ELLIPTIC REGULARITY Analysis 3 Report

ELLIPTIC REGULARITY

Analysis 3 Report

Maxim Jeﬀs

December 12, 2016

Introduction

In applications of Mathematics, elliptic partial differential equations typically describe stationary or steady-state pro- cesses. Laplace’s equation for instance, ubiquitous in Mathematical Physics, describes phenomena as diverse as steady fluid flows, static charge distributions and gravitating systems. In these applications, the physical interpretation would lead one to expect solutions to be extremely smooth.

Mathematically, ellipticity is defined as a seemingly innocuous condition on the coefficients of the highest-order partial derivatives. This condition has remarkable consequences, implying for instance that all weak solutions are in fact infinitely differentiable. Informally, this arises because the ellipticity condition allows one to construct a parametrix, a suitable substitute for an inverse operator. In order to perform this precisely, it is necessary to make the notion of a partial differential operator somewhat more robust, leading to the generalised notion of a pseudodifferential operator, which we introduce in the third section and use to prove the central results for elliptic partial differential equations.

The Symbol and Ellipticity

A partial differential operator of order m is a linear map P on smooth functions from RN to Cn in the form ∑ P = Aα(x)∂α |α|≤m where the Aα are smooth matrix-valued functions. The most important part of a partial differential operator P is its principal symbol. To motivative this definition, let u : RN → Cn be a Schwartz function. The Fourier inversion theorem then tells us that ∫ ∑ − · | | (P u)(x) = (2π) N/2 eix p i α Aα(x)û(p)pα dp (1) RN |α|≤m so that the action of P is largely determined by its principal symbol: ∑ σ˜(P )(x, p) := im Aα(x)pα |α|=m which is simply a matrix-valued polynomial. The elliptic operators are defined to be those for which the principal symbol is invertible (as a matrix) for all x and p ≠ 0. For instance, the scalar Laplacian

∑N ∂2 ∆ = − ∂x 2 i=1 i

1 has principal symbol σ˜(x, p) = |p|2, so that it is indeed elliptic. Admittedly, the significance of the ellipticity condition may well seem somewhat obscure. The idea is that we can attempt to invert an elliptic partial differential operator by defining an operator ∫ − (Qu)(x) = (2π)−N/2 eix·p [˜σ(P )(x, p)] 1 uˆ(p) pα dp RN Unfortunately, this does not actually work: the composition of operators is quite a bit more subtle. However, what one can do is consider a formal series of generalised integral operators that will give an inverse for the operator, up to some compact, infinitely smoothing operator. This shall be our programme for the subsequent sections.

Pseudodiﬀerential Operators

Equation (1) should be suggestive – what if we were to replace the polynomial expression by a more general function? We ought to impose some growth requirements, since we would not want our function to grow too much faster than a polynomial. Formally, we say that a smooth matrix-valued function σ(x, p) is a (total) symbol of order m if for all | α β | ≤ | | m−|β| multi-indices α, β, there exists a constant C with Dx Dp σ(x, p) C(1 + p ) . We then define the associated pseudodifferential operator of order m by ∫ (P u)(x) = (2π)−N/2 eix·pσ(x, p)û(p) dp RN It should at least be clear that if u is a Schwartz function and σ(x, p) has compact x-support, then the resulting function P u will also be Schwartz, so in this respect it will still behave like a differential operator. At this point, the reader may wonder what happens if we formally rewrite P as an integral operator using ∫ ∫ ∫ (P u)(x) = (2π)−N/2 ei(x−y)·pσ(x, p)u(y) dp dy “ = ” K(x, y)u(y) dy RN RN RN where we have set ∫ − − · K(x, y) = (2π) N/2 ei(x y) pσ(x, p) dp (2) RN This, of course, is complete nonsense: K(x, y) does not even converge in a meaningful sense if σ(x, p) is a polynomial. However, being analysts, we are not deterred: the original expression for P still makes sense. We may regard K as being a formal kernel. Equation (2) is an example of an oscillatory integral, and the theory of pseudodifferential operators has deep connections to harmonic analysis and the theory of distributions, which provide the language in which it is properly formulated (see [Nic10]). We will follow a more lowbrow route; instead of directly trying to give meaning to the kernel K, we will cleverly avoid the matter entirely by working formally with the symbols of the operators rather than the operators themselves!

Firstly, note that if we take m above to be negative, then we would expect a symbol of order m to yield an operator that is ‘smoothing’ in some sense (to be made precise later), with the amount of smoothing increasing as m → −∞. This corresponds to the observation that when m is sufficiently negative, the corresponding formal kernel will actually be integrable and we will be able to take a number of derivatives before it becomes singular again. Because of this, we say that ∑∞ σ ∼ σj j=1 → −∞ ∈ N is a formal development∑ for a symbol σ if each σj is a symbol of order mj with mj , and if for every m , the − n − symbol σ j=1 σj is of order m for all sufficiently large n. Returning to the idea of kernels, this may be regarded as an appropriate substitute for ‘convergence’ of the kernels. Although this analogy is somewhat difficult to make precise, it is the central idea of the symbol calculus for pseudodifferential operators. To get the theory off the ground, it is necessary to develop some fairly technical lemmata which we sketch in the next sections, omitting the routine verifications and illustrating the main ideas that make the theory work. In the sequel, we will often omit factors of (2π)−N with the understanding that the arguments are essentially unchanged.

2 The Symbol Calculus

We begin with the fundamental:

Theorem 1. Every formal series is the formal development of some pseudodiﬀerential operator.

Proof. The proof of this theorem is not difficult, but rather unenlightening. The idea is to take a smooth cutoff function ϕ : [0, ∞) → [0, 1] with ϕ ≡ 0 on [0, 1] and ϕ ≡ 1 on [2, ∞), and then ‘glue’ the symbols together using ∞ ( ) ∑ |p| σ(x, p) = ϕ σ (x, p) r j j=1 j where rj are some real constants tending to infinity that are to be chosen appropriately. There is no problem with convergence here, since the sum will be finite for any given point. The problem comes in managing the slightly tricky estimates to make sure that this does not grow too fast. ■

The next important question to answer is when we can regard formal kernels as pseudodiﬀerential operators. This is answered by

Lemma 1. Suppose A(x, y, p) is a smooth matrix-valued function with compact x, y support, such that, for ﬁxed m and given | α β γ | ≤ | | m−|γ| α, β, γ multi-indices, there is some constant C such that Dx Dy Dp A(x, y, p) C(1 + p ) . Then the operator ∫ ∫ (T u)(x) = (2π)−N ei(x−y)·pA(x, y, p)u(y) dy dp RN RN is a pseduodiﬀerential operator with symbol

∑ i|α| σ ∼ (DαDαA)(x, x, p) α! p u α Proof. As usual, what we really want to say here is that the kernel ∫ K(x, y) = ei(x−y)·pA(x, y, p) dp RN can be expanded as a ‘Taylor series’. We get around the possible singularity of the kernel K by an ingenious calculation, defining a principal symbol by ∫ ∫ ′ σ(x, q) = eix·(p−q)Aˆ(x, p − q, p) dp = eix·p Aˆ(x, p′, p′ + q) dp′ RN RN where the Fourier transform of A is taken in the second variable, so that ∫ ∫ ∫ ∫ ∫ (T u)(x) = ei(x−y)·pA(x, y, p)u(y) dy dp = eix·pAˆ(x, p − q, p)û(q) dq dp = eix·qσ(x, q)û(q) dq RN RN RN RN RN using the convolution theorem and the fact that the changes in the order of integration can actually be justified this time. Now we can simply expand Aˆ using Taylor’s formula!

∑ i|α| Aˆ(x, p′, p′ + q) = (DαAˆ)(x, p′, q)(p′)α + R (x, p′, p′ + q) α! q ℓ |α|≤ℓ for some remainder function Rℓ, and then use the properties of the Fourier transform to conclude that

∑ i|α| k(x, q) = (DαDαA)(x, x, p) + r (x, q) α! p u ℓ |α|≤ℓ for a remainder term rℓ(x, q) that we can easily show to have small order as ℓ → ∞ using the bounds on A. ■

3 For ‘genuine’ diﬀerential operators, the support of P u is always a subset of the support of u. One might ask whether this holds for pseudodiﬀerential operators also. The answer is negative in general, but we do have

Proposition 1. Suppose P is a pseudodiﬀerential operator with symbol σ(x, p) that has compact x-support. Then for all ε > 0, there exists a pseudodiﬀerential operator P ′ with symbol that is a formal development of σ such that any point in the support of P ′u is a distance of at most ε from the support of u, for any Schwartz function u.

Proof. Take a smooth function ϕ : RN × RN → R such that ϕ(x, y) ≡ 1 near the diagonal set {x = y} and is identically zero if |x − y| > ε. Then we can consider the integral operator ∫ ∫ (P ′u)(x) = (2π)−N e−i(x−y)·pϕ(x, y)σ(x, p)u(y) dy dp RN RN that certainly satisﬁes the requirements. Applying Lemma 1 shows that it must have symbol that is a formal development of σ. ■

Interaction of Symbols

With this under our belts, we can swiftly dispose of the crucial question of how symbols of operators interact:

Proposition 2. Suppose P,Q are pseudodiﬀerential operators such that there exists some compact K ⊂ RN with

1. the support of Qu and P u is a subset of K for all compactly supported smooth functions u;

2. P u = Qu = 0 whenever the support of u is disjoint from K. ∗ ∗ 2 RN ∗ Given P ,Q , that are L ( ) formal adjoints of P,Q, then they are also pseudodiﬀerential operators, where the symbol σP has formal development

∑ i|α| σ∗ ∼ DαDα(σ )† P α! p x P α

Similarly, P ◦ Q is a pseudodiﬀerential operator with symbol having formal development

∑ |α| i α α σ ◦ ∼ D σ (x, p)D σ (x, p) P Q α! p P x Q α

Proof. We have done most of the work in proving Lemma 1. Take a smooth cutoﬀ function ϕ : RN → Rn supported in K and any other smooth function u with support in K. We can see that the adjoint of P must be given by: ∫ ∫ ∫ ∫ ∗ i(y−x)·p † i(y−x)·p † (P u)(y) = e σP (x, p) dx dp = e ϕ(y)σP (x, p) dx dp RN RN RN RN

† The result now follows immediately by applying Lemma 1 and the fact that ϕσP = σP , since σP (x, p) must be supported for x ∈ K. We can deduce the second statement from the ﬁrst by writing Q = (Q∗)∗: ∫ ∫ i(x−y)·p † (Qu)(x) = e σQ∗ (y, p) u(y) dy dp RN RN and since this is simply an inverse Fourier transform of a Schwartz function (by our compactness assumptions), we have ∫ ∫ ∫ ix·p c i(x−y)·p † (P Qu)(x) = e σP (x, p)Qu(p) dp = e σP (x, p)σQ(y, p) u(y) dy dp RN RN RN

Applying Lemma 1, the product rule and the previous part of the proposition yields the claimed formula. ■

4 Sobolev Spaces

In functional analysis, bounded operators are typically the easiest to analyse. We would like to consider a class of spaces where differential operators become bounded operators. To this end, we define the Sobolev k-norm to be ∑ ∫ || ||2 | α |2 u k = ∂ u(x) dx (3) RN |α|≤k and the transformed Sobolev k-norm to be ∫ || ||2′ | | 2k| |2 u k = (1 + p ) uˆ(p) dp RN RN → Cn 2 RN for Schwartz functions u : . We define the Sobolev space Lk( ) to be the completion of the space of Schwartz N n functions u : R → R in the Sobolev k-norm. However, since there exists constants C1,C2 such that

2k 2 2k 2k C1(1 + |p|) ≤ 1 + |p| + ··· + |p| ≤ C2(1 + |p|) it follows from Fourier inversion that the k-norm and the transformed-k-norm are equivalent for Schwartz functions and 2 RN hence yield equivalent norms on the space Lk( ). Now suppose L is a partial differential operator of order m such that all of the coefficient matrices have all derivatives bounded. Then for all Schwartz functions u : RN → Rn we have   ∫ 2 ∫ ∫ ∑ ∑ ∑ ∑ 2 ∑ || ||2 α  β β  ≤ α+β ≤ ′ | γ |2 Lu k = D A D u M D u M D u RN RN RN |α|≤k |β|≤m |α|≤k |β|≤m |γ|≤k+m 2 RN Since the Schwartz functions are dense in Lk( ), by construction, this implies that L must extend uniquely to a bounded 2 RN → 2 RN linear map Lk+m( ) Lk( ) for all positive integers k. More generally, suppose that P is a pseudodifferential operator that has symbol σ(x, p) with compact x support. Then we have ∫ ∫ || ||2′ | | 2s| |2| |2 ≤ | | 2s+2m| |2 || ||2′ P u s = (1 + p ) σ(x, p) uˆ(p) dp (1 + p ) uˆ(p) dp = u s+m RN RN where we have used the assumption of compact support in order to apply Fourier inversion. Therefore P must also 2 RN → 2 RN extend uniquely to a bounded linear operator Ls+m( ) Ls( ).

Weak Solutions

2 RN It is useful to have a more concrete description of the spaces Lk( ). This can be done using the formalism of weak derivatives. In previous sections we were able to give a meaning to the integral operators defined by singular kernels by using a formal computation involving changing the order of integration where one side would be well-defined even when the other side failed to exist. Similarly, looking at the integration by parts formula ∫ ∫ (∂αf)ϕ = (−1)|α| f∂αϕ RN RN we can observe that the right hand side of the equation makes sense even when f is only integrable. This suggests that we might try to define the derivative ∂αf to be a function for which this equality is satisfied and hope that this definition matches our usual notion of derivative. More formally, if we are given f, g locally integrable functions on RN , we say that g is the α-weak derivative of f if for every compactly supported smooth function ϕ, ∫ ∫ gϕ = (−1)|α| f∂αϕ RN RN A simple argument shows that weak derivatives are essentially unique, if they exist, and if f is actually has a derivative ∂αf, then this will indeed be its weak derivative. However, weak derivatives need not exist in general, as one can see 2 RN by considering the simple case of a step function. We define Wk ( ) to be the space of all locally integrable functions with weak partial derivatives up to order k, with norm given by (3) with the partial derivatives replaced by the weak 2 RN derivatives. We claim that this is actually the space Lk( ), which will follow from:

5 2 RN Proposition 3. The space Wk ( ) is complete and the Schwartz functions form a dense subset. 2 RN α 2 RN | | ≤ Proof. Firstly, if fn is a Cauchy sequence in Wk ( ), then ∂ f is a Cauchy sequence in L ( ) for all α k. By the 2 N α α 2 N completeness of L (R ), they must converge to functions f := limn→∞ fn and g := limn→∞ ∂ fn in L (R ). We claim that ∂αf = gα. We know that ∫ ∫ α |α| α ∂ fnϕ = (−1) fn∂ ϕ RN RM for all compactly supported smooth functions ϕ. Applying Hölder’s inequality and taking limits of both sides then gives the desired conclusion. For the second part, we begin with some Lemmata: α Lemma 2. Suppose f, g are locally integrable with ∂ f = g weakly and let uδ be the standard molliﬁer. Then the actual α partial derivative ∂ (uδ ∗ f) is equal to uδ ∗ g for all δ > 0.

Proof. This is just a computation: ∫ ∫ ∫ α ∗ α − − |α| α − − ∗ ∂x (uδ f)(x) = ∂x uδ(x y)f(y) dy = ( 1) ∂y uδ(x y)f(y) dy = g(y)uδ(x y) dy = (uδ g)(x) RN RN RN ■

N Lemma 3. Let φr : R → R be a smooth cutoﬀ function, constant on Br(0) and supported on Br+1(0), with all derivatives ∈ 2 RN 2 RN → ∞ bounded. Then for all f Wk ( ), the sequence φrf converges to f in Wk ( ) as r . Proof. Suppose k = 1 and note that for all compactly supported smooth functions ϕ ∫ ∫ ∫

(∂ϕ)φrf = (∂(ϕφr) − ϕ∂φr) f = − ϕgφr + ϕ(∂φr)f RN RN RN so that the weak partial derivative of φrf is gφr + f∂φr. Now note that ∂φr → 0 almost everywhere as r → ∞, and 2 N since |(∂φr)f| ≤ M|f|, the dominated convergence theorem implies that (∂φr)f converges to zero in L (R ). The 2 N dominated convergence theorem also implies that φrf → f and gφr → g in L (R ) as r → ∞, hence implying that → 2 RN ■ φrf f in Wk ( ). The general case now follows by induction on k. ∈ 2 RN 2 RN Now, take f Wk ( ) and a sequence fn of compactly supported functions in Wk ( ) converging to f in 2 RN ∗ Wk ( ). We can then take mollifications uδ fn of these functions, which will be smooth, compactly supported and 2 N will converge to f in L (R ) as δ → 0. Then Lemma 2 implies that the mollifications uδ ∗ fn also converge to fn in the 2 RN → ■ space Wk ( ) as δ 0. Taking a diagonal subsequence completes the proof. 2 RN From this, we see that we can explicitly define the action of a partial differential operator L on the space Lk( ) ∈ 2 RN using the weak partial derivatives in place of the actual derivatives. We say that f Lk( ) is a weak solution if Lf = 0. Now we may ask: when is a weak solution actually differentiable?

Sobolev Theorems

The ﬁrst step towards an answer to this question is given by:

Theorem 2. (Sobolev Embedding) Consider the Banach space Ck(RN ) with the uniform norm ∑ || ||2 | α |2 u Ck := sup D u(x) ∈RN x |α|≤k

Then for all j > k +N/2, there is a constant C such that ||u|| k ≤ ||u|| 2 for all Schwartz functions u, and thus a continuous C Lj 2 RN ⊂ k RN embedding Lj ( ) C ( ).

6 Proof. Suppose |α| < j − N/2, so that we have ∫ ∫ |Dαu(x)| ≤ C (1 + |p|)|α||uˆ(p)| dp = C (1 + |p|)|α|−j(1 + |p|)j|uˆ(p)| dp RN RN

Applying Cauchy-Schwarz gives (∫ )(∫ ) |Dαu(x)|2 ≤ C2 (1 + |p|)2|α|−2j dp (1 + |p|)2j|uˆ(p)|2 dp RN RN

α 2 Since 2|α| − 2j < −N, the ﬁrst integral is ﬁnite and so |D u| ≤ K||u|| 2 . Summing over all α now gives the result. Lj ■

In particular, we can now justify why pseudodifferential operators of negative order are smoothing: they increase the differentiability of the functions they act upon. Furthermore, we see that if an operator P has a symbol that has order −m for all integers m, then it must be infinitely smoothing, that is, have image in C∞(RN ). Hence two pseudodifferential operators with the same formal development must differ only by an infinitely smoothing operator.

′ One can also ask when one Sobolev space is a subspace of another. If s′ < s, then since (1 + |p|)2s ≤ (1 + |p|)2s, we || || ′ ≤ || || 2 RN 2 RN have the inequality u s u s and hence Ls( ) embeds continuously in the larger space Ls′ ( ). In fact, one can say more: ∈ 2 RN Theorem 3. (Rellich) Suppose uj Ls( ) is a sequence of functions supported in B1(0) and uniformly bounded in the ′ 2 RN 2 RN → 2 RN s-norm. Then for all s < s, there is a subsequence that converges in Ls′ ( ). That is, the inclusion Ls( ) Ls′ ( ) is compact on functions that are uniformly compactly supported.

Proof. We begin by choosing ϕ a smooth function identically equal to 1 on B1(0) and supported in B2(0), so that uj = ϕuj for all j. This in turn implies that uˆj = ϕˆ ∗ uˆj, since all of the uj are integrable, and hence ∫ ∫ α α α −s s ∂ uˆj(p) = ∂ ϕˆ(p − q)ˆuj(q) dq = ∂ ϕˆ(p − q)(1 + |q|) uˆj(q)(1 + |q|) dq RN RN

| α |2 ≤ || ||2 The Cauchy-Schwarz inequality implies therefore that ∂ uˆj Kα(p) uj s for some continuous function Kα(p) independent of j. Therefore on a compact set, all derivatives of the uˆj are uniformly bounded in the supremum norm and so, by the Arzelà-Ascoli theorem, uˆj has a subsequence that is uniformly convergent on compact sets. Now ∫ ∫ || − ||2 | | −2(s−s′) | | 2s| − |2 | | 2s′ | − |2 uj uk s′ = (1 + p ) (1 + p ) uˆj(p) uˆk(p) dp + (1 + p ) uˆj(p) uˆk(p) | | | |≤ p >r ∫ p r || − ||2 uj uk s 2s′ 2 C ≤ + (1 + |p|) |uˆ (p) − uˆ (p)| ≤ + K(r)||uˆ − uˆ ||∞ 2(s−s′) j k 2(s−s′) j k (1 + r) |p|≤r (1 + r)

Take the first term less than ε/2 by taking r sufficiently large and then take the second term less than ε/2 by taking j, k sufficiently large. ■

p N All of this theory can be developed in the more general setting of the Sobolev spaces Lm(R ), defined in the analogous manner. Now one has the subtlety that one can play off m against increasing the value of p to gain more p N regularity. If we define the strength of the Sobolev space Lm(R ) by ΣN (m, p) = m − N/p, then we may regard p N functions in Lm(R ) as being ‘ΣN (m, p)-times differentiable’, justified formally by the theorem:

p N k N Theorem 4. (Morrey) Suppose k = ΣN (m, p); then Lm(R ) embeds continuously into C (R ).

Now the relationships between the various Sobolev spaces become rather more complicated so we cannot use the elementary and rather elegant proofs given above (see [Nic07]). However, these extensions become indispensable in the functional-analytic study of non-linear elliptic problems. We will not require these ideas to prove our main result:

7 The Parametrix

Elliptic regularity is fundamentally a deep theorem: other approaches use diﬃcult analytic results from harmonic analysis or the theory of distributions. We can now use the machinery we have developed to give a fairly straightforward and quite enlightening proof by establishing the existence of a parametrix, from which all of the fundamental theorems for elliptic partial diﬀerential operators will be very rapidly deduced.

Theorem 5. Suppose P is an elliptic partial differential operator of order m with compactly supported symbol. Then there exists a pseudodifferential operator Q of order −m such that PQ = I − S and QP = I − T , where S, T are infinitely smoothing operators. We call Q a parametrix for P .

Proof. Firstly, note that if P is an elliptic differential operator, then its total symbol σ(x, p) must also be invertible for all |p| ≥ C for some constant C > 0. This is because the principal symbol becomes the dominant part of the symbol as p → ∞ and invertibility is an open condition. Furthermore, the inverse must satisfy |σ(x, p)−1| ≤ C′(1+|p|)−m for some constant C′ which we can take to be equal to C. Now we follow the idea sketched in the introduction. Take a smooth −1 cutoff function ϕ : [0, ∞) → [0, 1] such that ϕ ≡ 0 on [0,C] and ϕ ≡ 1 on [2C, ∞) and set τ0(x, p) = ϕ(|p|)σ(x, p) . A simple argument then shows that τ0(x, p) is indeed a symbol of order −m. Let Q0 be the pseudodifferential operator with symbol τ0.

Now, Proposition 2 tells us that composition of pseudodiﬀerential operators is much more complicated than just multiplication of symbols:

∑ |α| i α α σ ◦ = (D τ(x, p))(D σ(x, p)) Q0 P α! p x α yet we see that since τ0σ − 1 = 0 for all |p| ≥ 2C, the above equation implies that the symbol of Q0P − 1 must at least have order m − 1. Therefore we may proceed inductively, writing ∑∞ τ ∼ τk k=0 for τk a symbol of order m − k, and then apply Theorem 1 to produce a pseudodifferential operator Q of order −m with the above formal expansion, such that QP − 1 has symbol of order −j for every j ≥ 0. Of course, Proposition 2 has several compactness assumptions and this is where we may use Proposition 1 to replace P,Q by ‘local’ operators with symbols that have the same formal developments and therefore only differ from P,Q by infinitely smoothing operators. A symmetric argument shows that there also exists Q′ of order −m with PQ′ − 1 infinitely smoothing. That we may take Q′ = Q is then a formal consequence. ■

The Fundamental Results

From Theorem 6 we may deduce much of the theory of elliptic partial diﬀerential operators as a corollary:

Corollary 1. Suppose P is an elliptic operator of order m with compactly supported symbol. Then ∈ 2 RN 1. For any u Lk( ), if P u is smooth, then u is smooth. In particular any weak solution or eigenfunction of P is smooth; 2 RN → 2 RN 2. For all k, P : Lk( ) Lk−m( ) is a Fredholm operator whose index is independent of k;

3. (Fundamental Elliptic Estimate) For all k there is a constant Ck such that ||u||k ≤ Ck(||u||k−m + ||P u||k−m) for ∈ 2 RN || || ≤ ′|| || ′ all u Ls( ). In particular if u is a weak solution of P u = 0, then u Ck C u L2 for some constant C depending only on k.

Proof. (1) is immediate, for if P u is smooth, then u = QP u + T u is also smooth. For any λ ∈ C, we can apply the same argument to P − λI to deduce the second statement.

8 (2) follows from the standard fact in Hilbert space theory that operators that are invertible modulo compact operators are Fredholm, along with the observation from the Rellich Theorem that inﬁnitely smoothing operators must be compact on compact sets. The invariance of the index follows from the fact that all elements of ker(P ) are automatically smooth and hence are in every Sobolev space, as well as the observation from Proposition 2 that the adjoint of an elliptic operator is also elliptic, so that all elements of coker(P ) = ker(P ∗) are also smooth.

The fundamental estimate in (3) is a result of ||u||s ≤ ||QP u||s + ||T u||s ≤ C(||P u||s−m + ||u||s−m), using the 2 RN → 2 RN Rellich theorem to conclude that T is a bounded operator Ls( ) Ls−m( ), at least on compact sets. The second ′ statement follows from ‘bootstrapping’ the elliptic estimate ||u||k ≤ Ck||u||k−m by writing, for any p a multiple of m:

||u||p′ ≤ C1||u||p′−m ≤ C2||u||p′−2m ≤ · · · ≤ Cp′ ||u||L2 and then noting that, for any given k, the Sobolev embedding theorem implies there exists some suﬃciently large p′ so || || ≤ || || ■ that u Ck C u L2 . p′

Final Remarks

What is so remarkable about the above corollary is that one would of course not expect any of these properties to hold for an arbitrary differential operator – elliptic operators are extremely special. The fact that they play such a fundamental role in Mathematical Physics and Differential Geometry certainly points to some deep connection between Mathematics and Physics. Finally, one should note that many of the compactness assumptions above can be removed if one is willing to do more work; the compact case is however sufficient for many applications.

References

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[Nic07] L.I. Nicolaescu, Lectures on the geometry of manifolds, 2nd ed., World Scientiﬁc, 2007.

[Nic10] , Pseudodiﬀerential operators and their geometric applications, Spring 2010 Topics in Topology, 2010.