(p.203) Appendix I. Some analysis topics
(p.203) Appendix I. Some analysis topics
AI.1 Some integrals and constants
We list some of the integrals that enter the calculation of the best constants in the smoothing effect.

(i) EULER'S GAMMA FUNCTION is defined as
$$\text{\Gamma}\left(p\right)={\displaystyle {\int}_{0}^{\infty}{t}^{p1}{e}^{1}dt,}\text{}p>0.$$$$\text{\Gamma}\left(p\right)\sim {\left(p/e\right)}^{p}{\left(2\pi p\right)}^{1/2}.$$ 
(ii) EULER'S BETA FUNCTION is defined for p, q > 0 as
$$B\left(p,q\right)={\displaystyle {\int}_{0}^{1}{t}^{p1}{\left(1t\right)}^{q1}}dt=2{\displaystyle {\int}_{0}^{1}{s}^{2p1}{\left(1{s}^{2}\right)}^{q1}}ds.$$$$B\left(p,q\right)=\frac{\text{\Gamma}\left(p\right)\text{\Gamma}\left(q\right)}{\text{\Gamma}\left(p+q\right)},$$$$B\left(p,q\right)=r{\displaystyle {\int}_{0}^{1}{s}^{rp1}{\left(1{s}^{r}\right)}^{q1}ds=}r{\displaystyle {\int}_{0}^{\infty}\frac{{x}^{rq1}}{{\left(1+{x}^{r}\right)}^{p+q}}dx}.$$ 
(iii) VOLUME OF THE UNIT SPHERE: It is well known that
$${\omega}_{n}=\frac{2{\pi}^{n/2}}{n\text{\Gamma}\left(n/2\right)}=\frac{{\pi}^{n/2}}{\text{\Gamma}\left(n/2+1\right)}.$$
(p.204) AI.2 More on Marcinkiewicz spaces. Lorentz spaces
From the definition of Marcinkiewicz norm in Section 1.1, formula (1.13), we immediately obtain an estimate for the distribution function of f of the form:
In this norm the explicit function U _{p}(x) = A x^{−n/p} of Sections 3.3 and 5.4 is not only the most concentrated one in the class of rearranged functions of equal norm, it is even the greatest element up to rearrangement. The existence of greatest element is an additional simplification for comparison arguments.
The same definition and norms apply when the base space is a domain of R ^{n}, or even a measurable subset K. One of the important functional features of M ^{p}(R ^{n}) or M ^{p}(Ω) is the fact that they are not separable. According to [BS88], every separable Banach function space is the closure of the set of bounded functions with support in a set of finite measure. This is not the case here, and in fact, when Ω is bounded, L ^{p}(Ω) is a nondense subset of M ^{p}(Ω). When Ω is unbounded or R ^{n}, then ${L}_{\text{loc}}^{p}\left(\Omega \right)\cap {M}^{p}\left(\Omega \right)$is a nondense subset.
In order to clarify this fact, we introduce the following functional in M ^{p} (Ω):
Lemma AI.1 N_{p} is a continuous seminorm in M ^{p}(Ω). Given f ∈ M ^{p}(Ω) we have ${N}_{p}\left(f\right)=0\text{}if\text{}f\in {M}_{0}^{p}\left(\Omega \right)$, which is defined as the closure of ${C}_{c}^{1}\left(\Omega \right)$ in M ^{p} (Ω). If Ω is bounded, the converse is true.
Proof The fact that N_{p} is a seminorm and is continuous are both clear. Let us prove next that N_{p}(f) = 0 for every $f\in {M}_{0}^{p}\left(\Omega \right)$. In the case where f is ${C}_{c}^{1}$ it is bounded, and then N_{p} (f) = 0. The result follows for ${M}_{0}^{p}\left(\Omega \right)$ by density.
On the contrary, if N _{p}(f) = 0, for every ε > 0 there is a value k such that ∥f _{k}∥ Mp ≤ ε with f _{k} = (f − k)_{+} sign (f). Now, f − f _{k} ∈ L ^{∞} (Ω) ⊂ M ^{p} (Ω) ⊂ L ^{p}(Ω), since Ω has bounded measure. The result follows.
Note that the limit N_{p}(f) is not zero in the case of example (1.14); actually, the limit is still N_{p}(U _{p}) = Ak _{p}.
(p.205) Lorentz spaces L ^{p, q} (Ω), 1 ≤ p, q < ∞, are defined by means of the norms
It is easy to see that L ^{p, p}(Ω) coincides with L ^{p}(Ω). The L ^{p,q}(Ω) form an increasing family of separable spaces as q increases. In the limit q → ∞ the norm becomes
AI.3 Morrey spaces
We have seen these in the study of extinction. Here is the definition.
Definition AI.1 Let p ∈ (1, ∞) and λ ∈ (0, n). The function $f\in {L}_{\text{loc}}^{1}\left(\Omega \right)$ is said to belong to the Morrey space M̃ ^{p, λ} (Ω) if
AI.4 Maximal monotone graphs
The nonlinearity in the filtration equation is in principle a C ^{1} monotone function, so that the equation may be written as
One of the main advantages of this generality is the fact that the inverse of a m.m.g. is again a m.m.g.; actually, both graphs are symmetric with respect to the main bisectrix in R ^{2}. The standard and somewhat awkward notation when using multivalued operators is set inclusion, so that when (a, b) is a point in the graph ϕ we write b ∈ ϕ(a) instead of b = ϕ(a), since generally ϕ(a) is not a singleton.
Maximal monotone operators in Hilbert spaces have been studied in full detail by Brezis in [Br73]. A review of the application of maximal monotone operators to the theory of the PME is contained in Chapter 10 of [V06].