# Tagged Questions

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### How to get this inequality given another inequality (norms and logarithms)

Suppose I have proved for all $f$ positive with $\lVert f \rVert_{L^2} = 1$ that $$\int f^2\log(f) \leq -C_1\log(\epsilon) + \epsilon\lVert \nabla f \rVert_{L^2}^2.$$ I want to prove that for all $f$ ...
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### 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 ...
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### Apolloniusâ€™ Identity inner product space

$||z-x||^2+||z-y||^2=\frac{1}{2}||x-y||^2+2||z-\frac{x+y}{2}||^2$ I proved it by expanding both sides and i found both sides are equal. Are there any easy way to prove it?
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### An inequality for inner product space: $\|x-z\|.\|y-t\|\leq \|x-y\|.\|z-t\|+\|y-z\|.\|x-t\|$

In a inner product space show that the following inequality holds. $\|x-z\|.\|y-t\|\leq \|x-y\|.\|z-t\|+\|y-z\|.\|x-t\|$ I am stuck in proving this inequality
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### DKW-style $\ell_{\infty}$ bounds for sum of i.i.d. random functions: $\to [0,1]$

Let $\mathbf{G}$ be the set of (edit: convex) functions $g: X \to [0,1]$, where $X$ is a compact subset of $\mathbb{R}^d$ or something like that. Suppose I have a distribution $D$ on $\mathbf{G}$. ...
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### 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 ...
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### 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?
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$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
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### 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 ...
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### 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 ...
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### $L^{\infty}$ 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 ...
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### 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 ...
$\Omega\subset\mathbb{R}^3$ 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 $$... 1answer 138 views ### fraction power vector-norm inequality If X is a Banach Space and x,y\in X. Then by the definition of a Banach algebra we know$$\|x.y\|\leq\|x\|\|y\|$$and thats how we can have relation for any positive power. i.e. n\in N, ... 2answers 115 views ### Inequality involving Lp space, Holder Space, Sobolev Space Do we have the following inequality or some variation of this: ||u||_{L^{p}(U)} \leq ||u||_{C^{0,\gamma}(\bar{U})} \leq ||u||_{W^{1,p}(U)} for n < p \leq \infty for u \in W^{1,p}(U) where U ... 2answers 79 views ### Using Results in Sobolev Spaces I have two questions about using results in Sobolev Spaces and a last one on an inequality involving Holder spaces: 1.The Gagliardo-Nirenberg-Sobolev Inequality states the following: Assume 1 \leq p ... 0answers 51 views ### Explanation on the proof of the continuous Hardy inequality Here there is a proof of the continuous Hardy inequality (theorem 2). I would like a explanation on the following passage. ... 1answer 61 views ### Prove x(t) is bounded given a integral inequality I want to answer the following question: x=x(t) is defined and continuous on [0,T) and satisfies an integral inequality$$1 \leq x(t) \leq A_1 + A_2\int_0^t x(s)\big(1+\log x(s)\big) ds$$for ... 2answers 121 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. ... 2answers 146 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 ... 0answers 18 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 = ... 1answer 113 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 = ... 0answers 124 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 ... 0answers 34 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} ... 0answers 31 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: ... 2answers 144 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 ... 1answer 78 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 ... 1answer 85 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), ... 2answers 546 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 ... 1answer 147 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$$ ...
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.