# $L^2$ and $L^\infty$ normed inequality for PDE solution: Which one is more informative and why?

I have the following inequalities

$$max_{t \in [0,T]} \lVert u_1(t, \cdot)-u_2(t, \cdot) \rVert_{L^2(\mathbb{R})} \leq C \lVert g_1(x) - g_2(x) \rVert_{L^2(\mathbb{R})}$$

and

$$max_{t \in [0,T]} \lVert U_1(t, \cdot)-U_2(t, \cdot) \rVert_{L^{\infty}(\mathbb{R})} \leq C \lVert g_1(x) - g_2(x) \rVert_{L^{\infty}(\mathbb{R})}$$

where for $x \in \mathbb{R}, t \in [0, T], T \in \mathbb{R}$ the function $U_j(t,x)$ is the solution to some PDE with different initial condition $g_j(x) = \mathbb{1}_{[K_j, \infty)}(x)$ with $K_1 < K_2$ and $x \in \mathbb{R}$ and $u_j(t,x)$ is the solution to the same PDE in weak formulation.

Which inequality is more informative and why? I am a bit confused because I read that on the one hand for $p<q$ that $$\lVert x \rVert _{L^q} \leq \lVert x \rVert _{L^p}$$ but also that $$\lVert x \rVert _{L^p} \leq n^{(\frac{1}{p}-\frac{1}{q})}\lVert x \rVert _{L^q}$$ Does that mean that both inequalities are equally informative in my case with $n=1$? And what about other cases with $n>1$?

EDIT Nr. 2 (the first edit concerned the whole question) Could it be that the first inequality is more informative because it tells us that $g(x) \in L^2(\mathbb{R})$ when g(x) is not known?

• I think the inequality $|x|_{L^p} \le n^{\frac{1}{p}-\frac{1}{q}}.|x|_{L^q}$ is not valid for any $n$. It must be saying there exists $n$ s.t. ... . And in your case, $n$ is not equal to 1. – crbah Mar 22 '16 at 10:16

## 1 Answer

The inequalities you cited only hold for the sequence version of the $L^p$-spaces, which are typically called $\ell^p$. In fact, the last one only holds for $x \in \mathbb{R}^n$.

For your function space $L^p(\mathbb{R})$ these inequalities do not hold: the function which is constant $1$ belongs to $L^\infty(\mathbb{R})$, but not to $L^2(\mathbb{R})$; and for a function with a "suitable" singularity it might be the other way round.

Hence, none of your inequalities is "more informative".

• Thanks for your answer. The inequalities should hold according to my lecture notes, so I edited the question and provided more detailed information. – mwater Mar 22 '16 at 12:41
• @mwater: No, I mean the inequalities $\|x\|_{L^q} \le \|x\|_{L^p}$ and $\|x\|_{L^p} \le n^{1/p-1/q} \, \|x\|_{L^q}$. These inequalities do not hold for $L^r(\mathbb{R})$! – gerw Mar 22 '16 at 15:33
• Ok, thanks for clarifying. So does that mean none of the inequalities is "more informative", even with the updated information? What do you think of my second edit (see last line of the question)? – mwater Mar 22 '16 at 15:48