4
votes
0answers
25 views
+50

Does the Gagliardo–Nirenberg interpolation inequality hold on compact closed manifolds?

The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on ...
0
votes
1answer
26 views

An application of Holder's inequality to show one norm is smaller than another

Let $p(s) = r(s) + m-1$ where $r:[0,T) \to [q,\infty)$ where $q \geq 2$ and $m > 1$ is fixed. Let $\text{Vol}(\Omega) = 1$. Then can we show that $$\lVert u \rVert_{L^{r(s)}(\Omega)} \leq ...
3
votes
1answer
34 views

need to prove an inequality with absolute value to the power of positive number

I need help to prove the inequalities in the following cases $ ||x|^p-|y|^p|\leq \begin{cases} |x-y|^p & \mathrm{if} \, 0<p<1\\ p|x-y|(x^{p-1}+y^{p-1}) & \mathrm{if} \, 1\leq p<\infty ...
0
votes
1answer
35 views

Minkowski inequality for $l_p$ norm.

I'm trying to prove the Minkowski inequality for the $l_p$ norm: $$ \| f + g\|_p \le \|f\|_p + \|g\|_p $$ where $f,g : \mathbb{R}^n \rightarrow \mathbb{R}$ are Lebesgue measurable functions and $p ...
0
votes
0answers
25 views

How about integral version of Holder's inequality?

In light of the fact that Minkowski's inequality have integral version, I thought there might be one for Holder's as well. I cannot find any through searching (there is an infinite product version in ...
1
vote
0answers
49 views

Hardy Littlewood maximal function and integral comparison.

Define the Hardy Littlewood maximal function $$g^*(y)=\sup \left\{\frac{1}{|B|}\int_B|g(x)|dx:B\text{ is any open ball containing y}\right\}.$$ For given $x_i,r_i,a_i$, first I have shown that ...
5
votes
1answer
62 views

An inequality of $L^p$ norms of linear combinations of characteristic functions of balls

Let $1<p<\infty$. Let $(a_n)_{n=1}^\infty$ be a sequence of nonnegative real numbers and $\{B_{r_i}(x_i)\}_{i=1}^\infty$ be a sequence of open balls in $\mathbb{R}^n$. Prove that there exists ...
1
vote
1answer
21 views

Prove that $||f||_2 \le \sqrt{2 \pi} ||f || _{\infty}$

Let $||f||_2=\sqrt{\int_{-\pi}^{\pi} f^2(x) dx}$ $||f||_{\infty}=\sup \{ |f(x)| \mid x \in [-\pi,\pi]\}$. Suppose $f: \mathbb{R} \to \mathbb{R}$ an in the space of piecewise continuous functions ...
4
votes
3answers
70 views

How to apply the Hölder's inequality in a clever way?

Here is the problem: Let $f\in L^p(\mathbb R^n)\cap L^q(\mathbb R^n)$ and $s\in[p,q]$. Show that $f\in L^s(\mathbb R^n)$ I'm almost sure that this is a simple exercise on Hölder's inequality yet ...
1
vote
0answers
57 views

Don't understand a PDE argument involving $L^p$ norms and inequalities

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see ...
0
votes
0answers
15 views

Does the inequality $\int_{\Omega}(-\Delta)^{\frac 12}G(w(x))(u(x)-C)^+ \geq 0$ hold? If not, can we bound it from above in a particular way?

Let $G$ be a locally Lipschitz function such that $G(0)=0=G'(0)$ and $G$ is also increasing. I want to know if $$\int_{\Omega}(-\Delta)^{\frac 12}(G(w(x))(u(x)-C)^+ \geq 0$$ where $C$ is a constant. ...
0
votes
1answer
36 views

Is it true $({\Bbb E|X|^p})^{\frac{1}{p}} \leq ({\Bbb E|X|^q})^{\frac{1}{q}}$, if $p \leq q$?

I ran into this post which shows $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$, $p \leq q$. So I guess it's true $({\Bbb E|X|^p})^{\frac{1}{p}} \leq ({\Bbb E|X|^q})^{\frac{1}{q}}$, if $p \leq q$, ...
0
votes
1answer
146 views

Cauchy–Schwarz inequality on vector-valued L2 space

Let $f$ and $g$ be square-integrable, $\mathbb{R}^n$-valued functions, i.e., $$ \| f \|_2^2 = \int \|f(t)\|^2 dt < \infty $$ where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$. I am looking ...
2
votes
1answer
57 views

Young's inequality

Let $U \in L^1(\mathbb{R}^d)$ and $\rho \in L^1(\mathbb{R}^d)$ such that $\rho \ge 0$ and the support of $\rho$ is included in $B(0,1)$ (the euclidean unit ball of $\mathbb{R}^d$). Is there a way to ...
0
votes
0answers
41 views

Expression for Lp Norm [duplicate]

Use Holder inequality $(|\sum_{i=1}^n x_n\cdot y_n| \le ||x||_p\cdot ||y||_q)$ to prove that for each $x\in \Bbb R^n$: $$||x||_p=\sup_{||y||_q\le 1} {|\sum_{i=1}^n x_n\cdot y_n|} $$ tried to find for ...
-2
votes
1answer
35 views

Is it true that,$ \|f\|^p_{L_p} \le 2^{p-1}\|u'\|^p_{L_p}+2^{p-1}|u(\zeta)|^p $?

Is it true that,$$ \|f\|^p_{L_p} \le 2^{p-1}\|f'\|^p_{L_p}+2^{p-1}|f(\zeta)|^p $$ for $f$ is $C^1$ in $[0,1]$ and $\zeta$ in [0,1]
4
votes
1answer
55 views

Is it true that $2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}$?

For any $C^1$ function defined in $(0,1)$, is it true that $$ 2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty} $$ If it is true, how to prove it?
0
votes
1answer
43 views

Is it true that $2^p\|f\|^p_{L_p} >2\|f\|^p_{L_\infty}$?

For any $C^1$ function defined in $(0,1)$, is it true that $$ 2^p\|f\|^p_{L_p} >2\|f\|^p_{L_\infty} $$ If it is true, how to prove it?
1
vote
2answers
80 views

What is name/references of inequality bounding sup-norm by $L_2$ norm (or a similar variant of this)?

I think you have something like the following inequality in most finite dimensional spaces or sufficiently restricted infinite dimensional space: $$ \|g\|_{\infty}\lt C\|g\|_2$$ where $C$ would ...
4
votes
1answer
67 views

Proving an inequality about an $L^2$ function

Let $u \in L^2(\mathbb{R}^2)$ be a function of two variables $x$ and $y$. I want to know if there is a relation between the Fourier tranform (with respect to $x$) of the $L^2$ norm (with respect to ...
1
vote
0answers
38 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): ...
3
votes
1answer
240 views

Jensen's inequality and $L^p$ norms

Let $(X,\Sigma,\mu)$ be a probability space; in particular, $\mu(X)=1$. The integral form of Jensen's inequality can be phrased in terms of permuting a convex function $\varphi$ (say, with the ...
3
votes
2answers
194 views

Two proofs concerning Hölder's inequality

I am studying functional analysis and I have come across two statements which can be proven by using Hölder's inequality, but I don't know how/why. Reminder: let $a\in l^p$ and $b\in l^q$, then: $$ ...
4
votes
0answers
119 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 ...
2
votes
1answer
305 views

Upper and lower bounds of a ratio involving vector norms

I'm working on a signal processing problem and need to analyze the following expression $$ G = \frac{n}{\sum\limits_{i=1}^n |w_i|} \frac{ \sum\limits_{i=1}^n g_i w_i^2}{\sum\limits_{i=1}^n g_i |w_i|} ...
2
votes
1answer
87 views

How can I give a bound on the $L^2$ norm of this function?

I came across this question in an old qualifying exam, but I am stumped on how to approach it: For $f\in L^p((1,\infty), m)$ ($m$ is the Lebesgue measure), $2<p<4$, let $$(Vf)(x) = ...
1
vote
1answer
92 views

Estimating the integral in norm.

I want to estimate the integral $$\int k(x,y)f(y)dy$$assuming the fact that $k(x,y), f(y)$ are in $L^p, L^q$ respectively. But I want to bound the the whole integral in $L^r$, $r\in [1,\infty]$. I ...
4
votes
1answer
224 views

Proof of an inequality of $L^p$ norms

For a general measure space, we define : $\|f\|_p= \left(\int\vert f\vert^p du\right)^{1/p}$. Let $0 < a < b < c < \infty$ and prove the following: $$ \|f\|_b \leqslant \max\{\|f\|_a, ...
4
votes
1answer
517 views

Liapunov's Inequality for $L_p$ spaces

Let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in L_p \cap L_q $, then $$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q} \tag{*}$$ ...
4
votes
1answer
244 views

$L^2$ norm inequality

I need some help with this homework question. I was asked to provide an example of a $n$-dimensional subspace $W$ of $L^2[0,1]$ such that all functions in that subspace with $L^2$ norm equal to $1$ ...
7
votes
1answer
1k views

Inequalities in $l_p$ norm

I'm having difficulty with the following problem. Any help would be appreciated. Problem: Consider the sequence spaces $l_p$ with the usual norm. If $1\le p\le q\le \infty$, I want to show the ...