 So I'm pleased to introduce Duc Viet Vieu, who's going to talk about complex motion peri equations in finite, finite pluricomplex entity. Okay, yeah, go ahead. Yeah, that's a lot for the iteration. Yeah. So let me start with my, my talking. So in the first part of the first line of my talk, I will record use about motion peri equation and some of the deleted notion. So let me first fix the setting where we work with motion peri equation. So, so as usual, we consider x omega is a compact. We call it manifold of dimension M. And we normalize over gas with volume from equal to one. As usually. And here, we recall that an omega is a function on AC is a function, which you locally is a sum of a smooth function and a blue is a public function. And that person satisfies you the inequality you see here in the slide in the sense of the current. Okay. So now it's a famous complex motion peri equation you saw you see here in the slide. So AC is plus omega power n equal to mule. Here mules is a probability measure. And use is an omega as a budget function. So, so let me recall to one important thing. Because the left hand side is not always verified for any use so we have to, we have to give a chance to that's a formula. So that's current years. For all of the talk is understood as a non pre-produced cell products of the current DDC user plus omega. So, and so that's a, that's not pre-produced product was introduced by the death of the loss in the local shipping. And the global shipping is by guests and he's a and there's collaborators. Okay. So he has a, those two, those for the convenience, I, I briefly recall how we define that non pre-produced non pre-produced in, in a compartmental manifold. So for, so if our function is about it, then the definition is classical, which you need to pay for that long. And here's a formula. You see in the slide. And, and because you see, you see about it. So, so that the current here is very fine. And so we can take it and so on. In general, so when our function is not, it's not, it's not about it. Then we just takes the, we first take the maximum of you with a small, small constant method K. Then we form the current here with about a potential. And then we take a limit and K can be 30. So, so this is the notion is in also an immediate fight. But it is a, it's a direct consequence of the proof, proof in the approaches of motion operators. And then, and secondly, the fact that the current is close, you're so not trivial, so we have to prove it. And then we move on to the new tool. Okay. Now we move on. So, here's I give a very quick summaries of the, of the existence of the solution. So, let me start with a smooth solution. Yeah. So when our, our measures is a volume form. The solution is smooth and unique. This is new to Joe. And this is. And secondly, if we decrease a bit the, the regularities of the measures, then we get a unique holder solution, hold a continuous solution. If our measures is hold a continuous. If we view it as a function nodes on the space of omega piece of function. So this results due to a number of, of authors. I give here's a list of some. Maybe it's helpful to give some examples of how the continuous measures. If you consider a measure with a healthy the cities with peace in a better than one, then it's a, it's a, it's a whole those continuous measures. Or in the, in the geometric way, we can take any, any volume form in the generic. And if you consider the Cauchy Siemens and some many phones, then that gives you a holder continuous measures. And now we, if we decrease one more, once more, the depravities of the measures, then we get a unique continuous solution. And here's a, I, I, I recall here's the formulation new to close in. So, if our measures is donated by capacities, then we get a continuous solution. And finally, when our measures is only not three problems. So that means that it has no mass on the positive intercept, then we also get a solution, which you need to. But the solution is not bounded in the general rules. And that's the result that you do is again in the local setting and the rest of the ideas and the news in the, in the, in the setting we, we, we consider in the talk. Okay, now, we go to a bit more about about with none, none three polar measures. So we need some more notion. The notion we need is a positive compression energy, which was introduced by certain in local setting, and by, by guests and most of us in the global city. Okay, so to be able to do to defy that notion, we need some, some observations about the monotonicity of the product. In case of clear setting is quite simple. Yeah, I will show you now as you see in the slide. Because the, that's integral. That means we compute the mass of that, of that motion best measure. Then by definition, you see that the mass is a small one. So, so that so you see that that doesn't not put a lot of loss of masses in the general zone. And we just motivate us to defy that's a set of E you see here, in a set of all of the omega basic function, which negative and satisfy the equality here. Okay, that's me the mass of the motion pressure measures of you equal to the maximum value. And now we consider the notion of weight. So we will be noted by W's minus here, the set of conveys increasing function key from the set of a negative real numbers to do to itself. So that's the key zero equal to zero itself, and, and the key money in 50 equal to money in 50. And, and the next one, the next set of the weight is the, is the following, we fix a big constant M positive. And we do not buy W plus M, the set of concave increasing functions. So we also will also have that the equality at the visual and the infinity money infinity. And, and data that greater satisfy this inequality use you see in the slide. So in general, we record a weight is an element of that. That's true set of we will say introduce. This is due to. And, and it's a, it's a local setting that's against because it does only some, some a special weight, as you saw is a example is lying there. They are there are the most basic examples and also the important ones. So if you consider the, the, the, the, the, this example, the, the, I'm sorry, the, the minus T power, power P with P here, if P between a diesel and one, you get a weight in, in W minus, and if P greater than one, you get a weight in the, in the, and this is a type of one from here. It will be. Okay, just about say one thing. So yeah. My understanding is that these two weight classes, go back even earlier. So these two weight classes, I think, originate from or let's space theory so in order it spaces. people work a lot with these two classes of weights. Yeah, yeah, yeah. I think that's how they came into sort of the complex world as well. Yeah, I'm quite sure that you may consider this great in the oral spaces in the real analysis here. And in the Pugli-Porgiasso theories computer communities, I guess so again, and the best of ladies. Yes, yes, yes. People introduce this weight here. Okay. Okay. So now we move on to, now we're gonna talk about, we're gonna speak of finite energy. Because the key is a weight as we saw in the last slide. And we did those by big E key, the set of heels in the big E. With finite energy, finite key energy. So here the key energy defined by this formula. So you see that because the heels is negative and the key is negative. So this here to give you a positive numbers. And we asked that the numbers is finite. This is what we asked. And here is an observation due to a guess as already. They observe that the set of the big E here can be sorted by the sequence of finite energy. And as follows, they have the terminology. Because a big E key is a lower energy and for key in W minus. And higher energy if the key in W plus and big M. So here is some remark to see why these spaces are important in practice. So the high energy spaces are important if you want to study motion paper where the measure certifies a good level of these. For example, it has the LP DCDs with a piece greater than one. And on the other hand, the low energy subspecies is also important if we want to work with the new, which is only the number three polar. So in that sense, it's a, it certifies a minimal level of these assumptions. So that's a notion of, so in that setting that the notion of energy is a give you a quantitative way to measure the fact that our measures is only the number three polar. And also the last remark is about that. It's about the practicals application. Even if we only work on with specific rate, like the T power P here, in practice we also need to approximate this weight by some smooth weight. And of course, this smooth weight has no specific form. So that's some of the reasons why we should be interested in working with general weight than some specific class of weight. Okay, now we move on with the question. Does that motivate the main result in the talk? Sorry, okay. So here's the problems we see here in the slide. So it can be formulated as follows. We have a two-megapacitive function, U and D, and as a motion pair of use is dominated by the motion pair of D. And we asked something quite a general question. So if V certifies some, some level of these properties, P, the same, and we asked whether the opposite use is like the same properties. And here's a list of, of course we have to specify P's in practice. So here is a list of P we work with. So if our property P's is that V is holder or continuous or powdered, then the question is asked by Closier. And we could also ask for, for example, P is that the V of finite P energy, then in that case whether opposite use certifies the same properties. And if you go a bit far away, then with the current of a higher body breeze and we have a similar factor about the super potential, but I also want to note it here. And here's some factor. So these problems is quite, it's widely open for continuity and the public authorities. There are some workers due to Closier and his collaborators, but I think the question is still widely open. And up to now, we have known the answers for holder of the property and for finite key energy. So the holder of the property was known a few years ago and with a list of people here and for the finite energies was recently result which we will speak of in the talk. And now we will talk more about that finite energy. So here's the result I did with Do. We give a complete characterization of measures with finite key energy. So, because as before, Mew is a probability measures on the icing and key is weight and here's what we obtain. Then Mew is a motion based measure with a potential in the finite energy classes. If and only if our measure certifies that inequality. So you don't have to read the whole inequality, but the main point is that the condition is a kind of inter-gurability condition. So this is useful with practice there. And here's some remark that results was known for that special weight T power P which is due to the solution and the guess of the idea. And the proof is in an essential way the special form of the weight. And this cannot extend to our setting. And there was some, that will work on you to bring up in the local setting for convective weight which is similar to our result. But somehow as I see the proof is in a crucial way. The fact that it's in the local setting. So I don't see a clear way to extend the work to the local setting. And the more in the work of the guess of the idea they pose a question which is somehow to give a characterization of a measure with finite energy. They formulated their question in a more concrete way, but I believe what they wanted is to as I interpreted it. Okay, now I will talk about some consequence of that result and how to prove it. And later in the talk I will show you how to extend that result to the bigger go music class setting. Okay, so here's the first consequences, direct one. This is what we spoke of in the last two slides about the colloquial structure. So if our measure music is dominated by some motion pair of some function in finite energy classes, then music is also a motion pair of some function in the same classes. So this is what I mentioned in the last slide. And now I want to mention the key ingredient in our proof. So this is something about a mixed energy estimates. The same as you see here, if they have fine in the finite energy class system with this normalization. Then they sit across from sea, such as for any peace sign in that big IP class system, then we have this inequality. So again, that's mixed energy estimates and what's known for that special weight. But again, the proof is of the special form of the weight. So here's our result, a hope for any weight, without any reticent, yeah. Can I have a quick, quick question here? Yeah. So did you spend some time to understand if this exponent here is optimal? I guess, I guess, I have never talked of the optimality here. But the exponent here comes while national reason is the proof. I mean, there are those artificial manipulation there. Thank you. Yeah. Okay. So now as I will show you the very big difference how we obtained the main result, the factorization and how we, some comment about that, the proof of that mixed energy estimates, only some comment there. So the first thing I want to tell you is that in the, is that the factorization of that we obtained, so it's essentially we solve a motion equation. So with the proof, we just follow the standard strategy. And the main difficulty is to control the key energies of the approximate of the solutions. And this is the point where we need that mixed energy is estimated. And now, so how do we prove that mixed energy estimates? So the argument is actually quite simple here. It's based on some kind of monotonicity argument and proofing the properties of the motion operators. Although the proof is quite simple, but I like to mention that we can use the same idea to prove the complexities of that finite energy classes. And that complexity has some history. It was characterized by both some assiduals and guess again is a long time ago in 2010. And that was solved by Dapas by using his work on the envelopes of prescribes and here's we can use that same idea. We can use the ideas of that mixed energy estimate to give another proof of that of that complexity. And also our proof also work in a more general setting with a different non-productive product. Okay. So here's, I just give one line or how we obtain that the mixed energy estimate. The first thing is that we decompose the integrals in the two parts that you see here. The first part with a b-side bigger than the fine and the second part with b-side less than fine. And for each part we use monotonicities of energy to control. And the point is that we make a good choice of the consistency here to obtain the estimate. Okay. Now this is all I want to talk about the Kelly setting. Now I'm going to move on with the bigger komoji classes. So here's a, so I will review the standard setting with bigger komoji classes. So because the data is a smooth close on one form in a big class on fine, okay? And fine is a modeled data pressure function. So that's mean the motion pair of fine is positive and fine equal to is the rooftop envelopes defined by these formulas. So that notion of models is what is introduced by Dapas in design new and also there was some work by Ross and Wilson about that's an envelope here. And here's a formulation of motion pair equation in our setting. We consider that equation here, the same one, but with you less than fine. So we control the similarity of you by that of fine. And now as a music with a mass equal to motion pair of fine. So here's some list of work about data equation. About data equation. When fine, the model, the basis function of a minimal security in our file, then that equation was also in the papers of the BOSTROM, SCDOS, and GASERD. And moreover, when the 445 genozo that it was sold by Dapas and his collaborator data, the condition is quite natural and also somehow necessary. And here's what we know so far. This equation has a unique solution. It's a glass, because big E here and if the function, sorry, the measure mu has LP density, then the solution of the same thing all the time as fine. So here's a new to the work of Dapas in as I knew. So now, if we think of the calisthenic, we have the characterization for measure of finite energy. And now with the equation, with the motion equation with the pre-cry signal D5, you can pose a similar question and actually we have the same kind of result. I will show you now. So here, firstly, we have the similar notion for energy defined by that formula. So again, it's less than fine. So this guy is negative. So with minus here, so that gives you a positive number. So we have a similar, a quite similar statement for bigger glasses setting. I don't know go through the statement here, but you can see in the slide. And now I just want to give a comment. If you try to prove that result, then everything goes in the same ways as in the calisthenic. So as long as we have a good enough integration by partner, this is a key point that we have to keep in mind. I will talk about that integration by partner now. So here's the statement that we need in the proofer. So it's a bit technical, but I will try to go through it. So we have a current by degree of one, sorry, by degree of M minus one, M minus one, and having no mass on the three-folder set. And V and W, they are differences of quasi-basic functions. We have V1, V2, W1, W2, they are quasi-basic functions. And we assume that V and W are bounded in our manifold ison. And key is a smooth function. And then we have that integration by part formula. So it's look complicated, but it's actually just like a smooth function here. So here I will go slowly with the location here. So here, if V and W, they are smooth, we could not ignore the bracket here, and the dot here, the wedge dot there. And if that's a smooth setting here, we have everything like normal, we have integration by partner. And so when our V and W, as in the statement, it's a hypothesis, then we have to, there's something we have to notice. The first thing is that we have the dot reaction. So here we can ignore that dot. We can define that product in a similar way as a non-free product. So if you don't know about that product, we will ignore this notation here. Think of it as a non-free product. As the important thing in that formula is in this term, the last term is a formula. It's the term about EW, WESA, ECV, and WESA-T. So these terms it was not known before. So that means to prove that formula, firstly, we have to define that third term. And once we can define it, we prove that we have the equality. So in this case of two steps, yeah. And now here's some remarker. The first thing is that the assumptions that the WESA-T is somehow optimal because when they are not bounded, I don't think we can make sense of this formula, the DV-WESA-T ECV. To see an example, we can think of when V equal to log Z with Z in C is a complex plane. Then in that case, the D log Z and WESA-T EC log Z is not of bounded mass in the complex plane. It's also a composite of complex planes. Okay, this is the first thing. And second thing is that these formulas were known long ago under the condition data. The function V1, V2, and W1, W2 have a small, unbounded locus there. That's been, they are locally bounded outside some closed, complete, truly polysept. So this is due to both some ACDs and gaseous ABD. So I just want to remark that that proof and the proof of this genus of results are different here. And before the appearance of these formulas, many, some, not many, but some worker is a complex motion of the equation. How to assume that the function V, the modal function, should have small unbounded locus because they need that interversion by path formulas proof in the people of both some and collaborators. But with data that new and more than those formula here, we can get rid of that assumption here. And also there are some version, some weaker versions due to lube and the sear. But they don't have the, there are some, there are some point, but the most important one, the most important thing is that they don't have the terms about EW regression, EV, ECV here. And this term is a cusso in many applications for example, if you want to run the variation of the method for complex motion of the equation with pre-registered securities, then you need that complete formulation here. And without that total, then it will not work. So here's a thing, as I mentioned in the last slide, as long as we have good enough integration by path, then everything goes through without any difficulties, as is tele-occasion. So this is what I meant with that formula. And this is the proof in the big class setting. I think I stopped here, yeah. Thank you. Thank you. So questions, questions from you. I had one question or comment, maybe two. Okay, yeah. Yes, so without understanding much of these technical parts of the proof, when I look at the statement of your main theorem, that kind of looks like a properness condition to me. May I go back to that one, Finland? Yeah, yeah, please. That's here? Yeah, so one thing this statement tells us is that if the energy is low, then the, well this integral on the left-hand side is low. So some kind of L1, L1 properness, but with a chi involved. I'm not quite familiar with the properness here. You mentioned that. So what I want to, or I guess what you want with properness is to say that the sub-level sets of your functional, the energy in this sense or have some kind of compactness properties. And I mean, I guess this doesn't say that, but it's still, if we assume that the energy is low, and let's say we want to minimize the energy, and then we assume the energy is low, then this statement gives us like a property for psi, some kind of boundedness on psi. I suspect it's the other way around here, exactly. So it's somehow not done. So for me, this is about when the energy is bigger. Oh, okay. So maybe I got the sign. So here's the, it's probably lambda here. It's important that it's cool so that lambda is a lesson one. Okay. So here lambda equal one, one half is the key, conveys an M over M plus one. So it's always less than one. So it's the key, the key part, yeah. Okay. So that doesn't mean that when the energy of psi very big, then the left-hand side cannot go to other if we're rated as energy. Mm-hmm, okay. It goes a little slowly, yeah. Right. So here you don't bound the energy from below, but from above, which somehow not quite there. Yeah, yeah, it is a form from the proof of, that's what we need here. Not perhaps another question here. So you mentioned this question of Gejizariyahi on the bottom of the slide. If I remember correctly here, for Gejizariyahi, they specifically conjecture that you can take lambda to be equal to one, right? No, no. I think that they, they those specifically is that inequality here. So they just ask for that the left-hand side is always finite, I don't know what you have. They only ask whether we can have the, we can bound that left-hand side by some function. I see. Which grows less than the energy of the side. Yeah, somehow it's there. I see, I see. So it's even weaker than what I just said. Yeah, yeah. Okay, thank you. Okay, yeah. I guess I will remember correctly, yeah. So I just like one small question. Yeah. So in the statement theorem, we have that supremum of size is always equal to minus one. And this seems to suggest that for such functions U, wait, right? So we can always, right? So for such U, we would have that it's energy, E chi energy always be bounded between some region. Yeah, okay. Yeah, yeah. It's just kind of there. When we normalize the P side as we do here, with the supremum is equal to minus one, this is not an essential factor. We only need it for clinical purpose, yeah. So that's when because the P side, which is less than one, minus one, then the energy is always about from below by one. Yeah. I see. So that's a, that's a, we just want to forget about the small energy and we work with the big energy, yeah. All right. The energy will also be bounded from above, right? Sorry, again? The energy will also be bounded from above if the supremum of size is equal to minus one, like from this equation, from... No, no, the energy can be as much as we want to. So I will show you again the formula. So here's the energy. Right. If our use is less than or minus one. Right. If the energy is about from below by something bigger than zero. Right. But it can be very bigger. Right. All right, but the supremum of U is minus one, then the energy is finite, bounded, that's what I'm trying to... About from below, yeah. Always about from below. Okay. By some constant, yeah, fixed constant. Okay. Yeah. Any more questions? Let's thank the speaker then. Okay, okay. Thanks again. Thank you. Thank you. Thank you.