# Tagged Questions

For questions about $L^p$ spaces, that is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence class of measurable functions with $p$-th power of the absolute value integrable. Question can be about properties of elements of these spaces, or when the ambient space on a ...

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### Real Analysis, Folland Problem 6.1.14 $L^p$ spaces

Problem 6.1.14 - If $g\in L^{\infty}$, the operator $T$ defined by $Tf = fg$ is bounded on $L^p$ for $1\leq p \leq \infty$. Its operator norm is at most $\|g\|_{\infty}$, with equality if $\mu$ is ...
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### Real Analysis, Folland Problem 6.1.2 $L^p$ spaces

Background Information: In this chapter we work on a fixed measure space $(X,M,\mu)$. If $f$ is measurable on $X$ and $0 < p < \infty$, we define \|f\|_{L^p} = \left[\int |f|^p d\mu\right]^{1/...
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### Sobolev Space $W_0^{1,p}(I)$ and the boundary of $I$

Given $I \subset\mathbb{R}$ an open interval, the Sobolev Space $W_0^{1,p}(I)$ is defined as $W_0^{1,p}(I)=\overline{C^1_c(I)}^{W^{1,p}(I)}$ (The closure of $C^1_c(I)$ on the space $W^{1,p}(I)$) . ...
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### What is the Stein Interpolation Theorem for any strip?

Stein Interpolation Theorem: Assume $T_z$ is an operator depending analytically on $z$ in the strip $0\leq Re (z) \leq 1.$ Suppose $T_z$ is bounded from $L^{p_0}$ to $L^{r_0}$ when $Re (z)= 0,$ and ...
### Sobolev embedding fails for $p=n$
As everyone knows, the Sobolev embedding fails fails for $n\ge 2$ if we assume $p=n$. The standard example is the function $u(x)=\log \log \bigl(1+\tfrac{1}{x}\bigr)$. This function is obviously ...
I wondered if there is, in general, a way to estimate the error (in $L^2$) between a Sobolev function and its mollified version. Let's say $\Omega\subset\mathbb{R}^n$ is bounded with smooth boundary ...