# Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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### Invariance under diffeomorphisms of the Hajlasz-Sobolev spaces

In this post it was shown that if $\Omega$ and $\Omega'$ are diffeomorphic non-empty open domains in some Euclidean space then the corresponding local Sobolev spaces are diffeomorphic with ...
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### Bound on the ratio of harmonic and arithmetic mean

Let $a_i>0$ for $i=1,...,n$. It is well-known that $A\ge H$, where $A$ and $H$ are the arithmetic mean and harmonic mean of the vector $(a_i)$, respectively. Is any lower bound on $H/A$ known?
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### Reference request for (weak*) metrizability of a bounded space of signed Radon measures on a compact set

I know the following is true and I know how to prove it (cf. exercise 50 on page 171 in Folland, Theorem 7.18 in Folland), but per my adviser's instructions, it would be better to find a source to ...
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### Low-degree polynomial approximation of the piecewise-linear function $x \mapsto \max(x, 0)$ on an interval $x \in [-R,R]$

For $R > 0$, consider the piecewise-linear function $\sigma_R: [-R,R] \rightarrow \mathbb R^+$, defined by $\sigma_R(x) := \max(x,0)$. Question Given $\epsilon> 0$, find a "low-degree" ...
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### Holder inequality with respect to convex function

Given $a, T>0$, by Holder's inequality we have $$\int_T^{T+a}f(s)ds\leq \left(\int_0^{T+a}|f(s)|^2ds\right)^{1/2}\cdot\sqrt{a}.$$ Do we have similar result if we replace $|x|^2$ by some convex ...
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### Is a bounded sequence of $H^1(\Omega)$ tight?

Assume $\Omega$ is a bounded subset of $\Bbb R^d$ and $(u_n)_n$ is a bounded sequence of the Sobolev space $H^1(\Omega)$. Question: Can we say that $(u_n)_n$ is tight in $L^2(\Omega)$ namely: ...