the proud of Dustin Hardy, Hal Silver, the Lone Ranger. His faithful Indian companion, Toto, the daring and resourceful mask rider of the planes, led the fight for law and order in the early western United States. Nowhere in the pages of history can one find a greater champion of justice. Return with us now to those thrilling days of yesteryear. Mile of the past, and the thundering hoofbeats of the great horse, Silver. The Lone Ranger rides again. Oh, Silver, the storming foe! A deputy came out of the sheriff's office with a hammer, a few horseshoe nails, and a new handbill. Hey, lefty, give me a hand with this. Got a nailer up where everyone can see it. Who's the critter this wanted? Name's Branigan. Now, just hold this in place while I put a couple of nails in. This gives a pretty fair description of the critter. Yep. Wanted for a string of robberies. Don't say nothing about murders. I guess the critter steered clear of hanging crimes. With this description, it won't be hard to recognize Branigan if he comes this way. Sheriff Comstock figures he'll pass close to this town and his way to the border. Is he'll head for the border, huh? Yep. I guess that'll hold her in place. See, look over there. Express office. There he goes, running for a horse. Here comes the sheriff on the dead run. Lefty, get the horses. Right. Express office has been robbed. We saw the critter right away. It's Branigan. Great day. Get them horses.  It won't take time to square each of you as a deputy. You gotta catch Branigan before he can cross the border into Mexico. Lefty, you take five men and ride direct to the border. Right.  The rest of you men will come with me. We'll follow the trail. Come on. You'll find this way. Right. Good. Get up there.   Between the town and the border, an Indian had dismounted near a waterhole. Heard hoofs approaching and looked up as Branigan came close. Hi there, engine. Oh, you ride plenty fast. Yeah, and if I'm seen, I'll be shot on sight. I'm taking your jacket in place of my coat. Huh? Are you outlaw? Yeah, I got no time to argue. If I can get about one minute more. The lawmen were not far behind the outlaw Branigan. The sheriff and those with him rode from one direction, while lefty and the others came from the south. Between the two, a man lay on the ground. There he is, boys. He stretched out on the ground. It's a trick. You sure that's Branigan? Look at the coat. That's the bright tan when ranch told about. There comes lefty from the other side. Good. Look out for tricks. You may be playing possum. Keep your guns ready. Lefty, every boys keep their guns on Branigan. He's playing some sort of a game. Now come on, Branigan. Stand up and stop playing possum.  We know you ain't asleep. Wake up. Stand on your feet. Hey, that ain't Branigan. It's an Indian. He's hurt. What's the matter, Indian? Who's got a canteen hand here? Here's one. Here, take a drink, Indian. Steady now, Indian. Me. Me tanto. Now, take it easy, fella. We can savvy. Branigan got you and changed clothes. Is that what happened? That's right. No, wait a minute. That is all the blazes. I seen what I thought was a red skin. He was riding south. For the border? Yeah. It must have been Branigan. We can't touch him in Mexico. Me. Sorry. If it wasn't for your clothes, he'd never got away. Well, hold on, Steve. Take it easy. But this Indian here... It's Banshan Indian, but it ain't his fault. It's our own fault for not capturing Branigan sooner. You... You see, fella? I see him all right. Where an Indian's clothes. Him ride horse? Yeah. It paint horse? Oh, it was a chestnut. Oh, that good. Why is it good? For him not get scout. Who? Scout. Horse, a tanto. Maybe tanto. Maybe scout fetch a friend here. Look here, Tondo, or whatever your name is. I'm Sheriff Comstock. I'm the man who's to blame for that killer being free today. It's my job to run him down and I failed at it. I have heard of an Indian named Tondo, and he said to write a paint called Scout. That right? What's more, this Indian I speak of is supposed to have a friend. That right? Maybe I'm wrong, maybe I ain't. Anyhow, I'm going to tell you something. If this paint horse brings a friend you mention, then by any chance that friend is masked. Huh? What are you driving, Ed Comstock? What do you mean, a masked friend of this engine? Boys, I'm talking to Tondo. As I was saying, Tondo, you might mention to your friend that Branigan has crossed the border and there ain't no way I can get him now. I'd be downright grateful for some help. Savvy? Ah, Tonto Savvy. Thanks. Is there anything you need now? No. All right, boys, we'll head back for town. Ain't you going to follow the trail no more? Maybe we can catch him. You giving up, Sheriff? I says we're heading back. Ain't no sense in riding farther south. Come on now. Get up there. The lone ranger waiting in camp for Tonto was surprised when Scout came in without a rider. He soon realized that something had happened to his friend. Here's Silver. It was the worker but a few moments to saddle up. Then the masked man set out to follow Scout's back trail. Let's go, Scouts. One, Silver.  You come and bring Scout to the big fella. Tondo, what happened to you? Wait here. We know you come soon. Those clothes. A fella catch Tonto off guard. You're hurt. It, it's not much. Lawman come here. Him say, hope you help. Lawman? Ah, Sheriff named Cumbstock. Cumbstock? He's a good Sheriff.  No. No, it, it's fella named Branigan. Branigan? Is he around here? No. Him skate to border. Him wear Tonto's clothes. He gets across a Rio Grande. The law can't touch him. That's what Sheriff say. Sheriff, hope you help. I see.  Well, how about you, Tonto? You think you can follow his trail? Ah, me be all right. Let me look at that wound and see how badly you hurt. Not bad. We can pick up Branigan's trail south of the Rio Grande. Ah, me find it. Wounds not very deep. We go now. Find trail, huh? Sheriff got plenty faith than you. All right, Tonto. We'll try to earn a good lawman's confidence today. Look, he's not steady.  Oh, Tonto! The masked man and Tonto followed the trail to the Rio Grande where it disappeared in shallow water. On the southern shore of the Muddy River, Branigan made no effort to conceal his tracks. Him go shore right here. Come on, Tonto.  Right up, Tonto. Oh, Tonto. Come on, brother. Come on, brother. Cray him easy from now on. It's a good thing, Tonto. He goes from here on. I'll have to go without your help. Why do you say that? Tonto, I've been thinking of Branigan. As long as he stays on this side of the river, he can't be touched by the law in Texas. That's right. He mustn't know that I found you in his clothes. Why? I can't go into it now, Kimosabe. I've got to shove on before it gets dark. Oh. I want you to find Shadow Comstock. Tonto turned back toward Texas, while the lone ranger continued on the outlaw's trail. He saw that the trail led directly to a small doby house. Here, he rained up. Oh, boy. Listen to me. Is that a big fella? Wonder if a bluff will work with a man like Branigan. I see. Hello there. Coming in. Hold it. You're covered. It's all right. Tonto, who said that? I said it. After the bright sun I can hardly see in this doby. Where are you? Right here. You're covered. Huh? Well, I thought you were Tonto. You're injustice, he was. Yeah? Those are his clothes. What about it? Where'd you get them? Where did you get that mask you were wearing? At least I didn't steal it from an Indian. Where's the Indian who was to meet me here? Look here, mister, before we talk anymore, I'll take those guns you're packing. Before we talk anymore, you'll put away the gun you're holding on me. That's big talk. Especially for the man that's under gun sights. Hand over those guns. I'd be more willing to talk if you'd hand over yours. Oh, you would, eh? Or, uh, must I take it away from you? You might have lived and, stranger, just you try it. Very well. Oh, boy. Ah, that hurt, didn't it? You, you... The bullet probably wrecked your gun. You should have done as I asked. I never saw anything like a draw of yours. Oh, we'll talk things over. Shall we sit down? Hand hurts like fury. How about my Indian friend, Bursey? You know, we're around here. Don't tell me that. We had a big deal lined up. I was to meet him. Well, he, uh, that is... Where'd you leave him? I, uh... Answer me. In Texas, north of the border. Oh. You left him in Texas, huh? I, uh, suppose you want me to go and find him. Well, I... That's too thin. The Texas law can act in this country.  You tricked me into crossing the border. Is that it? No. No lie to me. This is the Texas lawman sent you down here to meet me. None of them. How did you find out about the plans I had? See here. If you're dodging the Texas law, maybe we can get together. You, Tin Horn, go on back and tell the law that your scheme didn't work. Tell them that they can sit on the north shore of the Rio Grande until they grow whiskers a yard long. They won't get me. Oh, wait a minute. I ain't working with no lawman. My name's Branigan. You must have heard of me. Why should anyone have heard of you? Well, I'm on the dodge, my own self. Save your life for someone else. It's the truth. Now, why don't the two of us team up? If you've got something good in the fire, I can do anything that Indian Pal of yours can do. Big talk. I can. Just try me out. I can't handle a gun as fast as you did, but I'm no slouch. Well, you know about silver mines. Silver mines? I'm not interested in two-bit plays like sticking up a stagecoach or robbing an express office. I'm after something big. Listen, I can help. I know I can.  Oh, Mexicans. The mines on this side of the border. See that swell? That's first rate. Tell me more about it. I asked if you could boss a bunch of braids. Sure I can. I wonder. How much is this thing good for? How much?  The curtain falls on the first act of our Lone Ranger story. Before the next exciting scenes, please permit us to pause for just a few moments. Now to continue our story. Sheriff Comstock sat in his office with the deputy call left here, Steve came in with word of the man who'd worked in the express office. Well, here he's now, Lefty. Steve! Well, I just come from the Sawbones. Said the poor jant that Branigan clubbed will be as good as new in a few days. Well, it's not Branigan's fault that he's not dead. There was no call for him to knock him down after he got the money. Branigan must like to crack men on the head. He did the same thing with that engine. Sheriff, speaking of heads, folks in town that are going to be howling for your scalp if you don't do something. Do something? Well, Branigan, everyone says you're laying down on the job. Can't touch him while he's across the border. Any fool knows that. Sheriff Comstock, I got a few comments for himself. What you hear, Lefty? Some of the men are grumbling about the way you gave up the trail. Grumbling, are they? Well, let's have it, Lefty. Well, they figure we should have followed Branigan to the border. For all we know, we may still be on this side. Not Branigan. The dock figures you should have arrested the Indian as an accessory to his escape. Why? Because the poor critter got wrapped on the head by Branigan? Well, if the crook hadn't been wearing the engine's jacket, I wouldn't have let him get past me. So that's how the dock figures, eh? Yeah. If that's his line of reasoning, this town's mighty lucky that the dock ain't a lawman. Why? Now, listen, you two. Let people talk all they want. They don't know who that redskin is. He said his name was Tonto. Yeah. And you heard what I told him, didn't you? About taking word to some friend. That's right. Look out that window. What? Ain't that Tonto raining up outside? Well, sure enough. So it is. Boys, you just sit tight and see if the sea dive planner don't begin to show sproff. Hi there, Tonto. Oh. I figured you'd be coming here sooner or later. Did that friend find you? Uh, him find Tonto. You passed my message on to him? That's right. Him ride south. Him cross Rio Grande. Yeah? Did he send any word to me? Ah, him say, come here, see you. Ask you to help Tonto. Good. Help you? How? We need plenty men. Maybe 10, 12 men. Count on him. What else do you need? Get plenty shovel. Get horse. And come dig and ground. All right. You hear that, boys?  Dig what? Why? Never mind asking questions. Go get spades and start to dig wherever Tonto shows us. Taking orders from an engine. I don't like it. Stop your grumbling and do what you're told. Any man that don't like it can hand in his badge right now. Well, Sheriff, an engine given orders. I'm giving the orders, not the engine. What's more, maybe you don't know who that engine is. Unless I am mighty mistaken, what he says comes from the Lone Ranger. The Lone Ranger? Great scum. Mask rider. The one to say so in the first place. Leap and hand. Now get spades and shovels. I'll get more men. I'll get horses hit stuff. Anything the Lone Ranger says is right with me. Spades and shovels were soon loaded on wagons and the men followed Tonto out of town. It was a dry season and the real Grand was in many places nearly dry. Tonto led the way to a place where a mere trickle of shallow muddy water marked the boundary line between two nations. Here, under the Indian's direction, the lawman began to dig. I never heard of making a river running a new channel. He got a paw for a lot of digging to do. If you think you're working hard, look at the way that engine's making the dirt fly. Hey, Tonto, what's the bag of all this digging? What's the plan? Make river run through gully.  The Arroyo is where the river flowed a good many years ago. When we're done, it'll flow there again. Then what? Then the river won't flow along the border. It'll be north of the borderline. Branigan is to be brought near here by that man. He'll figure he's safe as long as he stays south of the river. But the truth is, he'll be on Texas ground. I sure hope it works out that way. If you're mighty surprised, don't break. Now bend to them shovels. We've got to get things ready. How's the man plan to get Branigan here? Leave that to the Lone Ranger. A Lone Ranger had convinced Branigan that it'd be wise to wait until darkness fell before leaving the small adobe house. It was a long, dragging afternoon with little conversation. We're to be partners. I don't see why you can't let me have one of your guns. I don't trust you that far, Branigan. But you smashed my gun with a bullet. You shouldn't have gone on me. That was before you talked about a silver mine. We weren't partners then. You sure you can find this place in the dark? Oh, I'll find the place all right. Ah, still a couple of hours to wait. Darkness was an hour away when Branigan looked out the window and saw two people. Those two are pals of mine. They are? Yeah, Moose Lacey and Breed Cortez. Are they wanted in the States? Yeah, same as I am. They're the ones who told me about this vacant house. Makes a first-rate hideout. They've seen our horses in the shed.  All right, boys. Come on in. I got someone you should go to. What's in the bag? I've got a bag of semi-glow. Yeah, but be cool, too. The only trouble is we have to kill two men. Who's that? Mess. Ah, take it easy, boys. He's all right. Yeah. He's letting me in on something big. Why don't you take the mask off? That's my business. Get over there, Breed, on the far side of the room. See? That's it. Well, he can't cover up. Don't count on that. Now, hold on, boys. There's no need to be suspicious of each other. We're all in the same boat. Oh, no, we're not, Rennegan. These friends of yours are wanted for murder on the side of the border. We're caught with them. We'll hang with them. So you don't like our company, eh? No. The Mexican authorities can follow your tracks. I'll like it even less. Now, wait, pal. We can let them know. Silver mine? What silver mine? Most I didn't make no argument about the mask because this gent's gonna let me take the place of his partner who didn't show up. He's gonna take me to a place where there's silver. Oh, he is, eh? I didn't promise that. Oh, don't hold out on my friends. Is, uh, this place near the border? Yeah, he said it was. Why? Rennegan, let me tell you something. While me and the breeder on the way here, we pass close to a place where the Rio Grande is mighty narrow and shallow. Well? What do you think we've seen? Well, how would I know? There was a dozen men on the Texas side. They was digging like fury. They was digging to change the course of the Rio. So as they'd have run in an old bed some distance north of the border. What about it? Those men were lawmen. I recognized the sheriff and some of his deputies. Now, let me tell you something else. Yeah? I once had a pal that thought he was safe over here and makes it calm. But he met up with a man that was gonna do him a favor. You see him? This man took him close to the Rio Grande. My pal figured to be safe as long as he stayed south of the Rio. But that's where he got fooled. The river was diverted to a new bed that Texas law nailed him while he was still south of the Rio Grande. Hey, mask man.  I'll tell you some more. The man that tricked my friend was tall. He wore two guns and a mask. Caramba, he did it. That smoy, he rode a white horse just like the one I said. Why are you... The horse was called Silver. That's what you call your horse. The man was called the Lone Ranger. Hold it. Looks like a stalemate. Doesn't it, Moose? Yeah. We got to drop on each other but breach on the other side of the room. What are you gonna do about him? I have the gun on you, senor. Watch him. You might get one of us. But the other will get you. Oh, it was all a trick, huh? You were going to take me over to the border instead of to a silver mine. You'd be a lot better off, Bennigan. Texas, you're wanted for robbery. That means jail. Yeah. If you're caught with these friends of yours on this side of the border, you'll face a hangman, a firing squad. I wasn't with them on the killing. You're with them now. You'll be caught with them. You'll die with them. Never mind the talk. Take his guns, Brady, and me. Don't try it. The law's already on the way here. Try more tricks, eh? No, Moose. No tricks. You made the mistake that criminals often make. You underestimated the lawman. They have followed your tracks. That is true. I can see them through the window. He's right. They are coming here. Your local fool you left tracks they could follow. That is true. That's it. Oh! My arm! You too. No! I'll show you. Good. The law is coming. They're nearly here. They'll get us. I fix you for that. I fix you. I cut you to pieces. No, you don't. I get my hand free. I fix you. Better give you a lullaby. The lawman can bring those two back to consciousness. How about you, Branigan? No, no. Don't hit me. Don't give me what you gave those two. Let me out of here. The law will get me. It sure will. I'm here with a stolen money. Those saddlebags. They'll hang me. I'm not a killer. I never killed anyone. Would you like to cross the border? Yes, yes. I don't want to die here. Give me the law in Texas. Come on then. This way. They'll get us. They'll come with me. They're shooting. Come on, Branigan. We're going to Texas. Come on.   You mean to say we dig all this digging for nothing? Branigan crossed with his own will. I've never seen the like of it. Hey, boys. Sheriff Consock says that we can pack the spades and go back to town. Branigan has come back to town. Branigan has confessed everything. And he crossed the border of his own free will and without being dragged across? Well, he sure crossed of his own free will like a dozen demons was on his tail. Well, I... But I reckon the masked man did a little persuading. I assume it's home. What you have just heard is a copyrighted feature of the Lone Ranger Incorporated.


What I'm actually doing here, I can't remember, oh yeah. Haha! Getting on with the campaign! Hold up, what's that there? The Gathering Legion, Legion event starts in 2 minutes 40 seconds. Let's go! What is this? I've actually read about this already, but I've never seen one or caught one in time, so that's what we're doing now! Gimme that goodches! Gimme that goodches! Gimme that goodches! Gimme that goodches! I mean like I've played this now, level 35. I mean I did see my horse at one stage. Haven't been back for him though. I saw my man at one stage somewhere early on. But I haven't been back for him so yeah, I do need to do that I think. I do need to do that. Oh my days, I've been slowed. I'm gonna miss this on the way out. Get out my way Beast! Ho ho ho! Up a second! What do we have here? Ho ho ho! I'm getting battered. I'm getting battered. No, leave me alone Beast, I need this event. Oh whoa!  Just leave me alone man. Jesus. Thing is just whirled. Come on, I gotta be close. I'm gonna miss this thing. Oh, there's a horse rider! Yay! We are there! Alright, has it started? Oh no. We've reached a destination. What's going on here? Oh it begins. I just saw that there. Right inside. Just saw that there. We have 30 seconds. Get out of my camp. Who said hello then?  Oh there he is on the screen. Volko says hello so it must have been another player. I suppose the dude I just killed. Oh we're back to start. We're back to start. Okay. I'm so glad this people here. Whoa he just got mini-mouted. I mean, I'm not sure I can see what's going down. Yeah this is wild man. It reminds me of my New World days. Those massive events and that were amazing. I mean, that can't be it. Surely. Oh yes, look at your map David. Look at your map. Look at your map when you see those red dots. That's where the enemy is. Okay, okay, okay. This is quite cool though. I love how it's like you come to these events and there's lots of people here. Sort of like a public event and destiny. Pretty cool.  I'm just literally just fine. I'm just fine. I'm not even sure what I'm fighting here. These guys that are here are just absolutely obliterating stuff. I'm glad to be a part of it. I appreciate the carry. Oh dear. Things are just getting annihilated under someone else here. Playing as that natural monster. The guy's got style. The guy has style. Oh my silly days. I'm just taking a backseat here. I ain't getting involved unless I have to. These guys are going down. What this guy has. I can't even tell how many people is here. I'm not going to lie. It's like three or four maybe. You can't damage him can you? Can we now? Oh I think we can now. Use my ultimate on him. Send my big old goal I mean. Why this thing goes on quite a bit to be honest. I will just take the loot. I ain't got to do much work. That's where we got that goal. That's where we'll go people. I think my army is doing well though because they're all getting bowed. I can see that. I can definitely see that. That's what I'm making of me.   That's what you gotta do. Get him goal him. Do him over. Do that thing over. Where's he running to? He ain't going nowhere. He just... He's getting his help. I don't even know what happened. I must have blinked. Are you down now B.H? Yes I know I know. Oh I'm tired you know. This game is taking the army man. I mean I'm getting no sleep because of this thing. Is that it now then? Do we go and get our loot now? No, no, no. Get him dead.  I don't know what we're doing.  Shoot him from afar. This is what we do. Let's start his house. See my minions air in the fight. A little bit of health left. I think I'm going to stay in there. I don't want to be in that. I don't want to be in that. I don't want to be in that. I don't want to be in that. This lady's dude.    There we go. I got last hit. Pretty loot. Pretty loot. Legion event completed. They gathered legions. Quite cool. Quite cool. I enjoyed it. I enjoyed that shiz. I enjoyed that shiz. Greetings. Let's actually have a look at that loot. I'm in this garbage. That event gave me nothing. That event gave me nothing. Now my day is just my luck guys. Just my luck. But I'll stop making this into a video. Hope you guys enjoyed. If you did leave a like really helps. And if you like what you see and want to see more Diablo 4. Be sure to subscribe and hopefully guys. I will see you on that next one.


Upgrade your smart phone with Appalachian Wireless. Get $300 off selected models. Capture memories. Stream and stay connected. Put the latest technology. Visit one of our stores or AppalachianWireless.com right now. Appalachian Citizens Law Center will be hosting a black lung town hall Monday, July 24th at Letcher County Central. The meeting is open to minors and their families and will feature speakers from the National Institute of Occupational Safety and local researchers. What we've noticed in the last couple of years is, you know, looking back five years ago is that a lot of minors were coming out with this complicated black lung, which is essentially saying they were a lot sicker than we'd ever seen before. And so we thought it was really important to bring folks out and to really express what was going on in the community because I feel like sometimes this isn't often talked about. And then fast forwarding to today, we'd actually tried to do something like this last year and obviously the flood put a lot of plans on hold. And so we're really excited that we were able to host NIOSH, the National Institute of Occupational Safety and Health, and then also Dr. Krum, who's local at a high level, and just bring an opportunity to bring the community together to hear from these experts that oftentimes we are able to hear from, but the public isn't always access to. One in five veteran coal miners in Appalachia suffer from black lung. While the disease is commonly known to affect older generations, it is now affecting minors in their 30s. Central Appalachia specifically is seeing this very complicated black lung in very young minors. And the reason why I think it's so important for folks to know about that is because it's not just coal dust that's impacting them now. It's really silica dust. Obviously, IMSA just recently brought out a new silica rule, but as of right now, even with that rule being presented, minors are exposed to more silica dust than any other occupation in the United States. But it's very hard to see someone who's in their late 30s, early 40s, now with a disabling disease that will never be cured. And so we really think it's important to take a moment to just say that we're going to give you the information you need and what is your next step from here. So NIOSH will also be bringing, they'll be presenting opportunities for folks to get screened in the next couple of months after the public town hall. So that'll be a really great opportunity for folks to come out as well. The meeting will be held from 4 to 6 p.m. with a reception to follow. For more information, contact Appalachian Citizens Law Center at<PHONE_NUMBER>. Reporting for Mountain Top News, I'm Brianna Robinson.


What's up guys vibing in the closet again, so it actually has the best acoustics. It just has terrible optics I wanted to just make a little announcement and this is not a contentful video So don't complain, but I want to make an announcement. I am giving up Absolutely giving up. I've been totally defeated. I am totally giving up my war against bloat. Okay? I'm not talking about I'm not talking about software bloat. I just want to be clear I'm talking about the word bloat because if you're on in the know on this channel, you know that for months Literally months.  No one sees it Okay, so Why did I do this? firstly, I Hold, you know, I I'm a I'm the kind of guy. I like fresh memes. Okay. I don't like stale memes I don't like people saying the same stuff over and over and over again. Okay. I Now don't get me wrong when a meme is good I will use it till it ends there I'm still putting soy jacks in every other video thumbnail because I I think I can still milk it some more But eventually there comes a point where a meme is just dead You just can't do anything with it and bloat it happened a wall It just happened a long time ago. Frankly. That that's what I think. Okay, because Every single comment on my channel would be this is bloat. That's bloated bloat bloat bloat bloat bloat And I just wish I just wish people had more bloated vocabularies I wish they could express themselves in different ways and that's that's what was going through my mind when I first Banned these words. Okay. I was like, oh, well, I'm just gonna ban these words and I'm sure I'll just go a week and Deprived of getting up comies for saying the same thing over and over again people will just you know, like Pavlovian dogs Deprived of reinforcement see saying these things. Okay, that's not actually how Pavlov's dogs work, but you know, whatever Okay, if if people don't get incentives for saying it all the time getting likes getting comments if they don't get that They'll stop saying it and I can unblock it But it went on for months. Okay, I looked up my spam filter today And I realized I need to give up because people have I mean, they were literally Thousands of comments like entire videos worth of comments that have been blocked because of this and I realize it obviously is not Change it. It's it's not making people more creative. It's not making them cleverer Okay, and honestly, I guess it is pretty libtarded to like ban a word that triggers me Doesn't really trigger me. Again. I just want I just wanted to voice it in the thing that people say. Okay, that's it But so I figured okay, whatever. I'm just gonna unban these words people if people want to call everything bloat I'm sure they will I'm sure people are calling things in this video bloat.  I'm so clever for this. I bet that that is have I It's almost that's what annoys me about it. I can always see it a million miles away.  They're actually both educational you should check them out Maybe I'll if I remember I'll put links to them in the video description or something like that But one is called GNU is bloated and one is called bash is bloated. They're both very good videos They explain for example the features that bash added on to the POSIX compliance shell to you know That might confuse you so you should check out bash is bloated for that Or if you want to learn about the added features to the GNU core utils that might mess up scripts that you write If you want them to run functionally on Mac OS or BSD you might want to check out GNU is bloated They're both very good videos but I feel since then especially the meme took off on my channel and Now I go to other channels and it's all bloat bloat bloat bloat bloat bloat bloat And I don't know I guess I guess I can't I guess I might as well be on the right side of history You know, I guess there's just nothing so it just must be a solid meme I don't know. Maybe I just need to learn to love it or something like that But that's it. In fact, you know, I'm now I I already told people I'll probably do a video on Cody Which is an extremely bloated program. I'll probably record that tomorrow then I'll do a video on no JS I think that'll be next and then I'll make a video on my top five favorite electron apps And then I'll do I don't know what what else I'm not I'm not even I'm sure there are people watching this like wow I can't wait not knowing that it's it's a joke. I just I just want that to be clear. Okay, okay? We don't you we don't use anything related to electron on this channel. Okay, but the Cody one isn't a joke I'll probably actually do I actually do it's it's nice for normies. Okay, whatever Yeah, but that's about it so I give up I give up the war on bloat so I don't have anything else funny to say goodbye


So I looked on Wikipedia, apparently. Fraunhofer was a German physicist. But he died in the 1820s. And he was a glass maker. So he was making lenses. And he started, and he worked in spectroscopy basically. And so the formula, I've heard of it as the Fraunhofer formula. But based on Google, it's not so clear exactly I mean, it is called this way. At least by some people, I don't know exactly why. And I thought he derived it for the Schrodinger equation. Now it has to do with, well, the wavelength in diffraction. So it's not completely out of topic either. One other remark about, so I mentioned the results on bounds on HS norms for the cubic analysis on T and R. And I mentioned, so my references were maybe not quite right. At least I forgot the work by Capellar and Greber, who also worked on that. And finally, so this is a remark that Professor Zetkov made to me yesterday that you can, in fact, think of, let's say, the cubic analysis on this space. And at least I would understand it in the following way. So these are essentially functions which are compactly supported in the first direction. These are essentially functions which are compactly supported in the second direction. And these are essentially functions which are compactly supported. So if you go far away in infinity in the y direction, then this one stands alone. And the same thing holds for that one. So what you could do is to evolve this one and that one independently through your equation. And then you plug in the equation that this gives you for that one. And it gives you an interesting non-linear equation in that space. So then you can really start to study it. But I not really realized this thing that you get essentially that these two are independent. And they just come in as some form of potential in the non-linearity for this one. And I thought it was something nice because or at least a connection is that it's connected to some outstanding problem which is the stability of the two lines solid on 4K P2. And so if you have nothing to do, sometime today you can go on YouTube and you can go look at the lecture of, or a lecture of Professor Mizumachi in one of the previous workshops. So he didn't quite talk about the stability of the two line solid tons, but he has some nice pictures about them in his first slides. So just for motivation. Okay, well maybe I can keep this. So now the goal for today is to start entering a bit more in the study of the Schrodinger equation on our space. And so last time we saw some basic facts about how to understand the linear flow at least on RD. And now today we're going to get, well we'll continue with linear estimates, but this time it will be strict arts estimates. And so which are the things, the basic tools that we're going to use to extend our solutions. And so this is, now I forgot, I think 2B. And what I would like to present is, well I'll give you two estimates about linear solutions. Now to state them precisely, we need to restrict two frequencies and now you have two notions, you have two natural derivatives, so derivatives in the x direction, the derivatives in the y directions. And they play somewhat different role and it would be especially important for later. So I'll tell you what it is later, but this is basically the little bit projector on frequency capital X in the direction X on frequency capital Y in the direction Y. And so the claim is that if you only integrate on an interval of time of size one, then you can have pretty nice estimates. And you can have, depending on the regularity either in X or in Y. So you can choose whatever you want. Now just a little remark about this. So first, at least this one is the best that you can do in the sense that this should hold for very concentrated solution. And so in particular they should imply the one on R3 and on R3 you should lose one quarter of a derivative. Sorry, what is this few capital X? Yeah, that's right, no, you're right. So it is just something that restricts your function to the frequencies that are of size capital X in the X dual, in the dual of R and capital Y in the dual of T2 and K. And this is essentially a function which is essentially one between, if this is of size between one and two and then zero, if this is too far from that. So essentially it forces C to be of size capital X, K to be of size capital Y, and so this is essentially like one quarter of a derivative in the Y direction. And so here this is not quite optimal in the sense that the total number of derivative is one quarter plus epsilon, but at least you need to pay A price in Y. So maybe you could make it one quarter minus epsilon that I don't know, but you cannot completely avoid the loss in Y. Okay, so essentially we'll prove the first one and the second one is not so different. So, okay. And for the first one, because essentially I don't want to work with this projector, we're going to prove something slightly stronger that implies that, which is that if you forget one dimension, then you have it with no loss. And if this is true, then just by taking out the frequencies in the G2 direction, you get the other one for free. Now this estimating dates back to work by Takaoka and Zetkov. And I would say it's fantastic because it's something really striking that you can estimate this norm, the L4 norm, which you could precisely estimate by the L2 norm in R2. And here you have the exact same exponent on something which is much less dispersive than R2. So, well, essentially the first time I saw it, I found it hard to believe. So this is why I think we're going to prove this one, but just by changing a bit the proof, you could get this one in a similar way. Now, on the other hand once you believe it, it's not very difficult. So, let's do it. So, and what we're going to use is the fact that the L4 norm is just the L2 norm of the square. So now let's look not exactly at the square of that, but at the product of two free evolutions and because, well, we'll see why this is good later. So, it suffices to consider, or maybe let's consider, we're going to take the product of two solutions, and maybe let me call them psi and phi, and then estimate it by duality, by multiplying by some function in L2. And so, the idea is that this will be in L2, this should be in L2. Now, let's compute that. So, remember that now we're on R cross T. So, and to understand, well, the free evolutions, this is something much easier to understand in the Fourier space, so let's write it as the Fourier transform of its Fourier transform. So, this one is going to be psi hat K of Xie. So now I have E and E to the I T Xie. So, I have C squared plus K squared, well, in fact, that's just one dimensional. Well, I'm sorry, maybe to fit it in one line out, write down the phases all together, and then the amplitude. So, I have C squared plus K squared, and then somehow here I have E I Xie Xie I K Y, and then C hat of K Xie. Now, this is this one. I'll have to do it with the other one, E to the I eta X, E to the I P Y, C hat P of eta, H of X, Y, T, D X, D Y, D T, D X, D E eta, and I have forgotten to integrate. All right, so now that you do that, you can see that, in fact, you could integrate in X, Y and T and just get the Fourier transform of your function H. So, in fact, you simplify that to something like that of phi K of Xie phi P hat of eta, and then H is Fourier transform in all of the variables. So, in X, I have Xie plus eta in Y, I have K plus P, and then the trace of the evolution is in the Fourier transform in time, which is Xie squared plus eta squared plus K squared plus P squared, D Xie, D eta. Okay, so this is really just Parseval, or Planchevel, here. But now you see that if this is a function in L2, this is a function in L2, their variables are completely independent and I would like to be able to choose any of them. Then I can probably do nothing better than evaluate that in L2. And so now what I really have to do is estimate the norm of this one in L2. Now, this is a little problematic at first sight because it's a function that is in L2. That means that it can, and it's in L2 of three variables, X, Y, and T. And so once I fill in all of these entries, I can integrate over three guys, but now I have a fourth integral. So there has to be something in this change of variable that cancels one of the variables. All right, so suffices to show. But so now you see it's really something in some sense about the Jacobian estimate because for any L2 function, I have to estimate this by its L2 norm. So that this thing is smaller than H in L2 of X, Y, and T. And here is... So in this generality, all right, here is where I have to remember one thing, which is that I'm not integrating over all time, but I'm integrating only over a bounded interval and how am I going to use that? Well, I'm going to use that in the sense that if I... So instead of integrating over an interval of size one, I could integrate over an interval of size epsilon for some fixed epsilon. And as a result, this, the Fourier transform in time, is going to be of slow variation. And this is what I have to use. So for all H, so this is... Okay, so well, let's do it on the other hand. Well, we'll have to choose this one as one of the variables, this one as one of the variables, and then make do with what is left. So we're going to call this one sigma and this one m. And we can take this out because those are the ones that are going to be nicely integrated directly. And then we have the left over. So sum over k integral over c of h hat of sigma m.  And there is... I don't know, I guess it's okay. Where omega is... Well, let's see. So I have c and then I have sigma minus c. So I'm going to have 2 c squared minus c sigma plus sigma squared. And then the same thing. I have k and I have m minus k.  And so now we have seen that the problem is that we have two integrals essentially for one variable. And the way out of this is going to be to add one more sum to that. So sum over m integral over sigma of sum over l sum over k integral over c. Because what I want to use is the fact that this might be complicated, but in fact this function doesn't really depend on omega. Or at least if I change omega by a bit, the function is not going to change too much. So 1 and then sum... No, that's h of sigma m omega c squared d sigma. And now what I want to say is that... Well, in fact, this thing I can... It doesn't really depend on omega in this interval. And so I replace this point-wise value in omega by its average on that interval. And so if I do that, I can write it this way. And so in general, this function, I can only bound it by some constant times its average. And then I have the supremum over l of the sum over k integral of sigma of that thing. So at least formality is correct. Now why was that such a good idea? So where we used since h hat... Let me forget about those two variables slowly varying. And so now you see how you should choose the epsilon so that at least you can make sure that this function is constant on the intervals of size 1. Okay, so this you can always do. And in fact you could always do whatever the width of the interval that you're choosing. But here is something that is really surprising that happens in this case. And it's something that is a nice computation. So this, I would let you do it in the ROR when you go back home. And it is the fact that this is bounded.  And in fact super l, sigma and m is uniformly bounded. Now yes, some number. So now I'll let you prove that. What is there? It's the fact that if you look... So you're integrating over z times r. And so what is this? Well you can, for every value, for every integer value you have one line. And what you're really doing is counting or looking at the Lebesgue measure of when this quantity is between l and l plus one. So what that essentially means is that you're taking any center inside your domain and you're looking at any radius, any circle of a given radius r which is going to be something like l plus sigma squared plus m squared, the square root of that. And you're looking at the area of your domain. The total length of those lines which are between r and r plus one over r. And the surprising thing is that this is in fact uniformly bounded. Here you could observe that if instead of integrating over the line I was just integrating over the whole of r2, so I would have to compute the area of this domain then it would be no problem. Why is it surprising that when I integrate over z cross r I still get something uniformly bounded? Because if I were just counting, if instead of having z cross r as my domain I was trying to do the same computation but now with only integer frequencies then this is no longer true. In fact you can find arbitrarily, I mean you can find, it's true on average of course but you can find some radius of, some radius arbitrarily large such that this grows like the, well at least it grows slowly like the divisor bound. And this is related to what Professor Parchesi said that if you look at the number of integer point on big circles then you could have arbitrarily many. And actually something that I learned recently which I thought was worth while this, which is that the divisor bound is sharp. There is, you can really saturate it with this exponential log log over log. So and now once we've made this observation then let's just observe that it's over because if this supremum is uniformly bounded by one I can just forget about it and now I have this sum that becomes just an integral and I'm in business. Alright, so a few observations about this. First, well you will treat this in a fairly obvious way to get this estimate except that there you will have to look at the number of integer points and this is where you get this way to the epsilon, this extra loss. Second, one thing which is worth noticing is that we had this estimate on r cross t and we did something completely trivial. We got an estimate on r cross c2 that is still optimal in the sense that it is scale invariant. And so essentially the way I understand it is that the dispersion really helps you once in one direction and it helps you when you integrate in the time direction regardless of how many entries you have here. So what you can do in higher dimension is lower this exponent because then you can try to... Alright, so that's it for the first estimate. Now one last thing that I would like you to observe from this is that it implies a stronger version of it which is a bilinear strict arts estimate. And maybe to make it simple let me... So this is again a nice exercise to do and I would encourage you to try to do it in two ways. One would be just to go through what we have been doing and then use the fact that we really had a bilinear estimate in these guys. Another one is that this is in fact in this case a formal consequence of that one if you use Galilean invariance and orthogonality properly. And well, I guess this is always true but it is mostly useful in this case. And what that means is that whenever you look at the product of two solutions, not only can you use this estimate to bound it but you can do better because the loss of derivatives you can always assume that it falls on the slower frequency, on the smaller frequency. Why prime is equal to y? I would expect to see again the y to the one fourth. No, because you get it twice. You should bound it by the L4 norm of it. And in fact, I think as maybe I learned I think from Professor Gérard is that just from this you can read directly that you should be locally well-posed in H1 half because you can, well, just from this. All right, and now I just... Well, maybe let's stay here. So this is it for the estimate on the linear solutions. Now what I would like to discuss is how you can use this to pass to estimate for non-linear solutions. And here you see that there is something to be understood because even though whenever you only used the regular strict arts estimate, then it's essentially clear how you're going to use them in your non-linear analysis because you're going to work in a space of functions such that this norm is going to be finite. However, if you want to use by linear strict arts estimate it becomes more complicated because what is the space of function is that if I take two I have a gain like that it's probably not going to be a nice linear space. So, well, a priori it's complicated except that you can remember that you're going to produce your solutions by iteration starting from linear solutions and so if you knew that your functions were always some form of superposition of linear solutions then you would get that for free and it would see be a linear space. And so this is the next thing that I would like to discuss and it's what I call a functional analytic trick, a cheat. So I'm going to talk about it but I'm not going to prove it and it's this fact that if you have something for linear solution then you have something for nonlinear solution. So if you can show an estimate like that so this is your space, let me call that M space and in our case it's going to be r plus t2. So if you can get some nice estimate for product of linear solutions then for all initial data you have zero in the ball of your space so let me not. So we try to keep it fairly general even though it's going to be false in food generality probably but I want to make a statement that is going to be true every time I'm going to use it. So in the ball of size so some constant let me forget about it one epsilon minus one in the ball in the space H there exists a unique solution of NLS which is continuous in your interval of time with values in your field of space H and in some more complicated space such that u of zero equals well u of zero. So in food generality as I said this is probably false however and I will again not prove it but I would like to motivate it a bit to get some idea where it comes from and why one could hope for something like that. So and what is the motivation for this? Well if you remember our equation and you can solve it using the variation of constant or the Duhamel formula and it's going to read something like this so this is just the usual Duhamel formula and so you see that better than the solution u itself this is something that is going to evolve a lot there is one thing that is supposed to be much more stable and it is u when you conjugate it by the free evolution so when you straighten it along those characteristics or this thing that I'm going to call v of t and so more stable and well then maybe the equation for v becomes just and this thing I will call is something that is going to happen all the time and I'll call that v i of the interaction of v v v or in other words u of zero and now you see that to make sense of your equation you really only have to make sense of this object now this is a function of t and let's assume that you would like to hope that v is in some Hilbert space or whatever in fact you probably want to evaluate something like that by duality so let's look at it in a product with something that is going to be in the dual space of what we want and because we can't forget about the other thing and so now let's write what it is it is the integral over time of integral from zero to t e to the minus is delta no plus to the minus is delta v of t e to the minus is delta v of t v of s is delta v of s h of t and then there is a t now I can so if I work in say L2 or something in L2 and this is going to be self-adjoint I can put it on the other side to get something which is a little more symmetric and now what I can do is switch those two time integrations because the only thing that at this stage really depends on t is that and now what I just get is the integral over s of e to the minus is delta v of s e to the minus is delta v of s and then e to the minus is delta h of t bar h of s ds where this is and let's assume that we work in L2 where h of s is the integral from t of h of s ds and now you start to see something that resembles very much what we are assuming control of and it would be exactly this if you could say that v of s is indeed very stable in the sense that it doesn't change too much and that it can be approximated essentially by a constant so the initial data let's say u0 u0 u0 and that you have the same property for this capital H but at least this capital H is some integral so even if you started with some weaker information on this little h then this has more chance to be true now to make it into a nonlinear solution you will have to do some fixed point argument etc so you think that you expect that one of these entries is going to be replaced by the difference of one p-care iteration and the next and so this is why in general you want to have three different functions now and one last remark is that you can also say that if this holds then at the very least your flow if it exists this is its third derivative at zero and this tells you that its third derivative at zero is bounded in the space H okay so now where is the cheat the cheat is in saying that well that you can form spaces X that are going to do this trick and so I think it started with some work of Pauquins and this is how we introduce the XSB spaces what I am going to use secretly is more some refinements of those that were introduced by I think the Tarou but at least that you can learn about in the article that I wanted to talk about yesterday and if I can give you my personal opinion so the XSB spaces are easier to define but there is a cost of entry if you want to work with them and my impression is that it's essentially the same cost of entry that you have if you want to understand what those are and how to work with them and for what we are going to for what we are going to use today both of these cases would work because we are never going to really be sharp in terms of the regularity we can always lose a little bit and I just want to say one last thing about this which is that once you understand that then it's good but it still leaves you with some question about how to choose this space H and there is something which is essentially equivalent to that but I personally find easier to guess what the right space H is when you are to get some kind of estimate and so this is the last thing I want to add is that it suffices in most cases because it depends obviously on what this space H is but let's say if it's a sobolev space something which is a little bit maybe less symmetric but that I find easier so let me not repeat this and so what you need to be able to do is to say that the two worst cases the two worst functions I'll estimate them in L2 and then the leftovers they are the ones that are going to define my space H and so once you start to see this then you start to see for example where H1 half is going to appear also a permutation but really this is just to make it more formal what happens is that you have to estimate the two worst term whatever that means in L2 in the weaker space and then if you can still get something bounded with the two nicer guys in your space H you have a nice theory in H all right so this was just a motivation but I don't think I want to talk more about this so if you have any question about that or a comment now is a good time to do it excuse me, are you in space for this lemma why you don't need the regularity of X to control the solution and why the first term itself can't control L4 well I guess we saw the proof so that's I think maybe you should pass this too I mean is it because you are restricted in 0 or 1 no so the point is that you get to choose to some extent which of the gradient you want to use to control your solution but certainly I mean all of this is bounded by the sum of the frequency to the one quarter this is just something a little bit more refined and it won't be important for us today but when we start looking at this modified scattering the whole point will be to be very economical with the regularity in the tourist direction and this is why it will be important for us to make sure that we only lose in one direction where we agreed from the beginning to lose a lot but on the direction where we want to have precise control we lose as little as possible okay and so now after this essentially with one or two ideas the proof that we are going to see today will rely a lot on estimating things like that so maybe let me do it once to show that you have local well-posedness in H one half and after that I hope you will just believe me for the computations just how much time do I have? 10-13 minutes okay alright so then at least we can do yeah at least let's do earlier on this maybe more sketchy in the modified scattering so now let me set this proposition and so you could get to one half I just don't want to be too technical in fact I would like so this is essentially a consequence of what we had before so what we need to do is to estimate something like this and then so you have this phi phi one phi two phi four phi x phi t and so where now I would like to forget about this distinction between the regularity in x or the regularity in y at this stage I could probably do it because I'm only going to use my L4 estimate that we had before that we're only hitting regularity in y however well after that I want to use a slight variant that has to mix both directions so and what is this so Pm is now the same thing as before except that it's the supremum of the two that is of size m and it's exactly of size m and so the integral that I have to use we need to control the sum choices of m1 m2 m3 m4 of i m1 m2 m3 m4 right because I can decompose my function into all of its Fourier coefficient etc and then I have to sum over all the possibilities and there is really one thing that we need to remark is that so first of all we can reorder them that is a little lie in the sense that some of them should have bars, some of them should not but just by changing I mean it won't make any difference all of the estimates that we have proved they could have bars or not it doesn't change anything and you could really just conjugate and change the time so once you accept this then it is clear that we can reorder them but there is one new thing that we have to remark that in fact the two highest frequencies they have to be comparable and why is that so let me just write it and why is that well just decompose this I mean write the Fourier inversion of all of this and what you're going to do is to integrate so this one would be at frequency c1 this one would be at frequency c2 c3 c4 and now you integrate that over x and so you're going to integrate e to the i c1 c2 c3 c4 x dx and well this is four more if I'm really on r cross t2 but it's really you can make sense of that and you see that this is going to be zero unless all of the frequencies add up to zero and now if one of them is much bigger than the three others then this can never happen unless two highest comparable and now once you know that the two highest are comparable well you cannot say anything about their difference they could be very small and so really this is the only thing that you can do in general but the amazing thing is that this is completely enough for us because now once we observe that then we have a product of four functions and we know that whenever we take the product of two we can estimate that in L2 and things go well and so okay so now or maybe let me just so assuming this order is going to be bigger and I'm going to add a all the time group one big frequency with one small frequency why because I have seen that whenever I have a product I can always the less on the lowest frequency and so if I have two terms I would like to forget both of the highest frequencies and now I'm going to use the estimate that I have of course just erased but that says that I can estimate that with by one half of the derivative of the smallest so in this case it would be m3 and in this case it would be m4 and now all of them are in L1 in L2 okay and now I would like to have an estimate like that and I remember that I'm not working in L2 space that would be too much I'm working in HS space so the two the two worst guys I'm going to keep them in L2 and the two lowest I'm going to start using the information about the space and so in particular I can start adding one half derivative but I can even add s derivatives so I'm going to have minus s here and then here HS and here HS and now that's it so why is that it well I'm going to so those two I only control them in L2 but they are twice the same essentially so it's really I can just do Cauchy-Schwarz and then I can sum over those frequencies those guys well I have to eat them all but they come with a negative power so I could really eat them all and get something bounded by the HS norm and that would give me local existence for small data now if I want to have local existence for large data I have to produce an epsilon in front of it but this is not something particularly difficult because at least for high frequencies this is already given directly by this estimate so let's say if the one of them M3 is bigger than some number I have something bounded by as we have seen phi 1 in L2 phi 2 in L2 in HS phi 4 in HS and if I only add the if I only sum over M3 with a negative power for M3 big then I it's a dyadic series I pick the last the first term and I'm getting A to the minus S-1 half so what that means is that if my third largest frequency is big then I'm good but on the other hand if my third largest frequency was small then I didn't really have any evolution I essentially just have a bounded function and so I could just use a cruder estimate for M3 small use instead and so I'm going to estimate one and then I have to gain some smallness somewhere but what I gain is the time and so I get T to the one half A to the three times phi 1 in L2 phi 3 in L2 and so you see that if I plug that back in I also leads to well now I'm going to add those numbers all the way to my threshold A and so I'm going to get T to the one half A to the three half times the functions and in fact I can do it for both terms so I could get that twice and so now when I add those two things I see that at least if I choose the time small enough well I'm going to choose this one A large to be able to produce a small epsilon and then I'm going to use to choose the time small enough so that I also have an epsilon here and I'll get the estimate so this tells you that whenever you start with an initial data in HS for any S bigger than one half you always have a global solution and so you always have a local solution on an interval of time which has a good chance of being fairly small and in particular if you start smooth then in particular in H1 then you know you can find a uniform bound on your H1 norm and so this gives you you can gain a little epsilon everywhere here can be made global if initial data is in H1 and now what I wanted to at least discuss a bit is what happens if you're not in H1 if your initial data is not in H1 okay so first this is this is the end of the introduction saying that whenever you're in H1 half you have really a nice local theory and everything can be solved by fixed points so this is really the best possible scenario and now the question and if your initial data is not in H1 then well at least you cannot really use the energy the fact that the energy gives you a uniform bound in fact we have seen that it is possible that any norm that it has is going to grow so you have to try to do something but still you can decompose so you can at least make the following observation that let's say your initial data you can decompose it into a part so first of all what is the difference between H1 half and H1 well you're not going to see this difference for the frequencies which are bounded because you're just putting a weight and if c is smaller than 1 then 1 plus c to the 1 or 1 plus c to the 10 is not going to make a difference so the only difference is in the tail of the decomposition so now let's try to decompose our function into something in the bulk where the particular topology in which we estimate it doesn't really change plus a tail so what we see is that the bulk gives me a nice H1 function so and if I imagine that the bulk contains most of the spectrum of my solution then I could just try to truncate it so could lead to a global solution but of course I have to I would have to correct for an error but this error is going to be important for us is that it's not in H1 that would be too good but it's much better than where I can make sure that these are errors so tail and so where can I make sure that these are errors well when I estimate them in H1 half or maybe with this proof in H1 half plus and in H1 half plus well if it's the high frequencies by a Chebyshev argument I can bound it by dh s norm and I get and so this is small for n large enough ok now I have a nice perturbation theory so I can make sure that this distinction between the two pieces is going to be valid for a small time and so what is something that could make me hopeful that by working a bit more I can really make it for a larger time is the fact that we have a decomposition in two guys but both of those guys get better in their property when s increases and of course if I were to reach one then it would be great because I could push n all the way to infinity and forget about this but what that means is that at the very least this distinction should get better and longer and longer time when n increases and then well at this stage you can say can you make it global or not and the point is that you can so now to make it global what you see is going to be crucial is to be able to track down well when something is going to leave the part of my solution which has frequency n and get in the tail to make it grow so that my tail which started small is not small anymore so to keep this decomposition we would like motion of frequencies and this is where you have to play a bit and so I think this kind of discussion was first proposed by Bourgain it could really prove that you can make sense of this in a very strong term in the sense that you revolve this part non-linearly you revolve this part linearly and then you can always keep some form of this decomposition provided s is large enough so s is close enough to 1 and what we're going to see is some variants that was proposed by the I team later to say that there is a way to tweak this a little bit which is to not abruptly remove the tail by using the so-called I method and so what is this so is that we would still be able to track down the bulk of our energy this way but we need to be more gentle with the tail and we would like to use in the best way possible the fact that we're working in HS so instead of looking at the projector which would essentially means that we're only looking at our function multiplied by that or the way I would think about it we're going to look at a multiplier which is well let me draw it so we also want something that is that gives us the same information in the bulk but we don't necessarily want to enforce that it becomes 0 too fast we instead want to enforce the fact that it decays at infinity like like this and so now we know that you want your function to be below this curve so below 1 for the first end frequencies and then decaying slowly so as to make sure that it remains pretty smooth then you essentially have one unique way and so and as we're going to see one other way to look at it is that instead of looking at your solution so you're going to look at the solution this times U and so they call that multiplier I because it's some form of integration and when you know that when you have good properties for bounds on IU the information that this gives you for U so if U if IU is bounded then U is below the following envelope so U is going to be IU divided by that and so as a result you have very good control on your solution for the first Fourier modes and you have worse and worse control as your frequency becomes larger and larger but somehow you want to make it to compensate it by the fact that as your frequency gets larger and larger because you're away from the space where you had a nice perturbative theory you also get some smallness from that from there and observe that this is much better than if you enforce your solution to only have control on this times your solution because this would give you that your solution but if you have control on p to the n times U then your solution is you only know that it is below this envelope and so here there is a way that it could go to infinity too fast for it to be under control and now the last preliminary remark is how are you going to control your solution and this is a way to get started the process that at least Professor Vichilia talked about and maybe also Daniel talked about is how are you going to control your solution solutions in HS or in fact in some norm you would like to say that this piece if it was in H1 then you would use it to think that the energy is controlled so you want to control the HS norm and so you want to look at an energy and you can start by looking at something like that and so at this stage you can now that we have U or IU you can play the same game either with one or the other whenever we do estimate will in fact work mostly and this is how I'm going to where I'm going to state the more precise results but just to say see why another way to get to the energy let me work with you because it is slightly nicer and so alright then I'll just spare you the computation I'll just give you the last result and then we'll see why why it was better to why it is good to control IU and so well once you have computed it then you get you can just plug that in and you get an explicit computation and you get this thing and then you get the same thing as above and now I can explain why this gets you started with a nice quantity so you chose I in such a way that it is one for low frequencies so now decompose all of those guys well okay maybe first we see that we have essentially so we have the same guy on the right hand side and then on the other side we have either Laplace of IU which is a pretty scary thing because we are assuming derivatives so much less than two or we have three copies of IU so this is always much better than that one so let's focus on this one now we can observe that if I was one then this should vanish and I is indeed equal to one if all of the frequencies are small so now we already know that for these two not be zero one of the guy is going to be forced to be a tail and so at this stage you start to gain a smallness factor but then the amazing thing is that you get one additional cancellation because if only one of the frequencies were small and all the other were if only one of the frequency was so you know that at least one frequency is big and all the others are small and now if only one of them were small then this would be one this would be one and you would have the same number of frequencies on both sides and the frequency of that term would be mostly the same as the frequency of that term so the I would fall on the biggest frequency and so you would get again another cancellation there so for all practical purposes you can force in this expression of four terms two terms to be at high frequencies and then after that it's turning the crank and I'll give you the two steps next time and then after that it's really just estimating carefully where those commutators arise so I'm sorry if I went a bit over time So the global result here is for S greater than or half? No the global result is only for five over six and you could but this is by no means sharp you could improve it if you were so this is just by using this thing the so called first generation I method and if you wanted you could improve it the important thing for us was what to get something that was below one that was global and was no more growing and then to say that in fact you just cannot do better and just when I see it in all of these estimates so the L4 norm is the only estimate that we use and so all of this would not see the would work in the same way if it was on the torus