2
votes
0answers
54 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
1
vote
2answers
20 views

How to prove this inequality for operator and function

How to prove this? $\sum_{k=1}^{\infty}|(Tf)_k|^2\leq ||T||^2||f||^2$ where $T$ is an operator and a function $f$. $(Tf)_k$ is the $k$-th coordinate of Tf. Should this involve Cauchy-Schwarz or the ...
1
vote
0answers
42 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 ...
2
votes
1answer
44 views

Poincaré inequality for a subspace of $H^2(\Omega)$

Suppose that $\Omega\subset\mathbb{R}^d$ is a smooth, bounded, and connected domain. Let \begin{equation} H=\{u\in H^2(\Omega):\int_\Omega u(x) dx=0 ~\text{and}~ \nabla u\cdot v=0~ ...
0
votes
0answers
53 views

Help with Gronwall's Inequality

I'm trying to solve an extra credit hw problem that has to do with Gronwall's Inequality. I understand (or think I do) how to find an upper bound when $p=0$, but I'm unsure of how to handle the extra ...
4
votes
1answer
102 views

A inequality of calculus [duplicate]

Let $f \in C^2[a,b]$ and $f(a) = f(b) = 0$, $f'(a) = 1$,$f'(b) = 0$, prove that $$\int_a^b|f''(x)|^2\,dx \geq \frac{4}{b-a}$$ Remark: This question is in the book functional analysis of Peking ...
0
votes
1answer
28 views

Find a counter-example for inequality

Let $C>1$ be a constant. I have to find polynomial $p(t)=a_0+a_1 t+\dots +a_n t^n$ such that: $$|a_0|+|a_1|+\dots + |a_n| \le C \sup_{t\in[0,1]} |p(t)|$$ doesn't hold. Any tip?
0
votes
1answer
34 views

when one can expect $\left \| |x|^{2}x- |y|^{2}y \right \| \leq \frac{1}{2} \left \| (x-y) \right \|$?

Suppose $(X, \left\|\cdot \right \|)$ is Banach algebra with the property that $\left\| (|x|^{2}x) \right\| \leq \left\|x\right \|^{3};$ for every $x\in X.$ Let $y_{0}\neq 0 \in X$ and fix it; and ...
0
votes
0answers
24 views

Why this relation in Hilbert space (with inner product $< >$) holds?

$c_k$, $f_k$ are sequences in Hilbert space, $g$ is a function. Why this relation below holds? How you derive it? $\sum_k|c_k\left \langle f_k,g \right \rangle|\leq(\sum_k|c_k|^2)^{1/2}(\sum_k|\left ...
0
votes
1answer
30 views

How to get this inequality using induction (analysis)

Consider the following functions $\theta:\Bbb R\to\Bbb R$ and $\Theta:\Bbb R^n\to\Bbb R$, sucha that: $$ \theta(x) := \begin{cases} 1-|x| & \text{if $|x|\le1$} \\ 0 & \text{if $1\le|x|$} \\ ...
2
votes
2answers
69 views

Directional derivative in a Sobolev-like inequality

I am trying to do the following problem: Let $\Omega \subset \subset \overline{\mathbb{R}_{+}^{n+1}} = \{ (x, x_{n+1}); \, x_{n+1}\geq 0\}$ (i.e., $\Omega$ is bounded inside the closed upper ...
2
votes
1answer
19 views

Integral Inequality Question: Effect of Exponents

Is it true that, if $\alpha q < (1-\alpha)q$ then $\int|k(x,y)|^{\alpha q}dy \leq \int|k(x,y)|^{(1-\alpha)q}dy$, where $k(x,y)$ is a positive measurable function?
2
votes
0answers
47 views

Unexplained “It Suffices to Show”

For $u \in C^{\infty}(\Omega)$, for $\Omega$ convex, and $p$, $q$ such that $\frac{1}{p}-\frac{1}{q}<n$, then in order to show that $$||u-u_{\Omega}||_{L^{q}} \leq c_{n} \left[ ...
1
vote
1answer
34 views

Need help filling in the details of proof of Jensen's Inequality

In a book on PDEs that I'm reading, I am trying to fill in the details of the proof of Jensen's Theorem, and am having a little trouble with the algebra. Here is the statement of Jensen's Theorem in ...
1
vote
1answer
86 views

Gagliardo Nirenberg Sobolev inequality for n >= 2

I have a quick question regarding the Gagliardo-Nirenberg-Sobolev inequality. It states that: Assume $1 \leq p < n$. There exists a constant $C$, depending only on $p$ and $n$, such that ...
1
vote
0answers
53 views

$L^{\infty}$ is p-concave

I want to show that the $L^{\infty}$ on a Banach lattice $X$ is $p$-concave with $M_{(p)}(L^{\infty})=1$. Where $L^\infty=L^\infty(X,\mathcal{M},\mu)$. Recall that a Banach lattice $X$ is said ...
0
votes
1answer
56 views

Using Cauchy Schwarz for functions in Sobolev Space

Hi I just want to confirm something simple and check that the following is allowed: The Cauchy-Schwarz inequality states if $A = ((a_{ij}))$ is a symmetric, non-negative $n \times n$ matrix then ...
1
vote
1answer
57 views

inequality about linear and piecewise constant interpolation?

$\Omega\subset\mathbb{R}^3$ is a bounded, and $u(\mathbf{x},t) \in C\big(0,T,L^2(\Omega)\big)$. We divide the interval $[0,T]$ in $N$ equal subintervals with the time step $\tau$. With the notaion $$ ...
2
votes
1answer
121 views

fraction power vector-norm inequality

If X is a Banach Space and $x,y\in X$. Then by the definition of a Banach algebra we know $$\|x.y\|\leq\|x\|\|y\|$$ and thats how we can have relation for any positive power. i.e. $n\in N$, ...
1
vote
2answers
82 views

Inequality involving Lp space, Holder Space, Sobolev Space

Do we have the following inequality or some variation of this: $||u||_{L^{p}(U)} \leq ||u||_{C^{0,\gamma}(\bar{U})} \leq ||u||_{W^{1,p}(U)}$ for $n < p \leq \infty$ for $u \in W^{1,p}(U)$ where $U$ ...
4
votes
2answers
73 views

Using Results in Sobolev Spaces

I have two questions about using results in Sobolev Spaces and a last one on an inequality involving Holder spaces: 1.The Gagliardo-Nirenberg-Sobolev Inequality states the following: Assume $1 \leq p ...
0
votes
0answers
44 views

Explanation on the proof of the continuous Hardy inequality

Here there is a proof of the continuous Hardy inequality (theorem 2). I would like a explanation on the following passage. ...
0
votes
1answer
60 views

Prove x(t) is bounded given a integral inequality

I want to answer the following question: $x=x(t)$ is defined and continuous on $[0,T)$ and satisfies an integral inequality $$1 \leq x(t) \leq A_1 + A_2\int_0^t x(s)\big(1+\log x(s)\big) ds$$ for ...
4
votes
2answers
83 views

Using the Extension Operator Theorem for Sobolev Spaces

I want to know if certain conditions hold after applying the Sobolev Extension Theorem: Assume $U$ is a bounded open subset of $\mathbb{R}^{n}$ and $\partial U$ is $C^{1}$. Suppose $1 \leq p < n$. ...
0
votes
2answers
85 views

How to prove Schwarz inequality for Hermitian forms?

I'm trying to do something like the proof of the Schwarz inequality for inner product. If $h(y,y)\neq 0$, then we can take $\alpha=-h(x,y)/h(y,y)$ and calculate $h(x+\alpha y,x+\alpha y)$ which is ...
0
votes
0answers
17 views

Using Sobolev-Gagliardo-Nirenberg [duplicate]

I am currently studying a proof of a General Sobolev Inequality. I have the following question: Consider the Sobolev Space $W^{k,p}(U)$. With the added assumption that $k > \frac{n}{p}$. Let $l = ...
1
vote
1answer
105 views

Using Sobolev-Nirenberg-Gagliardo

I am currently studying a proof of a General Sobolev Inequality. I have the following question: Consider the Sobolev Space $W^{k,p}(U)$. With the added assumption that $k > \frac{n}{p}$. Let $l = ...
2
votes
0answers
95 views

Using the Sobolev-Nirenberg-Gagliardo inequality in a proof

If $1 \leq p < n$. The Gagliardo-Nirenberg-Sobolev inequality states that there exists a constant C such that $||u||_{L^{p^{*}}(\mathbb{R}^{n})} \leq C||Du||_{L^{p}(\mathbb{R}^{n})}$ for all $u ...
0
votes
0answers
29 views

Lower-Upper bounds on the cardinality of a bounded set

Let $S$ be a finite set which is a subset of $\{(\alpha ,\beta ):\alpha , \beta \in \mathbb{Z}, \alpha\geq 0, \beta \geq 0\}$ and $ T(x,y)=\sum_{(\alpha ,\beta ) \in S} h_{\alpha, \beta} ...
0
votes
0answers
30 views

How to prove a duality of $L^p$ spaces? [duplicate]

Let $(\Omega,\Sigma,\mu)$ be a finite measure space and $f:\Omega\longrightarrow \mathbb{R}$ be a measuable function. Let $1\leq p< \infty$ and $1/p+1/q=1$. Prove that the following are equivalent: ...
3
votes
2answers
120 views

Poincare Inequality implies Equivalent Norms

I am currently working through the subject of Sobolev Spaces using the book 'Partial Differential Equations' by Lawrence Evans. After the result proving the Poincare Inequality it says the following ...
2
votes
1answer
77 views

How to prove $a_1^Ta_1+a_2^Ta_2\le b_1^Tb_1+b_2^Tb_2.$

Let $p,q > 0$, $a_1, b_1\in \mathbb{R}^m, a_2,b_2\in\mathbb{R}^n$ be vectors. Given that ...
0
votes
1answer
83 views

An inequality involving norms.

I have to know how we can show the following inequality: $\|u\|_{2}\leq\|u_{0}\|_{2}+\int_{0}^{t}\|u_{t}(t,x)\|_{2}dt$ where $\|u\|_{2}=\Big(\int_{\Omega}u^{2}dx\Big)^{1/2}$, $u_{0}=u(x,0)$, ...
1
vote
2answers
448 views

show operator norm submultiplicative

We had in our lecture on numerical analysis the following: Let $\mathrm{Lin}(X,Y)$ be the set of all linear maps $X\rightarrow Y$. Let $A\in\mathrm{Lin}(\mathbb R^l,\mathbb R^n)$ and ...
3
votes
1answer
143 views

Help with proving the Fatou's Lemma for discrete functions

I need help to understand two steps in this proof for the Fatou's lemma in its discrete version. Let $f_k:\mathbb{N} \rightarrow [0, \infty), k\in\mathbb{N}$, be a sequence of functions. Then $$ ...
0
votes
2answers
139 views

solution of the equation with exponential function

Let $m, n$ be an integers. Let $b \in R$. Solve the following equation for $n$. $$ \exp(m-\frac{2}{\pi}n)=n^{b/{\pi}}. $$ Thank you.
5
votes
1answer
61 views

A basic question on Type and Cotype theory

I'm studying basic theory of type and cotype of banach spaces, and I have a simple question. I'm using the definition using averages. All Banach spaces have type 1, that was easy to prove, using the ...
5
votes
1answer
118 views

An operator inequality

I would be most thankful if you could help me prove the following operator inequality. Let $A$ be an arbitrary linear operator on a Hilbert space, satisfying $$\left\|AA^{\ast} - A^{\ast}A\right\|\leq ...
0
votes
1answer
49 views

Operator inequality

I would be most thankful if you could help me prove the following inequality. Let $A$ be a linear Hermitian positive operator on a Hilbert space. Then show that $$\langle x,A^{2}x\rangle\geq \langle ...
1
vote
2answers
76 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 ...
7
votes
2answers
262 views

Understanding why Minkowski’s inequality doesn't hold true for $0 < p < 1$?

The triangle inequality given by $\left(\sum_{i=1}^n |x_i+y_i|^p\right)^{1/p}\leq \left(\sum_{i=1}^n |x_i|^p\right)^{1/p} + \left(\sum_{i=1}^n |y_i|^p\right)^{1/p}$ is known as “Minkowski’s ...
2
votes
0answers
70 views

Norm inequalities in a reflexive space

I am reading an article about reflexive spaces, with a specific example. The article mentions inequalities that I haven't been able to get around to. Here's the setup. The space $X = (\prod_n ...
1
vote
3answers
118 views

$\|f*g\|_q\leq \|g\|_q \|f\|_1$ and $\|f*g\|_\infty\leq \|g\|_q \|f\|_{q^{'}}$, $(1/q+1/q^{'}=1)$?

Now I'm reading the Young inequality. It says that if $f \in L^p(R)$, $g \in L^q(R)$, $1\leq p,q\leq \infty$, $1/p+1/q\geq 1$. Then how could we have the following inequalities: $$\|f*g\|_q\leq ...
2
votes
1answer
122 views

Inequality for norms

Let g(x, y) be function on $X\times Y$. Show that for all $p\geq q$ $$ \|\,\|g\|_{L^q(Y)}\,\|_{L^p(X)}\leq \|\,\|g\|_{L^p(X)}\,\|_{L^q(Y)} $$ Thsnk you.
2
votes
0answers
50 views

Inequality with $\|\cdot\|_p$ norm

Let $x_1, \ldots, x_{2m}$ be $\{0,1\}$ Bernoulli random variables, i.e. variables which takes values $0$ and $1$ with equal probability. Let $S_m$ be group of all permutations $\pi$ on $\{1, \ldots, ...
-1
votes
1answer
33 views

An inequality : $ \| u^3 \|_{H^3} \leq C ( \| u \|_{L^\infty}^2 + \| \partial_x u \|_{L^\infty}^2 ) \| u \|_{H^3}$

Let $u=u(x)$ be a real-valued function defined on $\mathbb R$. How does this inequality hold? $$ \| u^3 \|_{H^3} \leq C ( \| u \|_{L^\infty}^2 + \| \partial_x u \|_{L^\infty}^2 ) \| u \|_{H^3}$$ ...
1
vote
1answer
58 views

product of bounded linear operators

If I have 2 bounded linear operators $T_1,T_2$ such that $T_2:X\rightarrow Y$ and $T_1:Y\rightarrow Z$. I know that by boundedness, $||T_2(x)||\leq||T_2||\,||x||$ and using the norm of $T$ defined as ...
2
votes
1answer
66 views

Laplacian inequality in $L^\infty$

Let $\Omega$ be a bounded domain of $R^n$ and let $y\in H^2(\Omega)\cap H_0^1(\Omega)$ such that the set $[x\in \Omega/ y(x)\ne 0]$ has non nul measure and $ \; \frac{\Delta y}{y} 1_{\{x\in \Omega/ ...
2
votes
1answer
37 views

Inequality in inner product space

Given $V$ an inner product space with norm $(‖v‖_V)^2$=$∫_Ω(v^2 (x)+|∇v|^2 )dx$. Prove that $$(∫_Ω(|v||w|+|∇v||∇w|)dx)^2 ≤ ∫_Ω(|v|^2+|∇v|^2 )dx ∫_Ω(|w|^2+|∇w|^2 )dx=(‖v‖_V)^2(‖w‖_V)^2.$$ Any ...