# Tagged Questions

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### Is there a useful relationship between pointwise and $L^2$ distance?

It would be really convenient to get a bound on the point-wise closeness of functions by knowing their $L^2$ distance. Clearly, if two functions are close in the $L^2$ sense, you cannot get a general ...
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### limit of p norm as p goes to 0!

Suppose we have a measure $\mu$ and a space $X$ such that $\mu(X)=1$, and a function $f \in L^r$ for some $r > 0$, where $L^r$ is defined in the usual way even for numbers less than $1$. Show ...
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### Estimate for weak $L^{1}$ norm

Let the weak $L^{1}$ norm on $f$ be defined by $\|f\|_{\mathrm{WL}^{1}} = \sup_{t > 0}t D_{f}(t)$ where $D_{f}(t) = \mu(\{x \in \mathbb{R}: |f(x)| > t\})$ and $\mu$ is the standard Lebesgue ...
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### proofread $\left \| f -g\right \|_{p}^{p}=p(p-1)\int_{0}^{\infty}\int_{0}^{t}[m(f>t/g>s)+ m(g>t/f>s)]|t-s|^{p-2}dsdt$

Is this argument correct? $\left \| f -g\right \|_{p}^{p}=$ $\int_{0}^{\infty}m(|f-g|>t^{\frac{1}{p}}) dt\stackrel{c.o.v.}{=}$ $p\int_{0}^{\infty}m(|f-g|>t) t^{p-1}dt\stackrel{t=|u-s|}{=}$ ...
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### continuity of norms with respect to $p$

Let $f\in L^{\infty}(\Omega,\Sigma,\mu)\cap L^{1}(\Omega,\Sigma,\mu)$. Then $w(p)=||f||_p$ is continuous function of $p$ for any $p\in [1,\infty)$. How to prove this? I have obtained the proof that ...
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### Counterexample using counting measure

While proving that the norm of the mulplicative operator from $L^2(X) \to L^2(X)$ is the essential supremum of $|g|$ where $g \in L^\infty(X)$, I found that I need the $\sigma$-finiteness of the ...
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### essential supremum of a matrix multiplication operator

Suppose we have the space $L^p(R,R^n)$ where $1 \leq p < \infty$ (i.e the space of functions that take values in $R^n$ and are $L^p$ integrable) and suppose $T_m: L^p(R,R^n) \to L^p(R,R^n)$ is a ...