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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2
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0answers
24 views

Lebesgue-integrability of derivatives

Let $f:\mathbb R\to\mathbb [0,\infty)$ be a non-negative, twice-differentiable function. Suppose that $\int_{-\infty}^{\infty}f(x)\,\mathrm dx<\infty$, $\int_{-\infty}^{\infty}|f''(x)|\,\mathrm ...
1
vote
1answer
27 views

Why is $\frac{1}{x^{1/p} (\ln(x)^2+1)}$ in $L^1$ but not in $L^p$ for any $p>1$

From a practice qualifying exam, the goal is to find a function $f \geq 0$ on $(0,\infty))$ that $f \in L^p(0,\infty)$ iff $p=1$. One function suggested was: $$\frac{1}{x^{1/p} (\ln(x)^2+1)}$$ So ...
0
votes
0answers
13 views

continuity of some operator

Let $p \geq 1$ and let $A: l^{p + 1} \to l^{p + 1}$ be a continuous linear operator. Suppose that $A(x) \in l^{p}$ for all $x \in l^{p + 1}$. Is it true that the map $l^p \ni x \mapsto A(x) \in l^p $ ...
0
votes
1answer
56 views

The limit at infinity of a function with derivative in $L^2$

Suppose that $$\tag{1}f(x),f'(x),xf(x)\in L^{2}(\mathbb{R}),$$ then we have $$\tag{2}\frac{d}{dx}(x|f(x)|^{2})\in L^{1}(\mathbb{R}).$$ My question is can we get the following limit, and how ...
0
votes
1answer
30 views

$C_{c}^{\infty}(\Omega)$ is dense in $L^{\infty}(\Omega)$ with respect to the topology $\sigma(L^{\infty},L^{1})$

Show that $C_{c}^{\infty}(\Omega)$ is dense in $L^{\infty}(\Omega)$ with respect to the topology $\sigma(L^{\infty},L^{1})$, where $\Omega$ is an open subset of $\mathbb{R^n}$. My try: Let ...
3
votes
0answers
74 views

continuous linear functional on $l^{\infty}$ space

Let $l_{\infty}$ be the space of all bounded complex-valued sequences equipped with the supremum norm. Consider the natural standard basis $\{e_n\}_{n \in \mathbb{N}}$ of $l_{\infty}$. For any ...
2
votes
0answers
25 views

Show that there exists $ \lambda \ge 0$ such that $v=\lambda u$

Let $\Omega \subset \mathbb{R}^n$ be open. Let $u,v \in L^1_\text{loc}(\Omega)$ with $u \ne 0$ a.e on a set of positive measure. Assume that $$\phi \in C_c^\infty(\Omega), \int u\phi > 0 \implies ...
0
votes
1answer
17 views

Holders Inequality: Suppose $\int_{0}^\infty x^{-2}|f|^5 dx < \infty$. Prove that $\lim_{t \to 0} t^{-\frac{6}{5}} \int_0^t f(x)dx = 0$

I discovered last night that I have an error in my proof to the following problem and I need help fixing it (or need a new solution) $$ \text{Suppose that} \int_{0}^\infty x^{-2}|f|^5 dx < \infty. ...
0
votes
0answers
20 views

Minkowski Inequality when either $||f||_p = 0$ or $||g||_p = 0$.

I will recall that Minkowski Inequality says the following: Let $E$ be a measurable set and $p \in [1,\infty]$. If the functions $f$ and $g$ belong to $L^p(E)$, then so does their sum $f + g$ and, ...
0
votes
1answer
64 views

Weak formulation of a nonlinear problem with test functions in a dense subspace of $H_0^1$

I am reposting a question from Math Overflow, because it seems it gets no attention. Let $\Omega\subset \mathbb R^{d=3}$ is a bounded and Lipschitz domain. Let $u\in H_g^1(\Omega)$ satisfy the weak ...
1
vote
1answer
25 views

Problem about a compact operator $T:l^p\rightarrow l^p$

I have to solve this problem. Let $\{\lambda_n\}$ be a sequence of real number such that $\lim_{n\rightarrow\infty}\lambda_n=0$ and consider the operator $T:l^p\rightarrow l^p$, $1\leq p\leq ...
1
vote
0answers
36 views

What is the motivation for $l^p$ space?

I'm studying $L^p$,$l^p$ spaces recently, but I don't see the motivation for this. The only application I know is that $l^2$ can be used to characterize Hilbert spaces up to dimension. Is $L^p$ spaces ...
-1
votes
0answers
22 views

Show that step functions are dense in $L^1 (\Bbb R)$ [on hold]

A step function is, by definition, a finite linear combination of characteristic functions of bounded intervals in $\Bbb R$. Assume $f \in L^1( \Bbb R)$, and prove that there is a sequence $\{g_n\}$ ...
0
votes
0answers
9 views

Is $l^p(A)$ unique up to isomorphism?

Let $A, B$ be sets. I'm now studying Hilbert spaces and I learned that $l^2(A)\cong l^2(B)$ (isometrically isomorphic) iff $|A|=|B|$. This leads to natural questions. Let $1\leq p< ...
0
votes
0answers
29 views

Does convergence almost everywhere to an $L^p$ function and existence of a weakly convergent subsequence guarantee weak convergence?

By assumption, for $p \in (1,\infty)$, I have a bounded sequence of functions $f_n$ in $L^p$ (that is, $L^p$ norms of the functions are uniformly bounded) that converges almost everywhere to a ...
4
votes
2answers
43 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 < ...
2
votes
1answer
51 views

If $u_n^p \rightharpoonup v$ in $L^1$, then does it follow that $u_n \rightharpoonup v^{\frac 1p}$ in $L^p$?

Let $\Omega$ be a bounded domain. Suppose that $u_n^p \rightharpoonup v$ in $L^1(\Omega)$. Does it follow that $u_n \rightharpoonup v^{\frac 1p}$ in $L^p(\Omega)$?
0
votes
0answers
16 views

Show that $\lim_{p\rightarrow \infty}\|f\|_{L^p(X)} = \|f\|_{L^{\infty}(X)}$ [duplicate]

Suppose that $(X,M,\mu)$ is a measure space with $\mu(X) = 1$ and that $f\in L^{\infty}(X)$ show that $$\lim_{p\rightarrow \infty}\|f\|_{L^p(X)} = \|f\|_{L^{\infty}(X)}$$ Attempted proof - Let ...
2
votes
3answers
385 views

$L^1$ function is bounded almost everywhere

Suppose $f\in L^1(\mathbb{R})$. Is it true that $f$ is bounded (except for a set of measure zero)? I think it should be true, but can't show it formally. If $f$ is unbounded, why would we have ...
0
votes
1answer
152 views

Does Hilbert Transform commute with Function Multiplication modulo Compact on $L^p(R)$?

Define Hilbert Transform (HT) as the convolution with the function $1/x$. E. Stein proves in his book Singular Integrals and Differentiability Properties of Functions that HT, when understood as a ...
1
vote
3answers
46 views

$1\le p \lt \infty$ and $f_k$ nonnegative increasing. Then $f_k\to f$ in $L_p$ iff $\sup_k||f_k||_p \lt \infty$.

Let $1\le p \lt \infty$ and $0\le f_k$ increasing to $f$, and $f_k$ measurable. Then $f_k\to f$ in $L_p$ if and only if $\sup_k||f_k||_p \lt \infty$. I was able to show the if part, but I can't ...
2
votes
1answer
19 views

Equality for $\lvert\lvert f\rvert\rvert_1$

I am trying to solve the following exercise: Show that for $f\in L_1(\mu)$, $$\lvert\lvert f\rvert\rvert_1=\sup\Bigg\{\int fg d\mu : \lvert\lvert g\rvert\rvert_\infty\leq 1\Bigg\}$$ I know that as ...
0
votes
1answer
15 views

Covergence in $L^p$

If $f_k\rightarrow f$ in $L^p$, $1\leq p < \infty$, $g_k\rightarrow g$ pointwise, and $\|g_k\|_{\infty} \leq M$ for all $k$. Prove that $f_k g_k\rightarrow fg$ in $L^p$. Attempted proof - Let ...
1
vote
0answers
24 views

Real Analysis, Folland Problem 6.2.23 Dual of $L^p$

Problem 6.2.23 - Let $(X,M,\mu)$ be a measure space. A set $E\in M$ is called locally null if $\mu(E\cap F) = 0$ for every $F\in M$ such that $\mu(F) < \infty$. If $f: X\rightarrow \mathbb{C}$ ...
7
votes
3answers
379 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 $$ ...
1
vote
1answer
34 views

Showing this function is $0$ a.e.

I would like to show the following: Suppose that $g \in L^1(\mathbb{R}^n)$ and $\int fg \,d \mu = 0$ for any $f \in C_0(\mathbb{R}^n)$. Then $g = 0$ $\mu$-a.e. I'm stumped on trying to find an ...
2
votes
1answer
46 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} - ...
0
votes
0answers
19 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 $X$ ...
0
votes
0answers
9 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 ...
1
vote
0answers
16 views

Equality in Minkowski's inequality $L^p$ spaces Folland

I am re-posting this questions as the reference of possible duplicate provided did not serve me any insight. When does equality hold in Minkowski's inequality? (The answer is different for $p = ...
0
votes
1answer
15 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 ...
1
vote
1answer
55 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 ...
0
votes
1answer
25 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
vote
1answer
27 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 ...
1
vote
1answer
258 views

If $u \in W^{1,p}(U)$, prove that $Du=0$ a.e. on the set $\{u=0\}$.

Assume $1 \le p \le \infty$ and $U$ is bounded. (a) Prove that if $u \in W^{1,p}(U)$, then $|u| \in W^{1,p}(U)$. (b) Prove $u \in W^{1,p}(U)$ implies $u^+,u^- \in W^{1,p}(U)$, and ...
2
votes
1answer
62 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 ...
3
votes
1answer
44 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
votes
2answers
43 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 ...
1
vote
0answers
14 views

$\|h\|_{L^{p}} \leq C \|f\|_{L^{p}} \implies \|g\ast h \|_{L^{p}} \leq C_1 \|g\ast f\|_{L^{p}}$?

Suppose that $f, h \in L^{p}(\mathbb R) (1\leq p \leq \infty)$ so that $\|h\|_{L^{p}} \leq C \|f\|_{L^{p}}$ for some constant $C$. Take $g\in \mathcal{S}(\mathbb R^{d})$ (Schwartz Space). We note ...
0
votes
1answer
47 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) ...
0
votes
1answer
25 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. ...
2
votes
2answers
47 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 ...
3
votes
1answer
46 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 $$ ...
0
votes
0answers
10 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 ...
0
votes
1answer
17 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 ...
1
vote
2answers
48 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 ...
3
votes
1answer
39 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 ...
2
votes
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 ?
3
votes
1answer
268 views

Any positive linear functional $\phi$ on $\ell^\infty$ is a bounded linear operator and has $\|\phi \| = \phi((1,1,…)) $

This is a small exercise that I just can't seem to figure out. When I see it I'll probably go 'ahhh!', but so far I haven't made any progress. I'd like to prove that any linear functional $\phi$ on ...
0
votes
0answers
16 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 ...