Convergence of the norm in Lorentz space

Question

It's actually Exercise 1.4.11(b) in Classical Fourier Analysis. For $0<q,s\leq \infty$, if $\Vert g_n -g\Vert_{L^{q,s}}\to 0$, then we must have $\Vert g_n\Vert_{L^{q,s}}\to \Vert g\Vert_{L^{q,s}}$.

I have no idea how to deal with this, since in general, $L^{q,s}$ is equipped with a quasi-norm but not a norm.

I shall briefly introduce the $L^{p,q}$-norm. First define the distribution function $d_f(\alpha)=\mu(\{x : \vert f(x)\vert >\alpha \})$, and define the decreasing rearrangement of $f$ as $\begin{equation*} f^*(t)=\inf\{s>0 : d_f(s)\leq t\} \end{equation*}$. The property of $f^*$ is that it's decreasing and $d_{f^*}=d_f$, i.e., they have the same distribution function.

Finally we can define the $L^{p,q}$ norm as $\begin{equation*} \Vert f\Vert_{L^{p,q}}=\begin{cases} \Big[\int_0^{\infty} \Big(t^{\frac 1p} f^*(t) \Big)^q \frac {dt}t \Big]^{\frac 1q} & q<\infty \\ \sup_{t>0} t^{\frac 1p} f^*(t) & q=\infty \end{cases} \end{equation*}$

Answer 1

Let $0<\varepsilon<1$. The statement is equivalent to

$$\limsup_{n \to \infty} \|{f_n}\|_{p,q}\leq \|{f}\|_{p,q}\leq \liminf_{n\to \infty} \|{f_n}\|_{p,q}.$$

It is sufficient to prove the first inequality, since the second follows similarly. For this, we will use the fact that $$(f+g)^*(a+b)\leq f^*(a)+g^*(b)$$ (prove it!). For each $n\in \mathbb{N}$ and $t\in \mathbb{R}_{\geq 0}$, write \begin{equation} (*)\ \ \ \ (f_n)^*(t) = [f + (f_n-f)]^*((1-\varepsilon )t+\varepsilon t) \leq f^*((1-\varepsilon )t)+(f_n-f)^*(\varepsilon t). \end{equation} We will have to separate the proof into cases:

1. $q=\infty$. For $(*)$ \begin{align*} \|{f_n}\|_{p,\infty}&=\sup_{t>0} t^{1/p} (f_n)^*(t) \\ & \leq \sup_{t>0} t^{ 1/p} [f^*((1-\varepsilon )t)+(f_n-f)^*(\varepsilon t)] \\ &\leq \sup_{t>0} t^{1/p} f^ *((1-\varepsilon )t)+\sup_{s>0} s^{1/p}(f_n-f)^*(\varepsilon s) \\ &= (1-\varepsilon )^{-1/p}\sup_{u>0} u^{1/p}f^*(u) + \varepsilon^{-1/p}\sup_{w> 0} w^{1/p}(f_n-f)^*(w)\\ &=(1-\varepsilon )^{-1/p}\|{f}\|_{p,\infty} + \varepsilon^{-1/p}\|{f_n-f}\|_{p,\infty}. \end{align*} So \begin{align*} \limsup_{n \to \infty} \|{f_n}\|_{p,\infty}\leq (1-\varepsilon )^{-1/p}\|{f}\|_{p,\infty} + \varepsilon^{-1/p}\limsup_{n \to \infty}\|{f_n-f}\|_{p,\infty} = (1-\varepsilon )^{-1/p }\|{f}\|_{p,\infty}. \end{align*} Making $\varepsilon\to 0$, we get the desired inequality.

2. $1\leq q<\infty$. Using the equation $(*)$ again \begin{align} \|{f_n}\|_{p,q}&=\left(\int_{0}^{\infty} [(f_n)^*(t)t^{1/p}]^q\frac{dt }{t}\right)^{1/q} \\ &\leq \left(\int_{0}^{\infty} [t^{1/p}f^*((1-\varepsilon )t)+t ^{1/p}(f_n-f)^*(\varepsilon t)]^q\frac{dt}{t}\right)^{1/q}. \end{align} By the Minkowski inequality for $L^{q}(\mathbb{R}_{\geq 0}, \frac{dt}{t})$, we get \begin{align} \|{f_n}\|_{p,q}&\leq \left(\int_{0}^{\infty} [t^{1/p}f^*((1-\varepsilon )t)+t ^{1/p}(f_n-f)^*(\varepsilon t)]^q\frac{dt}{t}\right)^{1/q}\\&\leq \left(\int_{0} ^{\infty} [t^{1/p}f^*((1-\varepsilon )t)]^q\frac{dt}{t}\right)^{1/q}+ \left(\int_{0}^{\infty} [t^{1/p}(f_n-f)^*(\varepsilon t)]^q\frac{dt}{t}\right)^{1/q} \\ &= (1-\varepsilon)^{-1/p}\|{f}\|_{p,q}+\varepsilon^{-1/p}\|{f_n-f}\|_{p ,q}. \end{align} So \begin{multline} \limsup_{n \to \infty} \|{f_n}\|_{p,q}\leq (1-\varepsilon)^{-1/p}\|{f}\|_{p,q} +\varepsilon^{-1/p}\limsup_{n \to \infty}\|{f_n-f}\|_{p,q} = (1-\varepsilon)^{-1/p}\| {f}\|_{p,q}. \end{multline} Making $\varepsilon\to 0$, we get the desired inequality.

3. $0<q<1$. The mapping $H(x)=x^q$ is concave on $\mathbb{R}_{\geq 0}$. Moreover, since $H(0)=0$, $H$ is subadditive: $$[t^{1/p}f^*((1-\varepsilon )t)+t^{1/p}(f_n-f)^*(\varepsilon t)]^q \leq [t^ {1/p}f^*((1-\varepsilon )t)]^q+ [t^{1/p}(f_n-f)^*(\varepsilon t)]^q.$$ Therefore \begin{align*} \|{f_n}\|_{p,q}^q&\leq \int_{0}^{\infty} [t^{1/p}f^*((1-\varepsilon )t)+t^ {1/p}(f_n-f)^*(\varepsilon t)]^q\frac{dt}{t} \\ &\leq \int_{0}^{\infty} [t^{1/p} f^*((1-\varepsilon )t)]^q\frac{dt}{t}+\int_{0}^{\infty} [t^{1/p}(f_n-f)^*( \varepsilon t)]^q\frac{dt}{t} \\ &=(1-\varepsilon )^{-q/p}\|{f}\|_{p,q}^q+\varepsilon^{ -q/p}\|{f-f_n}\|_{p,q}^q \end{align*} and finally \begin{align*} (\limsup_{n \to \infty} \|f_n\|_{p,q})^q &=\limsup_{n \to \infty} (\|{f_n}\|_{p,q}^ q)\\ &\leq (1-\varepsilon )^{-q/p}\|{f}\|_{p,q}^q+\varepsilon^{-q/p}\limsup_{n \to \infty}(\|{f-f_n}\|_{p,q}^q) \\&= (1-\varepsilon )^{-q/p}\|{f}\|_{p, q}^q+\varepsilon^{-q/p}(\limsup_{n \to \infty}\|{f-f_n}\|_{p,q})^q\\&= (1-\varepsilon )^{ -q/p}\|{f}\|_{p,q}^q. \end{align*} Taking $\varepsilon \to 0$ and raising both sides of the inequality to the power $1/q$, we get the desired result.