# Tagged Questions

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### BMO functions are $L^p$ Loc for all $1<p<\infty$

In order to motivate my question, I'd like to remember that if $\Omega$ is a bounded domain and $f \in L^q(\Omega)$ for some $q>1$, by HÃ¶lder inequality $f \in L^p(\Omega)$ for $p \in (1,q]$ with ...
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### Limit of the p-norm of a function on subdomains equals the p-norm of the function on the union domain

Let $\Omega$ be an open subset in $\mathbb{R}^n$. Given a measurable function $f$, define $$||f||_{p,\Omega}=\inf_{a\in\mathbb{R}}||f-a||_{L^p{(\Omega})}.$$ Let $\{\Omega_n\}$ be a sequence of open ...
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### Can we expect, $h\ast \mu \in L^{2}(\mathbb R, (1+|x|^{2})^{s})$ for $h\in \mathcal{S}(\mathbb R), \mu\in M(\mathbb R)$ and $s>1/2$?

We put, $M(\mathbb R)=$ The space of complex bounded Borel measure on $\mathbb R$ [With each complex Borel measure $\mu$ on $\mathbb R$ there is associated a set function $|\mu|,$ the total variation ...
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### For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$.

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$. I am having a hard time starting. Any suggestions. I tried a straight forward approach. That ...
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### can we approximate $f,$ in $L^{p}$-norm, by a function $f+h$ which is constant in a some neighbourhood of the point?

Suppose $f\in L^{p}(\mathbb R), (1<p <\infty), \epsilon > 0, \gamma_{0}\in \mathbb R.$ Then My Question is: Can we expect to find, $h\in L^{p}(\mathbb R)$ such that ...
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### $\|\phi_{\lambda}- \phi_{\lambda} \ast f \|_{L^{2}(\mathbb R)}\to 0$ as $\lambda \to \infty$? ($\phi_{\lambda}(x)=\lambda^{-1} \phi(x/\lambda).$)

For $f\in L^{1}(\mathbb R),$ we define its Fourier transform as follows: $\hat{f}(t)=\int_{\mathbb R} f(x) e^{-ix\cdot t} dx ,(t\in \mathbb R).$ Suppose that $f\in L^{1}(\mathbb R)$ with ...
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### $\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

It is well-known that, for $f,g \in L^{1}(\mathbb R).$ Then, by Fubini's theorem, one can derive, $\widehat{f\ast g} = \hat{f} \cdot \hat{g},$ (that is, Fourier transform takes, convolution to point ...
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### Generalisation of vector-valued Marcinkiewicz interpolation theorem

Given a compatible couple $(X,Y)$ of Banach spaces and some measure space $\Omega$ Lions and Peetre identified the real interpolation space between the vector-valued Lebesgue spaces $L_1(\Omega;X)$ ...
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### Convergence in $L^1$ and $L^p$ [duplicate]

Assume: (a) $\{f_n\} \subset L^p$, (b) $f_n \to f \text{ }\mu\text{-a.e.}$ and (c) $\|f_n\|_{L^p} \to \|f\|_{L^p}\to0$. Then show that $$\|f_n - f \|_{L^p} \to 0$$ using Fatou's Lemma, first for ...
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### Condition for a product of two function sequences in $L^1$ to be in $L^1$

We have: 1. $\{f_n\} \subset L^1(E, \Sigma, \mu)$ 2. $g_n \subset L^\infty(E, \Sigma, \mu)$ 3. $\|f_n-f\|_{L^1}\to0$ 4. $g_n \to g \text{ }\mu\text{-a.e.}$ 5. $\{g_n\}$ is uniformly bounded. ...
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### Function compositions that are in $L^p$

We have $f \in L^p$. The goal is to show that $\exists \psi \in C(\mathbb{R^+}, \mathbb{R^+})$ such that $$\lim_{s \to +\infty} \frac{\phi(s)}{s}=+ \infty \text{ and } \phi(|f|) \in L^p$$ I neeed ...
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### Sequence of piecewise constant functions converging to any $L^2$ function

Let $\{P_i\}_{i=1}^\infty$ be a sequence of partitions of the interval $[0,1]$ with a vanishing mesh. Additionally $H_i$ be the space of piecewise constant functions (step functions) with pieces ...
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### Existence of a function in $L^{p}$ with a certain property

Is there a function $f \in L^{p}(\mathbb{R}^{n})$ such that $\|If\|_{L^{p}(\mathbb{R}^{n})} = \infty$ where $If(x) = \int_{\mathbb{R}^{n}}\frac{f(y)}{|x - y|^{n}}\, dy$? Such a case isn't covered by ...
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### Question about injection on an unbounded space

I have this space $$C_0((0,+\infty))=\left\lbrace u,u\in C((0,+\infty)),\lim_{t\rightarrow +\infty} u(t)=0\right\rbrace$$ with the norm $$||u||_{\infty}=\sup_{t\geq0}|u(t)|$$ how to prove that ...
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### Criteria to be in weak $L^{p}$ space

Let $X$ be a $\sigma$-finite measure space. Let $f : X \rightarrow \mathbb{C}$ be a measurable function and $1 < p < \infty$. Suppose for $f$ there is a constant $C$ such that ...
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### Relation between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$

As an exercice, I'm looking to find an inclusion or equality relationship between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$ when $w: x \to x^{-1/2}$. Actually, I think that we have the inclusion ...
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### what are closed sets in $L^{1}(\mathbb R)$?

Consider, $L^{1}(\mathbb R)$= The space of Lebesgue integrable functions on $\mathbb R$; for $f\in L^{1}(\mathbb R),$ we define its norm, by $\|f\|_{L^{1}}=\int_{\mathbb R}|f(x)| dx$; It is well-known ...
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### $f_{n}$ converges in $L^{1}$ and $\|g_{n}\|_{L^{1}(\mathbb R)}\leq \|f_{n}\|_{L^{1}(\mathbb R)}\implies {g_{n}}$ converges in $L^{1}(\mathbb R)$?

Let $f_{n}, g_{n}\in L^{1}(\mathbb R)$ (Lebesgue space). Suppose there exist $f\in L^{1}(\mathbb R)$ such that $\|f_{n}-f\|_{L^{1}(\mathbb R)}\to 0$ as $n\to \infty;$ that is, the sequence $\{f_{n}\}$ ...