6
votes
1answer
50 views

How to obtain the inequality $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p} $ from Jensen's inequality?

Let $f$ be a positive function with $\lVert f \rVert_{L^2}=1$. Let $p= 2n/(n-2)$. How to obtain $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p}$ from Jensen's inequality? Here all ...
1
vote
1answer
34 views

Reference for $f \in L^{p,\infty} \cap L^{q}$ then $f \in L^r$ for $p < r \leq q$

Okay, so I think I've shown that if $f \in L^{p,\infty} \cap L^{q}$ with $p < q$ then $f \in L^r$ for $p < r \leq q$ where $L^{p, \infty}$ denotes the weak $L^p$ space. what I did was I wrote $$ ...
0
votes
1answer
95 views

L1 convergence and Lp bounded implies Lq convergence

I have tried to solve this problem for almost a week and did not manage to, so I figured to ask it here: Let $(u_n)\to u$ in $L^1(0,1)$ strongly and let $\{u_n\}_{n\in\mathbb{N}}$ be bounded in ...
0
votes
1answer
149 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
61 views

Is $L^p \cap L^q$ dense in $L^r$?

It is known that $L^p \cap L^q \subset L^r$, where $1 \le p \le r \le q \le \infty$. Are all of these inclusions dense? I.e., do we have \begin{equation*} \overline{L^p \cap L^q} = L^r ...
5
votes
2answers
209 views

Are integrable, essentially bounded functions in L^p?

Given an arbitrary measure space (of possibly infinite measure), if $f \in L^1 \cap L^\infty$, then by Hölder's inequality, $f^2 \in L^1$, so $f \in L^2$. Intuition suggests that $f \in L^p$ even for ...
1
vote
1answer
105 views

If the weighted Lp norm of a measurable function is finite, is the weighted Lp norm of the antiderivative also finite?

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a measurable function such that $$ \int_{-\infty}^{\infty} |f|^p e^{-x^2} dx < \infty. $$ Define $g : \mathbb{R} \rightarrow \mathbb{R}$ to be $$ ...
1
vote
0answers
39 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
191 views

A variation of Jensens inequality .

The following version of Jensens inequality is used in lecture notes, But i don't seem to get it . if $\phi : \mathbb R^n \to R$ is a convex function and $f_i \in L^1 (\Omega) $ for all $i$ and ...
2
votes
2answers
576 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 + ...