Prove that $\|f\|_{p,q}= \inf\{\|g\|_p+\|h\|_q;f=g+h, g \in L^p(R), h \in L^q(R)\}$ is a norm on $L^p(R)+L^q(R) $

Question

Let $ 1\leq p,q < \infty $, I want to build a norm on $L^p(\mathbb{R})+L^q(\mathbb{R})$ and naturally, I define

$\|f\|_{p,q}= \inf\{\|g\|_p+\|h\|_q;f=g+h, g \in L^p(\mathbb{R}), h \in L^q(\mathbb{R})\}$.

Now, I need to prove that $\|f\|_{p,q}$ is a norm on $L^p(\mathbb{R})+L^q(\mathbb{R})$ but I unable to prove triangle inequality.

Could you please help me prove or give me some references for the proof of this result? I was trying to prove it but there is no progress.

Thank you so much.

Answer 1

Let $\epsilon >0$ and $f_1,f_2 \in L^{p} +L^{q}$. By definition of $||f_1||_{p,q}$ we can write $f_1 =g_1 +h_1$ with $||f_1||_{p,q}+\epsilon > ||g_1||_p +||h_1||_q$. Similarly we can write $f_2 =g_2 +h_2$ with $||f_2||_{p,q}+\epsilon > ||g_2||_p +||h_2||_q$. Now $f_1+f_2 =(g_1+g_2)+ (h_1+h_2)$ and $g_1+g_2 \in L^{p}$, $h_1+h_2 \in L^{q}$. Hence $||f_1+f_2||_{p,q} \leq ||g_1+g_2||_p + ||h_1+h_2||_q \leq ||g_1||_p +||g_2||_p+||h_1||_q +||h_2||_q < ||f_1||_{p,q} +||f_2||_{p,q} +2 \epsilon$.

Answer 2

Take $f_1,f_2\in L^p+L^q$. Take $g_1,g_2\in L^p$, $h_1,h_2\in L^q$ such that $g_1+h_1=f_1$ and $g_2+h_2=f_2$. Then $$ \|f_1+f_2\|_{p,q} \le \|g_1+g_2\|_p + \|h_1+h_2\|_q \le \|g_1\|_p + \|g_2\|_p + \|h_1\|_q + \|h_2\|_q . $$ This holds for all such $g_i$ and $h_i$. Hence, we can take infima $$ \|f_1+f_2\|_{p,q} \le \inf_{h_1+g_1=f_1,h_2+g_2=f_2}\left(\|g_1\|_p + \|g_2\|_p + \|h_1\|_q + \|h_2\|_q\right) \\ =\inf_{h_1+g_1=f_1}\left(\|g_1\|_p + \|h_1\|_q\right) + \inf_{h_2+g_2=f_2}\left(\|g_2\|_p +\|h_2\|_q\right) = \|f_1\|_{p,q} + \|f_2\|_{p,q}. $$

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This is a common trick: First prove a much weaker estimate, then apply infimum to get the claim.