# Questions tagged [lp-spaces]

For questions about $L^p$ spaces. That is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence classes of measurable functions such that $|f|^p$ is $\mu$-integrable. Questions can be about properties of functions in these spaces, or when the ambient space in a problem is an $L^p$ space.

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### Examples of dense $G_\delta$ sets in $L^p$ spaces

The Baire Category Theorem states that the countable intersections of open dense subsets of a complete metric space (called dense $G_\delta$ sets) are dense. Any open set is $G_\delta$, so any dense ...
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### If $f \in L^{1}(0,\infty)$ and $x \in (0, \infty)$, then $xf \in L^{1}(0,\infty)$? [closed]

Suppose $f \in L^{1}(0,\infty)$. Is it true that $$\int_{0}^{\infty}xf(x)dx < \infty \ \$$
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### $f=g+ih$ is $L^p$ if and only if $g$ and $h$ are $L^p$?

Let $X$ be a measure space, $f:X\to\mathbb{C}$ a measurable function, and $1<p<\infty$. $f$ is said to be $L^p$ if $|f|^p$ is integrable. Now, let $f=g+ih$, where $g$ and $h$ are real-valued. Is ...
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### Find function $f(x)$ so $x\hat{f}(x)\in L^1(\mathbb{R})$

Suppose we have $f\in L^1(\mathbb{R})$ so $\xi\mapsto \xi\hat{f}(\xi)$ is in $L^1(\mathbb{R})$, where $\hat{f}$ is the Fourier Transform for the function $f$. I'm trying to show that there exists some ...
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### Showing $\phi(x) = (1 − |x|)_+$ is $L^p$ for all $p$

Consider the function $\phi$ defined by $\phi(x) = (1 − |x|)_+$, I have been tasked by an exercise in my textbook to verify both $\phi,\phi'\in L^p([-2,2])$ and deduce $\phi\in W^{1,p}((−2, 2))$ for ...
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### Littlewood–Paley theory in case $p=\infty$

What happen with Littlewood–Paley theorem in the case $p=\infty$. Exists some result or paper for this case?, by https://en.wikipedia.org/wiki/Littlewood%E2%80%93Paley_theory we can estimate bound the ...
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### Connecting piecewise smooth functions using mollification.

Let's say I am given a function $f: \mathbb{R}\setminus (0,1) \to \mathbb{R}$ defined piecewise as $$f(x)= \begin{cases} g(x) \quad x \geq 1 \\ h(x) \quad x \leq 0 \end{cases},$$ where $g$ and $h$ ...
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### $L^2(\mathbb{R})$ is separable

I would like to prove that $L^2(\mathbb{R})$ is separable. Using the hint provided in Reed & Simon's book, I was able to show that: the set of simple functions on $[a,b]$ is dense in $L^2([a,b])$ ...
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### Is there a standard notational convention for $L_p$ ($\ell_p$) spaces, norms, and metrics?

In reading about $L_p$ spaces and $\ell_p$ norms and metrics, I have seen a wide variety of different notations for the same thing. Is there a standard convention for when to use lowercase $\ell$ vs ...
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