1
vote
1answer
26 views

$\|g\|_{L^{1}(\mathbb R)}=\sup \{ {|\int_{\mathbb R} fg|: f\in C_{c}^{\infty}(\mathbb R), \|f\|_{L^{\infty}(\mathbb R)}=1\}} ?$

I learn the following from the book: Fact: If $g\in L^{1}(\mathbb R),$ then $$\|g\|_{L^{1}(\mathbb R)}=\sup \{ {|\int_{\mathbb R} fg|: f\in L^{\infty}(\mathbb R), \|f\|_{L^{\infty}}}=1\}.$$ We put ...
2
votes
1answer
64 views

An amazing inequality of the integration of two functions.

Let $f:[0,1]\longrightarrow\mathbb{R}$ be measurable and $g\in L^1[0,1]$ such that for all $t>0$, $$ \int_{|f(x)|>t}|g(x)|~\mathrm{d}x\leq \frac{3}{t^2}. $$ Prove that for $1<p<2$, $$ ...
3
votes
1answer
38 views

a function with infinity L^p norm

Let $1\leq p<\infty$, $1/p+1/q=1$. For a function $f$ with $||f||_q=\infty$, can we write $$ ||f||_q=\sup_{g\in L^p(\Omega),||g||_p\neq 0}\frac{\int_\Omega |fg|}{||g||_p}? $$ or $$ ...
1
vote
1answer
36 views

If $\| \psi \|_2=1$ can I say something about $\| \psi' \|_2$?

If I have a differentiable $L^2$ function $\psi:\mathbb R\rightarrow \mathbb C$ which is normalised $$ \int |\psi(x)|^2\;\text d x = 1 $$ can I say anything about the order of $$ \int ...
0
votes
0answers
35 views

Dense subsets in $L^1(\mathbb{R})$

Which of the following are dense subsets in metrical space $L^1(\mathbb{R})$? set of smooth functions $C_0^{\infty}(\mathbb{R})$ with compact supports; set of above-mentioned functions' derivatives ...
1
vote
1answer
27 views

Equivalent Definitions of the $L_2$ inner product.

If $g \in L_2(\mathbb{R})$, then we can define the $L_2$ norm to have the following relationship: $\|g\|_2^2 = \int_\mathbb{R} g^2$. If $A\subseteq \mathbb{R}$, then we can define the norm of $L_2(A)$ ...
8
votes
0answers
87 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponenent allows for convenient ...
1
vote
1answer
32 views

$L^p$ norm of a measurable function is bounded by its operation on step functions

Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$ $$ \left|\int_0^1 fg d\mu\right|\leq \|g\|_q. $$ Prove that ...
2
votes
1answer
42 views

a condition given by step functions implies the condition holds for L^q space

Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$, $$ |\int_0^1 fg d\mu|\leq ||g||_q. $$ Prove $||f||_p\leq 1$. How ...
0
votes
1answer
47 views

negative Sobolev space contains $L^1$ for a compact domain

I'd like to use something like Aubin-Lions lemma for the following spaces: $$ C^{0, \alpha}(B) \subset L^1(B) \subset W^{-1, q}(B),$$ with $B \subset \mathbb{R}^n$ being a compact, say a closed ball ...
1
vote
1answer
33 views

Reference for $f \in L^{p,\infty} \cap L^{q}$ then $f \in L^r$ for $p < r \leq q$

Okay, so I think I've shown that if $f \in L^{p,\infty} \cap L^{q}$ with $p < q$ then $f \in L^r$ for $p < r \leq q$ where $L^{p, \infty}$ denotes the weak $L^p$ space. what I did was I wrote $$ ...
1
vote
1answer
25 views

questions about $L^p$ space with $0<p\leq 1$ parallel to the case $1<p$

Question (1). Riesz-Fischer Theorem: For $1\leq p\leq \infty$, $L^p(\mu)$ is complete. Corollary of proof: Let $1\leq p\leq \infty$. If $(f_n)_{n=1}^\infty$ is a sequence coverging to $f$ with ...
3
votes
1answer
63 views

can $L^p$ norm convergence and pointwise monotonic imply pointwise convergence?

Let $(f_n)_{n=1}^\infty$ be a sequence of measurable function such that $\lim_{n\to\infty}||f_n-f||_p=0$. If for any $x\in \Omega$, $\{f_{n}(x)\}_{n=1}^\infty$ is a monotonic sequence, can we deduce ...
5
votes
1answer
60 views

An inequality of $L^p$ norms of linear combinations of characteristic functions of balls

Let $1<p<\infty$. Let $(a_n)_{n=1}^\infty$ be a sequence of nonnegative real numbers and $\{B_{r_i}(x_i)\}_{i=1}^\infty$ be a sequence of open balls in $\mathbb{R}^n$. Prove that there exists ...
0
votes
2answers
12 views

Sequence of continuous functions with unitary norm

I want to construct a sequence $f_n$ of continuous functions in $[0,1]$ such that $||f_n||=1$ (so a bounded sequence) and $||f_n-f_m||=1$ (it doesn't have any convergent subsequences). The norm is ...
1
vote
1answer
32 views

a question about $L^p$ functions on domains in Euclidean spaces

Let $\Omega$ be an open set in $\mathbb{R}^n$ and $f\in L^p(\Omega)$, $1\leq p<\infty$. Define $||f||_{p,\Omega}=\inf\{||f-a||_p: a\in\mathbb{R}\}$. Prove that there exists $a\in\mathbb{R}$ such ...
2
votes
0answers
25 views

Proof verification-density of smooth compactly supported functions

I am trying to show that $C_{c}^{\infty}(\mathbb{R})$ (smooth compactly supported functions) is dense in $C_{c}(\mathbb{R})$ (in the $L^{p}$ sense). Can anyone check if my proof is correct? Let $f ...
0
votes
0answers
49 views

Generalisation of vector-valued Marcinkiewicz interpolation theorem

Given a compatible couple $(X,Y)$ of Banach spaces and some measure space $\Omega$ Lions and Peetre identified the real interpolation space between the vector-valued Lebesgue spaces $L_1(\Omega;X)$ ...
1
vote
0answers
33 views

Convergence in $L^1$ and $L^p$ [duplicate]

Assume: (a) $\{f_n\} \subset L^p$, (b) $f_n \to f \text{ }\mu\text{-a.e.}$ and (c) $\|f_n\|_{L^p} \to \|f\|_{L^p}\to0$. Then show that $$\|f_n - f \|_{L^p} \to 0$$ using Fatou's Lemma, first for ...
1
vote
0answers
46 views

Condition for a product of two function sequences in $L^1$ to be in $L^1$

We have: 1. $\{f_n\} \subset L^1(E, \Sigma, \mu)$ 2. $g_n \subset L^\infty(E, \Sigma, \mu)$ 3. $\|f_n-f\|_{L^1}\to0$ 4. $g_n \to g \text{ }\mu\text{-a.e.}$ 5. $\{g_n\}$ is uniformly bounded. ...
3
votes
1answer
69 views

Function compositions that are in $L^p$

We have $f \in L^p$. The goal is to show that $\exists \psi \in C(\mathbb{R^+}, \mathbb{R^+})$ such that $$ \lim_{s \to +\infty} \frac{\phi(s)}{s}=+ \infty \text{ and } \phi(|f|) \in L^p$$ I neeed ...
0
votes
1answer
33 views

Sequence of piecewise constant functions converging to any $L^2$ function

Let $\{P_i\}_{i=1}^\infty$ be a sequence of partitions of the interval $[0,1]$ with a vanishing mesh. Additionally $H_i$ be the space of piecewise constant functions (step functions) with pieces ...
1
vote
1answer
30 views

Existence of a function in $L^{p}$ with a certain property

Is there a function $f \in L^{p}(\mathbb{R}^{n})$ such that $\|If\|_{L^{p}(\mathbb{R}^{n})} = \infty$ where $If(x) = \int_{\mathbb{R}^{n}}\frac{f(y)}{|x - y|^{n}}\, dy$? Such a case isn't covered by ...
0
votes
1answer
43 views

Question about injection on an unbounded space

I have this space $$C_0((0,+\infty))=\left\lbrace u,u\in C((0,+\infty)),\lim_{t\rightarrow +\infty} u(t)=0\right\rbrace$$ with the norm $$||u||_{\infty}=\sup_{t\geq0}|u(t)|$$ how to prove that ...
0
votes
1answer
51 views

Criteria to be in weak $L^{p}$ space

Let $X$ be a $\sigma$-finite measure space. Let $f : X \rightarrow \mathbb{C}$ be a measurable function and $1 < p < \infty$. Suppose for $f$ there is a constant $C$ such that ...
1
vote
1answer
17 views

Relation between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$

As an exercice, I'm looking to find an inclusion or equality relationship between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$ when $w: x \to x^{-1/2}$. Actually, I think that we have the inclusion ...
0
votes
1answer
42 views

what are closed sets in $L^{1}(\mathbb R)$?

Consider, $L^{1}(\mathbb R)$= The space of Lebesgue integrable functions on $\mathbb R$; for $f\in L^{1}(\mathbb R),$ we define its norm, by $\|f\|_{L^{1}}=\int_{\mathbb R}|f(x)| dx$; It is well-known ...
2
votes
1answer
35 views

$f_{n}$ converges in $L^{1}$ and $\|g_{n}\|_{L^{1}(\mathbb R)}\leq \|f_{n}\|_{L^{1}(\mathbb R)}\implies {g_{n}}$ converges in $L^{1}(\mathbb R)$?

Let $f_{n}, g_{n}\in L^{1}(\mathbb R)$ (Lebesgue space). Suppose there exist $f\in L^{1}(\mathbb R)$ such that $\|f_{n}-f\|_{L^{1}(\mathbb R)}\to 0$ as $n\to \infty;$ that is, the sequence $\{f_{n}\}$ ...
1
vote
0answers
55 views

Convolution is continuous map

I can prove this when $f$ is assumed as continuous function but without assuming continuity i got confused. Suppose $ p \in (1, \infty) $ and $q$ is its conjugate exponent. Prove that if $f\in ...
1
vote
1answer
49 views

Product of weakly convergent sequence and sequence boundedly convergent in measure

Question: Let $\Omega \subset \mathbb{R}^d$ be open and bounded, $f, f_n \in L^2 (\Omega)$ and $f_n \rightarrow f$ boundedly in measure (meaning that $f_n \rightarrow f$ in measure and $sup\ ...
5
votes
0answers
87 views

Gradient on Sobolev spaces

Suppose I have a function $\Phi \in L_p(\Omega)$ and $\nabla\Phi \in W^{-\alpha}_p(\Omega)$, $1<p<\infty$ and where $\Omega\subset\mathbb{R}^n$ is a bounded domain. Can I conclude that $\Phi\in ...
0
votes
0answers
32 views

Conditions on $\alpha_n$ for $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ to be a norm on $l_p$

When $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ is a norm in $\mathcal{l}_p=\lbrace (x_k)^\infty_1 : \sum\vert x_k\vert ^p \lt\infty\rbrace $ and $\alpha\in\omega$. and $\omega$:space of ...
3
votes
0answers
70 views

Closure of Schwartz space in homogeneous Besov space

Let $\dot{B}^s_{\infty,\infty}(\mathbb{R}^d)$ denote the homogeneous Besov space of order $s$ with second and third index $\infty$, i. e. the homogeneous Zygmund space. Let $\mathcal{S}(\mathbb{R}^d)$ ...
1
vote
1answer
54 views

Equivalence of norms on weak $L^p$

I'm having trouble to prove the equivalence of two (quasi)norms on the weak $L^p$ space. Assertion: $$\sup_{\mu(A)<\infty}\mu(A)^{(1/p)-1}\int_A |f(x)|dx\leq ...
3
votes
2answers
103 views

Why is the set of all $\infty$-tuples with finitely many non-zero rational terms dense in $\ell_2$?

This statement has been given as an example in the book "Introductory real analysis" written by Kolmogorov and Fomin: The set of all points $x=(x_1,x_2,\cdots,x_n,\cdots)$ with only finitely ...
3
votes
1answer
41 views

$L^{p}$ inequality with a lower bound on measure

I am working on the following problem: Suppose $f \in L^{p}(X)$ for some $0 < p < \infty$ and the space $X$ is such that each set of positive measure has measure $\geq m$ for some $m > 0$. ...
1
vote
1answer
41 views

$\mathbb L^1 +$ a.s. convergence of sequence $(X_n)$ does not imply $\sup(x_n)$ is integrable

I'm looking for a counterexample. The setting is this: Given an probability space $(\Omega,\mathbb{F},\mathbb{P}) $, I look for sequence of random variables $(X_n)_n$ and a random variable $X$, all in ...
1
vote
0answers
76 views

Assume $\int_{D}fgdx=0\forall g\in C_{c}^{\infty}\left(D\right)$. Then $f = 0$ a.e. on D.

I want to prove this result: Let $D$ be an open subset of $\mathbb{R}^n$, $p \in[1,\infty)$ and $f$ be in $L^p(D)$. Assume $\int_{D}fgdx=0\forall g\in C_{c}^{\infty}\left(D\right)$. Then $f = 0$ ...
1
vote
1answer
33 views

Relation between convergences in $L^{p}$ for probability spaces.

I have read that for a probability space $(\Omega,\Sigma,P)$ it is true that $f \in L^{p}(\Omega,\Sigma,P)$ implies $f \in L^{q}(\Omega,\Sigma,P)$ if $p>q$, and hence $L^{2} \subset L^{1}$. I'm ...
2
votes
1answer
59 views

Subspace of Tempered Distributions

Let ${S_{h}}'(\mathbb{R}^{n})$ be the space of tempered distributions such that if $u\in {S_{h}}'(\mathbb{R}^{n})$, then $\lim_{\lambda\rightarrow \infty}{||\phi(\lambda D)u||_{\infty}} = 0$ for all ...
0
votes
1answer
41 views

Upper bound for the norm of inverse Fourier tansform

Recall Hausdorff-Young inequality: For any $f\in L^p(\mathbb{R}^n)$, we have $||\hat{f}||_q\le ||f||_p$, where $p$ and $q$ are conjugate exponents and $p\in[1,2]$. It seems to me that it follows ...
0
votes
2answers
89 views

Prove $\|f\|_{L^p}$ is not equivalent to $\|f\|_{\infty}$ in $C[a,b]$

Prove that in $C[a,b]$ the uniform norm is not equivalent to the $L^p$ norm for $(1\leq p < \infty)$ I am stuck on showing that the function below satifies the claim. I know that f is continuous ...
2
votes
1answer
204 views

Properties of $||f||_{\infty}$ - the infinity norm

Prove that $||f||_{\infty}$ is the smallest of all numbers of the form $\sup\{|g(x)|: x\in X\}$, where $f=g$ ($\mu$ almost everywhere). In addition, if $f$ is a continuous function on the measure ...
2
votes
1answer
107 views

Find an example of a sequence $\{f_k\}$ such that $f_k\in L^p$ for $1\le p <\infty$, $f_k\to0$ in $L^p$ for $1\le p < p_0$

Let $1<p_0<\infty$. Find an example of a sequence $\{f_k\}$ such that $f_k\in L^p$ for $1\le p <\infty$, $f_k\to0$ in $L^p$ for $1\le p < p_0$, but $f_k$ does not converge in $L^{p_0}$. ...
2
votes
3answers
77 views

Suppose $\{f_k\}$ is a sequence of $M$-measurable functions on $X$. Let $p_1$ and $p_2\in [1,\infty)$, and suppose $f_k\in L^{p_1}\cap L^{p_2}$.

Suppose $\{f_k\}$ is a sequence of $M$-measurable functions on $X$. Let $p_1$ and $p_2\in [1,\infty)$, and suppose $f_k\in L^{p_1}\cap L^{p_2}$. Also suppose there exist $g\in L^{p_1}$ and $h\in ...
4
votes
1answer
287 views

Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$.

Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$. So suppose $f\in L^p\cap L^q$. Then both $\int |f|^p d\mu$ and $\int|f|^q d\mu$ exist. For each $x$ in the domain ...
1
vote
2answers
119 views

convergence in $L^1$ for product of functions

If $f_n$ converges to $f$ in $L^1$ and $g_n$ converges to $g$ in $L^1$. Does it necessarily mean that $f_ng_n$ converges to $fg$ in $L^1$ for finite measure spaces.
0
votes
1answer
56 views

Examples of nonconvergent Cauchy sequences of functions

Let $f$ be a continuous, real valued function defined on a closed, bounded interval $I$, a subset of real numbers. Let $\{f_n\}$ be a Cauchy sequence in the $L^2$ norm. Give a counterexample that the ...
2
votes
1answer
614 views

Does $L^p$-convergence imply pointwise convergence for $C_0^\infty$ functions?

It is stated in my professor's notes that, given a sequence $\{f_j\}$ of $C_0^\infty(\Omega)$ functions (infinitely differentiable with compact support), and a function $g\in C_0^\infty(\Omega)$, all ...
2
votes
2answers
114 views

When is the logarithm of this function square integrable?

I was trying to prove $\log(1-|r(s)|^2)$ lies in $L^2$ when $r\in L^1\cap L^2$. How should I do this? Thank you!