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Spacing Films
Notes from the Maritimes
June 18, 2018 | By Sean Ruthen
At the end of May, the Royal Architectural Institute of Canada (RAIC) held their 2018 Festival of Architecture in Saint John, New Brunswick, and celebrated the venerable organization's 111th year. As a joint event held with the AANB for the first time since 1988, highlights included the Fellows Convocation and Keynote in the city's old Imperial Theatre, as well as the President's Dinner at which the RAIC Gold Medal was awarded to Montreal architects Saucier and Perrotte. With the Venice Biennale on right now (and read Aaron Betsky's take on it here), on a much smaller scale Saint John put on its best to host some four hundred architects from all across Canada, featuring three days of seminars, awards, and convocations. Day trips of the region were also offered along with walking tours of the old city, including visits to the sea caves at St. Martins and Hopewell Rocks.
It is a pivotal moment for the old port city of Saint John (spelled without the abbreviation to avoid confusion with the other St. John's in Newfoundland), for as up until 2016 the city's population had been in decline for four decades, with the city's most important asset, i.e. its young folk, leaving in droves for years to seek their fortunes elsewhere. With the opening three years ago of a new cruise ship terminal on the edge of the city's inner harbor, mere yards away from where the record setting Marco Polo was built and launched from in 1851, the city is experiencing a bit of a renaissance. With the new terminal able to berth three ships at a time, the downtown population now surges mid-week with thousands of curious visitors, many new to the old city and its history.
For those who come may not realize they are in one of Canada's oldest cities, incorporated in 1785, which tragically like so many other late 18th century cities, was half-consumed by a devastating fire in 1877. Saint John also has the dubious distinction that their fire was the most catastrophic, razing some 1600 structures and resulting in the need for a massive part of the busy port city needing to be quickly rebuilt. As part of the RAIC Festival, architects were toured around the downtown to look at the buildings constructed after the great fire, with a walking tour conducted by Gary Hughes, senior curator of the Museum of New Brunswick, and author of Music of the Sky: Architectural Drawings of Canada's First City 1822-1914.
Featured on his tour were several handsome buildings of stone, brick, and cast iron, all mostly built in the Second Empire style. The walk also featured one wooden structure – an old general store predating the 1877 fire, floated on a barge from 40 kilometers away up the Saint John River to its current location in Market Square after the great fire. Gary explained how quickly the new Saint John needed to be rebuilt, being one of the busiest ship building ports on the eastern seaboard, with a need to hastily rebuild the market on King's Square, along with the nearby Trinity Church, completed in 1881.
In addition to the seminar on the great fire, others included more topical subjects such as two on Passive House given by member's of Stantec's Vancouver office, along with two BIM seminars looking at interoperability and in-house training. The Festival's opening reception, following the AGM at the Hilton, was held in Market Square, an indoor atrium next to the harbour, between the civic convention centre and a re-purposed brick warehouse – once home to marine merchants, now a museum, library, and row of restaurants and local ale houses.
With the distance between the harbor and King's Square a comfortable three city blocks, Saint John is a very walkable city – on its north side lined with new modern buildings which march up the hill to the Square, built over the course of the 1970's and including Brunswick Square, comprised of a three-storey commercial podium from which rises a few mid-rise office towers, including the tallest tower in the city, all connected by the longest indoor pedway in Atlantic Canada. On the south side of King's street is the old city, with the original 1879 city hall along with several other heritage Second Empire structures, including the already mentioned Trinity Church, whose spire rises to 210 feet, making it one of Saint John's tallest structures. This older part of town is also where can be found some of the city's best restaurants, including a local coffee house called Rogue tucked down not one but two alleyways.
The South side is also home to a bit of a curiosity, a ghost of a building that was once part of the downtown fabric and is now gone. The site of the former Customs House on Prince William Street, directly across from the new cruise ship terminal, is now home to a surface parking lot. Called at the time of its construction "the finest building of its kind in America and second to only a few in the world" it is a travesty that all that remains now are some of the building's foundations excavated at the rear of the site. Constructed after the great fire to replace the previous customs house on the same spot, it stood until 1960 when it was abandoned as the government offices had moved into a newer, more modern building next door. Its demolition caused enough of a stir that it started the architectural heritage movement in Saint John, in many ways responsible for the preserved city one sees today.
Also on King's Square is the old Bank of Nova Scotia building, a proud post-1877 Great Fire structure whose ground floor hall is now used by the University of New Brunswick. This Grand Hall provided for the perfect locale to gather following the Fellows Convocation held on the third night of the RAIC Festival, itself held at the old 1913 Imperial Theatre across the square. Without a doubt, the highlight of the Fellows Convocation and the entire Festival was the keynote given by Francis Kéré, at which he expressed gratitude to all gathered to be able to show us his work. Along with his award winning 2007 Dano Secondary School in Burkina Faso, designed and built with his fellow villagers, he showed us more current projects including another school he did for the Aga Khan, as well as his design for last year's much celebrated Serpentine Gallery.
The most inspirational moment of Kéré's talk came when he put up a picture of himself and his villagers standing on a mock-up of a brick arch he had constructed with them, after showing them first how to dig, cut, and shape the brick. It is an astonishing story, made more so by his observation that as an oral culture, if it had collapsed his people would've passed the story on for generations. But of course, the arch held and became the barrel vault for the school he built with his village, and which still stands today.
Following Kéré's talk, the Fellows and others assembled at the Imperial were able to see outside firsthand the latest addition to the Saint John skyline, built on the remediated site of an old Irving filling station, the business that made the family its fortune. Rising from the corner site next to the Imperial, and now nearing completion, is the new eleven storey Irving tower, to be the new Saint John head office for the local family run corporation. With only a handful of towers in the city located between Market and Brunswick Square, it is the first tall building to be constructed in over a quarter century, and one of the few new buildings of its size to be built in some time. An exception is the city's new harborfront stadium, completed shortly before the cruise ship terminal opened in 2015. Itself standing on the site of the city's now demolished Canadian Pacific train station, which succumbed to the wrecking ball in the 1980's, its recent construction along with the new Irving tower is an indication that Saint John may at long last be emerging from its doldrums.
Leaving New Brunswick and the Bay of Fundy, traveling east into Nova Scotia and right to the edge of the Atlantic, the port city of Halifax is currently seeing a different kind of growth than Saint John. For some time, the downtown of Halifax has sat perched below the crest of Fort George, better known as Citadel Hill. On the hill above the old town clock, the city's most beloved (and photographed) landmark, the city has since time immemorial prevented the obstruction of the harbor from atop the Citadel, with the one rare exception being the construction of the Maritime Centre in 1977, a now Brutalist gem with a long vertical concrete slab, situated so as to minimize the obstruction of the harbor from the top of the city.
It would appear that with the recent construction of the new Halifax convention centre with its adjunct office towers (including BMO's new digs) has created a spot zoning free-for-all. I counted as many as three new residential towers under construction or already built in the city's downtown core, next to endangered Second Empire treasures on Barrington Street, and only blocks away from the old City Hall and St. Paul's church. It would appear Halifax is having its moment in the sun with business and residents alike moving back to the downtown core, but it appears it too is making the same mistakes that Vancouver did around English Bay in the 1960's, where large slabs of residential towers were allowed to be built and then were immediately regulated against.
All is not lost however in the Maritime's largest city, as a new downtown library by Danish firm Schmidt Hammer Lassen with local firm Fowler Bauld Mitchell has made quite the impression. Located next to the Dalhousie School of Architecture on Spring Garden Road, this new building was very busy during my visit, with a flurry of activity on an otherwise typical Monday afternoon, apparently a new popular hangout for students, locals, and tourists alike. And with a new building under construction on the waterfront next to the Maritime Museum by MacKay-Lyons Sweetapple, there is light at the end of the tunnel from Halifax's run in with spot-zoning.
And what trip to Halifax would be complete without a quick side trip to Peggy's Cove, a mere thirty minutes away from the downtown, where one is left speechless in the beauty and majesty of this tiny fishing village lashed to the rocks next to the relentless, pounding surf. With the lighthouse now automated like so many others around the world, it along with the small village is a reminder of a simpler time when architecture was an exercise in restraint, when every detail spoke volumes, and how the structures we erected were but passing signposts of our times, vulnerable to the flames of our rapid growth, and which we at best could hope to stand for a stint of time before all again is washed to the sea.
Sean Ruthen is a Metro Vancouver-based architect and writer.
Notes from Charlottetown
Book Review – Public Natures: Evolutionary Infrastructures
Book Review – Code and Clay, Data and Dirt: Five Thousand Years of Urban Media
Book Review: The Language of Cities
More posts by Sean Ruthen
Ben Levinson
Thank you Sean for such a thorough review. I did that tour last year and it was most enjoyable. Nice to see Canada thriving
Sean Best
I'm from the Maritimes and I was very impressed by the progressive changes mostly in Halifax on my 3 province tour last summer. Atlantic Canada is definitely a destination to add to the Canadian bucket list if you have not been!
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Inside Scoop: Facebook revamps News Feed
-Hello and welcome to Inside Scoop. I'm Sumi Das and joining me is Donna Tam, staff writer for CNET. Thanks for being with us, Donna. -No problem. -So, changes are coming to Facebook and you were at an event today where they were announced. We'll talk about those in just a second but first I wanna find out about the atmosphere at the event. What was it like? -Yeah. Well, there's a whole bunch of headquarters there. They filled out the room nicely. -At Facebook headquarters in Menlo Park. -At Facebook headquarters, right. And I think I heard from some folks who had been at the other event for the search announcement that-- -Graft search. -Graft search, there actually was less people there and maybe it's because we all kind of like heard enough rumors about what was going on. -Yeah. -And like, plus everyone knows like, you know, the News Feed, what it is. -Right. -And so it definitely-- -Before graft search, people were like it's gonna be a phone. It's gonna be a phone. -Yeah, exactly. -And this time, we knew it was gonna be about News Feed. -News Feed, right. -Okay. So, what were the announcements that were made? What are the changes that are coming? -So, basically, News Feed is now gonna have a content specific feeds, Which means that you can choose what type of feed you wanna browse. So, that could be photos or like music related post, -Okay. -which includes music your friends are listening to through other apps. Or like musicians that you follow or like on Facebook or bands that you like and then they'll have their posts rise up in this Music Feed. So, you could do that. There's also stuff for different feeds that have existed before like the Games Only Feed and the Close Friends Feeds, -Uh-hmm. -but now it's all like in one place. And basically, they've done a lot to the design in terms of making it less cluttered. -Right. -So, it's a very sweet design. -It did mention that many times. -Many times. -It's less clutter. -It's less clutter. -It's gonna be prettier and simpler. -Yeah. And basically, what they did was, they took a lot of inspiration from your mobile apps. -I noticed that they say that. -Yeah. -Instead of taking the desktop and trying to make that, you know, replicate that experience on mobile, they took the mobile version and are trying to make duplicate that, -Right. -and that's tough. -Right because they find it to be a version that has like faster navigation in a lot of ways because there's a constraint of space. And so, because there's a constraint, they were able to like think of ways to like make things function still with a little amount of space they have. -Uh-hmm. -And so they felt like that was appropriate for desktop because it makes it so that the images can be bigger. -Uh-hmm. -You know, the visual element is much more emphasized in its new design. -Right. So, yeah, let's talk about that because they did talk about how it's gonna be visual. They mentioned Pinterest a couple of times. They mentioned photos a couple of times. -Uh-hmm. -What they didn't mention as much was this little company that they own called Instagram. -Right. -What happened there? -I'm not sure what happened there but they-- definitely after the presentation was over, during the Q&A, someone asked him about that like what about Instagram? What about the treatment that they're gonna get? And Mark Zuckerberg said, you know, Instagram is gonna be treated like any other third party app that they have. So, Pinterest, like Pinterest, you know, Pinterest is gonna get the same amount of play and big break photos as an Instagram photo would. But I think that was his emphasis was like they kept repeating over and over again how old the content is treated the same. All the content will get this like new burst of, just new burst of like visual and like highlighting the elements that are important to that kind of content. -Right. It makes sense, you know, people like formats that are cleaner and easier to read and digest. But why are they really making these changes? -Well, I think, when you ask the people at Facebook, they'll say, you know, this is about like bringing a better experience to users and making easier for them to find the content they wanna find. -Uh-hmm. -I think in the end they also talk about how they realize it's also gonna bring more-- have people-- this will also have people spending more time on Facebook. -Right. -So, because they're gonna be looking, taking a deeper dive of things and spending times on different-- time on different feeds then, you know, it would result in spending more time. Advertisers probably love this because it's gonna be bigger ads as well. So, Facebook is very big and making sure that ads are a seamless part of the whole News Feed process. And advertisers have found that those that are put inside the News Feed definitely make more money and attract more attention, you know, once in the right-hand side where they kinda just pop in there, right? -Right. -So, in the end this will also mean that advertisers will get the nice treatment on their ads. They'll get bigger pictures, they'll get more emphasis on their links, you know, and so people again can see that within their News Feed and it'll be this-- as if they're, you know, looking at one of the content that maybe their friends provided or something. -That does seem handy. And could it be also-- I mean their has been some talk that they're sort of going after Twitter's business a little bit, with this new revamped or something. -Yeah. I think people have seen the similarities and that, you know, in the past, when they've talked about like how they do the News Feed, there's a filter, there's an algorithm, there's a way of filtering things out better, gonna be more relevant for you and people have been upset about, you know, reacting and people reacted, you know, negatively in some ways because they're thinking that they wanna be able to control what content they see and not having algorithm do it for them. So, when you have something like Twitter, which just kind of grows everything out there from all the feeds you follow. -Yeah. -You know, this is a way for Facebook to kinda replicate that as well. -Yeah. All right. And well, that's gonna be slow, so not everybody is gonna get it all at once. -Yeah. -They're gonna test it out, they get feedback and then gradually-- -Right. -get it to all of their users. -Correct. Yeah. -All right, Donna. Thank you so much. For Inside Scoop, I'm Sumi Das. Thanks for watching.
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Our sales and technical support teams will continue to be fully available to you via telephone and email. In addition, our Human Resources department will remain open. Service and repair work on your premises will be possible in urgent cases, however we will be subject to any national legislation imposed due to the corona crisis. All personal sales and consultation meetings as well as face-to-face training sessions are unfortunately suspended until further notice.
Many of our employees are currently working from home to help mitigate the spread of the pandemic. In those areas where this is unfortunately not possible, our workers are following strict safety and protection measures. The health and welfare of all our employees remains our highest priority. By adopting this package of measures, we hope to see through this difficult time with as little disruption as possible.
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Mets plan deGrom return next week after latest rehab start
By MIKE FITZPATRICK The Associated Press
FILE - New York Mets' Jacob deGrom delivers a pitch during the first inning of the first baseball game of a doubleheader against the Milwaukee Brewers Wednesday, July 7, 2021, in New York. New York Mets ace Jacob deGrom is finally ready to begin a minor league rehab assignment. Sidelined by injuries since last July, the two-time Cy Young Award winner is scheduled to start Sunday night, July 3, 2022 for Class A St. Lucie against the Jupiter Hammerheads.(AP Photo/Frank Franklin II, File)[ASSOCIATED PRESS/Frank Franklin II]
NEW YORK (AP) — Mets ace Jacob deGrom pitched four innings Wednesday for Triple-A Syracuse, finishing strong after a shaky start in what's expected to be his last minor league rehab outing before rejoining New York's rotation.
"The plan right now if everything progresses well is to pitch for us next time," Mets manager Buck Showalter said. "But that could change by what we hear tomorrow or the next day."
The two-time NL Cy Young Award winner would likely return to the Mets early next week in Washington. If he pitches Monday or Tuesday, he would be lined up on full rest to make a start in a potentially pivotal five-game series Aug. 4-7 at home against defending World Series champion Atlanta — although the NL East leaders have not committed to that timeline.
"We'll see where we slot that out," Showalter said.
New York entered Wednesday with a two-game edge in the division over the Braves.
Sidelined all season by a stress reaction in his right scapula, deGrom gave up four runs on a pair of homers and threw 41 of 67 pitches for strikes. He struck out six — twice fanning Kansas City All-Star catcher Salvador Perez — and walked three in a game that began at 12:05 p.m.
"It was good to see him get to almost 70 pitches. That was a good sign," Showalter said. "We'll get arms around it here in the next day or so, but we hope that his next outing's with us. We'll see."
Last week, Showalter said if deGrom needed another rehab start after this one, then that's what the team would do. He said deGrom is well aware of what it takes to be ready to pitch in the majors.
The 34-year-old right-hander hasn't appeared in a big league game since July 7, 2021. He missed the second half of last season with right forearm tightness and a sprained elbow, then went down late in spring training this year with the scapula injury.
He's made four minor league rehab starts totaling 12 2/3 innings, two at Class A St. Lucie and two with Syracuse. He's allowed five runs — four earned — and seven hits with 21 strikeouts and four walks, hitting 100 mph on the radar gun even in his first outing.
He threw a 60-pitch simulated game in Florida during the All-Star break, then a side session Sunday at Citi Field in New York. The simulated game was pushed back two days after deGrom experienced minor shoulder soreness.
The only hits deGrom gave up Wednesday were a solo homer to Brewer Hicklen and a three-run drive to Drew Waters that sandwiched consecutive walks and two strikeouts in the second inning against the Omaha Storm Chasers. Then deGrom struck out Perez, who's rehabbing a thumb injury, for the second time and retired his final seven batters.
"His arm felt really good," Showalter said.
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Delegation of the European Union to China
http://www.weibo.com/euinchina
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EEAS homepage > China > EU launches alumni engagement initiative in Beijing
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EU launches alumni engagement initiative in Beijing
Brussels, 03/12/2019 - 07:18, UNIQUE ID: 191203_2
The initiative helps participants in different EU cooperation initiatives stay in touch and keep their relationship with the EU
The EU has today launched the first EU alumni engagement initiative in Beijing, China, in which participants in different EU cooperation initiatives come together in a network. The goal is to help people stay in touch, exchange experiences and keep their relationship with the EU.
The network was launched at ceremony hosted by the EU Delegation in Beijing. Participants in EU programmes like Erasmus Mundus, Jean Monnet Professors and Centres and Marie Skłodowska Curie, as well as in the EU-China interpreter training project and in the EU visitor's project all took part in the event. H.E. Nicolas Chapuis, EU ambassador to China, and Ms Florika Fink-Hooijer, Director-General of the Commission's interpretation department, both took part in the event.
In his introductory remarks, Mr Chapuis, EU Ambassador to China said: "EU Alumni are active across all sectors of the EU-China relationship. I'm very happy that we are starting a project that will connect all the members of our "EU family", allowing us to stay in touch with them more systematically."
In her welcome speech, Ms Fink-Hooijer, Director-General of the European Commission's interpretation service, highlighted the importance of the EU alumni engagement initiative in the EU-China relations. Ms Fink-Hooijer said: "The main goal of the EU Alumni Engagement Initiative launched today is to strengthen the EU's worldwide relationships and partnerships with EU alumni, deepen the ties between the EU and its partners and foster a better understanding of each other. The Chinese Alumni act as connectors, building informal bridges between the EU and China."
The initiative has the potential to reach more than 4000 people from across China who have participated in EU-funded projects. The initiative foresees a cross-programme exchange – be it face-to-face like the alumni reunion organized on 3 December 2019, or in the future also an online web platform where the alumni network can stay connected.
The EU Alumni Engagement Initiative for EU Delegations aims at reinforcing EU's capacity to sustain a long term meaningful engagement with individuals in third countries who have already benefited from EU funded programmes and been in contact with the EU at one point in their lives. This action aims at upgrading the relationship between the Alumni and the EU into a long lasting partnership based on trust and broad mutual understanding.
Under the EUCITP cooperation, which started in 1986, over 500 Chinese interpreters were trained which today form part of a strong Alumni network. EUCITP participants include senior policy makers in a wide range of ministries and provincial governments, including a MOFCOM Vice-Minister and various Directors in the Ministries of Commerce, Foreign Affairs, Education, Science and Technology.
The EUCITP project is implemented by DG Interpretation in collaboration with the European Commission's department for Foreign Policy Instruments (FPI) - under the heading of "Public Diplomacy". The project, which has become a "brand" with an excellent reputation among Chinese partners, consists of two elements:
Cooperation with Chinese Ministries: Chinese officials come to Brussels for a five-month interpretation training course, which also gives them the opportunity to learn how the EU works.
Cooperation with Chinese universities: DG Interpretation provides teaching assistance to two partner universities in China (Beijing and Shanghai).
EU-China cooperation in the field of interpretation has a strong people-to-people component. It not only promotes multilingualism, but also contributes to a better understanding of the EU in China. At the same time, it engages with China as an international partner and facilitates interaction in areas such as trade, connectivity, research, security and foreign policy.
For the past 30 years, DG Interpretation has been contributing its know-how to many interpreter training projects, both within the EU and in countries around the world, such as China, Macao, Vietnam, Cuba, Russia and several African countries.
During her visit to Beijing, Ms Fink-Hooijer also met with representatives of the Ministry of Commerce and the University of International Business and Economics to discuss the continuous cooperation between the EU and China in the context of their joint interpreter training project (EUCITP). Ms Fink-Hooijer will also travel to Shanghai in the context of the 70th anniversary of the Shanghai International Studies University (SISU), an important partner under the EUCITP.
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India: High Representative/Vice-President Borrell meets with Prime Minister Modi and Minister of External Affairs Jaishankar in first visit to the country
Raisina Dialogue: High Representative/Vice President Josep Borrell addressed Raisina Dialogue and met Asian counterparts in New Delhi
Iran: High Representative/Vice-President Josep Borrell met with Foreign Minister of Iran Javad Zarif
India / Raisina Dialogue: High Representative/Vice-President Borrell in New Delhi for bilateral visit to India and to attend the Raisina Dialogue
Cultural Routes of Europe live painting performance by renowned Spanish artist Alberto Reguera
EU HRVP Federica Mogherini meeting with Premier Li Keqiang in Beijing Languages: 简体中文 /file/2jpg-90_en2.jpg This afternoon at the Great Hall of the People in Beijing HR/VP Federica Mogherini met with China's Premier Li Keqiang. Among the many topics discussed were the ongoing negotiations for an EU-China Comprehensive Agreement on Investment and the conclusion
HR/VP Federica Mogherini visits Beijing to discuss bilateral relations and foreign policy cooperation
12th Edition of the European Union Film Festival in China to open in Beijing on 28th Nov
Business leaders in Beijing discuss gender equality and corporate social responsibility
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First Fares Explained - Sort Of!
Those of you who are as old as me (over 24) might remember an American sitcom of the 70's called Soap. At the start of every programme they would run down the current synopsis involving a multitude of characters, finishing with the words "Confused? You won't be after watching this week's episode of 'Soap'!" After trying to explain the new First fares which apply from January I may need to use the same phrase, except I can't promise anyone will be none the wiser after I've attempted to make head or tail of today's announcement.
NI had already planned to make a post explaining why fares will still go up despite the recent crash in fuel prices. Quite simply the vast majority of firms buy their fuel a year in advance, known as "hedging", which means at the moment they will be paying a higher price for the fuel than it is at the pumps. Next year may well see firms benefit if prices remain low for a considerable amount of time, but don't expect that to be passed onto the passenger. There has been much spluttering and political style answers from various people in the industry I've spoken to regarding passing on fuel savings to the customer!
The ususal reaction when cheaper fares are mentioned!
So obviously being aware of this today First announced that many fares were being frozen, some were going down and those rising only going up the rate of inflation - "While some passengers may experience an increase in fares, we have tried to limit this as much as possible and keep any rises in line with inflation."- which right now stands at 1% due to oh yes cheaper fuel prices! OK so let's analyse that. Let's start in Norwich.
If you pay a single fare and currently pay £1.10, £2.10 or £3.00 then your fare is frozen. However if you pay £1.30 you might expect it to go up by a penny or so which is the inflation rate. Not so as it's going up 10p which is a rise of 7.69%. If you normally pay £2.50 that's going up to £2.60 which is still a 4% rise.
Where Norwich benefit are the multi trip tickets, with one glaring exception. A Norwich 2-trip ticket rises 10p to £4.30 (2.38%), however an Inner 2-Trip and Norwich Day ticket remain unchanged. But a Norwich Weekly drops from £19 to £18, a Norwich 10-Trip drops from £18.50 to £17, and a Norwich Monthly drops from £66 to £63. Now anyone getting on a First bus recently will have noticed a big green hand staring you in the face advertising the Hi-5 ticket - they are really pushing this ticket so of course in the Norwich area this has gone up from £7.50 to £8. A rise of 6.66%.
If you think Norwich was confusing you aint seen nothing yet! Starting with single fares the number of single fares in the Yarmouth area is doubling from 2 to 4. So if you currently pay £1.30 you will either pay £1.10 or £1.40, and if you currently pay £2.50 you'll either pay £2.10 (very good) or £2.60. Single and return fares outside the Yarmouth area will remain unchanged.
On the Multi-ticket side a Yarmouth Day ticket goes up 10p to £4.30 (2.38%), the Weekly remians at 315 but the much publicised Hi-5 goes up a quid to £7.50 - a rise of 14.4%.
Now it gets compicated. Currently there is a YarLow ticket and a YarLowXtra ticket which encompasses a wider area. So now only the YarLowXtra will remian, which means if you currently use a YarLow ticket you are in for a pleasant surprise. Although if you use a YarLow Day ticket you'll have to pay an extra 50p to £6.50 (8.3% rise) it is the same cost of a YarLowXtra ticket now. However if you currently buy a weekly YarLow at £20 that remains the same but you get the extra area, meaning a weekly Xtra ticket goes down a pound to £20. Even better is the 10-trip ticket which reduces to £19 from £20 YarLow and £21 Xtra. The bad news is the current YarLow Hi-5 goes up a whopping 25% to £10 but does include the wider area.
Still with me? Well done! Lowestoft is a mixed bag with single fares rising but returns falling. If you pay £1.20 single you'll now pay £1.30 (8.6% rise but still cheaper than Anglian's £1.40). The £2.00 single remains the same and the £2.40 single rises to £2.50 (4.3%). However the £2.00 return remains the same, the £3.30 return falls to £3.00 and the £3.80 return falls to £3.50.
The Lowestoft 2-Trip ticket is withdrawn and the Lowestoft Day ticket falls from £4.80 to £4.50. However the Weekly rises from £14.50 to £15.00 (3.45%) and the Hi-5 rises from £6.50 to £7.00 (7.2%).
If you live in Beccles it's not very good news. The Beccles - Lowestoft £3.00 return is rising to £3.50 (16.6%) and the Monthly X2 ticket currently £60 is being replaced by the YarLowXtra or Norwich Weekly at either £70 or £63 respectfully.
Basically if you live in Ipswich you are likely to benefit, if you don't you'll pay more. I think I'll just paste from the First website as it explains it all perfectly adquately. Suffice it to say there's not a single fare that rises by only 1%. My apologies for the slightly wayward layout below but it looks perfect on the draft page and even this took me far longer than it would have done to just describe the fares as I have in the above sections! Seemed like a good idea at the time but pdf to Blogger doesn't paste well!
Single and return fares to be revised as follows:
Adult Adult Child Child
Single Return Single Return
Current New Current New Current New Current New
n/a £1.00 n/a £1.60 n/a £0.70 n/a £1.20
£1.50 no change £2.20 no change £1.10 £1.00 £1.50 no change
£1.90 £2.00 n/a £3.50 £1.40 £1.50 n/a £2.50
£2.10 £2.00 £3.40 £3.50 £1.50 no change £2.30 £2.50
£2.60 no change n/a £4.70 £1.90 no change n/a £3.20
£2.80 £2.90 £4.70 £4.90 £2.00 £2.10 £3.20 £3.30
£3.40 £3.50 £5.70 £5.80 £2.40 £2.50 £4.00 no change
Endeavour fares adjusted accordingly
Return fares introduced on all sections of routes in zone 1 and Stowmarket
Period/multi-journey ticket prices to be revised as follows:
Adult Young Person
Current New Current New
Ipswich Zone 1 Day £5.00 no change £3.50 no change
Ipswich Zone 1 Week £15.00 no change £10.00 no change
Ipswich Zone 1 10-trip £15.00 no change £10.00 no change
Ipswich Zone 2 Day £7.00 no change £4.90 £5.00
Ipswich Zone 2 Week £21.00 £22.00 £14.00 £15.00
Felixstowe Day £3.80 £3.90 £2.70 no change
Felixstowe Week n/a £12.00 n/a £9.00
Ipswich Zone 1 Hi 5 £7.50 £8.00
Ipswich Zone 2 Hi 5 £9.50 £10.00
A new ticket for those travelling between Dereham and Peterborough is to be introduced, if they can understand the destination screens that is, and all Whole Network tickets are going up. In conclusion it could have been a whole lot worse, but there again not even First are mad enough to put fares up wholesale when fuel prices are crashing. They need to persuade people not to go back to their cars with the cheaper running costs. Will this fares package do the trick? I Doubt it as the longer distance traveller doesn't gain much at all, and they were the ones that needed the most sweet talking. Just one thing - don't make false claims. Not a single fares rise has been kept to the rate of inflation,and to claim so is a little bit naughty.
J C Schneider 24 December 2014 at 00:59
see www.busontheroad.net
smurfuk 24 December 2014 at 11:20
Ah, the old "if you can't convince them, confuse them"! I optimistically thought we might get an indication of how well (or ill) the margins and profits are doing. The conclusion: nobody has any idea. Some of the smaller co's seem to manage publish their fare tables, and a "fare calculator" for multiple journeys. Dear old Wirst have got themselves in such a muddle, that'd be impossible. Never mind, it could be worse, Essex can't sort out their timetables even after weeks of trying, constantly. Let's just put it down to an early excess of Christmas cheer! Merry Christmas, everybody.
First in Essex have got some fare revisions (increases) through the back door. The monthly tickets across the network are to be replaced with 4 weekly tickets. At face value, they represent great value but over a year, will cost more than the current set up. Fortunately, I'm in a position to buy an annual network ticket and they've been left alone for a couple of years.
Confused...... you will be .
Review Of The Year 2014 Part One
Feliz Natal a Todos
Quick Ipswich Update
Weekly News Round Up 21/12/14
Repaints Bonanza
69426 Arrives Home
Lurking In Lowestoft
Thameslink Transition Part Two
Where Are They Now And 33003
Thameslink Transition Part One
Petite Post
Norwich Rod Monochromacy
Weekly News Round Up 7/12/14
A Fantastic Friday Feast
I Don't Belieeeeeeve It!
Ipswich Arrivals, Returns And More
Final Ipswich Plans and Farewell To Yarmouth
Norwich Lampposts Debut And Other Bits


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text:
\section{Introduction}\label{intro}
We consider regularization processes to smooth out distributions on Riemannian and globally hyperbolic Lorentzian manifolds and investigate their compatibility with the wave-equation.
The Riemannian setting has been addressed in \cite{DHK:10} and we include an introduction
to this approach. For a globally hyperbolic manifold we pick a splitting of the metric as obtained by \cite{BS:05} which provides us with a globally defined time function and a foliation by space-like hypersurfaces.  The associated  Riemannian metric  naturally allows us to construct regularization  processes on distributional sections of tensor-bundles and differential forms. We show that these regularizations interact nicely with the wave-equation on the Lorentzian manifold and with the foliation provided by the metric splitting.

By a regularization process we mean a net of smoothing operators that assigns a net of
approximating smooth functions (or sections in a vector bundle) to any given distribution
(or distributional section). We are interested in preserving a maximal set of regularity
properties of the distribution in this process (support, singular support, wavefront set,
Sobolev regularity). To assign such properties to approximating nets of smooth objects
we employ the language of algebras of generalized functions in the sense of Colombeau
(\cite{C1,C2,O,GKOS}). In this approach any such regularization process provides an
embedding of distributions into a space $\ensuremath{{\mathcal G}}(M)$ of nonlinear generalized functions (given
by a quotient construction on spaces of approximating nets). This process
preserves the regularity and singularity structure of the distributions as described in Section \ref{CR} below.   The main interest in employing these regularizations is in studying non-smooth  curved space-times.

Our approach is motivated by work on wave equations on non-smooth curved space-times.  C.J.S.\ Clarke (\cite{Cl1,Cl2,Cl3}) suggested to study physical fields for understanding the singularity
structure of the space-time itself, i.e., to consider wave equations in low regularity.
His work gave rise to a number of further studies. In particular, in the framework of generalized
functions this line of research was pursued in \cite{VW,GMS:09,SV}. For a more detailed
introduction we refer to \cite{HKS}.

Thus let us consider a model  situation and  analyze the  generalized solutions to wave equations corresponding to a generalized Lorentz metric $\tilde{g}$ on the smooth manifold $M$ (cf.\ \cite{GKOS,KS:02,GMS:09}), i.e., a global Cauchy problem with initial data $a,b \in \ensuremath{{\mathcal D}}'(S)$ on a suitable initial value surface $S$ in the form
$$
  \Box_{\tilde{g}}\, u = 0, \quad u\! \mid_S\, = \iota(a),\;
\nabla_{\text{n}} u\! \mid_S\, = \iota(b),
$$
where $\iota$ is assumed to be an embedding of distributions into the algebra of generalized functions.

Suppose that the generalized Lorentzian structure induced by $\tilde{g}$ allows a splitting of the wave operator in the form
$\Box_{\tilde{g}} = \gamma \cdot (\Box_g + Q)$, where $g$ is a smooth ``background'' metric, $\gamma$ is a positive measure or a strictly positive generalized function, and $Q$ is a partial differential operator with distributions or generalized functions as coefficients, but whose coefficient singularities are concentrated in certain space-time regions. A simple example is a generalized Robertson-Walker space-time with $\tilde{g} = - dt^2 + (1 + \mu(t)) h$ on $M = \mb{R} \times S$, where $h$ is a smooth Riemannian metric on the smooth manifold $S$ and $\mu$ is a nonnegative generalized function. In this case $g = -dt^2 + h$ can serve as a smooth background metric and the above splitting would involve $\gamma = 1/(1 + \mu)$ and $Q = Q(t,\d_t)$, which acts only on the one-dimensional factor.

If we impose the splitting assumption as above, then we have for any generalized function $u \in \ensuremath{{\mathcal G}}(M)$ that the equation $\Box_{\tilde{g}}\, u = 0$ is equivalent to $\Box_g\, u + Q u = 0$. Let $u \in \ensuremath{{\mathcal G}}(M)$ be a solution to the Cauchy problem stated above. In attempting to extract its distributional aspects or assess its singularity structure we could adopt the following strategy. Let $v$ be the distributional solution (assuming that it exists and is unique) to
$$
  \Box_g\, v = 0, \qquad  v\! \mid_S\, = a, \quad
\nabla_{\text{n}} v\! \mid_S\, = b,
$$
and put $w := \iota(v) - u \in \ensuremath{{\mathcal G}}(M)$ as a comparison of the generalized function $u$ with its ``background distributional aspect''. Now, \emph{if the embedding $\iota$ commutes with restriction to $S$ and with $\Box_g$}, then simple manipulations allow us to draw the conclusion that $w$ satisfies the following Cauchy problem:
$$
   \Box_g w = Q u, \qquad  w\! \mid_S\, = 0,\quad
  \nabla_{\text{n}} w\! \mid_S\, = 0.
$$
Thus, $w$ satisfies an inhomogeneous Cauchy problem corresponding to the smooth background metric $g$ and with generalized functions occurring only on the right-hand side of the equation. In particular, $w$ will be smooth in regions where null geodesics corresponding to $g$ emanating from $S$ do not intersect the $\ensuremath{\mathcal{C}^\infty}$-singular support of $Q u$.

In the more general situation of a globally hyperbolic Lorentz manifold  provided with a foliation
into spacelike hypersurfaces via a splitting of the metric we shall construct  an embedding $\iota $  and show that the embedding almost commutes with $\Box_g$ and with restriction to space-like slices of the foliation.  The extent of failure to commute is roughly  measured by the divergence of the volume element on the slices along the time  direction.  Although we state our results in the scalar case, the extension to the case of  wave-equations on differential forms is obvious.

\section{Regularization of distributions on complete Riemannian manifolds}\label{CR}

In this section we describe a geometric regularization procedure for distributions  (or distributional sections of vector bundles) on complete Riemannian manifolds. This procedure encodes regularity and singularity features in terms of asymptotic behavior. Our approach is based on  \cite{DHK:10}, to which we refer for further details.

We first fix some notations from the theory of distributions.
Let $M$ be an orientable complete Riemannian manifold of dimension
$n$ with Riemannian metric $g$. The space $\ensuremath{{\mathcal D}}'(M)$ of Schwartz distributions on $M$
is the dual of the space $\Omega_c^n(M)$ of compactly supported $n$-forms
on $M$. Further, $\ensuremath{{\mathcal D}}(M)$ is the space of smooth compactly supported functions
on $M$. We identify $\ensuremath{{\mathcal D}}(M)$ with
$\Omega_c^n(M)$ via $f\mapsto f\cdot dg$, with $dg$
the Riemannian volume form induced by $g$. Thus $\ensuremath{{\mathcal D}}'(M)$ can be viewed as
the dual space of $\ensuremath{{\mathcal D}}(M)$. We consider $L^1_{\mathrm{loc}}(M)$ (hence in particular
$\ensuremath{\mathcal{C}^\infty}(M)$) a subspace of $M$ via $f\mapsto [\varphi \mapsto \int_M f\varphi dg]$.
If $E$ is a vector bundle over $M$ then $\ensuremath{{\mathcal D}}'(M:E)$, the space
of $E$-valued distributions on $M$ is defined by $\ensuremath{{\mathcal D}}'(M:E) = \ensuremath{{\mathcal D}}'(M)\otimes_{\ensuremath{\mathcal{C}^\infty}(M)}
\Gamma^\infty(M:E)$, with $\Gamma^\infty(M:E)$ the space of smooth sections
of $E$. By ${\mathcal E}'(M:E)$ we denote the space of compactly supported distributional
sections of $E$.
We shall assume that $E$ is endowed  with a Hermitian inner product so that the distributional sections $\mathcal{D}'(M:E)$ of $E$ can be identified with the dual $\Gamma_c^{\infty}(M:E)'$ of the space of compactly supported smooth sections.
Regularity properties of distributions we are interested in are, in particular,
encoded in the singular support $\mathrm{sing supp}(w)$, the wavefront set $\mathrm{WF}(w)$, and
the Sobolev regularity of any given $w\in \ensuremath{{\mathcal D}}'(M)$.

In order to be able to track these regularity properties in terms of regularizations of a
given distribution we need a conceptual framework that allows to assign geometrical and
analytical properties to regularizations, i.e., to nets of smooth functions. Algebras of generalized
functions in the sense of J.\ F.\ Colombeau (\cite{C1,C2,O})
indeed provide a well-developed theory of this
kind and below we shall briefly review some basic definitions, based mainly on \cite{GKOS,Gtop}.

The basic idea of Colombeau's approach is to assign to any given locally convex space $X$
a space $\ensuremath{{\mathcal G}}_X$ of generalized functions as follows. We define the space ${\mathcal M}_X$
of moderate nets to consist of those maps $\varepsilon\mapsto x_\varepsilon$ ($\varepsilon \in I:=(0,1]$) that are smooth
and for any seminorm $\rho$ of $X$ satisfy $\rho(X_{\varepsilon}) = O(\varepsilon^N)$
for some integer $N$ as
$\varepsilon\to 0$. Similarly we call a net  $x_{\varepsilon}$ negligible if for all seminorms $\rho$ and all integers $N$ the asymptotic relation $\rho(X_{\varepsilon})\sim O(\varepsilon^N)$ holds. The space $\mathcal{G}_X$ is defined as the quotient of the space of all moderate nets by the space
${\mathcal N}_X$ of all negligible nets. The class represented by a net $x_{\varepsilon}$ shall be denoted by $[x_{\varepsilon}]$.  In case $M$ is a smooth manifold we call $\mathcal{G}(M):=\mathcal{G}_{\mathcal{C}^{\infty}(M)}$
the standard (special) Colombeau algebra
of generalized functions on $M$ (\cite{C1,DD,GKOS}).
If $E\rightarrow M$ is a vector bundle then we set $\mathcal{G}(M:E):=\mathcal{G}_{\Gamma^{\infty}(M:E)}$.
For $E=\mathbb C$ the space $\mathcal{G}_{\mathbb C}$ inherits a  ring structure from $\mathbb C$. It is therefore
called the space of generalized numbers, denoted by $\tilde{\mathbb C}$.
Every space $\mathcal{G}_E$ is naturally a $\tilde{\mathbb C}$-module, and hence  is
called the $\tilde{\mathbb C}$-module associated with $E$ (\cite{Gtop}).

Similarly we shall also consider the subspace $\mathcal{G}^{\infty}_X$ of regular elements of $\ensuremath{{\mathcal G}}_X$, defined as the space of all those elements of $\mathcal{G}_X$ that can be represented by a net $x_{\varepsilon}$ for which one can find an integer $N$ in the above  relations independent of any seminorm $\rho$  on $X$. Thus $\mathcal{G}^{\infty}_X$  is the space of uniformly controlled asymptotics.  Again we set $\mathcal{G}^{\infty}(M):=\mathcal{G}^{\infty}_{\mathcal{C}^{\infty}(M)}$. In regularity theory,
$\ensuremath{{\mathcal G}}^\infty(M)$ is the analogue of $\ensuremath{\mathcal{C}^\infty}(M)$ in the theory of distributions (\cite{O,H,DPS,GH}).
This is based on the fundamental result (\cite{O}) that for open subsets $\Omega$ of $\mb{R}^n$,
$\iota(\ensuremath{{\mathcal D}}'(\Omega))\cap \ensuremath{{\mathcal G}}^\infty(\Omega) = \ensuremath{\mathcal{C}^\infty}(\Omega)$, where $\iota$ is the standard
embedding of $\ensuremath{{\mathcal D}}'$ into $\ensuremath{{\mathcal G}}$ via convolution with a standard mollifier. We will introduce
further notions of regularity theory based on $\ensuremath{{\mathcal G}}^\infty$ below.

The assignment $X\mapsto  \mathcal{G}_X$ is obviously functorial, so that  any continuous linear map $\phi:X\rightarrow Y$  naturally induces a map $\phi_*:\mathcal{G}_X\rightarrow \mathcal{G}_Y$. Thus a smooth map $f:M\rightarrow N$  gives rise to a pull-back $f^*:\mathcal{G}(M)\rightarrow \mathcal{G}(M)$. In particular $\mathcal{G}(M)$ defines a fine sheaf of algebras and similarly $\mathcal{G}(M:E)$ is a fine sheaf of $\mathcal{G}(M)$-modules.

We finally introduce the notion of association, which provides a concept of weak equality
between elements of  Colombeau spaces as well as a standardized way of assigning distributional
limits to certain Colombeau generalized functions. For $u$, $v\in \ensuremath{{\mathcal G}}(M:E)$ we say that
$u$ is associated (or weakly equal) to $v$, $u\approx v$ if $u_\varepsilon - v_\varepsilon\to 0$ in $\ensuremath{{\mathcal D}}'(M:E)$ for (some, hence any)
representatives of $u$, $v$. We say that $u$ possesses $w\in \ensuremath{{\mathcal D}}'(M:E)$ as a distributional
shadow if $u_\varepsilon \to w$ in $\ensuremath{{\mathcal D}}'(M:E)$.

To construct the desired regularization process on $\mathcal{D}'(M:E)$ we need two pieces of data:
\begin{enumerate}
 \item A Schwartz function $F\in \mathscr{S}(\mb{R})$ such that $F\equiv 1$ near the origin.
 \item An elliptic symmetric  differential operator $D$ on $E$ such that the speed of propagation  $C_D$, defined as
\begin{equation}\label{cd}
C_D=\operatorname{sup} \{\|\sigma_D(x,\xi)\|~|~ x\in M,\ \|\xi\|=1\}
\end{equation}
is finite.
Here $\sigma_D$ denotes the principal symbol of $D$.
Such an operator $D$ will be referred to as {\em admissible}.
\end{enumerate}
 As a consequence of the finite speed of propagation, the symmetric operator $D$ is essentially self-adjoint (\cite[10.2.11]{HR}).  Therefore the equation
 \begin{eqnarray}\label{transport_eqn}
 \frac{\partial}{\partial t}u=iDu\qquad u(\,.\,,0)=u_0,
 \end{eqnarray}
 has a unique solution for all times $t$ for any initial datum $u_0\in\Gamma_c^{\infty}(M:E)$.
 Indeed it follows from functional calculus that $e^{itD}u_0$ is a solution, and uniqueness
 can be established using energy estimates.

We note that functional calculus defines a map
$$
\mathscr{S}(\mathbb R)\ni f\mapsto f(D) \in \mathcal{B}(L^2(M:E))
$$
where the operator $f(D)$ can also be expressed by the Fourier inversion formula:
\begin{eqnarray}\label{FIF}
f(D):=\frac{1}{2\pi}\int_{\mathbb R}\hat{f}(s)e^{isD}ds
\end{eqnarray}
(in the sense of strong operator convergence).
 Since the operator $D$ is elliptic, it  follows that $f(D)$  has a smooth kernel by elliptic regularity.

 Let $F\in\mathscr{S}(\mathbb R)$   with $F\equiv 1$ near the origin and set $F_{\varepsilon}(x):=F(\varepsilon x)$. Then we shall obtain the desired regularizing procedure  from $F_{\varepsilon}(D)$. To do this we first take a closer look at  singularity properties available in generalized functions.

As has been mentioned above the space $\mathcal{G}(M:E)$ is a fine sheaf over $M$, hence provides a notion of support. Furthermore,
the notion of wave-front set can be defined for generalized functions both locally and in invariant global terms analogous to  distributions.
For $\Omega\subseteq \mb{R}^n$ open we call $u\in {\mathcal G}(\Omega)$
${\mathcal G}^\infty$-microlocally regular at $(x_0,\xi_0)\in T^*\Omega\setminus 0$ if there exists some
test function $\varphi
\in \ensuremath{{\mathcal D}}(\Omega)$ with $\varphi(x_0)=1$ and a conic neighborhood $\Gamma\subseteq \mb{R}^n\setminus 0$ of $\xi_0$ such that
the Fourier transform ${\mathcal F}(\varphi u)$ is rapidly decreasing in $\Gamma$, i.e., there exists $N$ such that for all $l$,
$$
 \sup_{\xi\in \Gamma} (1+|\xi|)^l|(\varphi u_\varepsilon)^\wedge(\xi)| = O(\varepsilon^{-N}) \qquad (\varepsilon\to 0).
$$
The generalized wave front set of $u$, $\mathrm{WF}_g(u)$, is the complement of the set of points $(x_0,\xi_0)$
where $u$ is ${\mathcal G}^\infty$-microlocally regular. By \cite{Ha}, for any $u\in \mathcal{G}(M)$, $\mathrm{WF}_g(u)$ can naturally be viewed as a subset of $T^*M\setminus0$ . It is  equivalently defined as  (cf.\cite{GH}),
\[
\mathrm{WF}_g(u)=\bigcap_{Pu\in\mathcal{G}^{\infty}(M)} \operatorname{char}(P)\qquad (P\in\Psi^0_{cl}(M)).
\]
Here $\operatorname{char}(P)\subseteq T^*M$ is the characteristic set of the
order  $0$ classical pseudodifferential operator $P$. The singular support of $u$, $\mathrm{sing supp}_g(u)$, is the complement of the maximal open set on which $u\in \ensuremath{{\mathcal G}}^\infty$. It then follows that $\mathrm{sing supp}_g(u) = \mathrm{pr}_1(\mathrm{WF}_g)$.

We are now ready to define  a class of regularization procedures
for distributional sections of a vector bundle.
\subsection{Regularizations}
\begin{defn}\label{bundle_embedding}
   A parametrized family $(T_{\varepsilon})_{\varepsilon\in I}$ of properly supported smoothing operators (in the sense
   of \cite[ch.\ 1.4]{CP}) is
 called an optimal regularization process if
 \begin{enumerate}
 \item \label{moderatevb}
 The regularization of any compactly supported distributional section $s\in\mathcal{E}'(M:E)$ is of moderate growth:
For any continuous seminorm $\rho$ on $\Gamma^{\infty}(M:E)$, there exists some integer $N$ such that
   \[
\rho(T_{\varepsilon}s) = O(\varepsilon^{N}) \qquad (\varepsilon\to 0).
\]
\item\label{Identityvb} The net $(T_{\varepsilon})$ is an approximate identity: for each
$s\in \mathscr{E}'(M:E)$,

\[\underset{\varepsilon \rightarrow 0}{\operatorname{lim}}T_{\varepsilon}s=s\quad \textrm{in}\,\,\mathcal{D}'(M:E).\]

\item\label{negligiblevb} If $u\in\Gamma_c^{\infty}(M:E)$ is a smooth compactly supported section of $E$ then
for all continuous seminorms $\rho$ and given any integer $m$,
\[
\rho(T_{\varepsilon}u-u) = O(\varepsilon^m).
\]

\item\label{singularityvb}
The  induced map $\iota_T: {\mathcal E}'(M:E)\rightarrow  \mathcal{G}(M:E)$  preserves  support, and extends to a sheaf map $\ensuremath{{\mathcal D}}'(M:E)\rightarrow \mathcal{G}(M:E)$ that satisfies,
\begin{eqnarray}\label{MO}
\iota_T(\ensuremath{{\mathcal D}}'(M:E))\cap \mathcal{G}^{\infty}(M:E)= \Gamma^{\infty}(M:E).
\end{eqnarray}
This implies in particular that $\iota_T$ preserves singular support.
\item The map $\iota_T$ preserves wave-front sets in the sense that for any distribution $s\in\ensuremath{{\mathcal D}}'(M:E)$
\[\mathrm{WF}(u)=\mathrm{WF}_g(\iota_T(u)).\]
\end{enumerate}
\end{defn}
As mentioned already we shall use spectral properties of the  elliptic differential operators
to obtain  regularizing processes satisfying the  above conditions.

Our main result in this section is the following.

\begin{thm}\label{operator_embedding}
Let  $F\in \mathscr{S}(\mathbb R)$ be a  Schwartz function such that $F\equiv 1$ near the origin. Let $F_{\varepsilon}(x):=F(\varepsilon x)$.
Given an admissible differential operator $D$,
the family of operators $(F_{\varepsilon}(D))_{\varepsilon\in I}$ provides an optimal regularization process in sense of Def.\ \ref{bundle_embedding}.
\end{thm}
The essential idea of the proof boils down to the following two steps.

First we prove  Theorem \ref{operator_embedding} under the assumption that  the underlying  manifold $M$ is compact. We then use finite propagation speed to extend the result to a general complete Riemannian manifold.

  Thus let us for the time being assume that $M$ is a closed  manifold. Then the space of smoothing operators $\Psi^{-\!\infty}(M:E)$ is a Frech\'et algebra.
 The functional calculus map defined by $D$  induces a smooth map $\phi_D:\mathscr{S}(\mathbb R)\rightarrow \Psi^{-\!\infty}(M:E)$, hence the estimate \ref{moderatevb} holds because $F_{\varepsilon}$ is a moderate net and $F_{\varepsilon}(D)=(\phi_D)_*(F_{\varepsilon})$, where $(\phi_D)_*$ is the induced map on the asymptotic spaces $\mathcal{G}_{\mathscr{S}(\mathbb R)}\rightarrow \mathcal{G}_{\Psi^{-\!\infty}(M)}$. Condition \ref{Identityvb} then follows as the ring map $\phi_D$ preserves approximate units.

When $M$ is compact the operator $D$ has discrete spectrum and the spectrum of $D^2$ satisfies  Weyl's law, namely
\begin{eqnarray}\label{weyl}
N_{D^2}(\lambda):=\#\{\lambda_i\in \operatorname{sp}(D^2)|~~\lambda_i\leq \lambda\}\sim C
\lambda^{\frac{\operatorname{dim}(M)}{2}}.
\end{eqnarray}
The Weyl estimates in conjunction with the fact that $F\equiv 1$ near the  origin now provides the compatibility condition \ref{negligiblevb} (see \cite{D}).

To describe the regularity of a distributional section  $u\in\ensuremath{{\mathcal D}}'(M:E)$ we first note that any distribution provides a map between two Frech\'et spaces, namely between the smoothing operators  $\Psi^{-\!\infty} (M:E)$ and the smooth sections  $\Gamma^{\infty}(M:E)$ by evaluation. More precisely, to any $u\in\mathcal{D}'(M:E)$ we associate the map
\[
\Theta_u: \Psi^{-\!\infty}(M) \rightarrow\Gamma^{\infty}(M:E)\qquad
\Theta_u(T):=T(u).
\]
 The mapping properties of the maps $\Theta_u$ imply that if $u\not\in H^k(M:E)$ for every $k>t$  then given any $\delta>0$,
$\|F_{\varepsilon}(D)u\|^2_{L^2(M:E)}$ is not $O(\varepsilon^{\frac{\operatorname{dim~M}}{2}+t+\delta})$ (see \cite[Lemma 7.4]{D}).
This direct description of Sobolev regularity is in fact stronger than condition 4 and therefore,
\[\iota_{F_{\varepsilon}(D)}(\ensuremath{{\mathcal D}}'(M:E))\cap \mathcal{G}^{\infty}(M:E)= \Gamma^{\infty}(M:E)).\]

The sheaf property of the embedding $\iota_{F_{\varepsilon}(D)}$ as well as the proof of the result for the more general case of a not necessarily compact complete manifold  rely on the finite speed of propagation of $D$, to which we turn next.
\subsection{Finite speed of propagation}
We shall reduce the computation of asymptotics on a complete Riemannian manifold to certain compact manifolds obtained as doubles of suitable compact submanifolds with boundaries.
If $X$ is a compact manifold with boundary,  a double of $X$,
denoted here by $DX$ is a closed manifold obtained by  gluing two copies of $X$  along the boundary $\partial X$ (see, e.g., \cite{Ko}).
If $X$ is a compact manifold with boundary embedded in a Riemannian manifold $M$ of the same dimension and
if $U$ is an open subset of $M$ such that $\bar{U}\subset \operatorname{interior}(X)$, then
there exists a
Riemannian metric on $DX$ such that the inclusion $j:U\hookrightarrow  DX$ becomes an isometric embedding.  Moreover, for any vector bundle $E\rightarrow M$ there exists a corresponding
vector bundle $E_X\rightarrow DX$ such that $E_X|_U$
is canonically isomorphic to $E_{|_U}$. Also, there exists a symmetric  elliptic operator
$D_X$ on $E_X$ that coincides with $D$ on $U$.

Let $u\in\mathcal{E}'(M:E)$ and fix a constant $c>0$.  Then by the Hopf Rinow theorem
the open ball $U:=B_{2c\cdot C_D}(\operatorname{supp}(u))$ (with $C_D$ as defined in (\ref{cd}))
is relatively compact and therefore
contained in a compact manifold
with boundary $X\subseteq M$. By the above $u$ can be identified with a distributional section of $E_X\rightarrow  DX$.

\begin{prop}\label{sheaf} With assumptions  on $u,c$ and $F$ as above $F(D)u$ and $F(D_X)u$ are both supported in $U$ and
\[F(D)u=F(D_X)u.\]
\end{prop}
\begin{proof}
The restrictions of $D$  and $D_X$  to the open set $U$ coincide, hence uniqueness of solutions to \eqref{transport_eqn}  implies that $e^{isD}u$ and $e^{isD_X}u$   agree for $s\le c$.  Hence the
claim follows from the Fourier Inversion Formula  \eqref{FIF}.
\end{proof}
From these observations the proof of Th.\ \ref{operator_embedding} can be deduced along
the following lines (cf.\ \cite[Sec.\ 4]{DHK:10}):

Let $\phi\in \ensuremath{{\mathcal D}}(-c,c)$ satisfy $\phi\equiv 1$ in a neighborhood of $0$ and let $u\in \ensuremath{{\mathcal E}}'(M:E)$. Then

$$
 F_{\varepsilon}(D)u =\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(s)\hat{F_\varepsilon}(s)e^{isD}u\,ds
  =j^*\left(\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(s)\hat{F_\varepsilon}(s)e^{isD_X}u\,ds\right)
$$

Based on this calculation we observe that the estimates  required for Definition \ref{bundle_embedding} in the general case of $F_{\varepsilon}(D)$ follow from that of $F_{\varepsilon}(D_X)$ which have already been established in case of the closed manifold $DX$. Furthermore the sheaf properties of $\iota_{F_{\varepsilon}(D)}$ can be obtained as a consequence of the calculations in Proposition \ref{sheaf}.
     The support of $u$ coincides with the generalized support of $[F_{\varepsilon}(D)u]$.  This implies that the embedding extends  to a sheaf morphism $i_{F_{\varepsilon}}:\ensuremath{{\mathcal D}}'(M:E)\rightarrow  \mathcal{G}(M:E)$.
Preservation of wavefront sets is more involved. We refer to \cite[Th.\ 3.10]{DHK:10} for a
complete proof of this property.
\begin{rem}
In the scalar case, an alternative proof (not relying on the above doubling-technique) of Theorem \ref{operator_embedding} can be found in \cite{DHK:10}, Section 3.
\end{rem}

\subsection{Applications} As a first application consider $E=\bigwedge^*M$, the exterior bundle over a Riemannian manifold $M$ and $D=d+d^*$ where $d^*$ is the Hodge adjoint of the de Rham differential $d$. Then $D$ is symmetric, elliptic and has propagation speed $C_D=1$ since $\sigma_D(x,\xi)^2= -\|\xi\|^2\mathrm{id}$, hence is essentially self-adjoint. Letting $\Delta:= D^2$ be the
Laplace-Beltrami operator on $\Omega^*(M)$, it follows from functional calculus that
$\cos(sD) = \cos(s\sqrt{\Delta})$ on $L^2(M)$.

Now let $F$ be an even Schwartz function which has germ $1$ at the origin. Then
$$
F(\sqrt{\Delta})=\frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{F}(s)\cos(s\sqrt{\Delta})\,ds
$$
as a Bochner integral in $\mathcal{B}(L^2(M))$.

Finally, let $c>0$ and pick an even test function $\phi_c$ with support in $(-2c,2c)$ and
such that $\phi_c \equiv 1$ on $(-c,c)$. Now set
\begin{equation} \label{tepsdef}
T_\varepsilon(\sqrt{\Delta}) := \frac{1}{2\pi} \int_{-\infty}^\infty \phi_c(s) (F_\varepsilon)^\wedge(s) \cos(s\sqrt{\Delta})\, ds.
\end{equation}
Then each $T_\varepsilon(\sqrt{\Delta})$ is a properly supported smoothing operator and
$(T_\varepsilon(\sqrt{\Delta}))_{\varepsilon\in I}$ is an optimal regularization process in the sense of
Def.\ \ref{bundle_embedding}.
From the construction of $T$ in terms of the functional calculus of the Laplace operator we
conclude the following invariance properties of the corresponding embedding $\iota_T$:
\begin{cor} \
\begin{itemize}
\item[(i)] Let $f:M\to M$ be an isometry. Then for any $u\in \ensuremath{{\mathcal D}}'(M)$, $\iota_T(f^*u) = f^*\iota_T(u)$.
\item[(ii)] If $\Psi$ is a pseudodifferential operator commuting with $\Delta$, then $\Psi$
commutes with $\iota_T$.
\end{itemize}
\end{cor}

For the special case $\mb{R}^n$ with the Euclidean metric the above construction gives
$$
  T_\varepsilon u = \mu_\varepsilon * u, \quad \text{with } \quad
    \FT{\mu_\varepsilon}(\xi) = \frac{1}{2\pi} \ensuremath{{\mathcal F}}\big(\phi \frac{1}{\varepsilon}
      \FT{F}(\frac{.}{\varepsilon})\big)(|\xi|).
$$
In particular, for the one-dimensional case $n=1$ we obtain $\mu_\varepsilon = \phi_c \cdot \FT{F}(./\varepsilon)/(2 \pi \varepsilon)$. Note that $\FT{F}(./\varepsilon)/(2 \pi \varepsilon)$ is a standard mollifier, i.e., a Schwartz function with unit integral and all higher moments vanishing. Thus the regularization
process reduces precisely to the usual Colombeau embedding via convolution (\cite{C2,O,GKOS}).

 A further  consequence of the above construction is that  it relates naturally to isomorphisms of vector bundles. Thus let $\phi:E_1\rightarrow E_2$ be a vector-bundle isomorphism (covering the identity map on $M$) and let $D_1$ be an order  one admissible operator on $E_1$. We  choose a Hermitian structure on $E_2$ which makes $\phi$ an isometry. Then the push-forward operator $D_2:=\phi D_1\phi^{-\!1}$ is isospectral to $D_1$.  The naturality of functional calculus then gives that for any Schwartz function $F$ with $F\equiv 1$ near the origin the embedding and the bundle map $\phi$  commute, that is
 \[\phi\circ F_{\varepsilon}(D_1)=F_{\varepsilon}(D_2)\circ\phi.\]
 We note that if $r_1+s_1=r_2+s_2$ then any Riemannian metric provides an isomorphism  of the tensor bundles $\mathfrak{T}^{r_1}_{s_1}\rightarrow \mathfrak{T}^{r_2}_{s_2}$  by `raising  or lowering of indices'. Thus if  we pick the  connection Laplace operators on the tensor bundles and an even Schwartz function $F$ as above we obtain a regularization process that is well behaved with respect to raising and lowering of indices.

\section{Regularization on globally hyperbolic space-times}

We now return to the situation described at the end of Section \ref{intro}.
Thus, throughout this section, $(M,g)$ will be a smooth space-time, i.e., a connected time-oriented Lorentz manifold. We first review the concept of metric splitting for globally hyperbolic
space-times. Building on this we construct invariant regularizations of distributions
on smooth globally hyperbolic space-times.

\subsection{The metric splitting of globally hyperbolic space-times and associated Riemann metrics}
The original definition, due to J. Leray of global hyperbolicity of a space-time $M$
appeared in \cite{L53}. It requires that the set of causal curves connecting two points $p$,
$q\in M$ be compact in the space of all rectifiable paths with respect to a suitable metric topology (\cite[Ch.\ XII, Sec.\ 8, 9]{CB}). In what follows, we will use an equivalent definition,
cf.\ \cite[Sec.\ 6.6]{HE}, and \cite[Ch.\ XII, Th.\ 10.2]{CB}.

 Thus we call a space-time \emph{globally hyperbolic} if it satisfies (a) strong causality and (b) for any $p,q \in M$ the intersection $J^+(p) \cap J^-(q)$ of the causal future $J^+(p)$ of $p$ with the causal past $J^-(q)$ of $q$ is a compact subset of $M$. Thanks to \cite{BS:07} condition (a) may be weakened to causality, i.e., non-existence of closed time-like curves.

As shown by Geroch (cf.\ \cite{Geroch:70}) global hyperbolicity is equivalent to the existence of a Cauchy hypersurface and in turn provides a foliation of $M$ by Cauchy hypersurfaces. Further development of these constructions and techniques led to the following result due to Bernal-S\'{a}nchez (cf.\ \cite{BS:05}) on the so-called metric splitting of a globally hyperbolic space-time $(M,g)$:
There exists a Cauchy hypersurface $S$ in $M$ and an isometry of $(M,g)$ with the Lorentz manifold $(\mb{R}\times S, \lambda)$ with Lorentz metric $\lambda$ given by
\begin{equation}\label{lambda}
    \lambda = - \beta\, dt^2 + h_t,
\end{equation}
where $\beta \in \ensuremath{\mathcal{C}^\infty}(\mb{R}\times S)$ is positive, $(h_t)_{t\in\mb{R}}$ is a smoothly parametrized family of Riemannian metrics on $S$, and $t$ denotes (slightly ambiguously) both the global time function $(t,x) \mapsto t$ and its values. In other words, in order to construct a regularization (or embedding) for distributions on the globally hyperbolic Lorentzian manifold $(M,g)$ we may as well
assume that $(M,g) = (\mb{R}\times S, \lambda)$.

The specific structure of the Lorentz metric $\lambda$ in \eqref{lambda}  suggests to associate with it the Riemann metric
\begin{equation}\label{rho}
  \rho := \beta\, dt^2 + h_t
\end{equation}
on $\mb{R} \times S$ and to simply use the regularization and embedding of distributions on $\mb{R}\times S$ based on this Riemann metric $\rho$. The regularization construction on Riemannian manifolds described above requires completeness of the Riemann metric $\rho$. It may happen that $\rho$ is not complete, however we may then introduce an appropriate conformal factor to obtain a complete Riemannian metric (cf.\ \cite{NO:61}). Note that the latter would amount to introducing the exact same conformal factor for the Lorentz metric $\lambda$ and would not change the class in the so-called causal hierarchy of space-times according to Minguzzi-S\'{a}nchez \cite{MS:08}. Therefore we assume henceforth that $\rho$ is complete.

Denoting by $\Delta_\rho$ the Laplacian w.r.t.\ $\rho$, for any $s\in \mb{R}$ and any $u\in \ensuremath{{\mathcal D}}(M)$ we set
$$
\|u\|_s := \| (1+\Delta_\rho)^{s/2} u \|_{L^2(M)}.
$$
The Sobolev space $H^s(M)$ of order $s$ is the completion of $\ensuremath{{\mathcal D}}(M)$ with respect to this norm.

\begin{ex} (i) Consider $S = \mb{R}$ and $\lambda = - dt^2 + \frac{dx^2}{1 + t^6 x^6}$ as Lorentz metric on $\mb{R}^2$. It is elementary to check that $(\mb{R}^2,\lambda)$ is globally hyperbolic, e.g., by showing that $\{0\} \times \mb{R}$ is a Cauchy hypersurface. The associated Riemann metric $\rho =  dt^2 + \frac{dx^2}{1 + t^6 x^6}$ is not complete, since the hypersurfaces $\{t\} \times \mb{R}$ with $t \neq 0$ are closed and bounded but not compact in $(\mb{R}^2,\rho)$. Multiplying $\rho$ by the function $\alpha \in \ensuremath{\mathcal{C}^\infty}(\mb{R}^2)$, $\alpha(t,x) = 1 + t^6 x^6$, yields a conformal metric which is complete (by the Hopf-Rinow theorem, since lengths of curves w.r.t.\ $\alpha\cdot \rho$ are greater than or equal to their Euclidean lengths).

(ii) Let $(S,h_0)$ be a connected Riemannian manifold. For the Robertson-Walker space-time $\mb{R} \times S$ with  Lorentz metric of the form $\lambda = -dt^2 + f(t)^2 h_0$, where $f \in \ensuremath{\mathcal{C}^\infty}(\mb{R})$ is positive, we have (cf.\ \cite[Lemma A.5.14]{BGP:07}):

\centerline{$(\mb{R}\times S,\lambda)$ is globally hyperbolic if and only if $(S,h_0)$ is complete.}

\noindent If this is the case, then the corresponding Riemannian metric $\rho = dt^2 + f(t)^2 h_0$ on $\mb{R}\times S$ is complete (see \cite{ON:83}, Lemma 7.40).

\end{ex}

\subsection{Regularization and embedding via the associated Riemannian structure}

Let $(T_\varepsilon)_{\varepsilon \in I}$ be the regularization (i.e., family of properly supported smoothing operators) and $\iota \colon \ensuremath{{\mathcal D}}'(\mb{R}\times S) \hookrightarrow \ensuremath{{\mathcal G}}(\mb{R}\times S)$ be the embedding associated with the complete Riemann metric $\rho$ as in (\ref{tepsdef}). We recall that the embedding  thus respects
the differential algebraic structure of $\ensuremath{\mathcal{C}^\infty}(\mb{R}\times S)$ and the wave front sets of distributions in the strong sense of generalized functions, that is, with respect to equality on the level of $\ensuremath{{\mathcal G}}(\mb{R}\times S)$. Furthermore, $\iota$ is also invariant under isometries of the Riemannian structure and commutes with the action of the Laplace operator $\Delta_\rho$ corresponding to $\rho$ on distributions and generalized functions, respectively.

Let $u \in \ensuremath{{\mathcal D}}'(\mb{R}\times S)$. Then according to \eqref{lambda}
the difference between the wave operator $\Box_\lambda$ and the Laplace operator $\Delta_\rho$ acts on $u$ as
\begin{equation}\label{Theta}
    \Box_\lambda u - \Delta_\rho u =
   \frac{-2}{\sqrt{\beta \det h_t}} \,\d_t
   \Big( \sqrt{\frac{\det h_t}{\beta}} \,\d_t u \Big) := - 2 \Theta u.
\end{equation}

From this equation and the commutation property of the embedding $\iota$ with $\Delta_\rho$ we obtain for any $u \in \ensuremath{{\mathcal D}}'(\mb{R}\times S)$
\begin{equation}\label{commutators}
  \iota(\Box_\lambda u) - \Box_\lambda \iota(u) =
  \iota((\Delta_\rho - 2 \Theta)u) - (\Delta_\rho - 2 \Theta) \iota(u) =
  2 \big(\Theta \iota(u) - \iota(\Theta u)\big).
\end{equation}
Thus, the precise invariance properties of the embedding with respect to the wave operator can be reduced to investigating the corresponding behavior upon interchanging $\iota$ with $\Theta$.

As a first simple, but useful, observation we point out that $\iota$ commutes with $\Box_\lambda$ in the distributional sense (i.e., in the sense of association in $\ensuremath{{\mathcal G}}$), which follows from the fact that $\Box_\lambda$ is a differential operator with smooth symbol. This proves the following statement.

\begin{prop} For any $u\in \ensuremath{{\mathcal D}}'(\mb{R}\times S)$ we have $\iota(\Box_\lambda u) \approx \Box_\lambda\, \iota(u)$.
\end{prop}

We will now show that under additional regularity assumptions on the distribution $u$ a stronger asymptotic property holds.

\begin{thm} \label{asymptth} If $u \in H^{3}(\mb{R} \times S)$ is compactly supported then
$$
\norm{\,[T_\varepsilon, \Box_\lambda] u\,}{L^2} =
\mathcal{O}(\varepsilon^2)  \qquad (\varepsilon \to 0).
$$
\end{thm}

\begin{proof}
As in \eqref{commutators} we have the basic relation
$$
    [T_\varepsilon,\Box_\lambda] u = 2 [\Theta, T_\varepsilon] u.
$$
Now recalling the Bochner-integral defining the action of $T_\varepsilon$ on $L^2$-functions we have
$$
  \Theta T_\varepsilon u - T_\varepsilon \Theta u = \frac{1}{2 \pi} \int_\mb{R}
    \phi_c(s) \frac{1}{\varepsilon} \FT{F}(\frac{s}{\varepsilon})
    \Big( \Theta \cos(s \sqrt{- \Delta_\rho}) u -
    \cos(s \sqrt{- \Delta_\rho}) \Theta u \Big)\, ds.
$$
Hence everything boils down to deriving asymptotic estimates for the commutator of $\Theta$ with the operator $S(s) := \cos(s \sqrt{- \Delta_\rho})$ ($s \in \mb{R}$). For any $v\in L^2(\mb{R}\times S)$ the function $w \in \ensuremath{\mathcal{C}^\infty}(\mb{R}, L^2(\mb{R}\times S))$, $w(s) := S(s) v$ ($s \in \mb{R}$), is the mild solution to the following Cauchy problem on $\mb{R} \times \mb{R} \times S$:
\begin{equation}\tag{$*$}
   \d^2_s w - \Delta_\rho w = 0, \quad w(0) = v,\quad \d_s w(0) = 0.
\end{equation}
In the above integral formula the term $\cos(s \sqrt{- \Delta_\rho}) \Theta u$ corresponds to the solution with $v = \Theta u$, whereas the term $\Theta \cos(s \sqrt{- \Delta_\rho}) u$ is just the
application of $\Theta$
to the solution $w(s)$ corresponding to $v = u$. Applying $\Theta$ to ($*$) we obtain
$$
  \d_s^2 \Theta w - \Delta_\rho \Theta w = [\Theta, \Delta_\rho] w =: f
$$
and
$$
 \Theta w \mid_{s = 0} = \Theta(w(0)) = \Theta u,
  \quad \d_s \Theta w \mid_{s=0} = \Theta (\d_s w(0)) = 0.
$$
Therefore the Duhamel principle yields
\begin{multline*}
   \Theta \cos(s \sqrt{- \Delta_\rho}) u =
   \Theta w(s)\\ =  \cos(s \sqrt{- \Delta_\rho}) \Theta u +
   \int_0^s (s-r) \ensuremath{\text{ sinc}}\big((s-r)\sqrt{-\Delta_\rho}\,\big)
   f(r)\, dr,
\end{multline*}
where $\ensuremath{\text{ sinc}} \colon \mb{R} \to \mb{R}$ is given by $\ensuremath{\text{ sinc}}(z) = \frac{\sin(z)}{z}$. In summary, we obtain
$$
   \Theta T_\varepsilon u - T_\varepsilon \Theta u = \frac{1}{2 \pi} \int_\mb{R}
    \phi_c(s) \frac{1}{\varepsilon} \FT{F}(\frac{s}{\varepsilon})
    \int_0^s (s-r) \ensuremath{\text{ sinc}}\big((s-r)\sqrt{-\Delta_\rho}\,\big)
   f(r)\, dr\, ds,
$$
where
$$
   f(r) =  [\Theta, \Delta_\rho] w(r) = [\Theta, \Delta_\rho]
   \cos(r \sqrt{- \Delta_\rho}) u.
$$
Since $[\Theta, \Delta_\rho]$ is of third order and maps $H^3_{\mathrm{comp}}$ into $L^2$ and
$\cos(r \sqrt{- \Delta_\rho})$ has operator norm $1$ on every Sobolev space there exists a constant $C_1 > 0$ such that $\norm{f(r)}{L^2} \leq C_1 \norm{u}{H^3}$ for every $r \in \mb{R}$. Furthermore, the operator norm of $(s-r) \ensuremath{\text{ sinc}}\big((s-r)\sqrt{-\Delta_\rho}\,\big)$ is bounded by $\sup_{z\in\mb{R}} |(s-r) \ensuremath{\text{ sinc}}((s-r)z)| = |s-r|$. Combining these upper bounds we estimate
\begin{multline*}
  \norm{\Theta T_\varepsilon u - T_\varepsilon \Theta u}{L^2} \leq
  \frac{C_1}{2 \pi} \int_\mb{R}
    |\phi_c(s)| \frac{1}{\varepsilon} |\FT{F}(\frac{s}{\varepsilon})|
    \frac{s^2}{2}\, ds \cdot \norm{u}{H^3}\\
    = \frac{C_1}{4 \pi} \norm{u}{H^3}
    \int_\mb{R} |\phi_c(\varepsilon \sigma)| |\FT{F}(\sigma)|
    \sigma^2 \, d\sigma \cdot \varepsilon^2
    \leq \frac{C_1}{4 \pi} \, \norm{u}{H^3} \norm{\phi_c}{L^\infty}
    \norm{\FT{F''}}{L^1}\cdot \varepsilon^2.
\end{multline*}
\end{proof}

\begin{rem}\label{comm_lemma} Applying the reasoning of the proof of Theorem \ref{asymptth}
to the operators $\d_t$ or $M_\alpha$ (multiplication by $\alpha$) instead of $\Theta$ gives
the following additional asymptotic properties:
\begin{itemize}
\item[(i)] If $u \in H^2(\mb{R}\times S)$ then  $\norm{\,T_\varepsilon \d_t u - \d_t T_\varepsilon u\,}{L^2} =
\mathcal{O}(\varepsilon^2)$  ($\varepsilon \to 0$).
\item[(ii)] Let $\alpha \in \ensuremath{\mathcal{C}^\infty}(\mb{R}\times S)$
and let $u \in H^1(\mb{R}\times S)$ have compact support. Then
$$
\norm{\,T_\varepsilon (\alpha u) - \alpha T_\varepsilon u\,}{L^2} =
\mathcal{O}(\varepsilon^2) \qquad (\varepsilon \to 0)
$$
\end{itemize}
\end{rem}
We have constructed the regularization operators $(T_\varepsilon)_{\varepsilon \in I}$ and the embedding $\iota$ of distributions on $\mb{R}\times S$ based on the Riemannian metric $\rho$ given by \eqref{rho}. However, the construction itself does not directly reflect the metric splitting in \eqref{rho} or \eqref{lambda} and so far we have not paid special attention to the foliation by the space-like Cauchy hypersurfaces $\{t\} \times S$ ($t \in \mb{R}$). However, this foliation becomes  essential in case of distributions that allow restriction to these hypersurfaces. The latter is true in particular for distributional solutions to the wave equation $\Box_\lambda u = f$, where $f \in \ensuremath{\mathcal{C}^\infty}(\mb{R}, \ensuremath{{\mathcal D}}'(S))$. In these situations we automatically have $u \in \ensuremath{\mathcal{C}^\infty}(\mb{R}, \ensuremath{{\mathcal D}}'(S))$, since $\ensuremath{\text{Char}}(\Box_\lambda)$ does not contain any elements of the form $(t,x;\pm 1, 0) \in T^*(\mb{R}\times S)$
(cf.\ \cite{D:93}, 23.65.5). Thus for every $t\in\mb{R}$ the value $u(t)$ is an element of $\ensuremath{{\mathcal D}}'(S)$ and the metric splitting \eqref{lambda} provides us with a Riemannian metric $h_t$ on $S$, which can be used to regularize or embed $u(t)$ according to our general construction. For every $t\in\mb{R}$ let $(T^{h_t}_{\varepsilon})_{\varepsilon \in I}$ denote the regularization obtained from the Riemann metric $h_t$ on $S$ and let $\iota_{h_t}$ denote the corresponding embedding $\ensuremath{{\mathcal D}}'(S) \hookrightarrow \ensuremath{{\mathcal G}}(S)$. The following statements compare these with the global constructions on $\mb{R}\times S$.

\begin{thm}
\noindent (i) The embedding $\iota \colon \ensuremath{{\mathcal D}}'(\mb{R}\times S) \hookrightarrow \ensuremath{{\mathcal G}}(\mb{R}\times S)$ respects the metric splitting \eqref{lambda} in the weak sense: If $u\in\ensuremath{\mathcal{C}^\infty}(\mb{R},\ensuremath{{\mathcal D}}'(S))$ then we have
$$
   \forall t \in \mb{R}:\; \iota(u)(t) \approx \iota_{h_t}(u(t)).
$$

\noindent (ii) If $u \in \ensuremath{\mathcal{C}^\infty}(\mb{R},H^2(S))$, then we have for every compact subset $Z \subseteq \mb{R}$
$$
   \sup_{t \in Z}
   \norm{\,(T_\varepsilon u)(t) - T^{h_t}_{\varepsilon} u(t)\,}{L^2(S)} =
\mathcal{O}(\varepsilon^2)  \quad (\varepsilon \to 0).
$$
\end{thm}

\begin{proof} (ii) Pick $\chi\in \ensuremath{{\mathcal D}}(\mb{R})$ such that $\chi\equiv 1$ near $Z$ and let
$\tilde u(t):= \chi(t)u(t)$. It then follows from \cite{DHK:10}, Prop.\ 3.7
that $(T_\varepsilon \tilde u - T_\varepsilon u)_\varepsilon$ is negligible on $\mathop{\mathrm{supp}}(\tilde
u - u)^c \supseteq Z$.
Thus we may without loss of generality assume that the support of $u$ is bounded
in $t$, so $u\in L^2(\mb{R}\times S)$. This allows us to employ the integral formulae
defining $T_\varepsilon$  and $T^{h_t}_\varepsilon$ and obtain
\begin{multline*}
  (T_\varepsilon u)(t) - T^{h_t}_{\varepsilon} u(t) = \\
  \frac{1}{2 \pi}
  \int_\mb{R} \phi_c(s) \frac{1}{\varepsilon} \FT{F}(\frac{s}{\varepsilon})
  \Big( \underbrace{(\cos(s \sqrt{-\Delta_\rho})u)(t)}_{=: w(s,t)}
  - \underbrace{\cos(s \sqrt{- \Delta_{h_t}}) u(t)}_{=: v(s,t)}
  \Big)\, ds,
\end{multline*}
where $v, w \in \ensuremath{\mathcal{C}^\infty}(\mb{R}^2, H^2(S))$ solve the following Cauchy problems, respectively:
\begin{align*}
   (\d_s^2 - \Delta_\rho) w &= 0
   \quad (\text{in } \ensuremath{{\mathcal D}}'(\mb{R}\times \mb{R}\times S)),&
   \quad w \mid_{s=0} &= u,& \quad \d_s w \mid_{s=0} &= 0,\\
   (\d_s^2 - \Delta_{h_t}) v(.,t) &= 0
   \quad(\text{in } \ensuremath{{\mathcal D}}'(\mb{R}\times S)),&
   \quad v(0,t)  &= u(t),& \quad \d_s v (0,t) &= 0.
\end{align*}
Therefore we have at arbitrary, but fixed $t \in \mb{R}$
$$
  \d_s^2 w(.,t) - \Delta_{h_t} w(.,t) = \Delta_\rho w(.,t) - \Delta_{h_t}w(.,t) =: f_t,
$$
where $f_t \in \ensuremath{\mathcal{C}^\infty}(\mb{R}, L^2(S))$, since $u$ (hence $w$) has values in $H^2(S)$. We observe that $\psi_t := w(.,t) - v(.,t)$ satisfies the Cauchy problem
$$
  (\d_s^2 - \Delta_{h_t}) \psi_t = f_t
   \quad(\text{in } \ensuremath{{\mathcal D}}'(\mb{R}\times S)),
   \quad \psi_t(0)  = 0, \quad \d_s \psi_t = 0,
$$
which implies
$$
  \psi_t (s) = \int_0^s (s-r)
  \ensuremath{\text{ sinc}}((s-r) \sqrt{- \Delta_{h_t}}) f_t(r) \, dr.
$$
Hence we obtain
$$
  \norm{\psi_t(s)}{L^2(S)} \leq |\int_0^s (s-r)
  \norm{f_t(r)}{L^2(S)}\, dr|
  \leq \frac{s^2}{2} \cdot \sup_{|\sigma| \leq |s|}
  \norm{f_t(\sigma)}{L^2(S)}
$$
and therefore arrive at
\begin{multline*}
  \norm{(T_\varepsilon u)(t) - T^{h_t}_{\varepsilon} u(t)}{L^2(S)} \leq
  \frac{1}{2 \pi}
  \int_\mb{R} |\phi_c(s)| \frac{1}{\varepsilon} |\FT{F}(\frac{s}{\varepsilon})|
  \norm{\psi_t(s)}{L^2(S)}\, ds\\
  \leq \frac{\varepsilon^2}{4 \pi}
  \int_{-2c/\varepsilon}^{2c/\varepsilon} |\phi_c(\varepsilon \tau)| |\FT{F}(\tau)|
  \tau^2 \sup_{|\sigma| \leq \varepsilon |\tau|}
  \norm{f_t(\sigma)}{L^2(S)}
  \, d\tau \\
  \leq \frac{\varepsilon^2}{4 \pi} \norm{\phi_c}{L^\infty}
  \sup_{|\sigma| \leq 2c} \norm{f_t(\sigma)}{L^2(S)}
  \norm{\FT{F''}}{L^1} = O(\varepsilon^2) \quad (\varepsilon \to 0)
\end{multline*}
uniformly when $t$ varies in a compact set.

(i) Since association is checked by action on test functions in $\ensuremath{{\mathcal D}}(S)$ we may reduce to the case that $\mathop{\mathrm{supp}}(u(t))$ is compact. Then $u(t) \in H^l(S)$ for some $l \in \mb{Z}$ and since $(1 - \Delta_{h_t})^{\pm l/2}$ commutes weakly with $\iota$ and strongly with $\iota_{h_t}$ the assertion follows from (ii).
\end{proof}

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redpajama_set_name: RedPajamaArXiv