Suppose that $f \in L^1(\mathbb{R}^n)$, $f \ge 0$, $\|f\|_{L^1} = 1$. How do I see that $\sup_{\xi\in\mathbb{R}^n} |\mathcal{F}(f)(\xi)| = 1$, and it is attained exactly at $0$?

Let $p,q\in(1,\infty)$ be conjugate exponents (i.e., $1/p+1/q=1$) and let $f:(0,\infty)\to\mathbb R_+$ be a Lebesgue-measurable function such that $\int_0^{\infty} f(x)^q\,\mathrm dx<\infty$. It follows fairly easily from Hölder’s inequality that $$\int_0^x f(t)\,\mathrm dt\leq x^{1/p}\cdot\|f\|_q\quad\forall x>0,$$ so that the (continuous) function $x\mapsto \int_0^x f(t)\,\mathrm dt$ must diverge no faster than $x^{1/p}$. (The notation $\int_0^x$ means Lebesgue integral on […]

I want to show that if $u_{m} \rightharpoonup u$ in $W^{1,\infty}(\Omega)$ then $u_{m} \rightarrow u$ in $L^{\infty}(\Omega)$. I know that I can’t directly use the compactness of Rellich Kondrachov Theorem since I am taking $p = \infty$. From Morrey’s Inequality I have $||u||_{C^{0,\alpha}} \leq ||u||_{W^{1,p}}$ where $\alpha = 1 – \frac{n}{p}$. Is it possible to […]

Let $(\mathcal{X}, \mathcal{A}, \mu)$ be a measure space. Take $f,g \in L^1(\mu)$. Prove that $\sqrt{f^2+g^2}$ and $\sqrt{\vert fg\vert}$ are in $L^1(\mu)$. First, I prove that $h = \max\{f,g\} \in L^1(\mu)$. Let $A = \{x \in \mathcal{X} \mid f(x) \geq g(x)\}$. Then: $$ \begin{align} \int_{\mathcal{X}} h \ \mathrm{d}\mu &= \int_A h \ \mathrm{d}\mu + \int_{A^c} h […]

Let $X=(x_{ij})_{i,j=1}^\infty$ be the matrix such that for all $a=(a_n)\in l_p$ and all $k\in\mathbb{N}$ the series $\sum_{n=1}^\infty x_{kn}a_n$ converges in $\mathbb{F}$. We assume that the sequence $Xa=(Xa)_k=\sum_{n=1}^\infty x_{kn}a_n$ is an element of $l_p$. Thus $X:l_p\rightarrow l_p$ is a linear map. Now fix $k,K\geq 1$. We define $X_k^K:l_p\rightarrow \mathbb{F}$ and $X_k:l_p\rightarrow \mathbb{F}$ by $$X_k^Ka=\sum_{n=1}^K x_{kn}a_n$$ and […]

Let $f:\mathbb{R^+}\to\mathbb{R}$ be an integrable function ($f\in L^1(\mathbb{R}^+,\mathbb{R})$). Do we have $$\lim_{h\to 0^+}\int_0^\infty|f(t+h)-f(t)|dt=0$$ ? How can we prove it ?

I think part a) can be solved by using $X_n=\frac{1}{n}\chi_{[0,n^2]}$ Not sure about part b).

I am trying to produce a sequence of sets $A_n \subseteq [0,1] $ such that their characteristic functions $\chi_{A_n}$ converge weakly in $L^2[0,1]$ to $\frac{1}{2}\chi_{[0,1]}$. The sequence of sets $$A_n = \bigcup\limits_{k=0}^{2^{n-1} – 1} \left[ \frac{2k}{2^n}, \frac{2k+1}{2^n} \right]$$ seems like it should work to me, as their characteristic functions look like they will “average out” […]

Let $A, B$ be two open sets in $\mathbb{R}^n, \mathbb{R}^m$ respectively and denote $\mathcal{C}_c(A\times B)$ the space of continuous functions with compact support in $A\times B.$ Is $\mathcal{C}_c(A\times B)$ dense in $L^p(A, L^q(B))$ for any $+\infty > q,p \geq 1 ?$ I believe that the answer is YES and I’m looking for a simple proof. […]

Is the set $\{ m \in L^2(0,1) : |m|_{L^\infty}\leq A \}$, (i.e. the set of $L^2$ functions with bounded $L^\infty$ norm) a closed subset of $L^2$? (Closed in the topology induced by the $L^2$-norm)

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