 Tako, da se da lahko se poslutimo. Očešla, da smo jaz naredili, da smo vse zelo vzvečili zelo všem vseh tukaj všeškej. In smo počutili še vseh tukaj vseškej. Proste, da smo vsega vsega in integracije vsega, da je tukaj, da je izgleda, da je izgleda, da je zelo vsega in in zelo vsega. In to je, to je tukaj, da tukaj skonstant je odličil. In, ok, analogosne pravosti možemo prišličiti, besej, da po vsej argument, po vsej argumenti možemo prišličiti, maximun, minimun prinsipul. Zato je, da, poured, način, da, if f times zero is greater than a constant or smaller than another constant, then this inequality is preserved at all later times. Stajamo bojo, zelo je tudi vmanselje. Pre Mutu, taj do kaj je, da so evenly posledilo raz VI, da počušam, da počušamo z z minus C, in prijemno odpočenim počušanje in tudi hrvosti f t vde m. Zdaj, možete vsezvati vsezvati vsezvati vsezvati vsezvati vsezvati na dojelni hr. A potem, a potem, ta dojelna je nekrativ, a potem, da se kako leži, da pokazan na dojelna z 0, to je 0, ok. Daj dobro prijemno se niste prijemno pripenje, da if you start from a function f0, which is non-negative and we integral of f0, the m is equal to 1, these two properties are preserved in time. And so these are the two properties which characterize probability densities. Tako, zelo vse možemo muze m, f0m, to je nema površenja v vse, nema površenja vse, nema površenja vse, nema površenja vse, in zelo, da bili nema površenja, da bomo vedno izgledati ft, nema površenja, da bomo daj, ta bila f tx, typically Markov, so would be just an index, then FTM will still be a probability measure, so it will be crucial, now, for the things we are going to say later on, that we understand in some sense this curve in a dual way, either as a curve in a two from this point of view, but also as a curve we will see in the space of probability measures, this will izgleda tudi je zelo vzelo. Prejmo se zelo vzelo vse zelo vzelo, vzelo vzelo vzelo, kako je bilo vzelo. Zelo več. Vsej zelo, ki boš tudi pričel, však je zelo, je entropija. Zelo vzelo vzelo, kako postahno otkaj vstahniju. Vse znači, da vse intervjučnem vsega. Tnji je odmah za vidjev. Tnji tnji je t只, tko je t z 0,5 vstahnji. Vzleda vzleda 1 plus log fT, fT. Zvonem vsega. Zato pa čas da lahko vzleda zazvalo 0, da si zelo tudi časno izgledi, da jaz vzleda pri vzleda. Zelo vzledaj, da lahko vzleda kaj pa teknik vzleda o logaritin, that fT is greater than a positive constant, ok. I'm working with the curve made by functions which are bounded from zero. So I have no regularity problem related to the logarithm here. And of course here I'm using that m is a finite measure. This is something I can't do in general, but this is one of the cases where working v krati m. Sreči m. Všeč na očetku pod vrste z vrste djih gradi, star gradi, star m. In zelo potrebo moguš vsega vrste ruri in je minus vsega vrste na nabla ft2 ftm star. Oko vse gradenje, vse derivatiče, je derivatiče relaksi, kaj je vse derivatiče, za kaj smo zeločili vse kalkulosti. Ok, zeloči me vse pravno. Zeločimo, da energičnjičnjičnjičnjičnjič je zeločila vse funkcionale. Vse funkcionale je vse vse interesanje. Pro always of all, again I can use the chain rule to write this in terms of chigar energy. It's equal to 4 times the derivative of the square root of f square vn, which is 8 times the chigar energy because the chigar energy has the normalization 1 over 2 of the square root of f. če v 2. 1. kar tu kraj je v 2. srečenje. Ali je to nekospečnju kuštizavlju, ki postošati umovr memories in vzpečnih prijev, k kaj je in konvexiti. Konvexiti je ne objav Cinemazov, kaj imam tvoje reprezentacije, da videl sem, da se boj iz tim, v srečen sprešelaj, če to je konvexita. To je bilo, da počutim konvexiti, da sem tukaj, vse ideje je, da je vse izgleda ta reprezentacija. To je ta reprezentacija, kaj je konvexiti. Zelo, da je izgleda, da ta reprezentacija je konvexita, da se je na R. Zelo, da ta reprezentacija je konvexita, je tukaj konvexita, In vsega pošljaj, da se zelo dvejkje težiči, da naprej vsega tukaj. In na konce, da jih ustajem, obježaj si pravne zelo v zelo. Tukaj je še in strančne, zelo kaj je dneča informacija, kaj je tukaj vsega informacija. I na vsega zelo je vsega in vsega je vsega, da je zelo kaj je pravno, zelo da je tukaj vsega, na hrvom. Ok. So, this more or less closes for the moment the so-called Eulerian point of view, or if you want the H side of the Sobolev theory. And now we start to discuss the W side of the Sobolev theory. Roughly speaking, what we see is the Lagrangian side. Ok. First of all, a few historical remarks. Well, of course the W definition became popular with the theory of distributions in the 50s, and it's just based on integration by parts. So, if you are in our N, so the weak derivatives are defined by the validity, by the validity of this integration by parts formula. But actually, it is less known that a lot of time before in 1901, Levy proposed a completely different, quite different approach to what we call now Sobolev spaces, which in the same years of the theory of distributions eventually was revisited by Fugleda. And what I'm going to do to define Sobolev spaces in metric measure spaces is to follow this point of view, and not this point of view. Why? Because in such sense this point of view is based on some regularity property of the measure. To have a good integration by parts formula, you need a good measure on your space. If I want to make a really general theory, and if you want to avoid this assumption, you should. So, basically the approach of Levy and then of Fugleda is to consider a kind of one-dimensional integration by parts formula, which is something that is much less restrictive, because it is related basically to the one-dimensional Lebesgue measure. And the one-dimensional integration by parts formula is that, I mean, in the abstract. So if you consider, you can write it classically as F of gamma 1 minus F of gamma 0, equal to the integral between 0 and 1, nabla F of gamma t, gamma prime of t dt. Ok. This is standard. And maybe in a more abstract way you could write it in this way, which also clarifies the structure of this formula that the integral on the boundary of gamma, if you think of gamma as a current, really, the integral on the boundary of gamma with an appropriate choice of the orientations is equal to the integral on gamma of the gradient of F. So, really, this corresponds to the integral on this current gamma of the one form associated to the gradient of F. And so, the definition of Levy and Fugleda is that F, so let's say between quotes, and then I will make it precise if, let's say, there exists a gradient vector field. Ok. Of course, the Levy and Fugleda definition was in our n, such that the integral on the boundary of gamma of F is equal to the integral on gamma of F for almost every. So, you can really view this again as an integration by past formula, you see, but is along the course and along almost all course, because, of course, since F is discontinuous and also, so, since this function might be discontinuous, it's not reasonable to impose regularity along all course. You may know that a Sobolev function is also continuous along almost every line, not along all lines. Ok, so now, the problem is how to define almost every, and this is where the contribution of Fugleda was very important. And then this approach has been revisited, let's say, roughly ten years ago by Shanmungalinga Koskela, Hainon and Koskela. And three years ago, we came to this problem with Gilles Savare and, again, we gave a new definition and a new concept of almost every curve, which is the one I'm going to adopt. A posteriori, our theory coincides with the other theories, but maybe I will not give details about this fact. Ok, what happens if you are on a metric space? If you are on a metric space, of course, you don't have vector fields, really. And so you turn this into an inequality. Ok, maybe I will call f, I will call the function g. So this function is in L2, and you require that this inequality is true. Remember that now this makes sense in a metric space, because this is usually the difference of the values of f at the end points of the curve, while the curvilinear integral can be still defined using the metric derivative. Here I say almost every absolutely continuous curve, this object is well defined, but still I have left the quotes because I have to tell you what we mean by almost every curve. Ok, and here really, in our paper, we put forward some ideas from optimal transportation with the concept of test plan. So take a probability measure pi space of AC2. We say that pi is a test plan if all marginals of pi, so Et sharp of pi, are bounded by some constant times m. So these are measures, and I want that all these measures should be bounded, have a bounded density, uniform in time with respect to m. So maybe sometimes I will use as a notation C or P to denote the smallest constant that you can put in this inequality. Ok, maybe I write this just as a reminder for these concepts explicitly. This means that if you integrate phi of gamma t d pi of gamma, you should be able to bound this integral of phi dm for any phi non-negative. Ok, this is the explicit translation of this inequality between measures. Ok, and then by duality we can say what is almost a very curve. So we say that p holds that the property p relative to curves holds almost everywhere, holds, let's say, 2 almost everywhere, if, so there is a power 2 here involved in this definition, if pi of all curves gamma such that p of gamma fails is equal to 0 for any test plan pi. Ok, but duality, so the property which is pi negligible for any pi test plan pi is said to be true almost everywhere. And so in this way we have given a precise meaning to this inequality. And so now this is a real definition. Well, because, you know, inside this definition there is an integrability, because a priori if you change, I mean, this is a part maybe of a more general theory where you could study h1p and w1p. So the role of this exponent appears here because you are taking probabilities concentrated on s2. Although, of course, the actual solid with the case p equal to I wanted to emphasize that this definition depends on the choice of an exponent. I will call these objects weak upper gradient. A function g having this property I will call weak upper gradient. The weak upper gradients are the class of functions g will satisfy this inequality for almost every curve. And then I can define how to solve a space. Let's say, let me put already the sumability here. So simply the space of functions for which you have at least one upper gradient in a weak upper gradient in L2. A few propositions, a few properties of weak upper gradients. So properties, well, is a lattice and is also convex closed set. And so because of these properties that I will not prove but they are not so difficult because of these properties you can take again the element with minimal norm and I will call it nabla df weak. To distinguish from the other one which was obtained and remembered by approximation with Lipschitz functions I will use w star. And again because of this lattice property this is not only minimizes the L2 norm but it minimizes also is minimal almost everywhere in the sense that I already described for relaxed gradients. And another property another interesting property is that is that if f is in w12 xdm then for almost every core gamma one is first of all one is first of all that the composition is w is a w12 function in 0, 1. So in such as we really go back to one dimensional sobole spaces on the real line the derivative in the sense of distributions of this real valued function on the real line almost everywhere for almost everything. So you see that really in this way we go back to a point wise description and maybe I will not give the proof of this fact for those who want to to work on it maybe I just give you the key technical lemma which allows to prove this so the proof is based on this so assume that you are in this situation you have a function h and you have a function g in l1 for 0, 1 and you know that you can bound the increments of h with g but this not for all almost every snt then this is a criterion for membership to w11 then h belongs to w11 0, 1 and the derivative in the sense of distributions of h is less than g for almost everywhere ok, once you prove this calculus lemma is not difficult to get but by the calculus lemma almost every curve to get to this ok and now we are going to discuss the relation between h and w spaces so the theorem which is surprisingly requires no assumption is that h is equal to w ok, a posteriori I mean these two spaces are the same and the gradients are the same ok, so now I start to discuss the two inequalities and the two inclusions ok, if you have in mind the classical theory there is always an e-s inclusion the e-s inclusion in the classical and also in this theory is that h is contained in w so the functions that you are able to approximate with smooth functions always satisfy integration by parts formula of course this comes with the proof that the gradient w is smaller than the gradient star almost everywhere so first of all let me start to discuss this e-c case while the other one we will see is much less trivial ok, so let's take a function in h in h12 so I know that f e I can write f as the limit in l2 of fh and remember that the second characterization of relaxed slopes I gave I can write nabla f star df star as the strong l2 limit so I can find an appropriate sequence for which f is equal to the limit of the where these are the slopes h are lip sheets in x now now the first observation is that the slopes are weaker per gradients lip sheets functions so if you have a function which is lip sheets then df belongs to wug of f let's say let me call this function h and this is easy to check because you have the point-wise inequality I mean you start from the point-wise inequality that d over dt of f composition with gamma this is really a point-wise inequality let's say at any point where the composition is differentiable you have that this is less than so at any point where the function is differentiable and there is the metric derivative you have really a point-wise inequality that you can easily check because the slope is really taking the supremum and then you integrate this inequality by integration you get that for any curve actually so not only a weaker per gradient but it is an upper gradient because this property is true along any curve in particular you have a weaker per gradient and now the key point to conclude is the stability of weaker per gradient which says ok, so let me put in this form that if fh converge to f in l2 and dfh and gh converge to wug fh and gh converge to g strongly in l2 I mean this is again like in the classical theory is a kind of closure property of the gradient operator belongs to wug of f actually the property is true like in the classical theory but ok for the purpose of this lecture it is sufficient for me to prove it with strong convergence but usual trick of convexification one can reduce to the case of strong convergence right but actually it is true under weak convergence ok, so for the moment I will give you later on the proof of stability let's see how we can use the stability to get this inclusion and this inequality ok, so now apply the stability with gh equal to dfh which we know are weak upper gradients to get that this gh are converging to the relaxer gradient to get that nabla df star belongs to the weak upper gradients and this implies at the same time that f is in w because I have found a weak upper gradient and the point wise minimality property tells me that the weak upper gradient is smaller than this object so now the key is to prove this stability property ok, the stability property is not difficult in lm ok, so now proof of stability again I need an intermediate observation which we might call fugled lm which tells me that from this convergence in l2 xm one obtains convergence on almost every curve so the convergence in l2 implies convergence on almost every curve and if you have this property ok you can also assume without loss of generality in case you extract a subsequence that this convergence also is true almost everywhere in case you extract a subsequence in order to achieve this property and now remember that e0 sharp of pi is really continuous with respect to m as a bounded density which implies if you have convergence almost everywhere that fn of gamma zero converges to f of gamma zero for pi almost every curve so if I fix a test plan because of the absolute continuity condition which you impose in test plans you have this convergence for pi almost every curve but pi is all arbitrary and so you get this also holds for 2 almost every curve because pi is arbitrary analogously you get that fn of gamma one converges to f of gamma one for 2 almost every curve for the same reason in so in so you see we have all the ingredients to pass to the limit in the weak up per gradient inequality because we simply write and then we pass to the limit along almost every curve and we recover the weak up per gradient property ok so this is the proof of stability and now I show you the short proof of Fugledes lem I convergence in L2 actually here I need also here I need a slight refinement so let me assume not only convergence but also that I can always achieve this pass into a subsequence ok this is what I am going to show I am going to show that if gh converges to g fastly in L2 in this sense then your convergence again this is sufficient for stability because you can always extract the subsequence in order to achieve this property ok so this is what I need to show now the idea is very simple because you start from a generic curve gamma and you apply other inequality times the integral in 0,1 of gamma dot square this is just held inequality ok now fix pi test plan and assume with no loss of generality this quantity is finite is L infinity that the action is bounded with respect to pi you can always assume this because pi is concentrated on course for which this is finite so any pi can be monotonically approximated by pi's which have this property so I need only to check my property is on this class this is a slightly smaller class of test plans for which this function is not L1 but L infinity not only L1 but L infinity ok and now I am going to integrate this inequality with respect to pi and so I get that the integral gamma can be bounded by the integral between 0 and 1 and now I am using fubini also now here there is a constant which comes from the fact that this object is bounded with respect to pi and then I apply also fubini here and now I have to use in a heavy way the fact that not only the marginals at time 0, at time 1 are bounded which is the property I used in the proof of stability property but also that all intermediate marginals are bounded so all these integrals can be bounded so these integrals are with t fixed so they can be bounded by a constant by another constant so here the constant c of pi comes into play the integral of g n minus g square dm ok and here I really use the c of pi for all times between 0 and 1 the proof is done because now I sum with respect to n and I get that the integral of the sum is finite remember that I was assuming that this series is finite with respect to n and this tells me that this is finite pi almost everywhere which means in particular that the integral on gamma of g n minus g converges to 0 pi almost everywhere ok and of course eventually I use the fact that pi is arbitrary and so the final statement is that this property is true so you really see in this proof how we used the axiomatization about test plan in a really crucial way ok, now we have to start to discuss the non-trivial implication and inclusion and of course this inclusion will come with the proof that the gradient star can be bounded by the gradient so let us go back to the classical proof mayor serring the proof of mayor serring is based on convolutions what will be the role of convolutions in this context the role of convolution is played by the heat flow although on linear we will see that it plays the same role so why in some sense this implication is more difficult because in some sense the only information that the function is in W and so the only information you have on your function is that on almost every curve we have some control of the oscillation of your function how to this what you have to do in order to build to show that F is in H we would like to build which are lip sheets which converge to F and such that the lean soup is less than the L2 norm of the weak gradient so this is what we really have to do and in fact I should put quotes here because our proof is not really constructive as we have seen in a moment we have a kind of indirect argument we don't have a really an explicit construction and I think an interesting problem is to see try to understand whether maybe inside our proof there is a kind of more explicit way of doing things our proof is not really constructive and let me start with some strategy strategy I still have some time before the break is to compare energy dissipation rates remember that we already computed exactly the energy dissipation rate of the entropy on the heat flow which was the integral and here we had the star gradient because the heat flow was done using the trigger energy this is the result I proved at the beginning of my lecture and what I am going to show using some ideas on optimal which come really from optimal transport is to show that there is a sharper estimates of energy dissipation rate you can bound it by one half of the of the star gradient plus one half of the relaxed gradient of the weak gradient so it says this estimate is sharper than this one because a priori for the moment we have this object is larger than the weak gradient so this is a better is better than this one and so in particular this implies that along the heat flow along the heat flow the two gradients coincide just comparing the two expressions and knowing that one is larger than the other and then there is a more delicate argument that I will not explain which tells you that we can extend this property up to time zero here I just want to illustrate the idea which leads to the fact that along the heat flow the two objects coincide and then a refinement of these arguments which however is beyond the scope of these lectures gives you that really you can reach time zero and so now I am going to show you and basically here in the proof of this fact there are two tools let's say Eulerian tulagranja and the first tool is the so-called superposition principle and the second tool is an important lem which we borrowed from a paper of Kubada so we started to call because of its importance we started to call it Kubada lem before the break let's illustrate what is the superposition principle and then we make a break let's start as usual I will start from the classical case in Rn and then we are going to see the metric version of this fact so in Rn what is the idea you start from a solution to the continuity equation again remember for me t is always an index so I have a velocity field time and space I have a time dependent family of probability measures that impose some integrability condition on the velocity the intuitive meaning of the continuity equation is that really you are describing the action, the evolution of this density of particles under the velocity under these particles move under the action velocity field so how can we make this precise how we can go from this description which is Eulerian to Lagrangian description which is the one I was giving before in words well there is a theorem which surprisingly does not require well this theorem has a long history I mean the first version is due to a C Young then it was revisited in the context of science by Smirnov and in the context which is relevant for me on solutions to the continuity equation was revisited again in my book with Gillian Savare so I will give you our version and the statement is that given mu t with t as before you can always find probability measure maybe I will call it pi on ac2 such that first of all precisely if you look to et for all times this is precisely mu t and the second property the second property is that pi is concentrated precisely on solutions to the ordinary differential equations so you see that in this way we can really make rigorous this interpretation and also the converse is to any pi with these two properties of course is not difficult to show gives rise to a distributional solutions to the continuity equation ok I think maybe we can make a break here and after the break I will start with the metric version of this statement so let's see now what is the metric counterpart the metric counterpart of this result well if you are in a metric space actually here only the metric is involved not the measure in this statement so there is no reference measure what is the replacement of this condition well this is replaced by the notion of absolutely continuous vix so it will be a curve of measures with values so in p of x precisely absolutely continuous if you take the vastest distance of that video was illustrating this morning so this object plays a carefully the role of this object in our N and in fact in our book where we were dealing in our N we precise to prove this description and this description are perfectly equivalent in our N but the advantage of this description is of course that it makes sense in any metric space and then the state so we are not given a velocity field a priori but then the state 20 is exactly the same so exist pi which is concentrated is a probability measures on ac2 with the right margin we have to at least to mimic a little bit the counter part of statement 2 and what we can show is that the integral so you see the velocity of course plays the role of v because at least the metric velocity of course so this gives me a function of t and this function of t is precisely the metric derivative of this family of course so this is a curve in the space of measures absolutely continuous so it takes its metric derivative and the metric derivative is precise square is precise in the mean value of the speed of course at time t ok this statement which we call metric superposition principle is due to a PhD student of Savare Lisini it appears Don Calcavar remember well in 2009 ok so in particular ok let's say where we are going to apply the Lisini theorem and let me make the connection with the test plans what we are going to do next we are going to apply the Sinitir to these curve measures with f0 and we will see in a moment that this so where f is the solution to the heat remember that these properties are retained in particular because of the infinity bound and because of this condition p of pi equal to mu t so this will be a test plan and so for my function in W I know that I am going to control the oscillation of the function along pi almost very cool ok so this is the plan this is what we are going to do but why are we allowed to apply the Sinitir and to the solution to the heat equation the reason is that now it comes into play which is the second key connection between Eulerian and Lagrangian so under the previous notation mu t equal to f t kubala lemma says that mu t belongs to ac2 let's say 0 let's say plus infinity because the heat flow is defined for all time with values in p of x w2 and what is even more important is that I can bound the metric derivative of this score precisely with the star gradient so here you will find again you will find again the fissure information ok so now let me give you the proof of the kubala lemma ok the proof is based on the duality formula in optimal transportation I don't know if we already we already stated it or yes so did we do state the duality formula in optimal transportation ok so I can let me just rewrite it let me remind you and I will write the duality formula in a convenient way so the duality formula tells me that w2 square between mu and mu is the supreme ok let me use here c in the in the nu minus ok maybe yeah c in the nu minus the integral of phi in the mu where the supremum runs among all pairs let's say c and phi and the constraint is of course that c minus phi c or y minus phi of x should be smaller than the cost function which in my case is always the square of the distance ok this is the water ok so it will be convenient for me to write this formula only maximizing with respect to phi so if I only maximize with respect to phi you see here I have to put the largest possible phi compatible with phi what is the largest possible phi compatible with phi it will be ok maybe I have this here the maximal phi given phi well you immediately see ok let me sorry let me there is a one half somewhere yeah ok let me put in this way to be more convenient for me to write in this way ok given phi the maximal c is the infimum x of phi of x plus here and actually this is a particular case is the case t equal 1 of a famous formula which is called the opflux formula so or a famous semi group which is the opflux semi group which is typically denoted by qt so let me put again a time variable here because it will be important qt of phi for me will be the infimum on x of phi of x plus 1 over 2t the square xy so a time zero and a time zero so this is really a semi group and so a time zero this is the identity and this formula corresponds to q1 ok and now I have written I write this formula in this way I am maximizing only in one variable phi and the integral of q1 phi d nu minus the integral of phi d nu and then what I would like to do now is to use this formula to estimate the versus and distance ok so let me give you an estimate between let's say mu zero so let's take mu zero equal to f0 m mu1 equal to f1 m so this will be my mu nu will be f1 where ft as usual is the solution to the heat flow and let's see how we can get an estimate of the distance using the fissure information to get this estimate I need also an additional property of the Hopf-Lachs semi group he is called Hopf-Lachs semi group use the definition and where this was a property well known in our n but surprisingly remains basically in any metric space that this family of functions solves a first order PDE in a point wise sense and precisely is the Hamilton-Jakobi equation so here you have the slope the point wise lip sheets constant so this is true point wise and actually well there are many refinements of these results to say that also there is the equality for almost a very x and almost a very t but for the purpose of this proof I only need the point wise inequality if you look in our papers you will find a more precise statement about also the equality case so we only use the fact that the Hopf-Lachs formula is a sub solution to the metric Hamilton-Jakobi equation metric because it involves the slope ok and then you see I fix a function phi in lip 1 x in lip x and then I estimate q1 phi dm1 minus phi dm0 ok so he is tempting here to write this in this way here with q0 phi dm0 and to imagine that there is a time variable in between right this is why I wrote this in this this expression in this way so this is the integral d over dt integral between 0 and 1 of d over dt of the integral of qt phi dmt and now I can use on one side of course here there is a kind of library rule to be applied so I can use on this side the Hamilton-Jakobi sub solution on this side I will use of course the heat equation so let's write explicitly the derivatives so this is the integral between 0 and 1 of d over dt qt phi dmt ok maybe maybe I will start to use also ft I mean I will use also the densities plus the integral of qt phi delta ft dm so I simply applied library rule which of course can be justified in this case and now here I am going to apply the integration by parts so this will be equal so I am going to apply on one side the integration by parts and on the other side I am going to use the sub solution property so this is less than the integral between 0 and 1 of the integral of minus 1 half dqt phi square ft dm and here I integrate by parts and here I get plus ok let me group these two now together plus and here I get the gradient this integration by parts of course involves the trigger calculus so here I should use the star derivatives dm dqt now I know that the star derivative is smaller than the slope again because the slope is an upper gradient for ellipsis function the star derivative is smaller than the slope by definition so this inequality remains true if I put a star here because I am putting a smaller object with a negative sign and then I apply the young inequality and I get one half the integral between 0 and 1 of the integral between of the integral of so you see dqt phi disappears because I am going to apply the young inequality and I get the integral of dft star over ft and remember the expression we started with ellipsis function phi eventually it disappeared from the estimate and then I can take the supremum so by taking the supremum with respect to x gives that one half w2 square between 0 and mu1 is less than one half integral between 0 and 1 and this is almost exactly the kvada lemma up to a time localization it's a kind of integral version it's a kind of integral version of kvada lemma I can remove of course the 2 if you apply this kind of inequality not between time 0 and time 1 but between 2 arbitrary times remember that you have the same group property so you can start from any time and continue for an arbitrary time you can also rescale time this inequality basically up to rescaling gives you that the time derivative square is less than the Fisher information there is a star here this is the point wise version this is the integral version but one can go from one to the other via a time scaling most of everything so this gives me kvada lemma ok and now I can combine these 2 to get the sharp energy dissipation rate which will complete my discussion about the equivalence of sobole spaces I rewrite the energy dissipation rate again I want to estimate from above minus the derivative of ft log ft ft is the usual solution to the equation with one half of the star derivative plus one half of the weak derivative ok, and the idea of the proof is actually a refinement the idea comes from a refinement of the estimate on the slope of the entropy you can find for instance in villainy's book so really here many ideas come also the idea of using the entropy of course comes from optimal transport and tomorrow we will see why all these things are related not only to sobole spaces but also to slope of the entropy and other relations ok, here I just wanted to mention the origin of the idea ok, so let's say that we fix we fix the two times s less than t and I want to estimate the oscillation of the entropy between these two times so it is ft log ft dm minus fs log of fs ds you see the signs are inverted precisely because I want to estimate minus the derivative ok, so first of all there is a point wise convexity inequality so the point wise convexity inequality is that b log b minus a log a just by convexity or the log of this function can be estimated by b plus 1 log b plus 1 sorry log 1 plus log b times b minus a ok, it is just a point wise inequality which comes out of convexity that is a sub differential inequality actually that you can check you just integrate this inequality almost every x and you get that first of all you can bound this oscillation with plus log 1 plus log of ft times ft minus fs dm ok now if you want since all these functions have integrally equal to 1 we can immediately drop this constant 1 which plays no role I mean the cancels and now I start to use all the tools that I introduced before so first of all I can view ft and fs as the marginals of pi so take now pi given by listenity you mean in this point wise inequality? no, I think this is true for any A and B I think this is a point wise inequality of course A and B positive I think this is a general inequality ok I have no point wise information between the order ft of x and fs of x ok so now let me pi given by listenity again I can apply listenity because I have kvada lemma which tells me that this curve is absolutely continuous so I can view ft as the projection of some pi and I can write this integral as the integral of 1 plus log of f of gamma t sorry, log of ft I keep this as a constant function and here I put instead f of gamma t minus f of gamma s ok maybe the proof becomes more transparent if I call if I give a name to this function let me call phi this function because the time t will be fixed for me so this is the integral of phi and so this will become phi of gamma t minus phi of gamma s d pi of gamma ok I simply use the fact that the marginals of pi are given by ft and fs and you see now I am precisely in a situation where I can apply the theory of weak upper gradients because I know that my function phi belongs to the w sobole space by chain rule and by chain rule I know that d phi w the chain rule is true also for the w sobole spaces is equal to times the weak gradient and so I can now estimate this integral with the integral of the integral between s and t d phi w of gamma r dot gamma r d r d pi of gamma ok and now I apply the young inequality and then I am going to estimate separately in different ways the two terms which come out of the young inequality so let me first write the kinetic term so this will be less than one half the integral of the integral between s and t I skip a step and I also apply fubini plus one half also here I apply fubini after applying young and here I get the integral d r and now remember that this is the kind of mean kinetic energy which by fubini by lisini theorem this is equal so I continue with the inequality but here I just replace one half the integral between s and t of the metric derivative square here I apply lisini theorem remember that lisini theorem tells me that precisely that this object is the metric derivative and here maybe here I could start using the chain rule to simplify ok now here I apply the fact that the marginal set time r are given by f r so here I write in this way so this pi now will disappear s and t of the integral of d phi w square times f r dm dr ok, so you see now pi disappeared because I used information that er ds so pi is equal to f r ok, now I can continue one more step but we are almost done because now kuvada to get here the relaxed gradient r, so here I simply estimated the derivative with the expression with the Fisher information even by kuvada lem and here now I apply the chain rule so we get plus one half integral between s and t of the integral of the weak gradient of f t divided by f t squared because this is the derivative squared of phi times f r dm dr ok and now let us examine what happens when s converges to t so you remember that we wanted to estimate the derivative of this quantity so think that we are dividing now t minus s and we are taking limits if you divide by t minus s and you take limits here you get what you wanted so one half of the star gradient and here you get a slightly different expression because you get if you divide here you get something which goes like f t squared r but as s goes to t also r goes to t so this is again and this can be proved rigorously by kind of we convergence arguments you can prove that this goes exactly like f w squared over f t as s converges to t and so you get precisely one half so the fish information but written with the weak gradient and so you get precisely the sharp energy dissipation estimate that I wrote ok so more or less this completes the proof of all these connections between sobolev different definitions of sobolev spaces actually by product of this proof more or less with similar arguments ok let me introduce now also in preparation of tomorrow's lecture let me introduce also another notion of slope which will play a role in tomorrow's lecture which is the so called descending slope remember that if you have a function y if you have a metric space y d y and you have had a function d g from y into r we define the slope of g at y as the lean soup as z goes to y of the modulus of g of z minus g of y divided by d z y in the theory of gradient flow so also in the theory I illustrated for maximal monoton operators in effects lecture there is another notion of slope which is more convenient which is the one where you take here the negative part so I will denote this by d minus it is the so called descending slope so this only measures how much you are able to decrease the value of your function so you do not care of how much you can increase it so it is simply again the lean soup as z goes to y but here I take the negative part which is not negative but according to the standard convention so so in particular if you have for instance the minimum point of g no matter if for instance if you have a corner that minimum points is equal to zero because you can't decrease even more your functional and this is particularly appropriate as we will see tomorrow in the theory of gradient flows also from the point of view the classical theory for instance of the Brezzese theory the Brezzese theory that I illustrated in the first lecture for instance because one can show that for a convex lower semi continuous functional the minimum norm speaking about convex lower semi continuous functional in Hilbert spaces and if you take the element with minimal norm which we call the gradient by definition this was called the gradient of phi this can also be shown to be equivalent to the not to the slope of the functional but to the descending slope of the function ok and now since tomorrow we are going to speak about the gradient flows also in metric spaces of course we can't use the sub differential we can't use scalar products and so on but this will be a good replacement this purely metric version will be relevant in the theory of gradient flows and it is perfectly consistent with the classical picture and so as I was saying by product of the proof I gave you is that we have a precise relation between Fisher information and slope of the entropy and the relation is that the Fisher information of course here now we can put star w because we know now that these two objects are the same that you want so the Fisher information is always smaller than the slope of the entropy in tomorrow's lecture where we will talk a little bit about the gradient flows and the CDKN theory we will examine more closely this connection of course without any other assumptions we don't expect equality here remember that we saw this kind of pathological examples where all these gradients trivialize so they are identical equal to zero so we cannot expect equality here in general but we will see that under curvature conditions this is true these two objects are the same and again you see this is an important connection between a Lagrangian point of view and the Eulerian point of view which is inside here actually here I should specify of course this is a functional in p of x and so when you are going to compute in slope you need to use the vastest and distance so here we are really taking as a functional gd entropy as a space y precisely the space of probability measures within the vastest and distance ok and I think maybe today I will stop just a few minutes earlier and tomorrow we will resume from a completely from a new subject