Show that there exist $f_1 \in L^{p_1}$ and $f_2 \in L^{p_2}$ such that $f = f_1 +f_2$.

Question

Let $1 \le p_1 < p \le p_2 \le \infty$ and let $f\in L^p$. Show that there exist $f_1 \in L^{p_1}$ and $f_2 \in L^{p_2}$ such that $f = f_1 +f_2$.

Just for clarify, we consider the $L^p$-spaces on $\mathbb{R}^n$. How can I start this exercise?

Answer 1

There are many ways to partition an function $f\in L_p$ as an element of $L_{p_1}+L_{p_2}$. The construction of Marcinkiewcz described below, allows also to add different "weights" to components in $L_{p1}$ or $L_{p_2}$.

Fixed $\lambda>0$. If $f\in L_p$ then $$\mu(|f|>\lambda)=\mu(|f|^p>\lambda^p)\leq\frac{1}{\lambda^p}\|f\|^p_p$$

Let $g_\lambda=f\mathbb{1}_{\{|f|>\lambda\}}$ and $h_\lambda=f-g_\lambda=f\mathbb{1}_{\{|f|\leq\lambda\}}$.

Notice that $|f|^{p_1}\in L_{p/p_1}$, and $\mathbb{1}_{\{|f|>\lambda\}}\in L_{p/(p-p_1)}$. An application of Höler's inequality leads to \begin{aligned} \int |g_\lambda|^{p_1}\,d\mu&=\int |f|^{p_1}\mathbb{1}_{\{|f|>\lambda\}}\,d\mu\leq \Big(\int |f|^p\,d\mu\Big)^{\tfrac{p_1}{p}}\Big(\int \mathbb{1}_{\{|f|>\lambda\}}\,d\mu\Big)^{1-\tfrac{p_1}{p}}\\ &\leq \|f\|^{p_1}_p\Big(\lambda^{-p}\|f\|^p_p\big)^{\tfrac{p-p_1}{p}}=\frac{1}{\lambda^{p-p_1}}\|f\|^p_p<\infty \end{aligned} That is, $g_\lambda\in L_{p_1}$.

For $h_\lambda$ we have that $|h_\lambda|\leq \lambda$ and so

$$|h|^{p_2} =|h_\lambda|^{p_2-p}|h_\lambda|^p\leq \lambda^{p_2-p}|h_\lambda|^p$$ which implies $$ \int|h_\lambda|^{p_2}\,d\mu\leq \lambda^{p_2-p}\int_{|f|\leq\lambda}|f|^p\,d\mu\leq \lambda^{p_2-p}\int|f|^p\,d\mu=\lambda^{p_2-p}\|f\|^p_p<\infty$$ That is, $h_\lambda\in L_{p_2}$.

Answer 2

Let $A:=\{ |f|>1\}$ and $g:=f_1\chi_A$, where $\chi_A$ is the characteristic function of the set $A$ and $f_2:=f-f_1$. Obviously $f=f_1+f_2$.

Also since $|f(x)|^{p_1}< |f(x)|^p$, for every $x\in A$, $$\| f_1\|_{p_1}^{p_1}=\int_{\mathbb{R}^n} |f_1|^{p_1} = \int_{\mathbb{R}^n} |f\chi_A|^{p_1}= \int_A |f|^{p_1} < \int_A |f|^{p} < \int_{\mathbb{R}^n} |f|^{p} = \| f\|_{p}^{p}< \infty$$

Now using pretty much the same reasoning (observe that $f_2=f\chi_{\mathbb{R}^n \setminus A}$), we can show that $$\| f_2\|_{p_2}^{p_2} < \| f\|_{p}^p<\infty$$

Thus, $f_1 \in L^{p_1}$ and $f_2 \in L^{p_2}$.