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 ...
1
vote
2answers
184 views
Proof of Clarkson's Inequality
Trying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that $$\left|\left|f+g\right|\right|_p^p + ...
2
votes
1answer
61 views
Continuous function bounded in $L^\infty$
Is a continuous (real-valued) function in $L^\infty$ a (everywhere-)bounded function?
1
vote
1answer
73 views
Need help in showing that $F(x)/x^{1/q}$ goes to $0$ as $x$ goes to $0$ and $\infty$.
$1<p<\infty$, $f\in L^{p}(0,\infty)$, $p^{-1}+q^{-1}=1$, define $$F(x)=\int_{0}^{x}f(t)dt,$$ then I need to show that $\frac{F(x)}{x^{\frac{1}{q}}}\rightarrow 0$ as $x\rightarrow 0$ and ...
1
vote
1answer
39 views
Extension of Fourier Transform
We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...
1
vote
1answer
14 views
Remainder of a series converges uniformly?
Let $B \subset \Bbb R^{\Bbb N}$ and $p \geq 1$. Suppose
$$
\sup_{u\in B}\sum_{n=0}^\infty |u_n|^p \leq 1,\qquad \sup_{u\in B}\sum_{n=0}^\infty |u_{n+1}-u_n|^p\leq 1
$$
Is it true that
$$
\sup_{u\in ...
1
vote
1answer
53 views
How to show that a certain function is in $L^p$?
How can I show that the function $ F(x)= \dfrac{|x| ^{-n+1}} { \log \frac{1}{|x|} } $, for $ 0 < |x| \leqslant \large\frac{1}{2} $ and $ F(x)=0 $, if $ |x|>\large\frac{1}{2} $, is in ...
0
votes
1answer
44 views
convergence in measure does not imply weak convergence
Suppose $\sup_n\|f_n\|_1<\infty$ and $f_n\rightarrow f$ a.e.. However it is not necessary that $f_n\rightarrow f$ weakly in $L^1$.
Can someone raise an example?
Thanks in advance.
0
votes
1answer
34 views
Square integrable function with bounded derivative goes to zero pointwise?
Prove the following: $f \in L^2(\mathbb{R})$ and $f'$ bounded $\Rightarrow$ $\lim_{|x| \to \infty}f(x)=0$.
In general, is it also true that $f \in H^1(\mathbb {R} ^n)$ $\Rightarrow \lim_{|x| \to ...
4
votes
0answers
130 views
Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.
Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$.
The following is from the book "Sobolev spaces" ...
4
votes
0answers
75 views
A question about functions in $L^p(E)$
I've been working on a problem. I found a couple of related questions on the site, but I was having a little bit of trouble clarifying everything. The goal is to show that given $f\notin L^p(E)$, ...
3
votes
0answers
74 views
The norm of an operator
Let $\rho(x)$ be a weight function in a unit sphere, such that
\begin{equation}
\begin{array}{l}
\displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\
\displaystyle 2. \rho(x)\in ...
3
votes
0answers
74 views
Weak $L^1$ as real interpolation space between $L^p$-spaces?
Let $\Omega$ be a measure space. We denote $L^{p,q}$ the usual Lorentz space. We use a real interpolation method $(\cdot,\cdot)_{\theta,q}$.
Suppose $1\leqslant p,q\leqslant \infty$. I know that if ...
2
votes
0answers
69 views
When does $|f*g|_{p}=|f|_{1}|g|_{p}$?
From Rudin, Real and Complex Analysis, 1st edition, Chapter 7, Problem 4
Suppose $1\le p\le \infty$, $f\in L^{1}(\mathbb{R}^{1})$, $g\in L^{p}(\mathbb{R}^{1})$. Show that the the integral defining ...
1
vote
0answers
23 views
Gauss–Ostrogradsky formula for Distributions
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $u\in W^{1,p}(\Omega)$, $v\in L^{p'}(\Omega)^N$ with $p\in (1,\infty)$. Let $\operatorname{div}(v)$ be the divergence of $v$ in the sense of ...
1
vote
0answers
24 views
Estimation of a scalar product
I encountered the following, which shouldn't be that hard, but I can't get my head around it.
The problem is the following estimate (part of a bigger equation, but here's just the difficult part):
...
1
vote
0answers
61 views
Inclusion maps on $L^p$
How to show for $1 \leq p < q < r$
the inclusion maps
$$L^p \cap L^r \rightarrow L^q$$
$$L^q \rightarrow L^p + L^r$$
are continuous.
where the norms are defined in the following:
$L^p$ ...
1
vote
0answers
161 views
Rademacher function and weak convergence
The function $r_{n}:[0,1]\rightarrow \{-1,1\}$ be defined by $r_{n}(t)=\operatorname{sgn}(\sin(2^{n}\pi t))$ is known as the $n$-th Rademacher function.a) Show that $r_{n}\xrightarrow{w}0$ in ...
1
vote
0answers
71 views
Uniform convergence in $L^p$-spaces
Let $f\in L^p(0,\infty)$, $p>1$. Show that $\int_0^\infty f(x)\frac{\sin xy}{x} dx$ converges uniformly in $y$ in every finite interval. Show also that $|g(t+y)-g(y)|\leqslant M|t|^{\frac{1}{p}}$.
...
1
vote
0answers
79 views
A continuous embedding.
If $2 \le q \le \infty$ and $2s >n$, then is there a continuous embedding $ H^s (\Bbb R^n ) \hookrightarrow L^q(\Bbb R^n) $ ? Here $H^s$ means a general Sobolev space for $s = 0,1,\cdots$.
1
vote
0answers
167 views
Convergent in $L^1(0,1)$ but not in $L^2(0,1)$ help understanding a paper from arxiv
http://arxiv.org/pdf/math/0205003v1
In around equation (1.1) the author says
"By necessity all authors have been led in one way or another to the natural approximation
$$F(n) := \sum_{a=1}^n \mu(a) ...
