 parameter, at a certain point it will be chosen. So a lot of these parameters actually are chosen along the way at a certain point. And as you can imagine this constant here now is going to depend on a hell of a lot of parameters on M0, on M0, on Delta, on whatever. But then, as soon as we choose all of them, they will actually become then geometric constants. Very good. So now, here we are, we have our ball. So here's P. Now pick up an optimal plane for the excess. So, let pi L be the plane which optimizes your excess. So this means that the excess of the graph U on the ball of radius L is actually the excess relative to this plane pi L. Okay, so now there's one thing which is kind of, I mean, there's one thing that you can conclude from either the proposition that your DU is C0-alpha or that you can actually directly conclude from the Georgia's excess decay by summing a certain estimate, which is done in the lecture, okay? But so one of the things that you can certainly believe is that since I proved that the derivative of U is C0-alpha, and since I essentially showed you that this optimal plane is kind of the average of the function DU over a ball of comparable size, right? So how much is this plane pi L tilted? So this plane pi L is tilted as most compared to the horizontal plane e to the power one-half, okay? So, okay, that is because the C0-alpha norm of DU is less or equal than e to the power one-half. So the C0 norm of DU is less or equal than e to the power one-half. And this optimal plane is essentially an average of the function DU over the corresponding scale, okay? Otherwise, you can prove it directly from the Georgia's excess decay and that is done in the lecture notes. Okay, very good. So now you apply this lemma that allows to pass from the cylindrical excess to the, from the spherical excess to the cylindrical excess, okay? So now take a cylinder of size square root of m, m0 L of L divided by 2, okay, center it at pl and oriented, I mean with base, actually, parallel to pi L. So this is this cylinder. Let us maybe call it cylinder cl, okay? And now this time conclude that the excess of the graph of U with respect to the cylinder and then taking, of course, the horizontal plane, I mean the plane parallel to the base of the cylinder as reference, okay? This one is actually less or equal than a constant. And then you will have L of L to the power 2 minus 2 delta, but remember that the cylindrical excess is not normalized. So it gets an extra power of L of L, which is the power m, which is related to the size of the base of the cylinder. Okay, so now what do I do with this fact? Well, I know that the excess is pretty small now at that scale and I can apply my Lipschitz approximation theorem, okay? So we denote by FL, say, in a slightly smaller cylinder on the cylinder of radius one-quarter. So this is going into the plane, which is orthogonal to pi L, so it's a function in this system of coordinates, okay? So we let this guy be the Lipschitz approximation of the theorem. So the Lipschitz approximation that we proved to exist in the second lecture. And so this means that the Lipschitz constant of FL will be less or equal than a constant. And then here I have e to the power gamma and then I have e to the 2 minus 2 delta times gamma, okay? And then I have the set where FL, so I can actually write it in the following way so that I don't have to bother about the coordinates. So if I take the graph of, so FL coincides actually with the function which is describing the graph of u in my new system of coordinates in order not to introduce too much notation. What you can actually say is that if I take the symmetric difference between the graph of FL and the graph of the function u inside the cylinder of radius one-quarter square root of m, m0 L of L centered on PL with base plane pi L, okay? This volume which is, of course, can be estimated using the Lipschitz constant of the two functions and the estimate on the set where they disagree. So the set where they disagree has area, so here I have e to the 1 plus gamma and then I have L of L to the power m, which is the unit, I mean, which is the volume of the corresponding base. And then here I have my super linear gain, which is L of L to the power 2 minus 2 delta and then here I have 1 plus gamma, okay? And the only thing that I want to kind of remark over here which will be heavily used in the rest of the proof. So the remark is that now gamma was a certain fixed constant in the Lipschitz approximation, which is only dimensional, okay? And this delta is up to me to choose. I cannot choose it equal to 0, but I can choose it pretty close to 0, okay? So choosing delta sufficiently small, I actually can get that 2 minus 2 delta times 1 plus gamma is strictly bigger than 2. It's in fact going to be something like 2 plus 2 beta, okay? So it's very important that we are gaining something over this 2. So this will be related to our C3 beta estimates, essentially. And okay, so this is a key point because if you go back to the proof of the Georges theorem, right? What we needed in the Georges theorem was a harmonic approximation, okay? Which is close as a little o of the excess. So we didn't need to gain a power, right? So I could have actually proved the Georges approximation theorem, sorry, the Georges decay theorem in a situation where instead of having this gamma as a gain on the sides of the coincidence set, I could have afforded something which is simply vanishing with respect to the excess, okay? So now here you see where actually this power gain is important, and it's important for the proof of Andgren's theorem. So it's important because if I didn't have a positive gamma, I wouldn't be able to play in this game and go above 2 in this estimate, okay? Good. So now what's next? So next I have my function fL. I actually smooth it by convolution with a convolution kernel which is of the size of the cube, L of L. So now let phi be a radial compactly supported smooth function. So compactly supported in, say, the ball of radius 1. Smooth, of course, with integral equal to 1. And let phi L be the corresponding family of mollifiers. Okay, so in particular, phi L of x is 1 over L to the power m phi of x divided by L. Okay, so now what do I do? I actually smooth fL by convolution with this convolution kernel. Okay, so define ZL to be the convolution of fL with phi L of L. Okay, so this is defined in a possibly, I mean, we will take as a domain of definition of this function something which is possibly smaller than this ball of radius 1 quarter, blah, blah, blah. And the reason is that when we are doing this smoothing we don't want to, we don't have to make estimates, we don't want to bother about what happens when the convolution actually eats the boundary of the domain, okay, of fL. Okay, so now we got a function in this tilted system of coordinates. Right, so here's phi L, here's phi L perp. And now this function here is giving you a certain graph in this system of coordinates. Okay, so now what I want to do is I want to tilt again the system of coordinates and get a function on my initial plane, phi 0, so on the horizontal plane. So I think if I remember correctly the notation, so the new function will be called gl. Okay, so this one is called ZL and then gl. So now we let gl be the function on some domain omega in my original system of coordinates by 0. So in this picture this cylinder is heavily tilted, right? It will actually not be the case. But what I therefore want is something like this. So here, down here, now I have my domain omega. And now this graph, I mean this function will be the graph of gl with respect to this domain omega. Okay, so now by doing all this operation we have, so we are taking this initial ball which is fairly large. Each time we are doing some of this operation it actually gets smaller. Okay, and then of course this omega will contain a certain ball centered on my central point XL. So omega contains a ball which is given by a certain small constant times L of L times this M0 centered on XL. Okay, and now the whole purpose of the, so first of all of course you have to take E sufficiently small so that when you have the bounds on the Lipschitz constant you are always allowed to change system of coordinates and assume that you are reparameterizing by a Lipschitz function. And then the second thing, what is the purpose of this M0? The purpose of this M0 is just to make the initial ball sufficiently large that even though we are always losing by making the tilting, by making whatever operation at the final step my cube L is actually contained in the domain of my function gl. And in fact maybe to be generous take a cube of twice the size and concentric. Okay, so now this fixes the choice of M0. So the parameter delta we have fixed to have this gain. The parameter M0 we are going to fix it as a geometric constant in such a way that the domain of gl contains twice the cube L. And now that we have fixed M0 this parameter M0 at the beginning is chosen so that when we are doing these operations these balls are always contained on the cylinder of radius 1. Okay, so fix M0 so that L' the cube concentric to L with twice side length is contained in the domain of gl. So this fixes the choice of M0 and correspondingly so fix a choice of N0 so that you are always sure that the ball of radius L which we used to define all this jazz is actually contained in the cylinder of radius 1 centered at 0 and with base pi 0. Okay, so now we are slowly getting there. So now what do I do? So I have these functions gl. So these functions gl are supposedly a good approximation of my original function. So why actually this is a good approximation of my original function? Well, because we use this Lipschitz approximation but the Lipschitz approximation is pretty close to the original function and then we are convolving. But we are convolving with a radial kernel something which was almost harmonic. So this almost harmonic function once I convolve it with a radial kernel will not get too far from itself. If it were harmonic the mean value property would guarantee that when I'm making the convolution it stays exactly the same. So now this is locally on my cube L a good approximation of my function. So what I'm going to do? I'm going to take a partition of unity and I'm going to glue all of these approximations together. So just to fix some notation, so let us give some definitions so that tomorrow I'm going to talk about these functions. So the terminology is pretty intuitive I think. So FL is going to be called is the tilted interpolating functions relative to the cube L, GL is the interpolating function. And now what I'm going to do? Well, I'm going to pick up a BUB function theta. So theta is a BUB function identically equal to 1 on say minus 1 1 to the power m and say theta is going to be seen finitely and compactly supported on some slightly larger cube. I don't know something like minus 9 divided by 8, 9 divided by 8 is probably going to work. I mean in the notes you actually get kind of the precise constants. And now what am I going to do? Well, I'm defining theta L. Well, theta L is what you can imagine is simply the rescaled BUB function which is going to be identically equal to 1 on the cube L. So theta L of Z has the following formula. This is going to be something like theta of Z minus XL. And then you have to divide by L of L and so it would have been smarter to take L of L to be the half, I mean half of the size of the cube. So probably this is the correct formula. Okay, and now you can imagine what they do. So I use this theta L to glue all the functions GL together. And these functions we will call, that's actually very unfortunate. So for this function I have the same notation as for the convolution kernel. Damn. Okay, I've got to fix this in the notes. Okay, so definition. So the glued interpolations are the functions psi k. Actually in the notes there are phi k, but now phi was the convolution kernel. So psi k, this is going to be the sum of GL times theta L divided by the sum of the theta L. Okay, so theta L divided by the sum of the theta L over all L in CK is a partition of unity for my initial grid. Okay, and now, so how am I going to prove the George's theorem? Well, sorry, Amgren's theorem. So I'm claiming, well one thing is pretty intuitive and I'm actually not going to bother about that. The function psi k, they are converging to my original function u. Okay, so that is actually pretty intuitive because after all I'm just, when I'm making the Liefitz approximation I'm throwing away a very small set, right, which becomes smaller and smaller as k goes to infinity. And at each scale I'm convolving with a smaller and smaller kernel. So I'm certainly converging uniformly. I mean each of these pieces GL is converging uniformly. So I claim the following theorem. The part is actually kind of trivial. So the part which is trivial is that psi L minus u in C0, sorry psi k minus u in C0 is converging to 0 as k goes to infinity. This is pretty obvious. The non-obvious thing is that now I will claim I have uniform C3 beta estimates on these approximating functions. And then the C3 beta regularity of u is actually an effect of the passage to the limit in this approximations psi k. So psi k C3 beta is actually less or equal than a constant and here there's going to be e to the power of one-half. And here the theorem claims that there exists a beta which is bigger than 0. This beta is related to the beta of sometimes ago. So if I've been lucky, you remember this 2 plus 2 beta. If I've been unlucky, this beta is equal to that beta. If I've been unlucky, I'm losing a factor 2, 4, 8, whatever. Good. And let me give you, so in the next lecture, first of all, I will not focus too much on this beta. I will mostly focus on giving you the idea for the C3 estimates. Then how you get the Heller exponent is just a matter of setting the epsilon and delta's kind of precise. It's not identical than others. But I want to give you a flavor for the estimates or somehow I want to give you a statement of the most important estimates that we have on these gls. Right? So because you can easily imagine the estimates on the psi k are actually related to estimates on the gls. Okay, so I have a key proposition and of course the key proposition is just telling you two things. The first thing is pretty intuitive. I'm gluing with the partition of unity some functions gl and I want C3 beta estimates for this glued approximation. So it better be that each single function that I'm gluing has a C3 beta estimate, otherwise it's certainly not possible. So this is the key proposition. So the first is that, so for every l in Ck I have that the C3 beta, actually if I have to be really precise this is an estimate on the derivative, right? Because I don't know what the function is actually put, right? So it might be that I have a very large constant and the function is far away compared to the excess. So this one has to be less than or equal to constant times e to the power of one-half with a constant which is independent on k and l. And then I need, however, the refined estimates of the following kind. So this estimate is going to be true for every i between 0, 1, 2, 3 and 4. Okay, so I take the i derivative of the function gl and I subtract the i derivative of a function gj where l and j are neighboring cubes. Okay, and I claim that this quantity is bounded by a constant in C0 is bounded by a constant e to the power of one-half. Let me write it over here. And then here I have the size of the cube. I have 3 plus beta minus i, right? So the fourth derivative, I mean, on the fourth derivative I have an estimate which is 1 over l of l to some power so it's a pretty crappy estimate. But for all the other derivatives, I know that you're nearby. Okay, so what is the idea? Well, the idea is that if you want uniform C3 beta estimates, right? So let us fix, so from proposition to theorem. So let us say that I want to estimate, for instance, the third derivative. So I want to estimate the third derivative of psi k. Okay, now you fix the point x and say which is in sub-cube l. Okay? Okay, then I have to differentiate something like gl and then I have the partition of unity, right? Which is, okay, so l, so let us say here I'm summing over j in Ck. Right, so here I have gj and then I have theta j divided by sum over all t-dust. Okay, so when I'm actually estimating these derivatives, of course I get the first term which I like very much which is the third derivative of gj and then here's something. Okay, but then I have all the other terms where actually the derivatives are actually following over here. So one term, for instance, is, one term is, for instance, sum d2 gj and then here I will have the derivative of theta j divided by the sum of the t-dust. Okay, and now this is kind of dangerous because the derivative of this guy is of the order l of l to the power minus one. Okay? So the idea is now that first of all, this summation is not over all j's because this point x belongs to l. Here in the partition of unity I will only see neighbors of l. Let me denote them by this. And then the idea is that if I see only neighbors of l, then actually since the sum of the theta j divided by the sum of the t-dust over the j's in the neighbor of l is identically equal to one, okay? I can rewrite this term in the following way. So I can rewrite it as d2 of gj minus d2 of gl times the derivative of this theta j divided by the sum of the t-dust. And that's because this derivative, d2 derivative of gl is not summed over the j, right? So I could actually pull it out over here. And now you see that these interaction estimates over here would tell me that this guy is less or equal than l of l to the power one plus beta. Okay? And even if I have l over l here at the denominator, so the product of the two, so all of this term would be estimated by l of l to the power beta. So it's a higher-order term. Okay? So these estimates between nearby cubes is then used to handle all the remainder terms in this derivative where I'm using the lightning school. Okay, sorry. But this is it.