$ in itself (if ...
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1answer
17 views

Integrability of $f(x)\sqrt{\frac{1}{x}}$ for $f\in\mathcal L^2$ and $\|f\|_2=1$

Is it true that for $f\in\mathcal L^2$ and $\|f\|_2=1$, $$\int_0^\infty f(x)x^{-1/2}dx<\infty?$$ I'm fairly stuck on this...(and I really hope it is true). In case it helps seeing a generalization,...
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1answer
27 views

Where does this inequality come from: $\frac{|f(x) - f(x-h)|}{h} \le ||f'||_{L^\infty(x, x-1)}$?

I came across this inequality today: $$\frac{|f(x) - f(x-h)|}{h} \le ||f'||_{L^\infty(x, x-1)}$$ I realise if we let $h \to 0$ we obtain the derivative on the left hand side so I can see it has ...
1
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1answer
30 views

Let $\mu $ be a $\sigma $-finite measure. Show that for any $f\in L_p(\mu)$, $\|f\|_1=\sup\{ \int fg\, d\mu :\|g\|_\infty \leq 1\}$

Let $\mu $ be a $\sigma $-finite measure. Show that for any $f\in L_p(\mu)$, $\|f\|_1=\sup\{ \int fg \, d\mu :\|g\|_\infty \leq 1\}$ I know that Holders inequality implies $\int fg \, d\mu \leq \|f\|...
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1answer
62 views

Real Analysis, Folland Problem 6.2.17 Dual of $L^p$

Theorem 6.14 - Let $p$ and $q$ be conjugate exponents. Suppose that $g$ is a measurable function on $X$ such that $fg\in L^1$ for all $f$ in the space $\sum$ of simple functions that vanish outside a ...
3
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1answer
47 views

Real Analysis, Folland Problem 6.1.16 $L^p$ spaces

Problem 6.1.16 - If $0 < p < 1$, the formula $\rho(f,g) = \int |f-g|^p$ defines a metric on $L^p$ that makes $L^p$ into a complete topological vector space. Attempted proof - Suppose $a,b > ...
2
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2answers
47 views

Real Analysis, Folland problem 6.1.11 $L^p$ spaces

Problem 6.1.11 - If $f$ is a measurable function on $X$, define the essential range $R_f$ of $f$ to be the set of all $z\in\mathbb{C}$ such that $\{x:|f(x) - z| < \epsilon \}$ has positive measure ...
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3answers
395 views

Triangle inequality fails in $L^{1,\infty}$

It can be proved that $\forall\varepsilon>0$ there exists $C(\epsilon)>0$ such that for all $f,g\in L^{1,\infty}(\Bbb R^n)$ we have that $$ ||f+g||_{1,\infty}\le(1+\varepsilon)||f||_{1,\infty}+...
2
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1answer
69 views

Real Analysis, Folland Problem 6.1.14 $L^p$ spaces

Problem 6.1.14 - If $g\in L^{\infty}$, the operator $T$ defined by $Tf = fg$ is bounded on $L^p$ for $1\leq p \leq \infty$. Its operator norm is at most $\|g\|_{\infty}$, with equality if $\mu$ is ...
2
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1answer
50 views

Real Analysis, Folland Problem 6.1.8 $L^p$ spaces

Problem 6.1.8 - Suppose $\mu(X) = 1$ and $f\in L^p$ for some $p > 0$, so that $f\in L^q$ for $0 < q < p$. a.) $\log\|f\|_{q}\geq \int \log |f|$ b.) $\left(\int |f|^{q} - 1\right)/q \...
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1answer
31 views

Proving that $c_c(\mathbb N)$ is a dense subset of $l^p(\mathbb N)$

Proving that $c_c(\mathbb N)$ is a dense subset of $l^p(\mathbb N)$ $c_c(\mathbb N)$-space of sequences which are zero after finitely many terms. $$l_p=\{(x_i)^{\infty}_{i=1}|\sum_{i=1}^{\infty}|x_i|^...
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1answer
48 views

Given that an integral containing $|x_{n}|$ is bounded by a constant, show that $x_{n}\to x$ in $L^{1}(0,\infty)$

This question is related to another question I asked earlier. For reference, this is the relevant part of that question: Let the sequence of continuous functions $\mathbf{\{x_{n}(t) \}_{n=1}^{\...
3
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1answer
56 views

Triangle inequality in weak $L^1$ space

I have some problems with this exercise: $\forall\varepsilon>0$ there exists $C(\epsilon)>0$ such that for all $f,g\in L^{1,\infty}(\Bbb R^n)$ we have that $$ ||f+g||_{1,\infty}\le(1+\varepsilon)...
2
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2answers
54 views

Let $f:\Bbb R \rightarrow \Bbb R$ be a Lebesgue measurable function in $L^2$. Show $F(x)=\int_0^x f(t)dt$ satisfies $|F(x)-F(y)|\leq C|x-y|^\frac 12$

Let $f:\Bbb R \rightarrow \Bbb R$ be a Lebesgue measurable function that is in $L^2$. Show $F(x)=\int_0^x f(t)dt$ satisfies $|F(x)-F(y)|\leq C|x-y|^\frac 12$. Here's what I have so far. $f\in L^2 \...
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0answers
11 views

Stability problem for finite difference scheme

u (x,y,a,t) a analytical solution of a PDE $$ u_t+u_a = \nabla_{x,y} \cdot \left(d(x,y,a,t)\nabla_{x,y}u\right) -\mu\left(x,y,a,t\right)u $$ $ u : \left[0,1\right]^2 \times \left[0,A\right] \times \...
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1answer
25 views

$L^p$-norm minimization under linear constraints: Does the optimum depend on $p$?

Consider the following norm minimization program: \begin{align} \label{1} &\min_{x \in \mathbb{R}^d} &&\lVert x - x_0 \rVert_p^p &(1)\\ &\text{subject to } &&Ax-b \ge 0 \...
4
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1answer
43 views

$f\in L^{1}[0,1]$ Show $\lim_{n\to\infty}\int_{0}^{1}|f(x)|^{\frac{1}{n}}dx = m(\left\{ {x:f(x)\neq 0}\right\} )$

The following is from a Sample Exam question I am studying from, and the question has stumped me. $$f\in L^{1}[0,1]$$ $$\lim_{n\to\infty}\int_{0}^{1}|f(x)|^{\frac{1}{n}}dx = m(\left\{ {x:f(x)\neq ...
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0answers
36 views

Can we show that each element of the Sobolev space $H^k(D)$, with $D\subseteq\mathbb R^d$ being a bounded domain, has a continuous representative?

Let $d\in\left\{2,3\right\}$ $D\subseteq\mathbb R^d$ be a bounded domain $\lambda$ be the Lebesgue measure on $D$ $H^k(D)$ be the Sobolev space Can we show that each element of $H^k(D)$ has a ...
2
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0answers
20 views

$f \in \mathcal l^{\infty}{'} $ ; $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty} $ is a sequence of non-negative terms ; is $f$ bounded? [duplicate]

Let $f:\mathcal l^{\infty} \to \mathbb R$ be a linear functional such that $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty} $ is a sequence with non-negative terms ; then is $f$ continuous ?
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1answer
20 views

$C(A)$ is $\|\cdot\|_2$-dense in $\ell_2(A)$

Let $A \neq \varnothing$ and $\cal {F}$$(A) = \{F \subset A \mid F$ is finite$\}$. Define $\ell_2 (A) =L^2(A, 2^A, \mu_C)$, with $\mu_C$ the counting measure. Let $C(A) = \{f: A \to \Bbb C, \exists ...
3
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1answer
55 views

What is duality argument for the operator on $L^p-$ spaces?

Suppose that the operator $T: L^{p}(\mathbb R) \to L^{p}(\mathbb R)$ (say for instance, some nice convolution operator) is bounded for $1\leq p \leq 2.$ At various, places we see that (for ...
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0answers
30 views

If $f_n$ is in $L^1$ and $f_n$ converges to $f$, is $f$ in $L^1$?

This question might come off as basic to most of you, but this isn't basic to me. If $f_n$ is in $L^1$ and $f_n$ converges to $f$ in $L^1$ ($||f_n-f||_1 \to 0$ as $n\to \infty$), does it necessarily ...
1
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2answers
54 views

Real Analysis, Folland Problem 6.1.12 $L^p$ spaces

Problem 6.1.12 - If $p\neq 2$, the $L^p$ norm does not arise from an inner product on $L^p$, except in trivial cases when $\dim(L^p) \leq 1$. (Show that the parallelogram law fails.) Attempted proof ...
1
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2answers
31 views

Is $\ell_q\subset\ell_p$? What's the “biggest” and “smallest” sequence space?

(By smaller and larger, I mean as subsets/supersets of complex sequence spaces.) I have a couple of questions about sequence spaces (over $\Bbb{C}$). I was proving that $(e^{(m)})$ (the constant $0$ ...
3
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1answer
106 views

Show that if $\,u \in W^{1,p}\left(I\right) \bigcap C_c\left(I\right)$, then $\,u \in W_{0}^{1,p}\left(I\right)$.

I want to show the following statement ($1 \leq p < \infty$), for an open interval $I$: If $u \in W^{1,p}\left(I\right) \bigcap C_c\left(I\right)$ then $u \in W_{0}^{1,p}\left(I\right) $. $W^...
0
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1answer
19 views

Showing an inequality in the proof of $L^p$ is a Banach space for $1\le p \lt \infty$.

This is part of the proof that $L^p$ is a Banach space from Folland's Real Analysis, but there is a part that I don't understand. Suppose $\{f_k\}\subset L^p$ and let $G_n=\sum_1^n |f_k|$ and $G=\...
0
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1answer
33 views

What are the differences between $l^p$ space and $L^p$ space?

I always thought small $l^p$ space, or so-called Banach space, is the space equipped with a norm similar to vector norm as $$||x||_p = \left( \sum_{i\in\mathbb{N}} |x_i|^p \right)^{\frac{1}{p}}$$ ...
0
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0answers
9 views

Diagonal argument involving relative compactness of densities

There is a claim from a paper which I do not understand: Let $D$ be a domain in $\mathbb{R}^d$. Let $(p^{\eta})_{\eta >0}$ be a family of densities for random variables on $(C[0,T], \mathbb{R}^...
5
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1answer
119 views

Real Analysis, Folland Problem 6.1.2 $L^p$ spaces

Background Information: In this chapter we work on a fixed measure space $(X,M,\mu)$. If $f$ is measurable on $X$ and $0 < p < \infty$, we define $$\|f\|_{L^p} = \left[\int |f|^p d\mu\right]^{1/...
1
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2answers
42 views

Sobolev Space $W_0^{1,p}(I)$ and the boundary of $I$

Given $I \subset\mathbb{R}$ an open interval, the Sobolev Space $W_0^{1,p}(I)$ is defined as $W_0^{1,p}(I)=\overline{C^1_c(I)}^{W^{1,p}(I)}$ (The closure of $C^1_c(I)$ on the space $W^{1,p}(I)$) . ...
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0answers
11 views

What is the Stein Interpolation Theorem for any strip?

Stein Interpolation Theorem: Assume $T_z$ is an operator depending analytically on $z$ in the strip $0\leq Re (z) \leq 1.$ Suppose $T_z$ is bounded from $L^{p_0}$ to $L^{r_0}$ when $Re (z)= 0,$ and ...
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0answers
30 views

Sobolev embedding fails for $p=n$

As everyone knows, the Sobolev embedding fails fails for $n\ge 2$ if we assume $p=n$. The standard example is the function $u(x)=\log \log \bigl(1+\tfrac{1}{x}\bigr)$. This function is obviously ...
2
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2answers
43 views

Approximate Sobolev function by smooth function - error estimate?

I wondered if there is, in general, a way to estimate the error (in $L^2$) between a Sobolev function and its mollified version. Let's say $\Omega\subset\mathbb{R}^n$ is bounded with smooth boundary ...
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0answers
18 views

Iterated convolutions w.r.t. different variables of a function

I do not understand a claim from a paper: Let $b:[0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}$ be a bounded function and let $$b^{n} (t,x) = b(t,x) \ast \psi_n(t) \ast\phi_n(x), $$ where $\...
3
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1answer
33 views

Show that $L^1$ function is $0$ a. e.

Suppose $f \in L_1(\mathcal{R})$ satisfies for every measurable $A \subset \mathcal{R}$ $$ |\int_A f| \leq [m(A)]^{(1+\epsilon)} $$ for some $\epsilon >0$. Prove $f=0$ a.e. This is a problem on ...
4
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1answer
64 views

$p$-norm inequality

Let $p \ge 2$ and $q$ such that$${1\over p} + {1\over q} = 1.$$Is it true that there exists a constant $c$ such that for all $x$, $y$ such that $\|x\|_q \le 1$ and $\|y\|_q \ge 1$$$\left\|x - {y\over{\...
1
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1answer
42 views

What does it mean “a Lebesgue point of $f$”?

Follan - Real Analysis p.243 Theorem 8.15 As far as I know, the term, Lebesgue point $x$, is defined for $L^1$ functions such that $\lim_{r\to 0} \frac{1}{m(B(x,r))} \int_{B(x,r)} |f - f(x)| dm =0$. ...
1
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1answer
37 views

Questions on proof that $\Vert \cdot \Vert_p$ is a norm when dealing with $L^p$ spaces

Unfortunately I never had an opportunity to take a functional analysis/PDE module so I keep running into steps I'm unsure of when I'm looking at proofs. I'm trying to fix that now with some books so ...
4
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2answers
45 views

Is $f(x)\exp(-x^2)$ summable if $f$ is square summable?

Suppose that $f \in L^2(\mathbb R)$; i.e. $$ \int_{- \infty}^\infty \vert f(x) \vert^2 dx < \infty. $$ Can we from this infer that $$ \int_{- \infty}^\infty \vert f(x)\vert e^{-x^2} \, dx < \...
0
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1answer
46 views

Intuitive explanation of p-norm in finite and infinite dimensinos

I am not a mathematician, so very rigorous treatment with things that only a math major learns will not suffice here. I want to learn about p-norms and i can't quite get the intuition behind them. I ...
5
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0answers
123 views

Integral operator is bounded on $L^p$ if it maps $L^p$ to itself

Here is a homework excercise. Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space,$1\leq p <\infty.$ and suppose that $k:X\times X\rightarrow \mathbb{C}$ is an $\Omega \times \Omega$ ...
0
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1answer
26 views

subspace of $L^2$

Let $(X,B(X),\mu)$ a measurable space, for a positive finite measure $\mu$, we consider $H=L^2(X,d\mu)$, Let $A$ a closed subspace of $H$, we know that $A$ is a hilbert space, can we say that it exist ...
2
votes
2answers
53 views

If $f \in L^2[-\pi, \pi]$, then $f \in L^1[-\pi, \pi]$ and$\|f\|_{L^1} \leq \sqrt{2\pi} \|f\|_{L^2}$

I'm learning about Fourier analysis and need help with the following problem (which is part of a subchapter on $L^p$ spaces): Using the Cauchy-Schwarz inequality show that if $f \in L^2[-\pi, \pi]...
0
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1answer
42 views

The Lebesgue Fundamental Theorem of Calculus

Let $f,g:\mathbb{R}\to\mathbb{R}$ be two Lebesgue integrable functions. If we have $$f(b)-f(a)=\int_a^bg(x)dx$$ for almost all $a,b\in \mathbb{R}$. How can we modify $f$ on a set of measure zero to ...
1
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1answer
49 views

If $||g_j-g||_{1,\infty}\to0$, then $||g_j||_{1,\infty}\to||g||_{1,\infty}$

I have some problems with my notes: my teacher wrote that if a sequence $\{g_j\}_j\subseteq L^{1,\infty}(\Bbb R^n)$ (which is the weak $L^1$ space, endowed with the quasinorm $||\cdot||_{1,\infty}$) ...
2
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0answers
47 views

If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$

Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$. Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
0
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1answer
19 views

Compactness Sobolev embedding for even functions on $\mathbb{R}$.

It is well-known from Lions's article,"Symétrie et compacité dans les espaces de Sobolev", that the subspace $H^s_r(\mathbb{R}^n)$ of the Sobolev space $H^s(\mathbb{R}^n)$ containing all radial ...
0
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1answer
35 views

$L^{\infty}$ convergence for random variable

I am slightly confused with this borderline case regarding $L^p$ convergence. In some probability books, they clearly state that $p<\infty$ whereas the online sources do not impose this ...
1
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1answer
61 views

If an $H^1$ function vanishes on a set of positive measure, its $L^2$ norm is controlled by the gradient

I am trying to solve question 15 from Evans' PDE book, chapter 5. You have a set of positive measure, subset of the unit ball $B$, such that $u$ is equal to zero on that set. Then, one can show that: $...