 And what we want to do, we want to complete our spaces so that we are in the framework of Banach spaces. And this is what is known as a Sobolev spaces. So in the simplest case, it assumes that we have a domain in Rn that is an open connected subset. We can define the space L2 of omega. So this is a space of all those functions which are square integrable plus an infinity. And you know very well from another score, so this is Banach space. In fact, Hilbert space. And what you also know is that in here, we have the space of smooth functions with compact support on omega. And it is dense in this subspace. So this means that I could define L2 by saying that I take the space C0 infinity omega and I close this with respect to the L2 norm. That's the approach I will take for the letter, for various generalization of this construction. So what we could do, we could define Lp omega. So this is just the closure of C infinity omega. But now I take the Lp norm. And what we also could do, we could construct spaces WKP omega. And this is the closure of smooth functions. But now with respect to the norm WKP. So let me define what it is. If I have a smooth function U, I can take its Lp norm. I can take its derivative, say, now U, can also take the Lp norm of Z, and so on, till I take all k's derivatives of U in the Lp norm. Well, sometimes, so this norm would be OK. But sometimes, so I want to take Lp, the norm to the power p. And then the p's root of that is sometimes better behaved. OK. Now the solve works also for many folds. So if m is a, say, compact, well, we don't need compact at this point. Just the Romanian manifold, I can define, if I have a function U, a smooth function with compact support on m, I can take its derivative. So this is a section of t star m. And since my manifold is Romanian, I have a querian derivative here, as a Levy-Chivita connection. So I can take the second derivative of the U, I can take the Z derivative, and so on. Right, and this is at hand. So we can define w kp on m in the same way. So I can take the space of smooth functions with compact support on m and close this with respect to the norm w kp. All right, and so what this gives us, this gives us a sequence of spaces for any p bigger than, say, 1. We have the space Lp m, which contains clearly w1p m. This contains w2p m, and so on, up to infinity. Now there is, so even more generally, what you could do, we could take a vector bundle E over m together with a connection. And we could define the stability of spaces for sections of this bundle just in the same way. So if you have a section, we know what the derivative of S is. We know what the second derivative is, and we can integrate this. So stability of spaces are also defined for sections. Now the basic fact in this theory is the following theorem, sometimes called subvalu embeddings theorem. I will take a combined version, which includes something which is known sometimes as contractual lemma, and so on. So the statement is this, the space wkp m, say, E, is contained in wmqem whenever provided k minus n over p is bigger or equal than m minus n over q. So let me assume for the whole theorem that our manifold m is, in fact, compact. So it's not important for all statements, but for some it will be crucial. So what this actually means is the following. It means that we have a natural operator which maps this space into this one. So this is on smooth functions given just by the identity map. And this extends to a bounded map between these subvalu spaces. In particular, part of the statement is that norm of u in wmq is smaller equal than c norm of u in wkp. And this is for all functions u. And c doesn't depend on this function. The second statement is that let's call this a j, an embedding operator. So j is compact provided we have the strengthening equality here. So let me write this again. k minus n over p is strictly bigger than m minus n over p. So by the way, n is the dimension of m. k is bigger than m. So that we don't lose any differentiability. The set statement is that we have a natural map from wkp into cr provided, again, k minus n over p is strictly bigger strictly bigger than r. And the fourth statement is that wkp mr, so that is a space of functions, is an algebra with respect to the point wise multiplication provided k minus n over p is bigger than 0. So as you see, this number k minus n over p appears over and over again. And it essentially tells you the properties of the functions in the so-called space. So I don't want to prove the theorem. The proof is pretty much involved. But I wanted to tell you just maybe one particular example where you can see what is actually going on. Are there any questions to the statement of the theorem? Yes? Yes, it's always the extension of the identity. So by the way, by saying that j is compact, what I mean is that whenever you have a bounded sequence in this space, its image in this space has a convergent subsequence. That's the definition of the compact operator. Now let me consider one example. Namely, we will take m to be the circle. So the simplest manifold. And let me take a smooth function on S1. So we'll multiply. So let u0 of theta be u of theta minus the integral of u theta. So from 0 to pi, 1 over 2 pi. So that the mean value of u0 is 0. And now by some of the argument, you know that there exists theta 0 in S1 such that u0 at theta 0 vanishes. Right, so let me take now u0 at the point theta minus u0 at the point theta 0. I can write this as an integral between theta 0 and theta u prime of, say, phi d phi. Right, this is, of course, also true for the absolute values. Now I can use a Cauchy-Schwarz inequality here to estimate this as an integral between theta 0 and theta norm u prime phi squared d phi to the power 1 half and integral theta 0 to theta 1 squared d phi squared. Now this is clearly an estimated further as a norm of, this is all u0 as a norm of u0 in w12 of S1 times, so this is no greater than the length of the circle, so this is square root of 2 pi. Now this is 0, and what we have, we have the bound on the C0 norm of u0 in terms of the w12 norm of u0. Right, and it's not hard from this to conclude that the norm of u in C0 is, in fact, bounded by a constant times norm of u in w12. So I assumed here that u was a smooth function, but now by an extension we have this inequality for all functions in w12, and this is precisely one of the embedding theorems that I had on the blackboard. If there are no questions to that, let us move to the next topic, and this will be elliptic operators. Here is a definition. I will say that L, so a map from smooth sections on M of some bundle E into smooth sections again over M in some bundle F is a differential operator of order, say L, which I will always assume is positive. If locally L of F can be written as the sum A alpha of x d alpha d of dx alpha applied to F. Right there, alpha is a multi-index, and the absolute value of alpha is most L. So what is going on here is this. I choose first local coordinates on M. These are x. So these are local coordinates on M. And I choose a local trivialization of both E and F so that I can think of sections here as functions F from u into, say, rk. And now what I'm saying is whenever I made those choices, I can represent L as a linear operator in the usual sense. So now the second part of the definition is that L is elliptic if the following holds. So I will define the symbol of L to be simply the highest order terms of L. That is, the absolute value of alpha is equal to L. And I take A alpha of x. And instead of taking here derivatives, I will replace this by a symbol. So I will take psi to the power alpha, where psi is in rn. What this means, perhaps more concretely, is the sum over alpha 1 alpha n. So z alpha 1 plus, well, they are all positive. So alpha 1 plus alpha n equals L, A alpha of x, psi 1 to the power alpha 1, and so on, psi n to the power alpha n. So this is called the symbol of the operator because of the obvious reason. And I say that L is elliptic if the symbol, if sigma psi of L is invertible for all psi non-zero and for all x. So the idea is very simple. If you have a differential operator, it's clear that some essential properties of this operator are encoded in the highest order terms. And we can encode this, so the higher order terms, in sigma psi. And if this is invertible, you may hope that in some sense our operator is invertible as well. Well, this is not quite true, but there is something left from there. So here is one example. If you take the Laplacian on Rn, that is just a sum of d2 over dx i squared, when i is from 1 to n, there's a minus sign. So the principal symbol of this operator delta is just minus sum xi i squared. And this is nothing else but just the squared norm of xi is a minus sign. So clearly, if xi is non-zero, this is non-zero, so we can invert numbers. Now here is perhaps a little more interesting example. Maybe before giving an example, so let me make a mark as that sigma of l as well defined as a homomorphism from pi star e into pi star f, where pi from g star m into m is a natural projection. So if I interpret here in the definition xi as a cotangent vector to my manifold m, the symbol for a fixed xi makes sense as a homomorphism between e and f. OK. Now perhaps one prime example. That is, we take m to be Romanian and oriented. Then we have, again, the Laplacian on function, say, this is minus star d star. And if you compute this in local coordinates, you will see that this is essentially the same Laplacian here plus some lower order terms. So this is, again, an elliptic operator. OK. So let us perhaps come to a more interesting example. So Dirac operators that we have seen the last time. If e is a Dirac bundle, we have defined, and I take a section of e. We have defined d of s to be the sum d over dx i, Clifford multiplied with Naples d over dx i of s. So i is here from 1 to n. And xi is, again, local coordinates on my manifold. Say it again. Wait. So I take d star and star again. Yeah. OK. But let us come, again, to the Dirac operator. So what we see here is that we have essentially the covariate. We have in local trivialization d over dx i plus 0's order terms. But 0 order terms are sort of immaterial for the definition of the principal symbol. And so what we see here is that the symbol, you know, if I write, say, xi, again, as xi 1 and so on xi n, so maybe beta sum xi i d xi, right? For the symbol of the Dirac operator, I have the sum i from 1 to n d over dx i Clifford multiplied. So let me know the Clifford multiplication by, say, rho of s times xi i. And this is just rho of, if you wish, xi i. So the sum xi i d over dx i. So this is essentially, again, xi. Now what we know is that Clifford multiplication with xi when squared is minus norm of xi squared. That is, rho of xi minus 1 is 1 over the norm of xi squared. So minus rho of xi. So in particular, the symbol is invertible, right? And therefore, this is an elliptic operator. And so we have seen that in dimension 4, Dirac operator splits into two parts, so the positive Dirac operator and the negative Dirac operator. And both parts are elliptic as well. The key property of elliptic operators is the following theorem. So if you have an operator L, this is a differential operator of order L between sections of vector bundles E and F, I can extend this as a bounded map from wk, say, k plus lp into wkp. And this is for any k and p. So the claim is that the norm of s in the wk plus lp norm is bounded by a constant times the norm of ls in the wkp norm plus norm of s in lp norm provided is elliptic. And this is sometimes called an elliptic estimate. So the proof of the theorem is not really very complicated, but it requires certain symbolic calculus, which I don't want to introduce. So it will take us just as granted. Anyway, but an important corollary from this statement is the following one. So by the way, I now always assume that m is compact. But let me, well, for this, it's not really that important. But here, if m is compact, elliptic, so let me denote now this by, say, star. The claim is that then star is a third whole map, which means that the kernel of l is finite dimensional. So let me write the dimension of the kernel of l less than infinity and the dimension of the core kernel of l. So this is the target space, so wkp divided by the image of l is also finite dimensional. So sometimes in the definition, you will also see that the requirement is that the image of l is a closed subspace in the target space. But you can show that this actually follows from these two conditions. And another, so one more part of the theorem is that the kernel of l consists of smooth sections only. OK, so let me give you an idea of the proof because some elements of this proof will play a role in the sequel. What we do is the following. So if u is an element of wkp and is in the kernel of l, by the elliptic estimate, we know then that u is in wkp for any k. And OK, anyway, but from this, by the so-called embedding theorem, we know that u is in cr for any r. And this tells us that u is a smooth section on m. So this proves the molar part. So let me prove that the dimension of the kernel is finite. So what we do is we take any sequence un in the kernel of l and we can normalize the sequence such that the norm of u in lp is 1. Now by the elliptic estimate, what we have is that the norm of un in wk plus l, well, say in wp is small or equal than c norm l u.