Measure Theory/Linfinity

The Relationship between Various L^p

There is just one loose-end that we left dangling a couple lessons before, which is the issue of when $p=1$. There we asked whether it was possible for to make sense of the idea that the conjugate of p is $p^{*}=\infty$, but for this to make sense we need to have some notion of $\|\cdot \|_{\infty }$ and $L^{\infty }(E)$.

It is also natural to wonder if there is a relationship between $L^{p}(E)$ and $L^{q}(E)$ if $1\leq p<q$. Indeed there is a simple relationship.

Moreover, if we let $p\to \infty$ then do we discover a notion of $\|\cdot \|_{\infty }$ which perhaps "plays nicely" with $\|\cdot \|_{1}$?

Exercise 1. A Strict Separation of $L^{p}(E)$ Spaces Let $f(x)=1/x$. Show that $f\in L^{2}([1,\infty ))\smallsetminus L^{1}([1,\infty ))$. Use this basic idea to show that for any $1\leq p<q$ there is always a function in $L^{q}([1,\infty ))\smallsetminus L^{p}([1,\infty ))$ In something like a converse, also show that there is always a function in $L^{p}((0,1))\smallsetminus L^{q}((0,1))$

The above exercise shows that, if there is to be some relationship between $L^{p}(E)$ and $L^{q}(E)$, it will not hold generally -- and it may depend on the nature of the subset E.

Exercise 2. $L^{q}((0,1))\subseteq L^{p}((0,1))$ 1. Let $1\leq p<q$ and show that there exists some constant $0<c$ (which depends on p and q) such that for every $f\in L^{q}((0,1))$, $\|f\|_{p}\leq c\|f\|_{q}$ Hint: Starting from the left side, write down a factor of 1 next to $|f|^{p}$ and apply the product bound. It may seem natural, when using the product bound, to take the p norm of $|f|^{p}$ but in fact this fails to produce any nice "cancellations". Instead, prove that $\||f|^{p}\|_{q}=\|f\|_{q}^{p}$ and work through the remaining details. 2. Now argue that $L^{q}((0,1))\subset L^{p}((0,1))$. Note that the containment is strict so you have two things to show: Both that the subset relation holds, and that there is some element in the set difference. 3. Finally, show how the above result can be generalized from the interval $(0,1)$ to some wider class of subsets of real numbers. At what point did you use any facts about $(0,1)$, and can you remove some details about this set which were unnecessary for the above proof?

L^∞

As in the previous exercise, we will assume that $\lambda (E)<\infty$. (This is the generalization that you should have found at the end.

We now have some sense of what happens as we let p in $L^{p}(E)$ grow larger. In particular, you get an ever shrinking sequence of spaces of functions.

What then is the set

$L^{\infty }(E)=\bigcap _{1\leq p}L^{p}(E)$

This is essentially the same question as finding a characterization of $\lim _{p\to \infty }\|f\|_{p}$.

Definition: $\|\cdot \|_{\infty },L^{\infty }(E)$

Let $E\subseteq {\mathbb {R}}$ be a subset of real numbers and $f\in L^{p}(E)$ for every $1\leq p$. Then define the infinity norm by

$\|f\|_{\infty }=\lim _{p\to \infty }\|f\|_{p}$

and the space of functions with finite infinity norm,

$L^{\infty }(E)=\{f:\|f\|_{\infty }<\infty \}$

Exercise 3. Experimenting with $\|\cdot \|_{\infty }$ 1. Set $E=(0,1)$. Compute $\lim _{p\to \infty }\|x\|_{p}$. Hint: If you're a little rusty with your tricky limits at infinity, use the "e to the ln" trick. 2. Set $E=(0,a)$ for some $0<a$. Compute $\lim _{p\to \infty }\|x\|_{p}$. 3. Set $E=(0,2)$. Compute $\lim _{p\to \infty }\|1-|x-1|\|_{p}$. 4. What happens if you multiply any of these functions, in any of these examples, by a constant? What happens if you sum any of these with a constant? Hint: Use the homogeneity of the norm for the constant multiple part -- don't make things harder than they have to be. For the constant sum part, if you consider ${\sqrt[{p}]{\int _{0}^{1}(x+c)^{p}}}$ it will be helpful to consider the following: Let $0\leq b<a$. Then $\lim _{p\to \infty }(a^{p}-b^{p})^{1/p}=a$ 5. Based on all of the experimentation above, what do all of these values have in common?

The above exercise hopefully suggests that $\|f\|_{\infty }$ is the maximum of the function ... er, well, maybe not exactly the maximum of the function. After all, the very well-behaved function $f(x)=x$ on the interval (0,1) has no maximum.

Ok, so maybe $\|f\|_{\infty }$ is really the supremum rather than the maximum. Well, but even this doesn't work, because on a null set we can mess around with the function values and not change the value of the $L^{p}(E)$ norm.

Therefore we need a definition like this but which is not sensitive to values on any null set. Rather than talking about a maximum, or a supremum, we talk about an "essential supremum".

Definition: essential bound, essential supremum

Let $f:E\to {\mathbb {R}}$ be any function defined on a subset of real numbers. Let $C\in {\mathbb {R}}$.

We say that C is an essential bound of f if there exists some null set $N\subseteq E$ such that $|f(x)|\leq C$ for every $x\in E\smallsetminus N$.

If ${\mathcal {C}}$ is the set of all essential bounds of f,

${\mathcal {C}}=\{C:C{\text{ is an essential bound of }}f\}$

then we define the essential supremum of f by the infimum of ${\mathcal {C}}$. That is to say, the essential supremum of f is

$\inf {\mathcal {C}}$

Exercise 4. ${\mathcal {S}}=\|f\|_{\infty }$ Let $\lambda (E)<\infty$ and let $f:E\to {\mathbb {R}}$ have a finite essential supremum, which we denote by ${\mathcal {S}}$. 1. Prove that ${\mathcal {S}}\leq \|f\|_{\infty }$. This will require using some analysis of integrals that we haven't done for a while. But it is natural to consider the indexed family of sets $A_{t}=\{x\in E:|f(x)|\geq t\}$ for any $t\in [0,{\mathcal {S}})$ With this one may apply the ML bound to obtain the desired result. 2. Prove $\|f\|_{\infty }\leq {\mathcal {S}}$. For the reverse inequality, let $\varepsilon \in {\mathbb {R}}^{+}$ be arbitrarily small. Let $B_{\varepsilon }=\{x\in E:{\mathcal {S}}-\varepsilon <|f(x)|\}$ Now prove that $\|f\|_{p}\leq ({\mathcal {S}}-\varepsilon )\lambda (A_{\varepsilon })^{1/p}$ Then take the limits $p\to \infty$ and then $\varepsilon \to 0$.

Exercise 5. Further Extending $L^{p}(E)$ Show that the notion of an $L^{p}$ space can be extended to include $L^{\infty }$. That is to say 1. Define the norm, space, and distance for $L^{\infty }(E)$. 2. Prove that it is a vector space. 3. Prove that the concept of the conjugate of a real number $1\leq p$ can be extended to include $p=\infty$, and that the product-to-sum bound still applies. 4. Prove that the integral product bound does not hold but the triangle inequality still holds when $p=\infty$, and that $L^{\infty }(E)$ is complete. Be careful about any conditions which must hold in order for theorems to hold -- they may not be exactly the same conditions for finite p.