All Questions
31
questions
0
votes
3
answers
88
views
Proving that $\iint f(x) g(x+y) dx dy \leq \int g^2(x) dx$
Let $D \subset \mathbb{R}^n$ be bounded, and $f,g: \mathbb{R}^n \rightarrow \mathbb{R}$. Furthermore let $f$ be nonnegative and such that $\int_D f dx= 1$.
I would like to prove that
$$\int_D \int_D f(...
- 6,277
1
vote
1
answer
95
views
How can I prove that $\Vert \int _Xfd\mu \Vert \leq \int _X\Vert f\Vert d\mu $?
Let $(X,\Sigma,\mu )$ be measure space. Denote by $\mathbb{R}^{m\times n}$ the set of the matrix $m\times n$. Suppose that $f:X\to \mathbb{R}^{m\times n}$ is a function whose coordinates are ...
- 1,209
2
votes
1
answer
79
views
Proving $\|fg\|_{L^1} \leq \|f\|_{L^p}^{\alpha} \|g\|_{L^q}^{1-\alpha}$
I am trying to prove the following inequality:
$$\|fg\|_{L^1} \leq \|f\|_{L^p}^{\alpha}\|g\|_{L^q}^{1-\alpha}$$
where
$$\quad 1= \frac1{p} +\frac1{q}, \quad 1 \leq p, q, \leq \infty.$$
In particular, ...
- 6,277
3
votes
0
answers
65
views
Understanding an the last step in proving the Poincaré inequality for smooth functions on $\mathbb{R}^n$.
I'm looking at the following: Let $u:X \to \mathbb{R}$ be an integrable and smooth function on a subspace $X \subseteq \mathbb{R}^n$ equipped with the Lebesgue measure $\lambda_n$, and let $B \...
- 491
1
vote
0
answers
26
views
$\int_{\mathbb{R}}|p(v-r,x)-p(u-r,x)|dx \leq C\frac{v-u}{u-r}$
Consider $p(u,x)=(4\pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x\in \mathbb{R}.$
Prove that there exists $C>0$ such that for all $0<u\leq v,r\in[u,v],\int_{\mathbb{R}}|p(v-r,x)-p(u-r,x)|dx \leq C\...
- 616
1
vote
0
answers
49
views
Hardy's Integral Inequality for $\alpha > -1$ for nonnegative functions with $p \in [1,\infty)$
I want to show that that for $d\mu(x) = x^\alpha dx, d\nu = x^{\alpha + p} dx$
We have
$$\int_0^\infty \left(\int_x^\infty f(t)dt\right)^p d\mu(x) \leq c \int_0^\infty f^p(x) d\nu(x) \quad \alpha < ...
- 39
6
votes
0
answers
135
views
If $f$ is real-valued bounded measurable and $\mu$ a complex measure, then $\left | \int_X f \mathrm d \mu \right | \le \int_X |f| \mathrm d |\mu|$
Let $(X, \mathcal X)$ be a measurable space. Let $\mu$ be a complex measure on $X$ and $|\mu|$ its variation. Then $|\mu|$ is a non-negative finite measure. By definition, $|\mu(B)| \le |\mu| (B)$ for ...
- 5,817
5
votes
3
answers
263
views
How to show that the integral inequality holds for vector-valued functions. [duplicate]
If I define the integral for $\int_a^b\textbf{F}(t)dt$ as $(\int_a^b F_1(t)dt, \int_a^bF_2(t)dt,\ldots, \int_a^bF_n(t)dt )$. How do I then show that
$$\left|\int_a^b\textbf{F}(t)dt \right|\le\int_a^b\...
- 1,262
3
votes
1
answer
226
views
For $0<p<1$ showing $\Big(\int_\Omega |f|^pd\mu\Big)^{1/p}\leq \frac{1}{R_0} + \Big(\int_\Omega |f_{R_0}|^p\Big)^{1/p} $
In this problem Limit of $L^p$ norm when $p\to0$, the writer states that for $0<p<1$ we have that
$$\Big(\int_\Omega |f|^pd\mu\Big)^{1/p}\leq \frac{1}{R_0} + \Big(\int_\Omega |f_{R_0}|^p\Big)^{1/...
- 2,567
3
votes
2
answers
105
views
$||E[X]|| \leq E[||X||]$
Let $X$ be a r.v. taking values in $\mathbb{R}^d,$ such that $X \in L^1.$ $||.||$ is an arbitrary norm on $\mathbb{R}^d.$ Prove that $$||E[X]||\leq E[||X||].$$
When $d=1,$ then letting $\theta=\frac{E[...
- 1,070
0
votes
1
answer
237
views
The Upper Bound of the $L^p$ norm of the maximal function
Let $f \in L^p(\mathbb{R}^n)$ Show for the maximal function that
$
||Mf||_{L^p} \leq 2 *5^{n/p}(p/(p-1))^{1/p} ||f||_{L^p}
$
if $1 < p < \infty$ and $||Mf||_{\infty} \leq ||f||_{\infty}$.
I ...
- 11
1
vote
1
answer
65
views
$\int_{3}^5 (1+|\xi|^2)^{s} d\xi \leq \int_{-1}^1 (1+|\xi|^2)^s d\xi$ for $s<0$?
Let $s<0$ and $D$ be a bounded subset of $\mathbb R^d$ and assume that $B_{D}\subset \mathbb R^d$ is the ball centered at the origin with $|D|=|B_{D}|$ (here $|S|$ denotes the Lebesgue measure of ...
- 337
0
votes
2
answers
112
views
Looking for an inequality relating $\int_Efg$ to the integrals $\int_Ef$ and $\int_Eg$
Let $f,g:E \to [0, \infty]$ be nonnegative, integrable functions. (I mean, Lebesgue integrable. $E \subseteq \mathbb{R}$.) Assume that $fg$ is also integrable.
I'm trying to look for an inequality ...
- 1,847
0
votes
2
answers
607
views
Show that for non-negative measurable functions $f,g$ with $fg \geq 1$ the inequality $(\int f^p)(\int g^p) \geq 1$ holds.
Let $\mu(\Omega)$ be a probability measure (i.e. $\mu(\Omega) = 1$), and let $f,g$ be non-negative measurable functions on $\Omega$ such that $fg \geq 1$. Show that $1 \leq (\int f^p )(\int g^p)$ for ...
- 2,662
1
vote
1
answer
62
views
Is $\int_{A_t} \frac{1}{|x_j|^{p}}\, dx \leq \sum_{j=1}^{d} \int_{A_{j,t}}\frac{1}{|x_j|^p}\, dx \leq \int_{A_t} \frac{1}{|x|^p} \,dx$?
Let $t>0$ and $A_t= \{ x\in \mathbb R^{d}: |x|>t\},$ $A_{j,t}=\{x\in A_t: |x_j|> |x_l| \ \text{for all} \ l\neq j \}.$
Question: (1) Can we say $A_t \subset \cup_{j=1}^d A_{j,t}$? (...
- 655