I’m following the book Measure and Integral of Richard L. Wheeden and Antoni Zygmund. This is the problem 4 of chapter 8. Consider $E\subseteq \mathbb{R}^n$ a measurable set. In the following all the integrals are taken over $E$, $1/p + 1/q=1$, with $1\lt p\lt \infty$. I’m trying to prove that $$\int \vert fg\vert =\Vert f […]

I’m wondering if it’s possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain). One can use interpolation to show that if a function is in two $L^p$ spaces, (e.g. $p_1$ and $p_2$,with $p_1 \leq p_2$ then it is […]

how to prove $\sup \{ \sin n \mid n\in \mathbb N \} =1$

Let $f(t)$ be a measurable and almost everywhere finite function, defined on the closed interval $E = [a, b]$. Prove the existence of a decreasing function $g (t)$, defined on [a, b], which satisfies the relation $m(E \cap \left \{ x: {g > x} \right \}) = m(E \cap \left \{ x: {f > x} […]

Let $\,f : \mathbb{R} \rightarrow \mathbb{R}$ be a infinitely differentiable function that is increasing and bounded. Then is it true that $\lim_{x\to \infty}f'(x)=0$?

This question already has an answer here: Continuity of $L^1$ functions with respect to translation 2 answers

Let $f:\mathbb R\to \mathbb R$ be a $C^\infty$-smooth function. Suppose that $f^{(k)}(0)=0$ for $k=0,\dots,n-1$. Prove that the function $g(x)=f(x)/x^n$ extends to a $C^\infty$-smooth function on $\mathbb R$. Comment: by l’Hôpital’s rule, $g$ has a finite limit at $0$, namely $f^{(n)}(0)/n!$. So, it extends to a continuous function on $\mathbb R$. However, I do not see […]

We know that if $ f: E \to \mathbb{R} $ is a Lebesgue-measurable function and $ g: \mathbb{R} \to \mathbb{R} $ is a continuous function, then $ g \circ f $ is Lebesgue-measurable. Can one replace the continuous function $ g $ by a Lebesgue-measurable function without affecting the validity of the previous result?

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f(0)=0$ for all real numbers $x$, $\left|f^\prime(x)\right|\leq\left|f(x)\right|$. Can $f$ be a function other than the constant zero function? I coudn’t find any other function satisfying the property. The bound on $f^\prime(x)$ may mean that $f(x)$ may not change too much but does it mean that $f$ is constant? […]

I have two questions which pertain to differentiability, connectivity and path connectivity. Ocasionally, I will encounter an author who defines connectivity in the following way: An open subset $U$ of $\mathbb{R}^n$ is said to be connected if and only if given two points $a$ and $b$ of $U$ there exists a differentiable mapping $\phi: \mathbb{R} […]

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