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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11
votes
1answer
81 views

Hardy's Inequality: Problems $3.14$ and $3.15$ in Rudin's RCA

In Problem $3.14$, we prove (a) Hardy's inequality, (b) the condition for equality, and I shall talk about (c), (d) below. Problem $3.15$ is the discrete case of Hardy's inequality. I have asked three ...
2
votes
2answers
67 views

Same on dense subspace implies same on whole space?

I read the whole proof of Theorem 13.18 Bruckner's Real Analysis book and I had no problem understanding the proof except for following two claims inside the proof that are stated without further ...
1
vote
0answers
29 views

$L_1$ convergence of holomorphic functions on closed disks to a continuous function

I'm trying to prove that if holomorphic sequence of functions $f_n$ converge to $f$ on closed disks in the $L_1(\Omega)$ sense on an open $\Omega$ and $f$ is continuous, then we can deduce that $f$ is ...
3
votes
1answer
49 views

Density of compactly supported smooth functions in Lp

This is an exercise in my Functional Analysis book; I tried to check my solution by referring to this site and elsewhere but nowhere have I found a similar argument to mine (which makes me skeptical ...
1
vote
2answers
65 views

Is $L^p$ linear for $0<p<1$?

The following is an exercise from Bruckner's Real Analysis: Show that for all $0 <p< \infty$ the collections $L^p$ of measurable functions defined on a measure space $(X, \mathcal{M},μ)$ such ...
1
vote
1answer
27 views

Function f such that $\oint_{B} \int_{\mathbb{R}^3} f(x+y) dx dy =0$.

Let $f \in L^1(\mathbb{R}^3)$ such that $$\oint_{B} \int_{\mathbb{R}^3} f(x+y) dx dy =0$$ for any bounded set $B$ of $\mathbb{R}^3$. I feel like the following is true : $$\int_{\mathbb{R}^3} f(x) dx =...
2
votes
0answers
31 views

Non-existence of supremum norm bound in terms of $L^2$ norm of gradient

Show there does not exist any constant $C>0$ such that, for any $\phi\in C^{\infty}_c(\mathbb{R}^2)$, the inequality $$\lVert\phi\rVert_\infty\leq C\lVert\nabla\phi\rVert_2$$ holds. Obviously, we ...
1
vote
0answers
62 views

Show that expression $f_n(t)=\int ^t_0 ne^{n(s-t)}f(s) ds$ such that $\lim_{n\to \infty} \|f_n-f\|_{L^p}=0$

Fix $p\in [1,+\infty)$ and let $f\in L_p[0,T]$. show that the expression $$f_n(t)=\int ^t_0 ne^{n(s-t)}f(s) ds\quad t\in[0,T], n\in \mathbb{N}$$ defines a sequence of continuous functions such that $\...
1
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0answers
75 views

Bounding operator $I^{\alpha} f(t) - \frac{1}{\Gamma (\alpha)} \int_0^t(t-\tau)^{\alpha - 1}f(\tau)d\tau.$

Let $p \in [1, \infty], \alpha > 0, T > 0$ and define $$I^{\alpha} f(t) - \frac{1}{\Gamma (\alpha)} \int_0^t(t-\tau)^{\alpha - 1}f(\tau)d\tau,$$ where $\Gamma$ is Gamma function. Show that $I^{\...
1
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0answers
43 views

Quotient space of $\ell^p$ is isometrically isomorphic to $\ell^p$

Show that $$M = \{ x = (x_k)_{k \in \mathbb{N}} \in \ell^p \mid x_{2k} = 0 \ \forall k \in \mathbb{N} \}$$ is a closed subspace of $\ell^p$. Further, $\text{codim}(M) = \infty$, $\ell^p \cong M$, and $...
2
votes
0answers
20 views

Approximating $f\in L_1\cap L_2$ with Schwartz functions

I am working on $\mathbb{R}^n$ for this problem. So every $L_p$ is actually $L_p(\mathbb{R}^n)$. I know that the Schwartz space $S$ is dense in $L_p$ for $1\leq p<\infty$. This is how I prove it : ...
0
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0answers
23 views

Proof $L^{p}(X,\mu)$ is a linear space normed with respect to the norm [duplicate]

I am studying Lp-Spaces and Banach-Spaces and the norm for $ p \in [1, \infty) $ is given by $$||f||_{L_{p}}=\left ( \int _{X} ||f(x))||^{p} \right )^{1/p}$$ but i didn't find how to test it. Do you ...
-1
votes
1answer
37 views

Exchange limit and essential supremum? [closed]

Suppose $f_n \rightarrow f$ uniformly. Does that imply that $\| f_n \|_\infty \rightarrow \| f \|_\infty$?
1
vote
1answer
61 views

On a proof of Hardy's inequality

I'm currently reading a proof of Hardy's inequality for integrals, which states the following: Let $p>1$ , $f \in L^{p}(0,\infty)$ and define $$ g(t) =\frac{1}{t}\int_{0}^{t}f(s)ds, $$ for $t > ...
0
votes
1answer
29 views

Why minimize squared L2 norm and not only the L2 norm?

I'm studying Inverse Problems and usually, they minimize the squared of the L2 norm($L_2, L_0, L_ \infty$), why don't minimize only the norm? if the goal is to have a measure of the distance between 2 ...
2
votes
0answers
40 views

$L^p$ norm of $\frac{1}{1-\overline{\alpha }e^{i\theta}}.$

I am trying to calculate the $L^p$ norm of the function $f_\alpha(e^{i\theta}) = \frac{1}{1-\overline{\alpha }e^{i\theta}},$ where $\alpha$ is a complex number with $|\alpha| < 1.$ Id est, for ...
2
votes
0answers
32 views

If $\Omega\subseteq\mathbb R^d$ is sufficiently nice, does it hold $L^\infty(\Omega)\subseteq H^1(\Omega)$?

Let $d\in\mathbb N$ and $\Omega\subseteq\mathbb R^d$ be bounded and open. Assume $\overline\Omega$ is a $d$-dimensional properly embedded $C^\infty$-submanifold of $\mathbb R^d$. Let$^1$ $(e_n)_{n\in\...
8
votes
2answers
111 views

When is an operator $T$ on $L^1(\mu)$ of the form $(Tf)(x)=\int\mu({\rm d}y)p(x,y)f(y)$?

Let $(E,\mathcal E,\mu)$ be a $\sigma$-finite measure space and $T\in\mathfrak L(\mathcal L^1(\mu))$. I would like to know which conditions on $T$ would be sufficient to conclude that there is a (...
3
votes
2answers
141 views

To prove continuity of $\varphi(p) = \int_X |f|^p\ d\mu$ on $E = \{p: \varphi(p) < \infty\}$ where $0 < p < \infty$

Suppose $f$ is a complex measurable function on $X$, $\mu$ is a positive measure on $X$, and $$\varphi(p) ~=~ \int_X |f|^p \; d\mu \quad (0 < p < \infty)$$ Let $E :=\{ p : \varphi(p) < \infty\...
1
vote
1answer
57 views

Show that for $f(s,t)=(s^2+t)^{-a}$ the function $f^b$ is integrable over $(0,1)\times(0,T)$ [closed]

This should be a rather simple question, but I wasn't able to figure out what's the trick to find a suitable upper bound: Given $a\in(1/2,3/2)$, $b\in[1,1/(a-1/2))$ and $T>0$, how can we show that ...
0
votes
1answer
84 views

If $(x_n)$ is a real sequence and $\forall (y_n) \in \ell_p$ we have $\sum\limits_{n=1}^{\infty}|x_n||y_n|<\infty$,then $(x_n) \in \ell_q $ [closed]

So, we consider $(x_n)$ is a real sequence and $1<p<\infty$. For every $y=(y_n) \in \ell_p$, we have $\sum\limits_{n=1}^{\infty}|x_n||y_n|<\infty$. We need to show that $x=(x_n) \in \ell_q$, ...
4
votes
1answer
44 views

Weak convergence in $L^2(\mathbb{R}^3,L_{loc}^2(\mathbb{R}^3))$ thanks to diagonal extraction.

Let $f_n$ be a sequence bounded in $L^2(\mathbb{R}^3,L_{loc}^2(\mathbb{R}^3))$ which means that, for any bounded set in $\mathbb{R}^3$, one has for any $n>0$ $$\int_{\mathbb{R}^3} \int_{B} |f_n(x,y)...
0
votes
0answers
31 views

Solutions to Mean $L^p$ Errors

$\DeclareMathOperator*{\argmin}{arg\,min}$Lets say we have a random variable $X$ which is absolute integrable. I.e., it is in $L^1$ space and so it is also in any $L^p$ space where $1< p\leq\infty$....
1
vote
0answers
37 views

Gronwall inequality for integrable functions

In the book "Some Gronwall type inequalities and applications" (Theorem 5): Let $x:[a,b] \to \mathbb{R}$ a continuous function that satisfies the inequality $$\frac{x^2(t)}{2} \leq \frac{x^...
1
vote
0answers
29 views

Vectors such that $\lVert x + y \rVert_p^p = \lVert x \rVert_p^p + \lVert y \rVert_p^p$

We fix $1 \leq p < 2$. What are the couple $({x},{y})$ of vectors in $\mathbb{R}^2$ (or more generally in $\mathbb{R}^n$) for which the following equality holds \begin{equation} \label{eq:here} \...
4
votes
1answer
52 views

Weak convergence of the vector norm a vector field?

Suppose $\Omega\subset\mathbb{R}^n$ and $\mathbf{v}^k:\Omega\rightarrow \mathbb{R}^m$ is a sequence of vector fields on $\Omega$ that converges weakly in $L^p(\Omega,\mathbb{R}^m)$ to $\mathbf{v}$. ...
2
votes
1answer
53 views

$f\in L^2(\mathbb{R})$ is absolutely continuous and $f' \in L^2(\mathbb{R})$ if and only if $\int |y\hat{f}(y)|^2dy<\infty$

This comes from Walter Rudin's Functional Analysis p. 388, exercise 21(c): But the domain I found on Engel's One-Parameter Semigroups for Linear Evolution Equations p. 66 is $$ D(A)=\{f\in L^2(\...
0
votes
1answer
40 views

Non-canonical examples of divergent sequences that are square summable? [closed]

The canonical example of a divergent sequence that is square summable, $\sum_{n = 1}^\infty a_n$ is finite, is the harmonic sequence: $\sum_{n = 1}^\infty 1/n$. Are there examples of sequences that ...
1
vote
0answers
22 views

Convergence in $L^2 (B(0,1))$ of $H^1 (\mathbb{R}^3)$ functions implies the existence of a dominating function?

I am currently reading an article, and there is a passage where I can't understand how to justify what the author says. I'll try to write what I think the relevant informations are, since a precise ...
0
votes
0answers
33 views

Can I generalize the classical Bernstein inequality?

I have know the proof of the classical Bernstein inequality as follows. Given some function $ f\in S(\mathbb{R}^d) $, where $ S(\mathbb{R}^d) $ denotes the Schwartz space of functions. Let $ \hat{f} $ ...
1
vote
0answers
31 views

Under what conditions is an infinite orthonormal set a basis of $L^2 (\mathbb{R})$?

I have an infinite orthonormal set in $L^2 (\mathbb{R})$ and want to know under which conditions it is as basis of $L^2 (\mathbb{R})$ and how to prove that. In the finite dimensional case, the ...
0
votes
3answers
39 views

$\overline{L^1(\mathbb{R}) \cap L^p(\mathbb{R})} = L^p$ for $1 < p < 2$? [closed]

Is $\displaystyle \overline{L^1(\mathbb{R}) \cap L^p(\mathbb{R})} = L^p(\mathbb{R})$ for $1 < p < 2$ ? I know this holds for $p = 2$. But, I am not able to find any source about the case when $1 ...
5
votes
1answer
42 views

The inclusion of $W^{1,p}$, $p\geq1$ into $C[0,1]$ with sup norm is bounded.

Fix $p \geq 1$, let $W([0,1])$ be the space of absolutely continuous functions such that for all $f \in W$ we have $\|f'\|_p^p <\infty$. Then this is a Banach Space with the norm; $\|f\|_W=\left(\|...
1
vote
0answers
25 views

Limit at $+\infty$ of $\mathcal C^1$-function whose derivative lies in $L_p$

Let $f:\mathbb R_{\geq 0}\to \mathbb R$ be a continuously differentiable and integrable function whose derivative $f'$ lies in $L_p$ for some $p\in[1,+\infty[$. The goal is to prove that $\lim_{x\to+\...
2
votes
2answers
35 views

Is $\mathcal{B}(L^p,L^1)$ strictly bigger than $L^q$?

Let $(\Omega, \mathscr{F}, \mu)$ be a $\sigma$-finite measure space and $1\leq q \leq \infty$. Denote by $p$ the conjugate exponent of $q$. We know, given a $g\in L^q (\Omega)$, \begin{equation} ...
0
votes
0answers
24 views

The set of bounded simple functions is dense in the space of square integrable functions (Stochastic integration)

In the classic book on stochastic integration by S. Cohen and R. Elliot "Stochastic calculus and applications" theorem 12.1.8 yields an extension of the stochastic integral from taking ...
0
votes
1answer
25 views

comparison of norms two sequences in $\ell_1$ space

Assume that we have two sequences of processes $X_{n} \in \ell_{1}$ and $Y_{n} \in \ell_{1}$ with non-negative components. Next, we know that $X_{n,i} \leq Y_{n,i}$ for all $i \in\mathbb{Z}_{+}$ and ...
0
votes
1answer
37 views

Orthogonal decomposition of the periodic space $L^2$

Let $L>0$ and define the space $$ L^2_{per}([0,L]):=\big\{ f: \mathbb{R} \rightarrow \mathbb{R} \; ; \; f \: \text{is periodic with period $L>0$ and} f|_{[0,L]} \in L^2([0,L]) \big\}. $$ The ...
0
votes
1answer
36 views

If $f\in L^2[0,1]$ such that $\int_0^x f(t)\ dt=0$ for all $x\in [0,1]$. Prove that $f=0$ in $L^2[0,1]$

Let $f\in L^2[0,1]$ such that $\int_0^x f(t)\ dt=0$ for all $x\in [0,1]$. Prove that $f=0$ I know that $L^2[0,1]\subset L^1[0,1]$ and for $f\in L^2[0,1]$, $f$ is zero iff $f= 0$ almost everywhere. ...
2
votes
1answer
31 views

A problem on weak convergence: weak convergence of a sequence implies the weak convergence of the square of this sequence?

Suppose that $\{u_m\}_m\subset H^1(\mathbb{R}^3)$ and $u_m\rightharpoonup u$ weakly in $H^1(\mathbb{R}^3)$. From the classic results, I know that there exists a subsequence such that $u_m\...
1
vote
0answers
25 views

About the proof of Minkowski generalized integral inequality

I am studying the generalized minkowski inequality for integrals: Let be $E\subset \mathbb{R^n}$, $F\subset \mathbb{R^m}$, $1\leq p \leq \infty$, $f$ measurable in $E\times F$. Then $$\left\| \...
0
votes
1answer
35 views

Showing relatively compact and finite dimension

I have a question where I have to show: Prove that a normed linear space is finite-dimensional if and only if every bounded subset is relatively compact. Show that the spaces $C[0,1]$ and $L^{2}(0,2 \...
3
votes
1answer
31 views

Definition of weak* convergence in $L^{\infty}$

I'm new to functional analysis and have (what I think is) a basic question on the definition of weak* convergence. Let $X$ be a normed linear space with dual $X^{\ast}$. According to the definition I ...
2
votes
1answer
22 views

Let $(a,b)$ and $f\in L_{\text{loc}}^1(a,b)$. For $x_0\in (a,b)$, $F(t)=\int_{x_0}^{t}f(s)ds$. Prove that, $DF=f$ (towards theoretical distribution).

Let $(a,b)$ and $f\in L_{\text{loc}}^1(a,b)$. For $x_0\in (a,b)$, consider $$ F(t)=\int_{x_0}^{t}f(s)ds. $$ Prove that, $DF=f$ (towards theoretical distribution). I thought of the following: Let $\...
1
vote
1answer
41 views

Let $u:\Omega\subset \mathbb{R}^n\rightarrow \mathbb{R}$ a measureble function. Prove that $u\in L^p(\Omega)$

The function $u:\Omega\rightarrow \mathbb{R}$ is such that $$ \sup\left\{\int_{\Omega}|u(x)|v(x): v(x)\geq 0 \text{ in }\Omega\text{ and }||v||_{L^q(\Omega)}\leq 1 \right\} $$ with $\frac{1}{p}+\frac{...
1
vote
1answer
63 views

How to show $\lim_{n \to \infty}f(x_n)=f(x)?$

consider the Banach space $\ell^1$, and let $e_i$ be the sequence $(0,\dots,1,0\dots)$, with $1$ in the $i$-th position. show that $\{e_i\}$ converges weakly to $0$ in $\ell^1$ but not strongly. My ...
0
votes
1answer
57 views

$X_n\stackrel{P}{\to} c$ implies $\mathbb{E}f(X_n){\to} f(c)$ for continuous (but possibly unbounded) $f$

Suppose that $X_n \stackrel{P}{\rightarrow} c$ as $n\to\infty,$ where $X_n, c$ are all positive. Let $f$ be a continuous (but possibly unbounded) function on $(0,\infty)$ such that $$\int_0^\infty |f(...
1
vote
1answer
30 views

Necessary and sufficient condition for $l_p$ space inclusion

I am trying to prove the following statement. Given $1<p,q<\infty$, $p<q$ iff $l^p \subset l^q$. The forward direction is easy. I having trouble with the opposite direction and I don't know ...
0
votes
1answer
51 views

Implication of Sobolev inequality

For $\varOmega \subset \mathbb{R}^n$, $n\ge2$, a bounded domain and $1 < q < 2 < p < \frac{2n}{(n-2)_+}$ I want to show that \begin{equation*} \| u \|_{L^p(\varOmega)}^2 \le C \left( \| \...
3
votes
2answers
73 views

$f\in L^1\implies \lim_{n\rightarrow\infty}f(n^2 x)=0$ a.e. $x\in\mathbb{R}$?

Can someone provide a hint for the following: $f\in L^1(\mathbb{R})\implies \lim_{n\rightarrow\infty}f(n^2 x)=0$ a.e. $x\in\mathbb{R}$ I can't really make heads or tails of it. Something makes me ...

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