2
votes
0answers
42 views

Integral inequality related to derivation

While trying to understand a proof, i have stumbled upon the following statement: Let $f \in L^p(a,b)$ be a $p$-integrable function. Then the inequality $$\liminf_{s \rightarrow t} \frac{1}{t-s} ...
3
votes
1answer
63 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 ...
2
votes
1answer
39 views

Find $\inf_{f > 0} T_f := \left(\int_A f \, d\mu\right)\left(\int_A \frac{1}{f} \, d\mu\right)$

This exercise gives me trouble: Let $F$ denote the collection of measurable functions which are positive $\mu$-a.e. and let $A \in \mathbb X$ satisfy $0 < \mu(A) < \infty$. For $f \in F$ let ...
0
votes
1answer
47 views

Some questions on the proof of Hoelders inequality.

I have some questions about the proof of Hoelder's inequality. Statement: Let $(X, \mathbb X, \mu)$ be a measure space. Let $p,q > 1$ with $1/p+1/q = 1$ and suppose that $f \in L_p(X)$ and $g \in ...
0
votes
2answers
79 views

An Hardy-Littlewood like inequality

Let $\mu$ be a finite Borel measure on $R$. Show that $|\{x\in R : \sup_{r>0} \frac{1}{2r} \mu ([x-r,x+r]) \ge \lambda \}| \le \frac{C}{\lambda} \mu(R)$ for any $\lambda > 0$ and some absolute ...
5
votes
2answers
215 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 ...
2
votes
1answer
114 views

continuity of norms with respect to $p$

Let $f\in L^{\infty}(\Omega,\Sigma,\mu)\cap L^{1}(\Omega,\Sigma,\mu)$. Then $w(p)=||f||_p$ is continuous function of $p$ for any $p\in [1,\infty)$. How to prove this? I have obtained the proof that ...
0
votes
1answer
46 views

Probability bounds

My question is the following: if my random variable $X$ has finite or bounded second moment $\mathbb{E}[X^2]\leq B$ can anyone develop any bounds on pdf of $X$. For example something like this ...
1
vote
1answer
106 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
1answer
138 views

Prove that $L^1$ is a Banach algebra with multiplication defined by convolution

To be more specific, prove that $L^1(\mathbb{R}^n)$ with multiplication defined by convolution: $$ (f\cdot g)(x)=\int_\mathbb{R^n}f(x-y)g(y)dy $$ is a Banach algebra. All the properties of Banach ...
18
votes
1answer
653 views

Do inequalities that hold for infinite sums hold for integrals too?

Let $\mathbb{R}_{\geq0}$ denote the set of non-negative reals and $+\infty$, and $\mathbb{Z}^+$ denote the set of positive integers. I will also let $\lambda$ denote the Lebesgue measure on ...
6
votes
1answer
856 views

Hölder inequality from Jensen inequality

I'm taking a course in Analysis in which the following exercise was given. Exercise Let $(\Omega, \mathcal{F}, \mu)$ be a probability space. Let $f\ge 0$ be a measurable function. Using Jensen's ...