# 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 6.1.5

Problem 6.1.5 - Suppose $0 < p < q < \infty$. Then $L^p \not\subset L^q$ if and only if $X$ contains sets of arbitrary small positive measure, and $L^q\not\subset L^p$ if and only if $X$ ...
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### Neumann Poincare operator maps $L^2$ in itself

How can I show that the Neumann-Poincare operator $$K_{\partial \Omega}[\phi](x) = \int_{\partial \Omega} \dfrac{(x-y) \cdot \nu(y)}{|x-y|^d} \phi(y) \ dy$$ maps $L^2(\partial \Omega)$ in itself (if ...
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### Integrability of $f(x)\sqrt{\frac{1}{x}}$ for $f\in\mathcal L^2$ and $\|f\|_2=1$

Is it true that for $f\in\mathcal L^2$ and $\|f\|_2=1$, $$\int_0^\infty f(x)x^{-1/2}dx<\infty?$$ I'm fairly stuck on this...(and I really hope it is true). In case it helps seeing a generalization,...
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### Where does this inequality come from: $\frac{|f(x) - f(x-h)|}{h} \le ||f'||_{L^\infty(x, x-1)}$?

I came across this inequality today: $$\frac{|f(x) - f(x-h)|}{h} \le ||f'||_{L^\infty(x, x-1)}$$ I realise if we let $h \to 0$ we obtain the derivative on the left hand side so I can see it has ...
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### Real Analysis, Folland problem 6.1.11 $L^p$ spaces

Problem 6.1.11 - If $f$ is a measurable function on $X$, define the essential range $R_f$ of $f$ to be the set of all $z\in\mathbb{C}$ such that $\{x:|f(x) - z| < \epsilon \}$ has positive measure ...
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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 ...
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### 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 ...
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### Approximate Sobolev function by smooth function - error estimate?

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 ...
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### Intuitive explanation of p-norm in finite and infinite dimensinos

I am not a mathematician, so very rigorous treatment with things that only a math major learns will not suffice here. I want to learn about p-norms and i can't quite get the intuition behind them. I ...
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### Integral operator is bounded on $L^p$ if it maps $L^p$ to itself

Here is a homework excercise. Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space,$1\leq p <\infty.$ and suppose that $k:X\times X\rightarrow \mathbb{C}$ is an $\Omega \times \Omega$ ...
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### subspace of $L^2$

Let $(X,B(X),\mu)$ a measurable space, for a positive finite measure $\mu$, we consider $H=L^2(X,d\mu)$, Let $A$ a closed subspace of $H$, we know that $A$ is a hilbert space, can we say that it exist ...
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### Compactness Sobolev embedding for even functions on $\mathbb{R}$.

It is well-known from Lions's article,"Symétrie et compacité dans les espaces de Sobolev", that the subspace $H^s_r(\mathbb{R}^n)$ of the Sobolev space $H^s(\mathbb{R}^n)$ containing all radial ...
### $L^{\infty}$ convergence for random variable
I am slightly confused with this borderline case regarding $L^p$ convergence. In some probability books, they clearly state that $p<\infty$ whereas the online sources do not impose this ...
### If an $H^1$ function vanishes on a set of positive measure, its $L^2$ norm is controlled by the gradient
I am trying to solve question 15 from Evans' PDE book, chapter 5. You have a set of positive measure, subset of the unit ball $B$, such that $u$ is equal to zero on that set. Then, one can show that: \$...