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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0answers
24 views

What powers of $|x|$ belong to $L^1$?

Prove that $|x|^ {−qp} \in L^{1}(U)$, where $U=B_{1}(0)\subset \mathbb{R}^{n}$. I think I could use polar coordinates to facilitate the work but not sure if it is useful.
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1answer
26 views

$1 \le p < q < \infty$ implies $L^q \subset L^p$

Suppose $1 \le p < q < \infty$ and $(X,\mu)$ is a Lebesgue measure space. Also suppose $X$ is of finite measure. Prove that $L^q \subset L^p$. First, we use Holder's inequality and find ...
1
vote
2answers
48 views

Volterra-like operator is bounded

Define $T:L^2(\mathbb R) \rightarrow L^2(\mathbb R)$ by $$(Tf)(x)=\int_{-\infty}^x e^{-(x-y)} f(y) \, dy.$$ I would like to show that $T$ is bounded and that $$\lambda = \frac{1}{1+iw}$$ is in its ...
2
votes
2answers
36 views

“Scalar product” of two Lp spaces

I was reading the book A. Lasotta and M. C. Mackey, "Chaos, Fractals, and Noise: Stochastic Aspects of Dynamic", Springer, 1991 On page 27, they defined a ``scalar product'' as follows. Let ...
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0answers
14 views

Uniform convergence of convolution using density argument [on hold]

An example in using $L^p$ norm inequality: Example1.24. If p=q=2, or more generally if q=p′,then $r=∞$. In this case, the result follows from the Cauchy-Schwartz inequality, since for all $x ∈ ...
4
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2answers
61 views

The density of polynomials in the space of continuous functions on the unit ball of $\ell^p$

Let $$B = \{a : \|a\|_p \le 1\} \subset \ell^p(\mathbb{N})$$be the unit ball, endowed with the weak topology. For which $p$, where $1 < p \le \infty$, are the functions of the form$$f(a) = q(a_0, ...
1
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1answer
30 views

Let $([0,1],\mathcal{B}([0,1]),\lambda)$, $\lambda$ Lebesgue measure in $[0,1]$.

Show that if $f$ is $p$-integrable then, for each $\epsilon>0$, exists a function $h$ which is continuous in $[0,1]$ s.t. $\|f-h\|_p\leq\epsilon$. Is there any simpler way to show it than ...
4
votes
2answers
56 views

If $\mu(|f_n|^p)$ is bounded and $f_n\to f$ in measure then $f_n\to f$ in $L^1$

Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of real measurable functions s.t., (a) The sequence $\displaystyle(\int |f_n|^p\ \mathsf d\mu)_{n\in\Bbb{N}}$ is bounded. (b) The sequence ...
1
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1answer
21 views

Integrability of Riesz potential

Given $f\in L^1(\mathbb{R}^3)$, define $$\phi(x)=\int_{\mathbb{R}^3}\frac{f(y)}{|y-x|}\,dy.$$ I was able to show that $\phi$ exists for almost all $x$ (I used the Lebesgue differentiation theorem). ...
0
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0answers
42 views

Convergence in $L^2$ on $C([0,1])$ implies convergence in $L^1$.

If $f_n \to 0$ in $L^2$ on $[0,1]$, show that $f_n\to 0$ in $L^1$ on $[0,1]$, providing that $f_n$ is continuous for each $n$. I'm feeling really stupid right now, I can't seem to figure this out. So ...
3
votes
1answer
25 views

Does $(\ell^{1}(\mathbb Z), \cdot)$ have a bounded approximate identity?

Put $\ell^{1}(\mathbb Z)=\{f:\mathbb Z \to \mathbb C: \|f\|_{\ell^{1}}:=\sum_{n\in \mathbb Z}|f(n)|< \infty \}$ and we note that $\ell^{1}(\mathbb Z)$ is an algebra under pointwise multiplication. ...
2
votes
1answer
71 views

Suppose $\mu$ is a finite measure on the Borel sets of $R$ such that $f(x) = \int_R f(x + t) \mu(dt)$ a.e., show $\mu(\{0\}) = 1$.

Problem statement: Suppose $\mu$ is a finite measure on the Borel sets of $R$ such that $f(x) = \int_R f(x + t) \mu(dt)$ a.e., whenever $f$ is real-valued, bounded, and integrable. Show $\mu(\{0\}) = ...
12
votes
1answer
138 views

$L^2(\mathbb{R})$ sequence such that $\sum_{n=1}^{\infty}\int_{\mathbb{R}}f_n(x)g(x)d\mu(x)=0$

I am currently studying for an analysis qualifying exam, and this problem has been bothering me: Suppose we have a sequence $\{f_n\}$ in $L^2(\mathbb{R})$ such that ...
0
votes
1answer
34 views

Conditions to ensure nice integrability of supremum of difference on neighborhood

Let $u\in L^1(\mathbb{R}^n)$. Based on that alone, can I say anything nice about the following integral? $$ \int\limits_{\mathbb{R}^n}{\sup\limits_{|y|\le h}{|u(x+y)-u(x)|} \text{ d}x} $$ Ideally, the ...
1
vote
1answer
29 views

Pointwise boundedness implies function included in $L^\infty$?

I am trying to show that $L^\infty$ is complete, and I am struggling to show that the limiting function I have come up with is in $L^\infty$. Here is what I have so far: We let $f_n$ be a Cauchy ...
3
votes
1answer
33 views

$\left \| \left \| f \right \|_{L^{p}} \right \|_{L^{q}} \leq \left \| \left \| f \right \|_{L^{q}} \right \|_{L^{p}} $ for $0<p\leq q$

Let f be bounded on $X\times Y$ measure space with $\mathbb{P}\times\mathbb{Q}$ probability measure, show that for $0<p\leq q$: $\left \| \left \| f \right \|_{L^{p}(\mathbb{P})} \right ...
0
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0answers
53 views

Metric-space complete?

My question is if a specific metric-space is complete, respectively under which conditions it is complete. I am rather a newby, but hope that the question is understandable. The metric-space is ...
0
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0answers
28 views

How to arrive at the answer $x_k= a^{-\frac{1}{p}}$?

My question is: How to arrive at the answer $x_k = a^{-\frac{1}{p}}$? I differentiated the expression Lagrangian that the author has defined w.r.t lambda and equated it to zero giving me summation ...
3
votes
1answer
38 views

Show that $\mu(f)\mu(1/f)\geq\mu(\Omega)^2$

Prove that $\mu(\Omega)^2\leq\int f \,d\mu\int\frac{1}{f}\,d\mu$. I don't know if that what I did is correct or if it will help to solve the problem, but here it is: Using the Hölder inequality ...
3
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2answers
53 views

For what $p$ is $\frac{1}{(x(1+\ln(x)^2))^p}$ Lebesgue integrable?

I'm trying to use the fact that given $f:[a,\infty)\to\mathbb{R}$ Riemann integrable for every closed interval $[c,d]\subset [a,\infty)$, then $f$ is Lebesgue integrable if, and only if, ...
4
votes
1answer
20 views

Weak $L^p$ implies strong $L^q$ for $q<p$

Another prelim question... Suppose $0<q<p<\infty$, and $E\subseteq \mathbb{R}^n$ has finite measure. Suppose $f$ is in weak $L^p$, i.e. $\lambda(|f| > t) \leq N/t^p$. Show $f \in L^q(E)$ ...
2
votes
1answer
56 views

On the space $L^0$ and $\lim_{p \to 0} \|f\|_p$

For $0 < p < \infty$, the definitions of the spaces $L^p$ are very natural. Then, we of course want $L^\infty$ and $L^0$ to be some kind of limits of $L^p$ spaces. What does the parameter $p$ ...
0
votes
1answer
29 views

How to show that $L^p$ norm is monotone increasing?

I am trying to solve the following (very standard) exercise: Let $(X,\mathcal M,\mu)$ be a measure space and $f\in L^r\cap L^\infty$ for some $1\leqslant r<\infty$. Then $f\in L^p$ for ...
1
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0answers
15 views

The convolution of compact $L^1$ weighted function is still $L^1$

Some old discussion can be seen here. Let $\omega\in L^1_{\text{loc}}(\mathbb R^N$) be given as a weight function. We further assume that $\omega\geq 1$, l.s.c, and satisfies, for a constant ...
3
votes
1answer
74 views

Taking the limit $\lim_{p\rightarrow \infty} \left( \frac{\|f\|_\infty}{\|f\|_p}\right)^p$

Taking the limit $$\lim_{p\rightarrow \infty} \left( \frac{\|f\|_\infty}{\|f\|_p}\right)^p$$ First I think the expression after taking the limit will depend on the function $f$. In my attempt, ...
3
votes
1answer
67 views

Functions by which one can multiply elements of $L^1_{\text{loc}}$

Let $u$, $\omega\in L^1_{\text{loc}}(\mathbb R^N$) be given. We further assume that $\omega\geq 1$, l.s.c, and satisfies, for a constant $C>0$, $$ \frac{1}{|B(x,r)|}\int_{B(x,r)}\omega(y)dy\leq ...
3
votes
0answers
43 views

Doob's inequalities for not necessarily right-continuous martingales

In Revuz and Yor, they denote $\mathbb{H}^2$ the space of $L^2$-bounded martingales, and $H^2$ the space of continuous $L^2$-bounded martingales. They state "... by Doob's inequality ... ...
5
votes
1answer
46 views

Assume that $f_n\to f$ in measure and $\sup_n\|f_n\|_{L^p(E)}<\infty$ for some $p>1$. Prove that $f_n$ converges to $f$ in $L^1$ norm.

Let $\{f_n\}$ and $f$ be Lebesgue measurable functions on $E$ where $|E|<\infty$. Assume that $f_n\to f$ in measure and $\sup_n\|f_n\|_{L^p(E)}<\infty$ for some $p>1$. (a) Prove that ...
0
votes
4answers
42 views

If $X_n\to X$ in $L^p$, then $E(X_n)^p \to E(X)^p$

Here's what I did: I want to prove that $|E(X_n)^p -E(X)^p| \to 0$. $|E(X_n)^p -E(X)^p| \leq E(|X_n^p-X^p|)$ But this isn't necessariliy $\leq E(|X_n-X|^p)$ (which converges to zero). Take as a ...
1
vote
1answer
22 views

“$f\ge 0$, $g>0$, $fg\in L^1(\mathbb{R})$ and $g\notin L^1(\mathbb{R})$” implies “$f$ is integrable over $[-r,r]$”?

Let $f\ge 0$ and $g>0$ be such that $fg\in L^1(\mathbb{R})$ but $g\notin L^1(\mathbb{R})$. Can we get the following conclusion: $f$ is integrable over $[-r,r]$ for any $r>0$ ? Intuitively I ...
1
vote
1answer
52 views

Why is $\mathfrak{L}^1$ not a vector space of functions that take infinity as a value

In measure and integration theory we sometimes work with measurable functions $$ f: (X, \mathfrak{A}) \to (\overline{\mathbb{R}}, \overline{\mathfrak{B}}),$$ where $(X,\mathfrak{A})$ denotes an ...
0
votes
1answer
41 views

Is the convolution pointwise bounded?

A problem from an old exam: Prove or disprove: if $p,q \in [1,\infty)$ such that $p^{-1}+q^{-1}=1$ and $f\in L^p, g\in L^q$, then the convolution $f*g$ is pointwise bounded. First of all: what ...
2
votes
0answers
46 views

Weak convergence and trace operator

Suppose that $u_j\rightharpoonup u$ in $W^{1,p}(\Omega)$ (notice the weak convergence), with $\Omega\subset \mathbb{R}^3$ regular enough. Let $v_j=Tu_j$, and $v=Tu$, where $T:W^{1,p}(\Omega)\to ...
-1
votes
1answer
61 views

Why is $f(x) = \sin(x)$ an element of $L^2(-\pi, \pi)$ not $L^2(a,b)$ [closed]

I am having some trouble understanding why some functions are members of $L^2(\mathbb{R})$ whereas other functions are members of some restricted subset of $\mathbb{R}$ such as $(-\pi, \pi)$ Can ...
2
votes
0answers
35 views

Factorization of $L^{p}$ spaces

Is the following theorem true and if so does it have a name? Theorem: Let $p$, $q$, $r\in\left(0, \infty\right]$ with $\frac{1}{q}=\frac{1}{p}+\frac{1}{r}$. Every $L^{q}$-function is the product of a ...
0
votes
1answer
36 views

Product of an $L^\infty$ function and an $L^1$ function is integrable

For every $f \in L^1(\mathbb{R^n})$ let $$ \hat{f} : \mathbb{R}^n \to \mathbb{C}, \quad \hat{f}(\xi) := \frac{1}{(2\pi)^{n/2}} \int_{\mathbb{R}^n} f(x) \exp{(-i \langle x, \xi \rangle)} ...
3
votes
2answers
89 views

pointwise almost everywhere convergent subsequence of $\{\sin (nx)\}$

Can you prove or disprove that the sequence $\{\sin (nx)\}$ has a pointwise convergent almost everywhere subsequence with respect to the Lebesgue measure on $\mathbb{R}$ ? Edit: I am adding my ...
2
votes
1answer
37 views

Specific question on $l^p$ spaces and its dual in weak * topology

I am covering now Lp spaces in my summer real analysis course and this problem from Folland related to the dual of Lp stumped me hard, it is problem 19 chapter 6 reads as follows: We define $ ...
3
votes
3answers
70 views

Multiplication operator on $L^1$

Let $\phi :X \rightarrow \mathbb{C}$ be measurable with respect to the measure space $(X,\mu)$. Suppose that $\phi f \in L^1(\mu)$ whenever $f \in L^1(\mu)$. Define $M_{\phi}(f)=\phi f$, for $f \in ...
2
votes
1answer
35 views

A form of Nash's inequality, $\|f\|_2\le C \|f\|_{1}^{\alpha} \|f'\|_2^\beta$

For $f\in \mathcal{S}(\mathbb{R})$ can anyone help me prove the following Nash inequality, $$\|f\|_2 \le C \|f\|_{1}^{\alpha} \|f'\|_2^\beta.$$ I believe $\alpha$ and $\beta$ should be $2/3$ and ...
4
votes
2answers
44 views

Why can the elements of $L^\infty$ be approximated in $L^p$ by $C^1$-functions?

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $f\in L^\infty(\Omega)$. From which theorem does the existence of $(f_k)_{k\in\mathbb N}\subseteq C^1(\overline\Omega)$ with ...
1
vote
2answers
38 views

Show that the series converges ($l^2$)

I know that $\sum_{k=1}^\infty|y_n|^2=S<\infty$. I also have that $\lambda >1$. I need to show that $$ \sum_{k=1}^\infty \left| \frac{y_1}{\lambda^k} + ...
0
votes
0answers
26 views

Continuity of translation property [duplicate]

Let $u \in L^{p}(U)$ where $1 \leq p \lt \infty$ & $U \subseteq \mathbb R^{n}$ . Define : $F : \mathbb R^{n} \to L^{p}(U) $ by $ F(y) := u(x+y)$ . Prove that: as a function of $y$ ; $F(y) $ is ...
3
votes
1answer
51 views

Question on $L^p$ spaces defining metric

First question here so really excited and hope you can help me, thanks! In my intro to functional analysis class we just now covered $L^p$ spaces and I was presented with this homework question: ...
0
votes
2answers
33 views

Absolutely continuous function whose derivative is in $L^2([0,1])$ etc., evaluate $\lim_{x\to0^+}\frac{f(x)}{\sqrt{x}}$

Suppose $f$ is absolutely continuous on $[0,1]$, $f(0)=0$, and $f'(x)\in L^2([0,1])$. Show that $$\lim_{x\to 0^+}\frac{f(x)}{\sqrt{x}}=0$$ So far I've got the following: Since $f$ is AC by the ...
2
votes
4answers
79 views

Are there relations between elements of $L^p$ spaces?

I have read about dual spaces and the relation $1/p+1/q=1$ as mentioned in the Wikipedia page. Are there any more theorems or relations that connect elements between the $L^p$ spaces for different ...
1
vote
1answer
43 views

$\{x \mapsto e^{2\pi i k x} \mid k \in \mathbb{N}\}$ is orthonormal basis of $L^2$

I want to show that $\{x \mapsto e^{2\pi i k x} \mid k \in \mathbb{N}\}$ is orthonormal basis of $L^2((0,1); \mathbb{C}) =: X$. Of course the only problem is to show completeness. In our lecture we ...
1
vote
0answers
47 views

How to apply Sobolev inequalities?

I'm struggling with an application of Sobolev inequalities in Evans' book. He presents his argument like this: For $4<p<5$ we have $2(p-1)=2(p-4)+6=2(p-4)+ 2^*$ and therefore $$\left( ...
4
votes
2answers
114 views

Are the Hermite-Gauss functions linearly dense in $L^1(\mathbb{R})$?

The Hermite-Gauss functions ($t\mapsto H_m(t)e^{-t^2/2}$) are known to be an orthonormal basis for $L^2(\mathbb{R})$, a fortiori linearly dense in $L^2(\mathbb{R})$, and all are in the Schwartz space ...
4
votes
1answer
59 views

Is $A=\{x \in \ell^2 \mid \sum_{n=1}^{\infty} \frac{x_n}{n}=0 \}$ dense in $\ell^2$

I think that the answer is no I thought quite a bit about this problem. My idea was to build a sequence $(y_n)_{n \in \mathbb{N}} \subset A$ such that given a $x \in \ell^2$ we pick the first N ...