 So, so this is Liz Vivas from Ohio State University, and she's going to talk about all the spaces for a class of domains in SIA. Thank you. Thank you for the introduction and for the invitation. I'm glad to be here at home giving you a talk about some recent work about how these spaces, which is actually some something that I have, I wasn't very familiar with until I started this project. So I hope I'll pass on to you the motivations and the reason why I got interested on this. All right. So, oh, I should say before I actually start that this is joint work with Bobby Gupta and Catherine Talger, and Loretta Malensani, okay. So what we were interested on was to find some specific reproducing kernel formulas for the heart of triangle. So the heart of triangle, it's a classic object in several complex variables. And so let me actually just remind you in case this is something not very familiar to you of what the heart of triangle is. Well, it's just basically, as you can see, geometrically here, I just drew in situ as the mod of Z and mod of W, but what this basically is is just simply, let's see. So this is mod of Z, less than mod of W, less than 1. And why is it a classical object? One reason that it is a classical object is because it's as an example of a pseudo-converting language, which is a natural space in which we study different properties of holomorphic functions in several variables, but also it's one of those spaces that have a very bad singularity in the boundary. So it's not as smooth as the origin. So here it looks kind of nice, leptchastive, but it's not really, because if you think about this, you have to rotate this in this direction to really have an idea of how does it look in situ. And if you do that, you'd notice that at this point, it's not even a graph. So at the origin, let me just write that down, and that by itself makes the study of the many properties of the hundred triangles difficult or different than they will be on other pseudo-convex domains that have a smooth boundary. Now what was, as I said, the motivation we're looking here, we were looking at different function of spaces here. And one thing that recently the heart of the triangle has come to light on there's different work very recent. The prayer is about the Berman projection of the heart, in the heart of the triangle. So what is the Berman projection to think about all the L2 functions on H and you projected basically to this piece, which is holomorphic functions on H that are L2. And when we talk about strictly pseudo-convex domains with smooth boundary, with infinity boundary, these are the kinds of domains that have the most regularity properties. So being a domain of this type gives you some regularity properties with respect to this type of projection. This is called the Berman projection. That's when it's not a smooth bounded domain, things are not so simple. And recently Chekrabarty and Seitunku proved that for the heart of the triangle LP regularity it's not a given for every P. So it is in the same infinity smooth boundary case for strongly pseudo-convex domains, and then you can relax those things, but for the heart of the triangle itself, even though it's pseudo-convex and nice, it's not so LP irregularity, to say, or maybe regularity for some piece, for a certain interval of P. But as opposed to LP regularity for the nice cases for P between 1 and infinity, and this kind of properties about the LP regularity or the LP regularity actually are a tool that allows us in several complex variables to do different things, but the LP regularity, it's a new phenomenon. All right, I will not continue talking about these Berman projection things, because when it comes to the Berman projection, there's like a lot of different works by different people. I will not exhaust the works and the names, but just say that it has become quite an active area of research to study, not only heart of triangles, but also generalize heart of triangles. Was there a question? Yeah, so can you explain a few words like what do you mean by LP regularity here? Yeah, so I mean that the Berman, so I can restrict my Berman projection to like LP instead of L2, and here it will be unbounded, an unbounded operator or not. That's what I meant. Okay, thank you. Yeah. Okay, so how are these spaces, or well, the idea here for how are these spaces, and what is the idea for heart of spaces in one variable is that you have somehow the boundary condition of a function or the boundary here of a holomorphic function and can recover on the inside. So let me actually just remind you a little bit of what are holomorphic heart of spaces defining C. Also, I should say, I should include this word holomorphic heart of spaces because this is something that people do use sometimes in other contexts, harmonic heart of spaces, etc. So let's talk about C. All right, so in C, holomorphic heart of spaces have been extensively studied in the following cases on disk. This is actually the classical holomorphic heart of space theory, basically, the upper half plane, of course, just being bi-holomorphic to this, but once you have it onto this, we have Riemann mapping theorem. So people also study holomorphic heart of spaces on simple connected domains. And now once you study them on simple connected domains, you take that idea of how do you do it in each one connected domains, and you can do it on multiple connected domains. So here, I mean, yeah, so somehow the study of heart of spaces have been extended to this. And there's different applications. I will not cover them now, but I will just mention the D-Var solutions, again, regularity of certain interoperators, characterization of life forms, etc. So there's a well-known theory here. And what about in CN? In CN, it has been done more in a case-by-case basis. And the reason is, because we don't really have a Riemann mapping theorem to have a general, for instance, set out for simple connected domains. And also, I will explain it in a bit, but there is not really a canonical definition of a heart of space. But what type of cases have been studied? Of course, if you generalize the disk to polydisk in CN, it has a standard heart of space, and also, say, the ball, interzero radius 1, CN. And others, actually, I should mention maybe some other cases that this has been studied on, say, tube domains and hyperconvex domains. With hyperconvex domains, say, I just mean that there is an exhaustive presumptive function that defines my domain. All right, so still the problem is, or as I was saying, the main reason why we don't have a theorem for the heart of triangle is because there is no canonical definition. I will explain what do I mean by this. What is the canonical definition in CN, but not in other spaces. All right, so I will say that the aim of my talk today is to construct heart spaces for certain classes of domains that will cover the heart of triangle. That was basically our motivation. So let's study now how the heart spaces have been constructed in one complex dimension. So let's start with the disk. OK, so here I have started writing some things. This might be very familiar to you or not, but let me just remind you a couple of facts of all different ways that we construct heart spaces in the disk in CN. So the classical way is that you start with functions that are holomorphic on D. So this is my notation, OK, all of these just mean holomorphic on D. And but not all of them will be inside my heart space. Only the ones whose average value on certain circles around the origin on every circle going towards the boundary around the origin. One that is basically radius is the average is around these little disks. So that's what one of the classical ways of defining this. This is like every homomorphic functions such that the supreme. I don't know why I'm writing here whenever to die. But OK, so I guess we do define them the size of our function like this. But so that's that's one way to define it. That's one way to define our L2 hearty space. But there is another boundary based definition of this in this setting. The hardest space are functions that are homomorphic in D, whereas this other way, it's another classical way to do it in which you're going to have that the functions are actually boundary values. So what we do here is you take the closure of AD. What is AD? AD is going to be simply holomorphic functions in D, continues up to the boundary. And I basically restrict these functions to the boundary of D and take the L2 completion because these are not L2 complete on the boundary of D with respect to which L2 measure with respect to just the length measure there. So this is another way to see it. And again, now this will be seen as more as a boundary based definition. And this one is actually more of a exhaustive exhaustion procedure. But indeed, there is no difference. These two things are exactly the same. You can characterize one with the other by just using your power series. One is going to live in the boundary and you check that it's going to have only a e to the i and theta for n positive. And then you just associate it to the same e to the z to the n h2. And this way of thinking of the L2 hardy space, well, people just really don't distinguish between these two because they are the same. But what allows us is also to have reproducing kernel of our space for functions in AD. So as a consequence, we're going to have that for f belongs to AD. We're going to have the following formula. You can deduce the value inside your domain by just knowing the values outside. This is many times what it's called the zecocernal. OK, so actually let me just write down the tiny proof. This might be the only proof that I write today. So we know that if f is continuous up to the boundary by Cauchy, it means that you can write f of z for z inside this integral f of w divided by w minus cdw. And if I want to restrict this to a measure on the boundary as the arc length measuring the boundary, I can use my w-bar, multiply inside. w-bar has equal to 1. And this is exactly equal to the integral on the boundary of d of this. Let me write that all together to one side. f of w multiplied by 1 over 2 pi i minus c times w-bar. Now here I will have my arc length measure for z. So this part over here is what's known many times as the zecocernal of d. And as you can see, it depends on two variables, z and w. And it's given me, given boundary values, I can obtain the function to complete the behavior of my function inside of z. OK, so before we move on, this might be something I missed. But if you would take analytic functions on the disk with finite L2 norm, will you integrate over the whole disk? Would you get something completely different or is it similar to this? I think you will get something similar to this, because in the radial direction, you will just have something finite. And the boundary values, if they're all homomorphic, the boundary values are the largest values. So they should be dominated by this L2 norm somehow. Thanks. You're welcome. All right, so this is all in D. And I mentioned before that a lot of this study can be translated to a simple kind of domain. But you cannot really translate it to things like puncture domains. And if we think about the heart of triangle, the heart of triangle is really by homomorphic to D cross D star. So is the D star part that it's giving us trouble? And so to really understand what's going to happen on the heart of domain, we really need to understand what is the problem in D star or what part of this study doesn't extend to D star. Which one of these things are not canonical there or like they are not well behaved or which one to choose from? It's probably going to be a different thing if I choose the first one, et cetera. So let me tell you a little bit about what happens in D star. D star, by that I just limit the puncture disk to this minus the origin. And I'm going to start with certain definition of, well, I think I'm going to try to call my H to the star, which is going to be again boundary values. Now the boundary is going to be the boundary of the disk union to origin. But since the origin is just a point, it would make sense to just think about my function as functions in the boundary of D. So here A, I'm being vague because I'm not telling you what A is, but it's going to be both. What are my choices for A? A could be as I chose it up there. I chose holomorphic functions on D star. Intersection continues up to the boundary of D star. And okay, whatever choices I have, I will show you here a couple of choices that we have. I'm going to have a similar scheme though, in which I restrict to the boundary of D star. And I completed with respect to L2 of the boundary of D star. Well, as I just said, the boundary of D star is two parts. And with the one point, there is no, you can just regard that for L2 measure. And there is my argument measure of the circle. All right, but if we are here, if we do this, well, these are basically functions that are continuous up to inside the disk. And therefore, these are just the same thing that we had before. So this is just A of D. Any questions? I was just wondering that the earlier choice that you made is the same as AD, but then you mentioned it. Yes, exactly. It is exactly the same AD, right? So this is not a good choice. Nothing new, basically. Another choice that we could have is, instead of doing, okay, so I definitely want to have holomorphic on D star. But now I'm not going to ask for continuity up to the origin, just continuity to the boundary of D. And what a problem arises here that I'm not showing you at the moment, but I'm kind of going to explain in a second why it doesn't work. So this space will have as a problem that point evaluations will not be bounded. So let me just write it down here. And what is happening really that every function that has an essential singularity at the origin belongs to A. That's sorry that it's continuously up to the boundary. It's going to belong to A. And basically those functions are going to give me trouble for trying to find reproducing kind of, reproducing kind of formulas. So I'm going to go back to this in a second to explain why this is not good, why point evaluations are not bounded is bad for my case, but now I'll let you know which one is the choice that we do choose here. So do you mean not bounded in L with respect to L2? Yes, I mean with respect to, so yes. If you start thinking about like the point of, because what I want to do is have a reproducing curve of the space, right? So I want to have something that it's bounded in one direction. So each point of the disk you pick point evaluation at that value may not be an L2 bounded functional, that's what you exactly, exactly. It's not going to be bounded by a constant independent. Yeah. Thank you. All right, so the third choice and the choice that we choose really here is, well, essential singularity is not good, but things that do we do allow are going to be poles. But of course poles have an order. So we're going to choose different poles up to different order case. And that's going to give me different, different, how these spaces. Okay, so the choice that we choose here is going to be A is going to be equal to what is going to be, well, first I better give some indexes. It is one of the following. And what am I going to choose here? Well, it's going to be basically functions that are holomorphic in the puncture disk. But when you multiply it by Z to the K, F, they are actually on the original AD. So continuous up to the boundary of both sides. Continuous up to zero that will give you the holomorphicity. So these are, again, we can think about these. So just simply all holomorphic functions we pose up to order K. And now I can construct my new definition of a hard space of order K will be something like this. HK to star will be constructed as the completion. And this KK to star with respect to the L2. And even though it looks a little clunky because you're choosing different case, this actually has a natural hierarchy. And of course we will have that H zero is nothing but like classical AD and that's actually going to be for the for the H one. So this is going to be included. This is going to be. The inclusion is strict. And we have that. That in each one of these, because of what I mentioned that the in this case point of evaluation will be bounded, you will obtain a kernel. You can obtain actually formulas for each one of these spaces. So we do have reproducing kernel. What we could call psycho kernels. Of each one of these. So you will have to. You have some function in each of these spaces. You can integrate with respect to this. And this is a nice exercise. Just check that it works for each one of these case. So you can see that it will be in my puncture disk, of course, and W will be in the boundaries. So we do obtain some reproducing psycho kernels here. And of course, if you are in business, each one, you have all the several kinds of each one, but also which two, which three, etc. And that's basically an indication of something that we notice. Okay, so you had a nice hard disk space on D. And you get some filtration of some type of hard disk spaces on the puncture disk. So with our idea was, well, we can really extend these. There's no reason why the puncture disk are special at all. If you have some hard disk spaces on a parental domain type of thing, there's going to be some inheritance scheming, which if you take away some, in this case, you take away a point because you're in C, but if you're in C, and you can take away basically a full co dimension one. Hypersurface type of thing, even like a union of those things and get a new space, which will be, which will have a hard disk space as filtration of hard disk spaces. And so that is our work. And I will explain to you how we cannot really see the hardest triangle as one of these spaces. It will not be, but it will be by holomorphic to one of them. And I can use the by holomorphisms and not always, not precisely how these spaces are not as permanent spaces are. They are not, they don't translate nicely by by holomorphism as always, but in many cases, they do, we are going to have a way to translate it back to our hard disk triangle. But again, that was our motivation to just study the process started ended up being that there were many other examples that were interesting that fitted on this category. So I'm not going to slow, but this next slide is all written up already. So let me just tell you what was our, how were our construction, how our construction goes. All right, so this is at the inheritance scheme. This is going to work for any CN for any and bigger cotton one. And this is what I'm going to call the barn space. Slide. Okay. So we have any CN domain. What am I going to be? What am I going to require of my domain? I'm going to require, first of all, let's look at which objects we will, we will think about. Well, some measure in the boundary. Supported on the boundary of my domain. Sorry. In the boundary of my domain and. Omega T. Should just go over here. It's just going to mean. Omega Union. So why. Sometimes we do care about just some parts of the boundary. So you're in. And the poly disk, this cross disk. Then it turns out that actually the boundary values that you're interested are actually not in the full boundary of the. And the full boundary of the cross D, but only in the boundaries this cross boundary of this, which is specific part of your boundary. So that's what I did not want to put the whole boundary. So I'm just going to go over here. I'm just going to go over here. I'm just going to go over here. Just depending on which case by case we go. The support of my measure will be just some part of the boundary. And these are the classic, the spaces that we introduce. A Omega D will be just simply. The holomorphic functions in Omega. Which are continuous up to the boundary in union T. And this is how we define our hardy space. So I'm going to measure as the completion again. Think about these restricts that at the boundary of the, or even at T. Respect well to of, of, of me. I'm a fine number of measure. Okay. So I do call these apparent space if. I like this part too. So I'm going to go over here. We call these apparent space if what? Well, first of all. It's something like point of elevation should be bounded, but well basically the extension of that will be that for any compact. In my Omega. I want that the uniform F. Okay. Should be bounded by CK. Then L to norm of F outside. We don't want things to. The second condition is a little bit technical. We don't want things to go to zero and not be zero in principle or accumulate weirdly on the boundary. And some points. So this is the second condition. If you have that FN. A sequence that is cause she. I'm going to go to new costume. Sorry. It is kind of hard to read. She say here. To me. And uniformly in compact is going to zero inside Omega. Then we want that this L to norm. Should be going to zero. If these things are satisfied and these are satisfied for total the examples that I show you about for the desk. And if they are satisfied. Then. We say H to Omega new is going to be. How this space of. Of. Perhaps, perhaps a quick question. So in case of your punctured disk. What was the T in that case. It was still my, my boundary, my boundary of the whole disk. Okay. I'm not the point. Not the point anymore. Yes. Yes. I did ignore to be. I did ignore my. So my bunch of this actually is not in this category of parent space. My punch for this will be on the category of. I'm going to do that. Now I'm going to take away something of the parent space. To be. Thanks. Yes. And I should say these conditions. Are. Are right now for a omega new. So I said here for any K. I want F all of these. For which FFs that belong to. These. And same thing here. But we're going to have other families maybe that. Have these are good conditions in which I can apply a similar. At the end. So do not repeat. Later on. Let me just write it over here. A function space. F. On. Omega T. Wait. F restricted T. You know to be. Strongly admissible. I mean, it's just a word that we. Decided to use for just basically again. Summarizing these two condition. If a and B hold. So in this case, basically I'm saying I'm going to have a harder space. As long as a to a omega new is a. It's a strongly admissible space. And. When this. When this. For some cases that omega has been studied, like say, see two small domains in CN. New being the Euclidean surface area measure. These coincides with the classical hard space. What, what people do with the classical hard space, which classical hard space. I mean. Functions that are homomorphic insights. Continuous up to the boundary. And just take the boundary values of these kinds of things. Complete. And the same thing is that. The one that. Coincides is with. If you do. Poli disks. Cross itself. And times you're in. And I think about the measure now. Find. I will have the one I mentioned, like the one on. The boundaries. The product of the boundaries. Then we will also have that. All right. So that was the parent space. And then we will have the data. The data that we have with the discourse to disk. What do we do? What is our. Or scheme. We take something that we called. But. The leader domain. Or hyper surface. The middle domain. So. Let me. Let's go. This is going to be. For this space. For these. Hypersurface. Okay. So you, you had the Omega up there. And what we do is we take. V. And V. It's going to be just an irreducible. Define and a little bit. Minimally defined on a little hypersurface and Omega closure. So I just, I want, I want, I want. A defining functions here. So that V is basically my defining function equal to zero. And. Okay. So who is V. And say we have a defining function called. Okay. So not always we can find these, but suppose we can find a global. And we can find a function to minimally defined it say here. So that V is basically five equal to zero. Or five is just in. The morphic of Omega union. Continuous of the boundary. So. So we're going to. Take away this. Hypersurface and obtain a new domain. Okay, so maybe I should do a little drawing here. This is Omega. You might take away some V. And whatever you are left with. It's going to be my Omega star. And the. The theorem that we have is that for each one of these Omega stars, I can find a good space. So I can find a good space as long as my original Omega had a good hard space as a was. Actually, I should have given you a couple more conditions because of course it matters what happens with the. I don't want my measure. Over here, my new measure over here. To. Be concentrated say on to support. The intersection boundary of Omega. So, okay, so maybe I should add here. So, okay, I'll add it in a second. Okay, so such that. If. Omega intersection being. Space here. Such that on my intersection view, not empty. Okay, and I also want. That's the support. The measure. On this intersection should be Spanish. So remember that my, my support of my measure is somewhere in the boundary. And we don't want again to have. Any measure, any concentrated measure there. Okay, so the theorem says that basically for each K. I'm going to have the same definition that I show you above basically remember it was with Z to the K times F. Now instead of Z to the K. My. Functions that I just multiply by this fight. Okay. You can change. Of course you're. Defining function and you should obtain the same. Same space. And this is strongly admissible. So remember that meant simply that. Both. A and B conditions are satisfied. And that gave me some way of defining a harder space basically. For this. Each K. To omega star. You. Which is just basically going to be a K. Omega star new. Restricted to the boundary. Well, actually to the support of menu. This is going to be. Are reproducing Carol. Of those days. So I do have also some. To. That I can use. And this is basically what we call the hearty space. Level K of Omega star. Okay, so. The. The example that we saw above was when. Omega was the disk. It was the space. It was the space. It was equal to zero. It was equal to Z, which of course didn't mean like the. The origin is the only. Hyper surface. I will think. But of course we can generalize these 40 for another examples. In that case we saw an strict inclusion. And. As I said above. It's not. It's not admissible. If you think about the. This is a second condition. And. And these can actually be generalized to. Well, you can actually have a union of different hyper surfaces. You just do each one at the same. Step by steps. Generalizes. To. Omega star. And. You can actually have a multi-index. You can actually have a multi-index. But of course, every time you do this, then you have a multi-index now because depending on what. Where you do it. You get a hard disk space for each. For multi-indexes now. All right. So let me actually show you. I'm almost out of time. So let me show you some examples of, of, of what kind of things we can cover. So first of all, for us, for this problem above, I'll show you some formulas in which you get the filtration. Like it's never stops, but in fact, we did find some cases in which the filtration does stop. So this is what's happening in general. The filtration goes in this direction. It's zero. Of course, it's just the original. Hard disk space of your. If it happens to be intersection. The support of your measure is empty. The inclusion will be always a strict. That's kind of what happened on the. On the problem that I show you. Well, but if you have to be intersection, the support of your measure is not empty. The above filtration might stabilize. So I just want to show you a little example here. This is a, this is a, this is a, this is a, this is a, this is a, this is a, this is a, this is a, this is a, this is a, this is a, this is a, this is a, this is a, this is a, this is the, this is the, this is the. I just want to show you a little example here. What's sometimes called egg domains. For any B. You're equal to one. You can. Draw this and. In this means. We can study different things. So I'm going to take away some. But depending on what measure do I choose, so I'm going to choose two different measures here. If first I could choose just the Euclidean surface, measure on the boundary of these, and then you're going to get that each HK star just stabilizes immediately and they are all equal to actually the H2 megapenium. So there's no difference, they all stay the same, but if we would have chosen instead the Mongeum pair operative measure, or the boundary related associated to this specific exhaustive function saving Mongeum pair measures, so yes, think about this as being your exhaustive function, then we do get actually something different with respect to this measure, I'm going to get something that does not stabilizes. Actually it stabilizes after a certain value P minus 1, and after that everything is the same for our key, for our key we're going to be. Okay so we also have reproducing carbonyl Hilbert spaces, I didn't write down the formulas, I did write down the formulas from D to D star in a similar way you can get formulas, if you have, let me just write it down maybe, that's a theorem, the last thing that we talk about, theorem three is that if say C omega was your sego kernel for D for omega, then we obtain actually a sego kernel for the k level, let's call it like this, and this is going to be sorry, so it's going to be 5w to the k divided by 5v to the k, I'm going to play by my initial sego kernel over here, yeah let me just end here, so this is going to be for omega minus v, all right and maybe just the last remark was that the hard to struggle is not of the type omega minus v, but D cross D star is, and then I can use a specific holomorphic function that takes me from, specifically by holomorphic function that takes one to the other, and you just use the classical map between each other, takes one by holomorphically to the other, now as I did mention by holomorphism it's not something that really makes hardy spaces canonically defined, but still we can use this one specific hard map to work on, all right sorry whenever time, thank you very much, you did not at all, actually our time limits are quite lax, oh wow all right, if there's a you know a couple more thoughts that you wanted to add and go ahead, sure yeah maybe I will mention that the hard toes triangle is one of the different type of spaces that have been studied, there is a generalization that has become really kind of started more recently because it has the same properties, it's also zero convex, but it's also the boundaries very barely behave and this is what it's called generalized hard toes triangles, and our our inheritance scheme works for these two, formulas become really like messy and then of course like the nice formulas on the on the disk become very very complicated, you start having like gamma functions and weird things like that, but they are concrete and so we do our goal of like finding very concrete formulas for functions that are defined on the boundary that how do you go inside was still something that it works, so that's that was probably something that we our motivation and now we do think that this can be in the settings, but yeah we haven't yet thought about that very much, yeah symmetric space, all the symmetric spaces probably things like that. Well definitely thank you, thank you very much for for the nice summary of these results and also I hope I definitely appreciated the quick introduction to hard G spaces, I'm sure the students appreciated it too. Why if my daughter is contributing from the background, she probably has opinions on hard G spaces too. She came running in and noticed your notes, I'm not sure she had an opinion on hard G spaces, but she definitely appreciated the colors, the colors. Yeah I actually was not very familiar either with hard G spaces before I am more familiar with barman spaces and but yeah now after having worked through these cases I realized why there's not really a canonical literature for this because it's very much case by case and really different people introduced their own different hard G spaces type of things, so there's different definitions floating out there and yeah so this this way for instance with the measure on the banner is associated to the, this is Poletsky, Poletsky's way of seeing most of these hard G spaces, so he uses this for not only for this case but for like any hyper convex domains type of situations. I do have a couple of quick questions here. First one is more sort of an expanding nature, so in your last main theorem I didn't really have enough time to understand all the quantities here in this equation, so what's this QK on the left? Can you remind me what this QK is? Yes so as I was saying like for the disk there's some reproducing formulas right, so you have the z's inside the disk and omega is going to be on the boundary of the disk and you have some kind of reproducing formula integral of f of omega, so here is going to be some kind of sevo kernel, this is what it's called sevo kernel in general whenever you have some type of formula of this type I shouldn't say the disk anymore now omega, so that's what happens in omega for which f does this work well for every f in the hard disk space, so this would be if we have this for omega then there should be some type of formula for omega v minus v and that's precisely what my QK meant that for those ones I'm going to have my integral now it's not c omega anymore at this way and that's what I was trying to describe here of course it's minus v or something like that but yeah for f I might have a sorry I didn't mean to interrupt but I meant like for f on that on that family specific family etc. I might have a question about this as well you said so if you underbi holomorphism does the the actual formula for the kernel transforming any any sort of reasonable way or yeah so um so yeah you can see I guess here on this specific part why the bihormorphisms are not sometimes good because you really need some to transform in a nice way you really need some bihormorphism up to the boundary and if you have it and it goes all the way up to the boundary then yes you can rather translate this f of z to some you know kind of chain rule situation inside it um when it's not like on this case z w comma w then it's um the the transformation is not as nice and so you have to introduce a couple more but um yeah I can write down what Sego Kernelta will think for each is it ends up being actually pretty messy oh but you got something explicit I think we did get something explicit yeah yeah we did get something explicit um sorry there's there's four coordinates here right because there's z and w h I mean here is like just z and w both in one dimension but here I have z w and z prime w prime and we did we did get something explicit z z prime y minus w yeah so that's that's the Sego Kernel we obtained on on h um and it's not symmetric it shouldn't be because the heart of struggle is not but yeah like this was for this this one but it as soon as we move on to to m and n then like all these nice constants become very complicated and we do get weird but we do get things that are explicit um yeah okay but perhaps another quick one this one's definitely very open-ended so uh regarding your your spaces of uh so maybe even two questions so I wasn't fully able to read between the lines enough here to understand so so the stabilization of this filtration is this good or bad like how is this uh perceived is this uh which one is more desirable um I guess uh I guess it does say something about some kind of something about the function of space that it stabilizes at some point or not right like uh I don't know if to say more desirable or not but I think that when when you have um inclusions like these which don't and that means that there is more and more space to have strange functions maybe that that have a behavior that doesn't really correspond to like the right the the original space without without the hypersurface deleted um I don't know if I will say one is more desirable than the other but I think the stabilization is a good thing because that gives you like a stopping point where from which you can really start working like okay this might be my or my the best hard space I can obtain right like it stabilizes at some point but it's still not has it hasn't been like the original one which didn't give me anything new now yeah but maybe like things like the disc the punch or this right it never stabilizes there and I think that tells you something right that there can be holomorphic functions that have infinitely many poles in some ways but yeah and then a connected question this is the open-ended one so so having uh so to a hypersurface one can associate a a line bundle and vanishing up to a specific order on a along a hypersurface can be interpreted as being a section of a specific line bundle so I just wonder if you know so along these lines these inclusions can be also sort of thought as you know like uh if you look at higher and higher powers of certain type of line bundles does the number of sections you have somehow or sorry the kind of sections you have it does it stabilize I wonder if that type of point of view is something that you know you you look that we thought about it we did think about that but the condition in which we weren't sure how to proceed if we thought about it in that way was what does it mean for something like say a line bundle to be on l2 of mu so I guess we had to be like a little more when we came to the L2-ness we had to make some choices and so things weren't as canonical anymore I don't think that that means that there's no way to see it on that on that point of view that would be more clear it's just that we at the time we did it we didn't couldn't see it directly so I mean it could be that it's not that very fruitful to to to transition to that it could also be that it's not right thank you thank you thank you what note making gap are you using by the way I love this good notes good notes good notes I'm looking