 Dobro, moj začo. Očne, tako je, da je tudi pričo zvuk, da se razvajamo, neko so pričo so pričo, da se pričo so pričo so pričo, neko so pričo so pričo, neko so pričo so pričo so pričo. Nekaj, da smo pričo so pričo, neko so pričo, neko so pričo. V 2010-2012. Tudi tudi povede delaj. Zato je tvojeh povede delaj, tebej tačne bo način v tem, nekako, zato je. Povede o dobro v metrične zelo, in pri teoreom hradi, druga povede je na metrične zelo z Rejmanianikov, ko je tukaj neko, ki so spetne in tukaj neko. Očnemo, ki se povede, da je izvahno, kako se neko v povede. Zgleda je, ki se povede, ki očnemo, ki se povede, ki se povede, ki se povede, ki se povede, ki se povede, ki se povede, ki se povede. tudi še predstavljati. In pa tudi sem pričel sem od tudi včetne delati o najbolj delavnega. Včešljamo, da bomo vstajati do vzvečenje delavatih, načine in zelo me zelo včetno učin sem zelo všim učinem, ali se pričel sem to nekaj v povrstu v povrstu. Vsrednji je, da smo izvijeli nekaj je ta vzvečen teori o Romanii zazdaj se celo drugi paper izgleda, je to dobro teori o rešenju, ljubovu obrženju, vzelo na metrične zvrste. Zato vzelo na metrične, vzelo na metrične zvrste in zelo vse zelo vzelo na metrične zvrste. X, D, M, M, za zelo na metrične zvrste. Ok, da se poči, poči sem izvrstvalo nekaj analog, Kaj je teori? Znamo, da se pričeš na moment, da se pričeš. Znamo, da se pričeš o metričnih spesih. Nelj je nekaj zelo, da se pričeš, da se pričeš, da se pričeš, da se pričeš, da se pričeš. Kaj je zelo zelo zelo, da se pričeš, da se pričeš, da se pričeš. Nelj je konstant, t1 minus t2, da se pričeš, da se pričeš. To je bilo počet, da se pričeš, da se pričeš, da se pričeš. Rešo si je zelo, da se pričeš, da se pričeš. Na svetu od večjelj počet. Kaj je tudi teori? Zelo je, da je skupo, da se pričeš. Tako, če vidimo, da je začelješ v terči nekaj kurvacij, in da je včetka z vrštima matrična zrata. Sreč niski, da ti je v remaniom menifoldu, kaj si zelo pače vsečnega rečnega zrata v klastika. Kratko, zremaniom menifoldu tudi je satisovat, Klassically it is satisfied, so by strongly consistent I mean this property. And the one of the merits is also that it is stable under Gromov Hausdorff limits. So you might start from classical spaces, smooth Riemannian manifold, so you may take a limit in the sense of Gromov Hausdorff, which is the natural notion of convergence for metric spaces. And then the limit object will be only in general a metric space. But you can still make a sense of this condition in your limit space, and I think it is pretty clear because this is a point-wise inequality that this condition is stable under Gromov Hausdorff convergence. OK, so this is an extremely successful theory, which can be used as a guideline for the rich theory, as we will see in a moment also the rich theory is based on suitable convexity property. So now what happens if we add the measure to the picture? Well, this ingredient is important for many applications, because for instance in many cases you can describe in limiting procedures some concentration effects of volume. So in the limit you have not only a distance, but you also have a limit volume measure. Of course you might think that in classical situation the measure is generated out of the distance. For instance on a Riemannian manifold you will take as m simply the volume measure of the manifold. Or maybe in some other spaces you might take as m a suitable Hausdorff measure. The Hausdorff measure, the k-dimensional Hausdorff measure associated to the metric space. But maybe in some cases you don't have a natural dimension. So you have no natural way to associate to the distance, the measure. And so we will see that it is much nicer and much more natural also from the point of view of differentiable calculus to consider them as decoupled. Even if in some situations they are coupled. Ok, so here there are basically two theories. The Bacry-Emery theory, which uses actually very little the distance as we will see in a moment, but more the measure. And the theory of Lot, Sturm and Villene, which uses actually both. Distance and measure are heavily used in this point of view. And ok, just also to start to use also the set terminology of Guido yesterday, we could say that this is an Eulerian theory and this is a kind of Lagrangian theory. And so in order to connect the two theories, of course where you are in a no smooth world, you have to develop new tools connecting the Eulerian and the Lagrangian viewpoint. And this is precisely, the connection is done precisely by the optimal transportation. Ok, so now in this overview I just say quickly what is the, let me start from the Bacry-Emery theory, which is I think much more intuitive than the Lot-Sturm-Villene theory. Well, the Bacry-Emery theory is based on a classical equality, which is called Bokner identity. And it says that the Laplacian, if you are, let's say, on a Riemania manifold, well, all these objects make sense. This is equal to the scalar product, of course here we should use the metric to compute the scalar product between gradients plus the norm of the action square plus the Ricci tensor applied to the gradient. So it's a quadratic function in F, which you can write in this way. Ok, now what happens? It happens that you can easily turn this identity into an inequality by saying that, I mean, if your manifold is n-dimensional, you can use the Schwarz inequality to bound this from below with the trace of this matrix, which is, again, Laplacian. And here you will get Laplacian of F squared over n. And if you know that the Ricci curvature is greater than k, I mean, in the sense of tensors, greater than k times the identity, here you get... And so in this way, in some sense, you get rid of the two dangerous terms, which you can't define, you can't define easily in a no-smoot word. Ok. While on the other end, we will see in a moment that in a no-smoot word all these objects, gradient of F, Laplacian of F, can be defined. So how hard these objects are defined in the Bacry-Emeri theory? Ok, anyhow, let me summarize now the Bacry-Emeri, the so-called Bacry-Emeri, the Bacry-Emeri condition is really amounts to right, maybe in an integral form, in a distributional form, the inequality that Laplacian minus, this is the Bacry-Emeri condition. Well, so very briefly, how I will not spend so much time on the Bacry-Emeri theory, but I want just to convey the essential ideas. In the Bacry-Emeri theory there is usually a semi-group, the heat flow, which is called the heat flow. So you start from a linear, self-adjoint, order-preserving semi-group. And so you see, to define this semi-group you typically need only these two ingredients. A priori you don't have any distance to start with. And then out of these you have the Laplacian, which is the infinitesimal generator. So it's a totally functional analytic approach. And then you have the Laplacian. And out of the Laplacian basically with an integration by parts, let me say formally, but this can be made rigorous. How can you define the gradient square? Just to integrate and this construction can be localized. So you see that even without having a distance, a priori you are able to define something which plays the role of the gradient. And then you see, you have all the ingredients to state this condition. This theory has many merits and particular, let's say, among the strong merits of this theory is the derivation in sharp form with the sharp constants of many functional or many, let's say, geometric and functional inequalities. So this is a very powerful theory from this point of view. And, for instance, we could mention the isoperimetric, the Poincaré inequality, the so-called Liya or parabolic inequality, and many others. I mean, all these inequalities can be totally derived using these calculus based on this weak Böckner inequality. So this is an extremely powerful theory and it's also very intuitive because it uses a formalism which is very close to the classical formalism you would use on a Riemannian manifolds. Maybe what are the limitations? Limitations are in some sense that some aspects, like localization, globalization and also some issues related to stability of this theory. For instance, what happens if you take a limit of spaces, a limit of semi-groups? Is it true that the limit semi-group satisfies the same condition? What is the limit semi-group? So many issues related to this, this localization, stabilization are not so clear. Maybe one of the reasons is that the role of the distance is in some how hidden. So you don't see really the distance in this point of view. Although eventually you can recover a distance, but I will not say how. Ok, now let me go instead to the lots-turm-Vilani side which is more recent. So this theory started in the 80s while the lots-turm-Vilani theory goes back to the last 10 years and ok, let me start again from a smooth picture. So let's say we are on a Riemannian manifold and D, of course, is the induced Riemannian distance and we consider these familial transform maps. Ok, you see, this is the familial transform maps which basically appear also in the in the Brene tier in the existence of optimal maps. For instance, in Rn this will be simply where phi is a c-concave map and so on. So here is the connection with the optimal transport. In general, if you want to describe optimal maps in Riemannian manifolds you should use the exponential map. Ok, and now the question is that Richi curvature appears not only in the Buckner identity but appears also when you try to compute the evolution so you see this as a family of maps from m to m indexed by t and then you can look for the evolution of the gradients. This is a Riccati equation that I will not write. What is more important for us is the evolution of this scalar equation which I will call Jt of x. So the Jacobian of this of the determinants the Jacobian determinants of these maps and then it turns out that this satisfies differential inequality t equal to zero this is greater than the Laplacian and here we really see some kind of analogy with the Bacry-Emeri condition. Again you see the Laplacian divided by the dimension plus Richi curvature and again you can say that Richi is greater than a constant and say Ok, so now the question is can we give the meaning to this differential inequality also on a metric measure space right? So the main difference you see is that we are not using the heat flow but we are using this kind of Jacobians. Well the answer is yes and we can use the convexity properties of the entropy. So let me introduce now the entropy so as we look for an integral form of this inequality yeah kato t there is a t Ah there is a t, right, right, sorry I simply forgot to write it sorry, this is a familiar map switch at time zero should give you the identity and very well so let me introduce now the entropy so the entropy functional actually for later uses more convenient to view this as a functional of measures so we put this functional equal to plus infinity if mu is not absolutely continuous with respect to m so mu will be a probability measure in my space I will use p for the space of probability measures as also Guido was doing and then in the case where density is the standard ok, this is also a good point to make a few standing assumptions that I will always do in my lectures so that with these standing assumptions all the objects that I am going to use I will define well, first of all I will assume for simplicity that this is compact and that m will be finite let's say even a probability measure so I will normalize it to be a probability measure and maybe I will always assume to simplify that the support of m is equal to x ok so under this assumption we know that the entropy is non-negative is a non-negative functional ok, so of course all the things that I am going to illustrate can be extended because it is important to include in the theory for instance the Euclidean space which is not compact or measures which are only finite above that set but this creates a lot of technical complications and so in my lecture I will always stick to this situation but none of them is essential and so let's say that we want to compute the density and go back to a Riemannian classical framework and we want I define now mu t equal to t t sharp over mu 0 for t small for t sufficiently small this is a geodesic and let us compute the value of the entropy along this geodesic ok, so first of all formula by the change of variables formula you can check that mu t has still a density mu t has still a density with respect to m but what is more important is that since we are in a smooth situation we can compute explicitly the density using the area formula and the density is precisely rho 0 over the Jacobian over j t with the map I call the j t composition with t t to the minus 1 so this is just change of variables formula actually you see one of the main problems when you do this in a metric space is that you don't have the change of variables formula so you have to bypass somehow this computation but here we can really do an explicit computation and then we compute the entropy the entropy will be will be equal by will be equal to again rho 0 over j t composition t t to the minus 1 log of rho 0 over j t composition with t t to the minus 1 the m ok, now we should change variables when we change variables you see this factor j t disappears and we get that this is the integral of rho 0 log of rho 0 over j t the x let's say to emphasize that here I am changing variables let me call x variable here and y the variable here ok so here I made the change of variables y equal to t t of x and then you see we get that this is equal to the initial entropy minus rho 0 log of j t and remember that minus log of j t was precisely the quantity which had this convexity property point the one which had this point wise convexity property I wrote before and so this leads to the so-called cd0 infinity condition for the moment I just stated with k equals 0 so for the moment infinity means that they are not putting any upper bound on the dimension and completely neglecting the Laplacian term and let's also to simplify even more because now I am just making a survey let me put k equals 0 so I am just saying that the reach is non-negative and then this means that I only used the information that this quantity is convex and so the cdk infinity condition is that the entropy is convex along let's say vast extent geodesics I mean the geodesics given by the optimal transportation distance the optimal transportation distance for me will always be the one with the quadratic cost so for me the cost in the general overview that Guido is doing we have to choose as a cost function simply the square of the distance and this is basically the lot will any so part of the lot will any stone because then we had to add the dependence on k the dependence on n but we will see this later on also all these parameters here I just want to emphasize that again this is based on a suitable convexity property very well ok so so what are now yeah what are now the limits again the merits and the limitations of this theory again the big merit is a strong consistency actually for any value so the parameters for the appropriate condition on Arimanian manifold it is satisfied if and only if it is satisfied in a classical sense we have stability measured Gromov house of limits so here we have to add measured because we are dealing with metric measure spaces so there is a variant of the Gromov house of theory which allows to deal also with limits of metric measure spaces ok but I will never have the time to illustrate this I just mention it ok, now let me go instead to the limitations what are the limitations of this theory well a technical limitation is that everything here is based on optimal transportation we will see on jodesics and so on and it turns out that many results in the so-called nonbranching assumption so nonbranching means that your metric space should have this property that it is impossible for a jodesic to split for a minimizing jodesic of course so you should do many many results for instance the derivation of the initial derivation of Poincaréian quality in this theory was based on this assumption this is a big limitation because this condition by itself is not stable so we are putting again into play an unstable condition for instance L infinity which is the typical example of a branching space L infinity in R2 is the limit in any reasonable sense of LP so all these spaces are strongly uniformly convex so they are nonbranching the limit is branching to play in describing another limitation on this theory well well this is maybe a bit controversial whether this is a limitation or not let me first state the result and then I will explain so let's say Rn and though with any norm so choose any norm in Rn and use this norm to generate the distance and then we take as a measure always the Lebesgue measure satisfies CD0 infinity and actually even the dimensional one but that we will see maybe later on this is a sense so in some sense this axiomatization is not sufficiently strong to distinguish between different norms so on one end one might see well ok it's a good point because in this theory we have many examples so from this point of view is not a limitation on the other end it is a limitation if you see it from the point of view of the what we might call the chigar caulding program so what is the chigar caulding program the program of chigar caulding was to to describe limits I mean gromovausdel for measure gromovaus limits or classical objects of Riemannian manifolds under let's say Richi plus volume or diameter bounds and Richi lower bounds plus volume other because of course we can use the fact that this theory is strongly consistent and stable to say that any limit in this program or trying to describe the closure of Riemannian manifolds the limit object satisfies the CD0 infinity condition on the other end if you look only to the CD0 infinity condition in principle L infinity might appear as limits of these manifolds well on the other hand using the so called splitting theorem and I think Nikola in the last week will speak about this result the splitting the approximate splitting theorem implies that for instance L infinity is not in the closure so it was already known that it is impossible to reach spaces like L infinity by closing Riemannian manifolds but you don't see this phenomenon using only the CD the kn theory and so in fact this was our starting point our starting point was precisely to try to whether to find a stronger axiomatizacion so the question that we want we answer is there a Riemannian CDK and theory which we use to to rule out L infinity and other examples and since it has to get a better description of the closure of Riemannian manifolds and how does it relate to the to the other theory to the Bacry Emery now the answer to these two questions basically depends on a deeper understanding calculus of the heat flow in metric measure spaces so in such as the technical part of my lectures were really going to give precise statements and some proofs will be dealing with this part and maybe only in the last hour we state the answers but unavoidably without proofs to these two questions ok, so this is my plan very well ok, so maybe I use now the 15 minutes before the break to introduce a little bit of terminology in basic tools so I already wrote my standing assumptions some basic tools well the first one is the notion of absolutely continuous map and metric derivative so let's say if I take an interval I into R and if I take a gamma from I into R I say that gamma is absolutely continuous if for some g in L1 I non-negative for which we have this inequality so you see in this way absolutely continuous is just a purely metric concept and then you may wonder what is the smallest function g that you can put in this inequality sorry, yeah sorry, yeah, right sorry, pardon ah, right sorry yeah ok, and then you may wonder ok, if you have some g so absolute continuity is defined by the existence of this function g you may wonder what is the minimal function g the minimal function g exists up to a negligible set and it is what we call the metric derivative so this is a notation right is reminiscent of the fact that in this classical context you will take the gamma prime which is a vector and the modulus of gamma prime here we are in a metric context so you should take this just as a notation and this is the limit as h goes to 0 ok, and the part of the statement is the existence of this limit almost everywhere so this limit exists almost everywhere and it provides you precisely the smallest function for which you have this inequality ok, and analogously you can define acp for p acp is simply defined with requiring g to be in lp so that for instance ac infinity corresponds to Lipschitz functions ok so this is in some sense the basic Lagrangian definition and the basic Eulerian definition is the notion of slope so we want to define basically the modulus of a grade let's say modulus well you have to be careful and distinguish even formally between gradients and differential so we use d just to remember differential and not gradients so let's say for a Lipschitz function this is the so called slope or also called in the literature point wise Lipschitz constant is simply the lean soup as y goes to x of the f of y minus f of x ok, this I don't need moment and when we will be speaking about the CDK and theory it will be convenient to use another formulation ok, let's say this is 2 another formulation of the optimal transport problem that I will not make I think Guido will not speak about this so I should introduce it briefly and then we can make a break so the so called time dependent dynamic formulation of the optimal transport problem ok, in the classical presentation of the optimal transport problem you only care about the initial and final position which you can imagine that they are glued by a map or by a plan as Guido was illustrating before yesterday but what happens in this geometric context start to be important also to know in some sense dynamically what happens in between so the path that you follow in going from x to y ok so how can we do this so let's say that mu and nu are probability measures in x so these are the typical data of the optimal transport problem the cost is one half d squared so let me introduce the so called action gamma the action of gamma before maybe I should introduce my space I will introduce the space of absolutely continuous maps AC2 and I view this as a subspace of this space so this is a polished space this is completely separable and this is a sigma closer subset of this space and so I have no problem in considering probability measures inside this space so strictly speaking these are probability measures in this space which are concentrated on this borel subset and I will call this dynamic dynamic transference plan or dynamic plan and you see by this probability measure I encode really the information also of the path that I am going to follow to go from x to y another notation that I will use is the evaluation map so this will be the evaluation at time t etio gamma will be simply gamma at time t sometimes I will also use this notation and then I will introduce the action very much like in Riemannian geometry the action of a of a curve in AC2 is simply the integral between maybe I will put one half here oh no maybe not t and notice that exactly like in the classical case one can show that this is always greater than the vector gamma square the proofs are exactly the same using the definitions I already gave of metric derivative absolute continuity with the quality if and only if gamma is a length to minimizing curve with constant speed and this is also greater of course than gamma of one gamma square between gamma one and gamma zero and now the idea is that in order to formulate the optimal transport problem in a dynamic way instead of minimizing this I minimize this so the dynamic optimal transport problem is you minimize and now eta will be a probability measure in AC2 and now how can I specify the initial and final constraint I will use just the evaluation maps so e0 of eta should be equal to mu and d1 of eta should be equal to mu and this is actually this is exactly the problem so there is this theorem which tells us that this is just a reformulation of the usual problem used by Guido so theorem if xd is a geodesic space let's say it's a length space so this means that given any two points there is a length minimizing curve between the two points with length equal to the distance then the minimum in the dynamic optimal transport problem again is equal to the minimum in the Kantorovich problem so it's the same problem it's only that it is more descriptive because now we can really say that eta eta is concentrated on geodesics with constant speed you see because of this inequality in the case of equality you get the test is concentrated on geodesics with constant speed and if you want to make also the connection in minimizers you will find that e0 e1 sharp of beta you see this is a probability measure in x cross x is a minimizer in the Kantorovich problem and that's why this is a complete description so instead given a minimizer in the Kantorovich problem if you connect any two points by constant speed geodesic you build an eta but it will be convenient to view we will see immediately maybe even today that it will be convenient to use this language or probability measures in the space of course to develop a calculus ok so I think I can stop here for 10 or 15 minutes ok so today I will only speak about the problem of sobole spaces on metric measure spaces so in success to develop some calculus it seems natural to develop a kind of theory of weekly differentiable functions ok so let me remind you also to make a kind of analogy classical analogy that in the already in the classical theory of sobole spaces which was developed in the 50s there were two competing definitions the H definition which is obtained by approximation with smooth functions I mean here I am speaking of course in Rn and then there is the so called WAP in space which is based by integration by parts and then there is a famous paper by Meijer Serrin 1960 whose title is precisely H equal W so they prove in quite general domains you see in convolutions partitions of unity that the two spaces coincide so what we are going to do basically today and also part of tomorrow is to revisit all these things in a metric measure space ok let me start from H definition the H definition was revisited by Chieger so the so this is the theory of relaxed gradients and this was revisited by introduced by Chieger in his famous paper on geometric and functional analysis about well his motivation was to develop a kind of Rademacher theorem in metric measure spaces so he had slightly different motivations but he proposed this definition he proposed this definition of sobole space so we say that f belongs to the space h1p xm xdm and now the roles of the distance are very clear in this definition how they interact if you can find if first of all if there exist fn which are lip sheets which converge to f in lp ok let me say that here for me p will be always between ok maybe I will stick to the case p equal to for simplicity because it's the only one relevant for my lectures fn converges to f in l2 and so the role of smooth functions is played here by lip sheets functions and the gradients the role of the gradient is played by the local lip sheets constant so you see here the interaction of distance and measure because to define the local lip sheets constant you need a distance and then you have to weight this quantity using your measure and then associated to this definition we can call a relaxed slope which is a non empty subset of l2 for any function in h1p all weak l2 limits sorry so associated to any family fn you have many a priori many weak limits of this quantity because this is a family bounded in l2 you take all weak limits you take all functions which are greater than a weak limit so here of course I mean a subsequence and for all possible approximations actually it's technically convenient I will not explain this but it is based on the so called mazur lemma on a convexification argument to show that this is equivalent also to this definition is equivalent to g is greater than a strong l2 limit of gn gn greater gfn well the passage from weak to strong convergence is due to a convexification argument you know that you can turn by convexification weakly convergent sequence into strongly convergent ok and this is also a useful characterization of the relaxed slopes ok and then again we want to identify a minimal object and here it comes into play the second characterization so first of all for instance you can use even the first one this set is convex is a convex subset of l2 but I mean it's difficult using this characterization to show the closure because you can't make diagonal arguments using the weak topology while you can do them using this characterization and so it turns out that rs of f is a closed and convex in l2 and so we can single out in this in this class an element with minimal norm which I will denote by df star which is the minimal relaxed gradient is the unique element with minimal l2 norm among all relaxed gradients and then I will call sugar energy simply the integral of the minimal relaxed gradient and so this object plays in the theory the role of the Dirichle energy in the more classical theories in fact one can show basically using again this kind of characterization that this is a convex and lower semi continuous energy by convention you can put it to plus infinity as in the classical theory of minimization of the Dirichle energy on l2 minus h12 ok, so when I mean lower semi continuous I mean l2 lower semi continuous so I view this as a functional in l2 and I set it to plus infinity on l2 minus h12 ok, and then let's see what we can do with this well there are now some elementary calculus rules which already show that some classical properties are retained in this setting so the basic facts are first of all the chain rule so we have a standard chain rule so here there is an inequality of course this object is defined up to negligible set because it is a function in l2 and with equality if phi prime is non-negative and in other useful factors we will see in a moment is so called decomposability or the class of relaxed slopes decomposability means that if I have any two relaxed slopes and if I take any borrow set in x I can combine the two so the function which will be equal to the characteristic function of b times g1 plus the characteristic function of the complement of b times g2 is a relaxed slope and for instance now using precessive decomposability one can show a point-wise minimality property point-wise minimality property which says that for any g which is in a relaxed slope so remember that we defined the minimally relaxed slope by minimizing an integral quantity the l2 norm but posteriori we can say that this quantity is minimally so for any relaxed slope we have this inequality y and the proof is simple let's say if this set has positive measure sorry if this set has positive measure and call this set equal to b then g tilde which is equal to g times the characteristic of b plus df star the characteristic of x minus b is a relaxed slope which is strictly smaller norm so first of all it is a relaxed slope by the decomposability property and the normal g tilde is strictly less than the norm which contradicts the minimality so because of the minimality property you get also the point-wise minimality property ok, these are good news but now let me say some sense some kind of bad news or say some warnings well the first one which actually will be very important for our axiomatization but in general the trigger energy is not quadratic so you are integrating the square of something but this object is not quadratic and in particular I didn't say that of course in h12 you can put the norm is not Hilbert so this is not an Hilbertian norm and actually one of the main goals of trigger paper is also to prove that even if it is not an Hilbert norm in some situations one can build an equivalent in Hilbert norm but in some cases the natural the natural norm in this space is not an Hilbert norm and so let me give you an example why this happens ok, let me say that this is equivalent to if ok, the best result is a result I got last year with the marino and Savare in Colombo so which improves a little trigger paper so under some assumption one can find an equivalent in Hilbert norm but not in general so let me give you an example of this phenomenon very simple, again we go back to L infinity maybe my notation of L infinity might have created some confusion I meant by L infinity I didn't mean the space of sequences I just meant N with the sup norm just a finite dimensional space ok, let's say even N equal 2 and you see, so in this case in some sense and here you see also the dual nature of the F because the distance is the infinity norm so since this is a normal differential should be dual and so you can easily compute that the slope in my relaxed slope is the L1 norm and so so this means that and so this is not a quadratic form if you integrate this you are not getting a quadratic form ok, it's too homogeneous but not a quadratic form so the notation can be a little misleading but keep in mind that in general we are not dealing with a quadratic form the second important warning so we are going to see very general result which basically identify again h equal to w and also many other results which almost require no assumption on the metric measure structure but the warning that in some cases all objects can be trivial so maybe they are identified but they are simply trivial so let me make a simple example take as your space at 0,1 as m you take Lebesgue measure I'm sorry, as distance you take the Euclidean distance instead of taking the Lebesgue measure you take a very concentrated measure which is where this is a the standard any enumeration of the rational numbers of 0,1 in this case one can show that the trigger energy is identically 0 so in this case h1,2 actually coincides with l2 and the trigger energy is identically 0 so this means that I mean as some stages some additional assumption on the metric measure structure should come into play to say that these objects are not trivial so how we can prove this well the idea is that ok maybe I give it just a very quick sketch of the construction so let's take any ellipsis function or c1 if you like and define fn as the integral fn of t as f of 0 plus the integral between 0 and t of f prime key n yes where key n is the characteristic function of an open set a n and these open sets contain the rational numbers and the Lebesgue measure of a n goes to 0 sorry is the 1 1 minus k n and so since you know fn converges to 0 in l1 or 0 1 clearly fn converges to f uniformly but fn prime is equal to I mean the slope of fn or the derivative of fn is equal to 0 on a n and so it is 0 almost everywhere and so this means that I have found an approximating sequence identically 0 and so and so my trigger energy is identically 0 ok and so you may wonder when these objects are not trivial for instance a positive result so I already said that the space has an equivalent norm if you have this doubling condition and one also the main result of this paper by trigger is that the doubling condition for for the measure plus the Poincaré inequality and maybe I write now on this board the Poincaré inequality I mean the local Poincaré inequality the local Poincaré inequality says that well there are many equivalent formulations but that you can control the deviation of f from its mean by a constant maybe in a slightly larger ball lambda greater than 1 and c greater than 0 which of course should be independent of r time the local Lipschitz constant so it is a kind of structural assumption on the metric measure structure which is satisfied in many many important cases so under the Poincaré inequality condition plus the doubling property for the measure one obtains that for instance and this should be thought really as a kind of Rademacher theorem these objects are not trivial for instance on Lipschitz functions they coincide with the local Lipschitz constant and it is clear that Lipschitz constant is not trivial in many spaces this is not a trivial object and so this is not trivial this is a very deep result I mean I just mentioned this for completeness ok, now let us start to look to the Laplacian still in this framework in the framework of the relaxed gradient of the H sobole spaces let us look to the Laplacian and heat flow ok in order to define both the Laplacian and the heat flow we have a very established theory which is the theory of maximal monocon operators which was developed by Komura Brezis and many others in the 70s so let me remind you the basic facts I mean the basic facts are that you have an Hilbert space which in our case will be L2 you have an energy functional phi from H into r union plus infinity convex and lower semi continuous and in our case the energy functional is the trigger energy with one half and given these ingredients so called the sub differential I will denote by d of phi the set of points x for which phi is finite which is a convex subset and again in our case the domain of phi will be h12 and for x in the domain of phi I will define the sub differential of phi at x as the collection of slopes so these are really dual objects this should be taught really as elements of the dual but of course I will identify them with vectors of h such that phi of y is greater than phi of x plus xi y minus x for any y k so is the set of directions of supporting hyperplanes so is the slope of a supporting hyperplane to the graph of phi at the point x so in our case we take as definition of the Laplacian sorry I should have added something this is a convex closed subset again it is easy to check that it is a convex closed subset so again I can take the element of minimal norm which I will denote by nabla phi this is really a vector a vector of my space and now in our case I define as Laplacian minus the gradient of the trigger energy and now the this definition so in our language Laplacian will be a vector and the vector with small less than 2 norm satisfying this property that the trigger energy of g let me write maybe in this way minus because xi is equal to minus minus h delta f dm for any h in each one so for us the Laplacian will be defined by this formula again an important warning delta again this is very misleading notation is non-linear in general is a non-linear operator because it is the gradient of something which is not quadratic in general so for instance by exercise you could compute what is the Laplacian in the infinity example you will find a second order non-linear operator of course unless if the trigger energy is quadratic then you get a linear operator this is for the definition of Laplacian and now what about the definition of the heat flow now I will state the basic theorem the basic theorem of the maximum amount operator's theory in initial condition x0 in the closure of the domain of phi phi remember was a convex lower semi-continuous function in my Hilbert space there is a unique solution xt to the Cauchy problem actually the Cauchy problem can be stated as a differential inclusion so you are just saying that for almost every time you go into opposite in a direction opposite to some element in some differential with x locally absolutely continuous x you see is only locally absolutely continuous because I am considering also initial conditions with infinite energy so in the case for instance of deli-clay energy you could start from any function of infinite energy you have only local absolute continuity and the initial condition is that the limit is t goes to zero so this Cauchy problem has a unique solution and then there are several regularizing effects which I can mention here only briefly but they are part of this classical theory the regularizing effects for instance the main ones for us first of all that for almost every time this unique solution selects really the gradient so you start with the differential inclusion but at the end you get an equation and the second important fact which plays sometimes a role is that the right derivative exists for any time so you get really a point wise equation provided you work with the right derivative not only almost a derivative but for any positive time you get so in our case with our notation we really get the heat flow so coming back to our particular case for any since the closure of H12 contains the closure of leap sheets which is a 2 so we can solve our problem for any initial condition well it requires some work the formal idea is this ok maybe I complete this statement and then I give you the formal idea coming back to our particular situation where we were working with the Chigar energy the closure of the sobole space is L2 and so for any F0 in L2 the solution Ft to the heat equation since our solution becomes a little tautological because delta was defined precisely in this way but we will see that this is the right point of view and the usual warning this is not a linear equation in general because the Laplacian is a nonlinear operator but for this nonlinear we have existence and uniqueness ok now coming back to your question the uniqueness comes from this monotonicity in fact this is where the name monotone operator comes so this operator which is a multivalued operator because in principle associated to any X you might have many difference lobes so this is a multivalued operator is monotone what does it mean this comes out of convexity this means that p-qx-y is always nonnegative whenever you choose p in the sub differential of phi at x and q in the sub differential of q or phi sorry at y this is a simple property that you can check just as a consequence of convexity ok once you have this monotonicity property take two different solutions let's try to differentiate their L2 norm sketch of uniqueness so take two different solutions starting from the same point so you know that this quantity is 0 at the origin and then you get here is prime 1 of t minus prime 2 of t x1 of t minus x2 of t but remember that this is now opposite to an element in the sub differential in this point and this is opposite to an element in the sub differential of this point and so you can use this monotonicity property to say that this is negative so in fact this is a case where you have not only uniqueness so the flow associated to the solutions is contractive so now we have our solution to the heat flow and what is important also for some applications is that we have an integration by parts formula this is a bit surprising because a priority operator is not linear but still we have an integration by parts formula which says that minus the integral log g delta f dm is less than you see this is the inequality we will have in a classical setting because after integrating by parts here you will get nabla g nabla f with equality if g is a function of f is a non decreasing function of f so for today maybe I will stop I will just give this proof and then we can stop let us fix an epsilon positive and we consider the bigger energy of f plus epsilon g but the definition of sub differential d is larger than the bigger energy of f minus epsilon here I just applied the definition of sub differential because minus delta f is in the sub differential of the bigger energy is by definition the element with minimal norm actually this proof would work with any element in the sub differential well on the other hand on the other hand one can easily check that df by convexity df star plus epsilon dg star is a relaxed slope of f plus epsilon g because if you can approximate f you can approximate g then combining the two approximations you get something which is still an upper bound a relaxed slope for f plus epsilon g so you can bound this by the definition of bigger energy we have put the minimal object the minimal relaxed slope and so here if you expand this square you get that d is the bigger energy plus the integral epsilon times df star dg star plus a quadratic term and so if in this chain of inequalities you simplify and you divide by epsilon you get just a refinement of this argument which is based on the fact that in the chain rule formula we know that we have a quality when phi is non decreasing just a refinement of this argument tells you that if g is a non decreasing function of f then you have a quality here in particular ok, maybe I say also this and then we stop in particular you can take g equal one so here we are in a compact case so I can take g equal one and this tells me that g of course will be a function of f and so this tells me that the integral of delta f is equal to zero but this is also the integral when you study the heat flow and so this means that is constant along the heat flow ok, and I think maybe tomorrow we can restart from here I need a few more properties of the heat flow and then we start to discuss the w sobal spaces