THIS IS FOR THE TEST ONLY. You can order the ebook from Cedar Fort (see link.) This course will explore the prevalence of children's media use, many of the media effects that are of most concern to those interested in the well-being of youth, and the social scientific theories that explain how children's brains cognitively and emotional process media content. Finally, the course identifies specific strategies that can be taken to help youth avoid the potentially negative effects, and enjoy the potentially positive effects, of media exposure.
This is the test and evaluation PDF file that goes with Sugar, Salt, & Fat, 2nd edition. It can only be purchased with the eBook.
THIS IS FOR THE TEST ONLY. The e-book can be purchased from Cedar Fort (see link below). It is estimated that 1.4 billion adults—one in three across the globe—are overweight. Registered dietitian and health writer Clare Fleishman explores surprising new suspects contributing to our planet's expanding girth, including air conditioning, drugs, pollution, and others which get little notice as lack of will power shoulders the blame. This important work examines how we live in a new century and why obesity has become a major killer from the American Midwest to the cities of Europe and across the deserts of Africa.


I thought the January Stamp of the Month was perfect for this layout. The title is so true. This baby is truly adored.
So I switched up photo sizes a bit from the original pattern. I bought these photo spots from Kiwi Lane, a place that sells plastic templates in borders, shapes, and photo mats. I love how it takes my creativity to the next level.


One of my addiction is to read and study different successful people and their success principles. I'm trying to apply their principles in my life.
First, Know yourself. It's very important to know where you stand and what are your values, it will be easier for you to know where you going and what you want to achieve in your life. Invest on yourself and work out on your talents and skills.
Third, Practice daily. Practice makes perfect. This is true, if you are newbie in the stock market try to have a demo account or trial account and practice or join PAMpayamang stocks of the week pick on PAMers FB group.
I will leave you the following questions and reflect on it.
Feel free to share your success principles and share it with your friends.


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Q: Extracting mulitple strings in pdf text I am currently encountering an issue with extracting text from a PDF using Python 3.34
I am using the below to extract ID1; This works fine after I tried running this.
text = pdf.getPage(x).extractText()
IDs = extractWhiteText('ID1:([\d]+?)\D', text)

When I want multiple ID's using below; nothing returns.
IDs = extractWhiteText('ID1:([\d]+?)\D' | ID2:([\d]+?)\D, text)

Please can someone assist me in amending the above code to extract multiple ID's.
It may help if I also include the definitions used;
def extractWhiteText(pattern, text):
    IDs = {}
    match = re.search(pattern, text)

    if match:
        IDs = {'cID': int(match.group(1))}

    return IDs

redpajama_set_name: RedPajamaStackExchange


I engaged in concrete batching plant sales for many years, the industry has been the praise of our customers, and explain to everyone below the concrete mixing station to buy knowledge, hoping to be helpful to our customers.
A concrete construction performance label; thereby to choose what the mixing console. Such as water conservancy project you must choose the forced mixing console. Addition, it should be based on available to stir concrete type of material optional ingredients stations and storage silo.
Concrete construction tasks have their duration; these two parameters to choose how much specifications for mixing station set amount of concrete total task M; pouring concrete number of days for T; daily number of hours worked for H; utilization factor K, you should use a mixing station specifications X = M / T * H * K, where K is 0.7-0.9. finished concrete should also be considered in the selection of the transport situation.: direct pumping or vehicle transport transportation vehicle the volume is also an important basis to determine Batching.
Construction environment and construction objects; choose to buy concrete mixing station, should give full consideration to the environmental impact of the construction of objects and construction, to ensure the smooth construction and construction quality in the following cases, we recommend that you best be prepared for.
4. Operating quality of personnel; Generally, small mixing station structure is relatively simple, the control system is also relatively simple, so the lower the requirements for operation and maintenance personnel. Larger station complex structure, high degree of automation, operator requirements this section also higher so you buy mixing station in addition to considering the previous several factors should be considered factors.
Formulated choose; Under normal circumstances, the manufacturer of a mature product formulation, specifications, quantities and varieties.


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The Eagle Regiment has won numerous Grand Champion honors at various local marching band competitions. The Regiment has also won awards for Best Music, Marching, General Effect, Percussion, Color Guard, and Drum Major, along with Straight Superior ratings at the Florida Bandmasters Association Marching Performance Assessment over the past 28 years. The Eagle Regiment Marching Band has performed statewide and nationally in the following events:
2018 Florida Marching Band Coalition State Finals, Class 5A Champion
2018 Bands of America Orlando Regional, 4th Place Finalist
2016 Florida Marching Band Coalition State Finals, Class 5A 1st Runner-Up
2016 Bands of America Atlanta Super Regional, 9th Place Finalist
2013 Florida Marching Band Coalition State Finals, Class 4A 2nd Runner-Up
2004 Bands of America Atlanta Regional, 10th Place Finalist
2003 Florida Marching Band Coalition State Finals, 1st Runner-Up
2003 Bands of America Florida Regional, 5th Place
2003 Bands of America Atlanta Regional, 13th Place
2002 Florida Marching Band Coalition State Finals, 2nd Runner-Up
1999 Tournament of Roses Parade, Pasadena, California
1999 Florida Inaugural Parade, Tallahassee, Florida
1997 Fiesta Bowl Parade
Winner of the 1997 NBC 6 Best Band Search, Miami, Florida
1996 Orange Bowl "King Orange Parade"
1996 Philadelphia Thanksgiving Day Parade
1995 Florida Marlins 1st Home Opener
1995 Peach Bowl Parade Grand Champion
1994 Orlando Thanksgiving Parade
The Stoneman Douglas Wind Orchestra a 2018 Performing Ensemble!
On Tuesday, May 22, 2018, the Stoneman
Douglas Jazz Bands presented "What We Play is Life" – a night of jazz. They were joined on stage by world renowned 14-time Grammy Award winning artist Herbie Hancock, Lincoln Center Jazz artist Wycliffe Gordon, and 14-time Grammy Award musician Paquito D'Rivera.
Eagle Regiment wins back-to-back FMBC State Championships!
The Eagle Regiment concluded the 2018 marching season on Saturday, November 17th, at the Tropicana Dome with a performance that was awarded the highest score of all marching bands at the FMBC State Championship Finals (92.95), including the top score in all three major captions (MUSIC, VISUAL, & GENERAL EFFECT). Stoneman Douglas was named the FMBC CLASS 5A STATE MARCHING BAND CHAMPION for the second year in a row!
Eagle Regiment wins Grand Championship with record score!
Even in the midst of a torrential downpour during the final 3 minutes of the performance, the Eagle Regiment took it to another level at its final FMBC regional of the 2018 season being named Grand Champions of the Park Vista Striking Cobra Invitational with a score of 92.9, the highest score Stoneman Douglas has ever earned at this point in the season! The Eagle Regiment won Best Music, Best Visual, Best General Effect, and Best Percussion. Our next stop is the State Championship Semifinals at the Tropicana Dome in St. Petersburg in just under two weeks.
Eagle Regiment wins Music, named BOA Regional Finalist!
Thirty-one marching bands from Florida, Georgia, Alabama, North Carolina and South Carolina made up a field of highly-competitive bands which included a BOA Grand National Champion as well as BOA Regional Grand Champions. After the Prelims Competition, Stoneman Douglas won BEST MUSIC in class, ended up ranked 3rd overall and was named a 2018 BOA Regional Finalist! After an even more exciting Finals performance, the Eagle Regiment ended up placing 3rd in Music and GE and 5th in Visual to earn an overall 4th Place finish, just .2 shy of 3rd Place! This is the highest finish at a BOA Regional in the history of our school! Click here to see
© 2023 Stoneman Douglas Marching Eagle Regiment.
Admin@Eagleregiment.com

redpajama_set_name: RedPajamaCommonCrawl


This service is currently available to all practices in Eastbourne Hailsham and Seaford CCG.
The service will see most non-urgent Cardiology patients.
The service is for patients aged 18 and over – under 18's must be referred to appropriate Hospital Service.
Patients with undiagnosed chest pain – these should be referred urgently to the Rapid Access Chest Pain Clinic.
Patients with NT-proBNP levels of more than 2000 pg/mL – these should be referred urgently to the Rapid Access Heart failure Clinic.
All referrals MUST include an ECG and relevant Blood Tests.
The clinicians working for The Community Cardiology Service will inform you of any necessary prescribing.
If a referral appears inappropriate, we will return it with suitable explanation. If the patient requires acute care, the referral will be sent to Eastbourne DGH to ensure no delay for the patient with feedback sent to the GP.
NB – We cannot accept referrals which are not on the referral form and do not have a recent ECG and relevant blood tests. This will delay patient triage and potentially affect waiting times for appointments.
Please let us know on the referral form of any special accessibility requirements, e.g. language spoken and if they will be able to bring own interpreter, signing requirements etc.
GPs are sent letters clearly stating final diagnosis, management plan and onward referral where necessary within five days of completion of investigations.
All patients are sent a personalised care plan within five working days following the completion of their investigation. This will reflect the discussions between the patient and clinician during their appointment at the clinic and is written in language that is easily understood.
In appropriate cases patients are given a six-week 'open door' to contact us directly without having to go through their GP if specified follow up or investigation is required. Case example is a patient referred with palpitations who, when seen has no symptoms but is given a further six weeks to have a monitor fitted for an appropriate length of time if their symptoms recur.


We can help you look after your desktop IT by providing a continuous service Health-Check through the following procedures.
Keeping your workstations healthy will enhance performance and reduce expensive downtime.
Managed Workstations consist of three separate levels.
We normally recommend Full Support, an all encompassing package which leaves you free to concentrate on running your business, secure in the knowledge that your desktops are in safe hands.
Set up is quick and easy. You can even opt for a free trial to see how it all works.
Just give our Managed Services team a call and we'll have you up and running in no time.


It's long been a mystery to me that computer science isn't being taught in U.S. schools. No, not computer literacy, which is also important, but often stops at the "how to use application x, y, or z" level. Why are we not teaching students how to program, master, and manage the most powerful aspects of the most important invention of the 20th and 21st century?
I believe there are two reasons, both based in fear.
1. Fear that adding a new "science" will take time away from "real" math and science. In my opinion, the US K-12 math and science curriculum has been frozen in time. It's not relevant or real anymore, and needs a vast overhaul. But there are lots of forces at work to keep the status quo definitions of what kids are taught. And I do mean to draw a distinction between what students are taught and what they learn. For too many young people, what they learn is that math is boring, difficult, and not relevant, and science is about memorizing arcane terms. This is just a shame and waste.
2. Fear that computer science is too hard to teach in K-12. People worry that teachers are already stressed and stretched, that there aren't enough computer science teachers, and that computer science is just something best left to colleges. That's just a cop out. There are lots of teachers who learn to teach all kinds of difficult subjects – no one is born ready to teach chemistry or how to play the oboe, but people learn to do it all the time. Plus, there are computer languages and development tools for all ages, and lots of support on the web for people to try them out.
Previous PostPrevious Will these new tech supplies get used? Yes!


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SEARCH FOR: ALEX BANIN
Top 50 Chicago Projects of 2021
Every year, artists hailing from Chicago prove exactly why this is a city that belongs in the upper echelon of music, and this fact grows more abundantly obvious with every release that is put out. Whether a musician is world renown or locally recognized, there is never a shortage of creativity, effort, or innovation, and whether the rest of the country notices this or not, local fans are well aware of the goldmine that we are sitting on here in the Midwest. Our goal here at Lyrical Lemonade has always been to shine a light on the remarkable sounds coming out of the city every single year, and even when we go in with certain expectations or presumptions, our minds are absolutely blown once we realize just how much dexterous and unique music was brought to life in the span of 12 short months. One thing that is inimitable within the local scene here is simply the sense of community because whether these talents are teaming up with one another or on entirely different levels of fame, everyone seems to show love to each other, ensuring their awareness of the eye-opening and head-turning styles that are being generated across the …
Realistic – [The O'My's] ft. [Alex Banin]
I have been a huge fan + supporter of The O'My's for a minute now, they have made some of my favorite music to come out of Chicago ever, and today they are back with a brand new music video for "Realistic". Everything that The O'My's have done throughout their career has been done with a precise vision and careful intent, which is why they are so well-respected and have built such an impressive catalog thus far. This new visual is no different, along with director Groupthink, The O'My's carefully crafted something beautiful and memorable. For a group that has had many great music videos over the years, this one definitely stands out to me, it just might be their best yet. Watch this new visual below and judge for yourself! Director/Editor: Groupthink
Chris Patrick: The Lemonade Stand Interview
A few months ago I had New Jersey native Chris Patrick come by the Lyrical Lemonade office while he was visiting Chicago and today I am happy to finally be putting out the final product! I spoke with Chris about plenty of subjects including growing up in NJ, his relationship with JID, being featured on the 2K soundtrack, his musical influences, his EP From The Heart, Vol. 2, working with Alex Banin, attending therapy, his goal to eat 5 Popeyes biscuits with no beverage and we even discussed a viral clip that shows him getting dunked on.
Forget About It – [Alex Banin]
Alex Banin is a name that I had only heard about a month ago when she came into the Lyrical Lemonade headquarters for a Lemonade Stand interview (which you should check out here), but ever since, I have made sure to deeply familiarize myself with all of the remarkable skills she has brought to the table in her blossoming music career. When I saw the homie Elliot write about her brand-new EP Did I Imagine that just came out, I went directly to her social media to find the link, and while I was there I saw that she teamed up with the other homie Michael Del Rosario to shoot a music video for the song "Forget About It" off of the project, so I figured I would double down on articles because this is a must-see video. As you make your way through the whole video, there are tons of abstract moments and ideas that I believe they decided to leave up to the individual viewer, with many characters dressed in what looks to be paper Mache bunny heads leading the way of this conceptual video. While this isn't a surprise considering one of these masks accompany her on …
Did I Imagine – [Alex Banin]
Alex Banin is a gem within Chicago's music community and she's been steadily improving since I first heard her music, and today, she is back on our website with her new EP called Did I Imagine. When it comes to unique/angelic vocals, there aren't many that can compete with Alex Banin, her vocal abilities are straight-up jaw-dropping, and she flexed these skills on this new EP. There are four new records on this tape and the run time is just under ten minutes, but it features some of Alex's best music to date. The upbeat introductory record "Closure" is my favorite of the bunch because I loved hearing her attack a more uptempo instrumental, it made for a different type of song than I was used to hearing from her, and I enjoyed hearing her in that bag. All of these songs are unbelievably good though, the third song "Forget About It" is a beautiful track that displays her range, and it actually just got a new Michael Del Rosario visual to match, so check that out after. Stream this brand new EP below and if you enjoy it, be sure to give Alex Banin a follow on Twitter here!
Alex Banin: The Lemonade Stand Interview
Alex Banin is a creative originally from New York who now stays in Chicago, and she has been making a name for herself over the past couple of years with her excellent music releases. A few months back, we brought Alex to the Lyrical Lemonade office for a brand new interview for The Lemonade Stand. We asked Alex about plenty of topics including her time being a collegiate fencer, considering herself a Chicago artist, working with KOTA The Friend + Chuck Inglish, what music she's currently working on musically, where she sees herself in five years, and much more. Alex Banin is a star in the making, you will be seeing her name more + more throughout the next couple of years, so you might as well get familiar with her now!
Watch Alex Banin's new performance on 1Take
One of my favorite new series is 1Take, a liver performance-based series that takes place right in Chicago, at the legendary spot, Classick Studios. Alex Banin is an artist that is from New York originally but resides here in Chicago and we claim her as an artist from the city because her musical ascension took place here. She is a multi-talented singer who I believe has a mass amount of potential, so when I saw that she was the latest guest on this growing series, I couldn't wait to watch it! Needless to say, she did a great job + really showed off her beautiful vocals. Watch this live performance below and be sure to subscribe to Classick Studios YouTube channel for more releases in the future! Produced by @vlushairlines
Hawthorne-[Alex Banin]
Just yesterday, I was asked by someone, "what has been your favorite song of 2020?" For most, this question would undeniably be a difficult one, but for me, the response was quick and simple: "Hawthorne" by Alex Banin. I've heard about a total of 10 songs in my lifetime that have completely changed the way I look at music and they're always the songs that make me feel more than I ever thought I could feel. It is common to listen to music that gets you to bop your head or snap your fingers, but when you come across a track that knocks you off of your feet, it is evident that it's a special one. Alex Banin's latest offering is so good that it really doesn't make any sense. From the moment I heard this girl sing earlier in the year, I knew that she was destined for greatness and "Hawthorne" solidified that belief in my mind all the more. The short, yet fulfilling offering does an amazing job of putting Alex's vocal talents on center stage. Her voice is powerful, yet soothing..drastically unique but familiar at the same time. Her personality exudes super star which is why I …
Premiere: About April (Music Video)-[Alex Banin]
Stellar music is hard to come by, so whenever I receive songs that completely blow me away, I'm endlessly grateful to say the least. Today, it's only right that I share with the readers what was shared with me and why I think it's so spectacular. If you're looking for your latest indulgence, look no further. An artist named Alex Banin is making an appearance on Lyrical Lemonade today for the release of her brand new song titled, "About April". The 22-year-old London born artist, grew up in New York, but now is based in Chicago, where she started making music. The new piece that dropped last week has been preforming well on streaming platforms and its only a matter of time before it catches fire and is heard by everyone. The subtle production was truly what caught my ear at first listen. It is strong but doesn't overpower Alex's beautiful voice which you truly won't want to miss out on. The best artists are those who can tell stories just as well as they can sing, and that perfectly encapsulates Alex in totality. In this peaceful and reflective piece, the up and comer takes us on an introspective journey …
Interview: Inside Kota the Friend's 'Everything'
Kota the Friend details the process behind his new album 'Everything', the impact of fatherhood on his life, and offers advice to artists trying to make it work on their own in our latest interview.


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African American Fraternities and Sororities
by Annie Nelson on June 16, 2014
African American Greek-letter fraternities and sororities created in the 20th century arose from a need for African American students on college campuses to build a support system of brotherhood, sisterhood, and service. The formation of African American Greek-letter societies was in defiance of the long view that African Americans were not intellectually capable of understanding Greek study, as well as to contest their exclusion from white Greek-letter fraternities and sororities. Since their inception, African American fraternities and sororities have played a key role in shaping generations of black leaders. Many of America's most prominent business leaders, scientists, politicians, entertainers, and athletes took their first steps toward creating a better world as a member of a fraternity or sorority.
Since the early 20th century, five major fraternities were created in the United States. Alpha Phi Alpha Fraternity, Inc. was the first African American Greek-letter fraternity created at Cornell University in 1906. Following suit, Kappa Alpha Psi Fraternity was founded (Indiana University, 1911), Omega Psi Phi Fraternity, Inc. (Howard University, 1911), Phi Beta Sigma Fraternity, Inc. (Howard University, 1914), and Iota Phi Theta Fraternity, Inc. (Morgan State University, 1963).
There are four major sororities, all of which were formed in the early 20th century. They include Alpha Kappa Alpha Sorority (Howard University, 1908), Delta Sigma Theta Sorority (Howard University, 1913), Zeta Phi Beta Sorority (Howard University, 1920), and Sigma Gamma Rho Sorority (Butler University, 1922).
Omega Psi Phi Fraternity, Chi Phi Chapter records, 1973-1997
Alpha Phi Alpha Fraternity, Inc.
Kappa Alpha Psi Fraternity, Inc.
Omega Psi Fraternity, Inc.
Phi Beta Sigma Fraternity, Inc.
Iota Phi Theta Fraternity, Inc.
Alpha Kappa Alpha Inc.
Delta Sigma Theta, Inc.
Sigma Gamma Rho Inc.
Zeta Phi Beta Inc.
Links, Inc.
National Pan-Hellenic Council, Inc.
Straight Black.com
by Abby on June 5, 2014


Free advertising analysis papers, essays, and research papers.
Advertising analysis essaysThe advertisement that I chose for this essay is for Absolut Vodka here we have a very popular line of advertisements, which has been running for many years now. The central theme to these ads is that the bottle seems to be always the main focal point. The advert its.


A LGBTQ Veteran-Active Duty Support Group will meet on the third Saturday of the month from 12-1 pm (first meeting will be January 19, 2019) at Identity. Just so you know, although we are calling the group "Veteran-Active Duty," we want this meeting to be for anyone who has ever served in the military in any capacity. This way we include people who may have been dishonorably discharged or served in the military but never achieved veteran status. Having this wider net of membership is one of the reasons why we want this meeting to take place off the VA campus.


The Florida Accreditation Council for Continuing Medical Education and the Florida Department of Health – Bureau of Epidemiology developed a free training program for healthcare providers titled "Recognition, Management and Reporting of Acute Pesticide Poisonings in Florida". The reporting information is specific to Florida, but the medical information is applicable anywhere.
Pesticide Information Resources provides over 150 links to websites containing general homeowner information on pesticides; children and pesticides; mosquitoes and other pests; lawn and garden care; alternatives to chemical pesticides; and other topics.
Annual reports summarizing pesticide poisoning surveillance data, with case narratives, are available.
What should you do if you've become sick due to a pesticide exposure?
If you have recently been exposed to a pesticide and are experiencing symptoms, contact Poison Control at 1-800-222-1222 or call your physician.
To report a misuse of pesticides, contact the Michigan Department of Agriculture and Rural Development (MDARD) at 1-800-292-3939.
For more information, contact the Pesticide Surveillance Coordinator at 1-800-MI-TOXIC (1-800-648-6942) or schwartza@michigan.gov.


we will reply you ASAP! for Modular Container House , Modern Container House , Mobile Container House , our products have won the trust of customers and been very salable both here and abroad.
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This modern and spacious one bedroom apartment, on the 3rd floor with elevator, is well located in the Marais. Fully renovated and equipped to a high standard, the apartment features large flat-screen TV, Miele combination washing machine/dryer, integrated dishwasher, full oven, 4 stove burner, microwave and integrated fridge and freezer.
The separate bedroom looks onto internal courtyard and the large shower opens onto the bedroom. The WC is located off the living area, at the entrance to the apartment.
The double dining (seating 6 people) and living room open onto internal courtyard, offering peace and quiet.
The original exposed beams and wooden floors, combine with the modern conveniences to make this is a charming and modern base, set in the heart of the sort-after upper Marais.


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Biden nominates Cindy McCain for ambassadorship with the UN food agency
Jake Lahut
Cindy McCain.
Pier Marco Tacca/Getty Images
President Biden has tapped Cindy McCain for a post at the United Nations.
Cindy McCain will represent the US at the UN's food agency if she's confirmed by the Senate.
She endorsed Biden during the 2020 campaign and is the widow of the late Sen. John McCain.
Cindy McCain, an Arizona businesswoman and the widow of the late Republican Sen. John McCain, has picked to take over as the US representative for the United Nations Agencies for Food and Agriculture.
The White House announced the nomination on Wednesday, with President Joe Biden making her one of the few Republicans set to enter the administration. Her nomination will require Senate confirmation, similar to other ambassadors.
McCain took heat from former President Donald Trump during the campaign for endorsing Biden, and even got censured by the Arizona GOP, which she called a "badge of honor."
The 67-year-old earned a bachelor's degree from the University of Southern California and started her career as a special education teacher before meeting her future husband, then a US Navy liaison officer to the Senate after coming home from Vietnam, where he was a prisoner of war.
Although the McCains were pitted against Biden and former President Barack Obama in the 2008 presidential campaign, both she and President Biden have described themselves as friends.
"I am deeply honored and look forward the work ahead," McCain tweeted Wednesday.
She also shared congratulatory messages she received, including from Democratic Sen. Krysten Sinema of Arizona.
—Cindy McCain (@cindymccain) June 23, 2021
The UN food agency has a mission to address global hunger and spur efforts to make agriculture more sustainable.
It also coordinates with governments following disasters to shore up their food supplies.
Biden also announced a slew of other ambassadorship picks on Wednesday, which had been delayed during the early months of the administration as they prioritized public health-related posts.
NOW WATCH: What it's like inside North Korea's controversial restaurant chain
White House cindy mccain United Nations


When my kids were tiny, we lived on a military base in Hawaii.
The first day, the baby got locked in the house with the keys.
In a panic, I called the MP's and they came out. They said I would have to wait for a new key to be made.
My neighbor came over. She calmly pushed up one of the pieces of glass in the paneled window, reached in and unlocked the knob.
Is that Savannah?! So so cute!!


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Home › Netflix News › When will 'After Ever Happy' be on Netflix?
When will 'After Ever Happy' be on Netflix?
The fourth After movie has just touched down on Netflix in select regions and others to come.
Published on November 24th, 2022, 9:31 am EST
Picture: Voltage Pictures
The fourth entry in the teen romance movie series After is called After Ever Happy and is headed to Netflix in multiple regions including the US in December 2022.
Based on the novels by Anna Todd, the After movies are often heavily criticized because of their poor screenplays and sometimes atrocious acting. Despite this, they are profitable and popular in many parts of the globe.
A fourth movie, following the three subsequent movies released annually since 2019, is due to be released in theaters on September 7th, 2022, after premiering in London in August 2022.
Beyond a fourth movie, Castille Landon teased that there may be other works telling Variety in August 2021, "I'm working on some other projects in the "After" universe and I'll keep that vague." A fifth movie has since been confirmed and is currently in development with the entry being called After Everything.
Voltage Pictures is the US distribution company behind the movie that sells it to Netflix as an Original in select regions and as a licensed title in others.
What to expect from After Ever Happy
Castille Landon is back for directing duties on the fourth entry with Sharon Soboil returning for writing duties.
The cast in the fourth movie includes Hero Fiennes Tiffin, Josephine Langford, Carter Jenkins, Louise Lombard, Rob Estes, Kiana Madeira, and Chance Perdomo.
A plot guide has been released, which gives us a good idea of what we can expect from the fourth movie:
"As a shocking truth about a couple's families emerges, the two lovers discover they are not so different from each other. Tessa is no longer the sweet, simple, good girl she was when she met Hardin — any more than he is the cruel, moody boy she fell so hard for."
When and will After Ever Happy be on Netflix?
Multiple regions are receiving After Ever Happy but not simultaneously.
For the past two films, we've seen Netflix in Asia receive the movie first before hitting the US and Canada much later.
A few regions of Netflix saw After Ever Happy arrive on Netflix in Asian regions on September 29th, 2022. The movie is labeled and marketed as a Netflix Original title in these regions.
That includes:
Netflix Hong Kong
Netflix India
Netflix Japan
Netflix Philippines
Picture: Netflix
Netflix US and Canada received After We Fell significantly later than Asia, with most regions getting the movie in October 2021, while the US had to wait until mid-January 2022.
In November 2022, Netflix confirmed that two more regions would receive the new movie on Christmas Day 2022.
Regions that are getting the new movie on December 25th includes:
Netflix USA
After Ever Happy — the fourth film in the After film series — starring Josephine Langford and Hero Fiennes Tiffin hits Netflix (in The US) on December 25. pic.twitter.com/wOLIKAE8Yt
— Netflix (@netflix) November 21, 2022
Amazon will carry After Ever Happy in other regions
The past two movies have been released on Amazon Prime Video in several locations, including the United Kingdom and France. That will once again be the case going into the fourth movie and that's despite the original movie, After, streaming in the UK as of September 2022.
Back in August 2021, Prime Video secured the rights to both After We Fell and After Ever Happy. Amazon Prime Video is debuting the movie in September 2022.
Are you looking forward to checking out the fourth After movie on Netflix? Let us know down below.
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\section{Introduction and statement of results}
For $d\geq 2$, suppose that $u,v$ satisfy the homogeneous wave equations
\[
\square u = \square v = 0,
\]
where $\square = \Delta_x - \partial_t^2$ with $(t,x) \in \mathbb{R} \times \mathbb{R}^{d}$, with initial conditions
\[
(u(0),\partial_tu(0)) = (u_0,u_1), \qquad  (v(0),\partial_tv(0)) = (v_0,v_1).
\]
We consider the bilinear global space-time estimates
\begin{equation}\label{FK}
\|D^{\beta_0}D_-^{\beta_-}D_+^{\beta_+}(uv)\|_{L^2({\mathbb R}^{d+1})}\leq C \|(u_0,u_1)\|_{\dot{H}^{\alpha_1}\times\dot{H}^{\alpha_1-1}}\|(v_0,v_1)\|_{\dot{H}^{\alpha_2}\times\dot{H}^{\alpha_2-1}},
\end{equation}
where $\dot{H}^s({\mathbb R}^d)$ denotes the usual homogeneous Sobolev space with norm
\[\|f\|_{\dot{H}^s({\mathbb R}^d)}:=\|D^sf\|_{L^2({\mathbb R}^d)}.\]
Here, we adopt the standard notation $D:=\sqrt{-\Delta_x}$, and the operators $D_+$ and $D_-$ are defined as
\[
\widetilde{D_{\pm}f}(\tau,\xi):=\big||\tau|\pm|\xi|\big|\widetilde{f}(\tau,\xi)
\]
using the space-time Fourier transform
\[
\widetilde{f}(\tau,\xi):=\int_{{\mathbb R}^{d+1}}e^{-it\tau-ix\cdot\xi}f(t,x)\,\mathrm{d}x\mathrm{d}t.
\]
Estimates \eqref{FK} go back to work of Beals in \cite{Beals} and Klainerman--Machedon in \cite{KM'93}; certain special cases were established and used in the study of the null forms for some nonlinear wave equations. Further special cases were proved, and some applications presented, in \cite{KM'95}, \cite{KM'96}, \cite{KM'96-2}, \cite{KM'97}, \cite{KM'97-2} and \cite{KS} among others, and the full range of exponents $(\beta_0,\beta_-,\beta_+,\alpha_1,\alpha_2)$ for which \eqref{FK} holds was found by Foschi--Klainerman in \cite{FK}.

Although not the focus of the present paper, we mention that generalisations of \eqref{FK} where $L^2$ norm on the left-hand side is replaced by a mixed space-time $L_t^qL_x^r$ norm have also been extensively studied. The first progress for $q = r < 2$ was obtained by Bourgain \cite{Bourgain2}, considering bilinear estimates without the multiplier weights but with separated frequency supports, and the sharp estimates (in the sense of exponents) were later obtained by Wolff \cite{Wolff} and subsequently Tao \cite{Tao} (generalised by Lee \cite{Lee}). With the multipler weights, the full range of exponents in the mixed-norm case was obtained (up to endpoints) by Lee--Vargas \cite{LV} for $d \geq 4$, and by Lee--Rogers--Vargas \cite{LRV} when $d=3$, leaving only some non-endpoint cases when $d=2$.

If $\square u = 0$ then we have the standard decomposition into ($+$) and ($-$) waves $u=e^{itD}f_++e^{-itD}f_-$, where $f_\pm$ are given by
\begin{equation}\label{upm}
u(0)=f_++f_-, \quad \partial_t u(0)=iD(f_+-f_-),
\end{equation}
and the half-wave propagator $e^{itD}$ is defined on the Schwartz space using (spatial) Fourier inversion to be
\[
e^{itD}f(x):=\frac{1}{(2\pi)^d}\int_{{\mathbb R}^d}e^{ix\cdot\xi}e^{it|\xi|}\widehat{f}(\xi)\,\mathrm{d}\xi, \qquad (t,x)\in{\mathbb R}\times{\mathbb R}^d.
\]
Using this decomposition, the inequalities \eqref{FK} can be reduced to the following bilinear estimates for the propagator $e^{itD}$:
\begin{equation}\label{FK1s}
\|D^{\beta_0}D_-^{\beta_-}D_+^{\beta_+}\left(e^{itD}f e^{itD}g\right)\|_{L^2({\mathbb R}^{d+1})}\leq C \|f\|_{\dot{H}^{\alpha_1}({\mathbb R}^d)}\|g\|_{\dot{H}^{\alpha_2}({\mathbb R}^d)},
\end{equation}
and
\begin{equation}\label{FK1sbar}
\|D^{\beta_0}D_-^{\beta_-}D_+^{\beta_+}\left(e^{itD}f \overline{e^{itD}g}\right)\|_{L^2({\mathbb R}^{d+1})}\leq C \|f\|_{\dot{H}^{\alpha_1}({\mathbb R}^d)}\|g\|_{\dot{H}^{\alpha_2}({\mathbb R}^d)}.
\end{equation}
In addition to the works cited above, estimates related to \eqref{FK1s} appear in the book \cite{Bourgain}, and applications to well-posedness problems for some nonlinear wave equations are also discussed there.

Closely related to the estimates \eqref{FK1s} and \eqref{FK1sbar}, at least in the symmetric cases $\beta_-=\beta_+$ and $\alpha_1=\alpha_2$, is the following inequality:
\begin{align}\label{BR}
& \|e^{itD} f e^{itD} g \|_{L^2({\mathbb R}^{d+1})}^2 \nonumber \\
& \leq    \frac{2^{\frac{5-7d}{2}} \pi^{\frac{2-5d}{2}}} {\Gamma(\frac{d}{2})}   \int_{\mathbb{R}^{2d}}|\widehat{f}(y_1)|^2|\widehat{g}(y_2)|^2 (|y_1||y_2|)^{\frac{d-1}{2}} (1-y_1^\prime\cdot y_2^\prime)^{\frac{d-3}{2}}\,\mathrm{d}y_1\mathrm{d}y_2,
\end{align}
where $y' := |y|^{-1}y \in \mathbb{S}^{d-1}$. When $d\geq 3$ the constant is sharp, and a full characterisation of extremising initial data is known; this was first proved by Foschi \cite{Foschi} in the case\footnote{This is stated for the case $f=g$, but the same argument in \cite{Foschi} allows $f$ and $g$ to be different.} $d=3$, and for arbitrary dimensions in \cite{BR}. The question of determining sharp constants and identifying extremisers for weighted linear and bilinear space-time estimates for dispersive and wave-like propagators has been studied in a number of recent papers. In addition to \cite{BR} and \cite{Foschi} already mentioned, see \cite{BBJP}, \cite{Carneiro}, \cite{HZ} and \cite{OT} (as well as \cite{Foschi}, again) for the Schr\"odinger equation, \cite{Jeavons}, \cite{OR2} and \cite{Quilodran} for the Klein--Gordon equation, \cite{BezJ} for the wave equation, and \cite{BezSugimoto}, \cite{BezSugimoto2} and \cite{OR} as well as further references contained in these papers, for more general propagators and spatial weights in the linear setting on $L^2$.

Our main result is the following one-parameter family of sharp bilinear inequalities in the case of ($+-$) waves. Before proceeding, we introduce some notation. For $\beta\in{\mathbb R}$, we write
\[
\mathcal{I}_\beta(f,g)=\int_{{\mathbb R}^{2d}}|\widehat{f}(y_1)|^2|\widehat{g}(y_2)|^2 (|y_1||y_2|)^{\frac{d-1}{2} + 2\beta} \left(1 -y_1'\cdot y_2'\right)^{\frac{d-3}{2}+2\beta}\,\mathrm{d}y_1\mathrm{d}y_2,
\]
defined for suitable functions $f$ and $g$. Also, we write $|\square| = D_- D_+$, so that
\[
\widetilde{|\square|^\beta u}(\tau,\xi):=|\tau^2-|\xi|^2|^{\beta}\widetilde{u}(\tau,\xi).
\]
Finally, we set $\beta_d := \max\{\frac{1-d}{4}, \frac{2-d}{2} \}$, so that $\beta_d = \frac{1-d}{4}$ when $d \geq 3$, and $\beta_2 = 0$.
\begin{theorem}\label{thm1}
Let $d\geq 2$ and $\beta>\frac{1-d}{4}$. Then
\begin{equation}\label{b-eq}
\||\square|^\beta(e^{itD}f\overline{e^{itD}g})\|_{L^2(\mathbb{R}^{d+1})}^2\leq \mathbf{W}(\beta,d)\mathcal{I}_\beta(f,g)
\end{equation}
holds with constant
\[
\mathbf{W}(\beta,d)= 2^{\frac{1-5d + 4\beta}{2}} \pi^{\frac{1-5d}{2}}  \frac{\Gamma( \frac{d-1}{2}  + 2\beta)}{\Gamma(d-1+2\beta)}.
\]
For any $\beta > \beta_d$, the constant is optimal and equality holds if and only if
\begin{equation}\label{maxdef}
\widehat{f}(\xi)=\lambda\widehat{g}(\xi)=\frac{e^{a|\xi|+b\cdot\xi+c}}{|\xi|}
\end{equation}
for some $\lambda, a, c\in\mathbb{C}$ with $\operatorname{Re}a<0$, and $b\in\mathbb{C}^d$ with $|\operatorname{Re}b|<-\operatorname{Re}a$.
\end{theorem}
Estimate \eqref{b-eq} implicitly holds for initial data such that $\mathcal{I}_\beta(f,g)$ is finite, and the necessity of the lower bound $\beta > \frac{1-d}{4}$ is evident from the expression for $\mathbf{W}(\beta,d)$. In order to state that the constant is sharp, the additional lower bound $\beta > \frac{2-d}{2}$ arises by evaluating $\mathcal{I}_\beta(f,g)$ on initial data satisfying \eqref{maxdef}.

When $\beta=0$, inequality \eqref{b-eq} recovers \eqref{BR} since the duplication formula for the gamma function gives
\[
\mathbf{W}(0,d) = 2^{\frac{1-5d}{2}} \pi^{\frac{1-5d}{2}}  \frac{\Gamma(\frac{d-1}{2})}{\Gamma(d-1)}  =  \frac{2^{\frac{5-7d}{2}} \pi^{\frac{2-5d}{2}}} {\Gamma(\frac{d}{2})},
\]
and the $(++)$ waves on the left-hand side can be freely switched with $(+-)$ waves. The case $\beta=\frac{3-d}{4}$ in Theorem \ref{thm1} is obviously special since the power of the angular weight defining $\mathcal{I}_\frac{3-d}{4}$ is zero and so by Plancherel's theorem,
\begin{equation} \label{e:threshold}
\mathcal{I}_{\frac{3-d}{4}}(f,g)=(2\pi)^{2d}\|f\|_{\dot{H}^{\frac{1}{2}}(\mathbb{R}^d)}^2\|g\|_{\dot{H}^{\frac{1}{2}}(\mathbb{R}^d)}^2.
\end{equation}
Therefore, \eqref{b-eq} recovers inequality \eqref{FK1sbar} in the case
$$
(\beta_0,\beta_-,\beta_+,\alpha_1,\alpha_2)=\left(0,\frac{3-d}{4},\frac{3-d}{4},\frac{1}{2},\frac{1}{2}\right)
$$
with sharp constant. In this sense Theorem \ref{thm1} unifies \eqref{BR} with certain cases of \eqref{FK1sbar}. Of course, one may also ask for a similar unification of \eqref{BR} with certain cases of \eqref{FK1s}, by establishing analogous sharp estimates to \eqref{b-eq} for $(++)$ waves. It turns out that such estimates are already essentially contained in \cite{BR} and the argument given there for $\beta = 0$ easily extends to general $\beta$. It is not the case that \eqref{b-eq} follows from the arguments in \cite{BR} and hence the main focus in this paper is the more difficult $(+-)$ case. For completeness, we provide a new proof of the analogous estimates to \eqref{b-eq} for $(++)$ waves in the final section, with an argument which clearly elucidates the additional technical difficulty in the $(+-)$ case.

Inequality \eqref{BR} has been shown to imply sharp forms of certain linear $L^4$ Sobolev--Strichartz estimates for the wave equation, by controlling the right-hand side of \eqref{BR} in a sharp manner in terms of classical homogeneous Sobolev norms; this is done in \cite{BR} in five space dimensions, and more recently in \cite{BezJ} in four space dimensions. Here we observe that if the initial data $f$ and $g$ are both radially symmetric, we can use polar coordinates to calculate
\[
\mathcal{I}_\beta(f,g) = 2^{\frac{7(d-1)}{2} + 2\beta} \pi^{\frac{4d-1}{2}} \frac{\Gamma(\frac{d}{2}) \Gamma(d-2+2\beta)}{\Gamma(\frac{3d-5}{2} + 2\beta)} \|f\|_{\dot{H}^{\frac{d-1}{4} + \beta}({\mathbb R}^d)}^2\|g\|_{\dot{H}^{\frac{d-1}{4} + \beta}({\mathbb R}^d)}^2
\]
and the following sharp estimates valid in \emph{all} dimensions may be obtained immediately from Theorem \ref{thm1}.
\begin{corollary} \label{c:radial}
Let $d \geq 2$, $\beta > \beta_d$ and let
\begin{equation} \label{e:Cdefn}
\mathbf{C}(\beta,d)=\frac{2^{d-3+4\beta}\Gamma\left(\frac{d}{2}\right)\Gamma\left(\frac{d-1}{2} + 2\beta\right)}{\pi^{\frac{d}{2}}(d-2+2\beta)\Gamma\left(\frac{3d-5}{2}+2\beta\right)}.
\end{equation}
If $f$ and $g$ are radially symmetric, then
\begin{equation} \label{e:radialall}
\left\||\square|^\beta (e^{itD}f\overline{e^{itD}g})\right\|_{L^2({\mathbb R}^{d+1})}^2\leq \mathbf{C}(\beta,d)\|f\|_{\dot{H}^{\frac{d-1}{4} + \beta}({\mathbb R}^d)}^2\|g\|_{\dot{H}^{\frac{d-1}{4} + \beta}({\mathbb R}^d)}^2,
\end{equation}
where the constant is sharp and equality holds if and only if $f=\lambda g$ satisfy \eqref{maxdef} with $\lambda, c\in\mathbb{C}$, and $a \in \mathbb{C}$ such that $\operatorname{Re}a < 0$, and $b=0$.
\end{corollary}
Without the additional hypothesis of radial symmetry, the estimate \eqref{e:radialall} fails to hold for \emph{any} finite constant when $\beta < \frac{3-d}{4}$.
\begin{proposition}\label{p:counter}
If $\beta<\frac{3-d}{4}$, then for any $A>0$ there exists $f\in\dot{H}^{\frac{d-1}{4}+\beta}({\mathbb R}^d)$ such that
\[\frac{\left\||\square|^\beta|e^{itD}f|^2\right\|_{L^2({\mathbb R}^{d+1})}}{\|f\|_{\dot{H}^{\frac{d-1}{4}+\beta}({\mathbb R}^d)}^2}>A.\]
\end{proposition}
Thus, \eqref{e:radialall} shows that the range of admissible exponents is widened when the initial data are radially symmetric. Moreover, our expression for the sharp constant $\mathbf{C}(\beta,d)$ clearly indicates that the range $\beta > \beta_d$ cannot be further widened, even under the radial hypothesis on the initial data. Results of this type for initial data restricted to the radial case for estimates similar to \eqref{FK} were proved by Klainerman--Machedon in \cite{KM'93}, and for the estimates \eqref{FK} themselves by Foschi in \cite{Foschi3} (in the case $\beta_- = \beta_+  = 0$). The fact that certain estimates, including Strichartz-type estimates, improve on radially symmetric data is well-known and has been studied more widely; see for example \cite{BGO}, \cite{CO}, \cite{FW}, \cite{Hidano}, \cite{HK}, \cite{Sogge}, \cite{Sterbenz}.

Regarding Proposition \ref{p:counter}, we remark that in Example 5.1 of \cite{FK} it is shown that $\beta_- \geq \frac{3-d}{4}$ is a necessary condition for \eqref{FK} to hold by showing that \eqref{FK1s} fails in the ($++$) case. To prove Propositon \ref{p:counter} we must consider the ($+-$) case, and we use a different argument (using initial data of the form \eqref{maxdef}) to establish this.

For a restricted range of $\beta$ we are able to extend Corollary \ref{c:radial} to general initial data without the assumption of radial symmetry, by applying some of the techniques from \cite{BezJ} to make a bound on the right-hand side of \eqref{b-eq} which does not incur an overall loss of sharpness in the constant. In order to state this, we introduce a collection of maps $T_\beta:\dot{H}^{\frac{d-1}{4}+\beta}({\mathbb R}^d)\rightarrow L^1(\mathbb{S}^{d-1})$ defined by
\[T_\beta f(\omega)=\frac{1}{(2\pi)^d}\int_0^\infty |\widehat{f}(r\omega)|^2 r^{\frac{3d-3}{2}+2\beta}\,\mathrm{d}r,\]
for $\omega\in\mathbb{S}^{d-1}$.
\begin{corollary}\label{cor4}
Let $d \geq 2$, $\beta \in (\beta_d, \tfrac{5-d}{4}]$ and let $\mathbf{C}(\beta,d)$ be given by \eqref{e:Cdefn}.

\emph{(i).} If $\beta>\frac{3-d}{4}$, then
\begin{equation}\label{e:alpha}
\left\||\square|^\beta|e^{itD}f|^2\right\|_{L^2({\mathbb R}^{d+1})}^2\leq \mathbf{C}(\beta,d)\|f\|_{\dot{H}^{\frac{d-1}{4} + \beta}({\mathbb R}^d)}^4
\end{equation}
and the constant is sharp, with equality if and only if $f$ satisfies \eqref{maxdef} with $c\in\mathbb{C}$, and $a \in \mathbb{C}, b \in \mathbb{C}^d$ are such that $\operatorname{Re}a < 0$ and $\operatorname{Re}b=0$.

\emph{(ii).} If $\beta\leq\frac{3-d}{4}$ and $p=\frac{2(d-1)}{3d-5+4\beta}$, then
\begin{equation}\label{spec}
\left\||\square|^\beta (e^{itD}f\overline{e^{itD}g})\right\|_{L^2({\mathbb R}^{d+1})}^2\leq \mathbf{C}(\beta,d)|\mathbb{S}^{d-1}|^{\frac{3-d-4\beta}{d-1}}\|T_\beta f\|_{L^p}\|T_\beta g\|_{L^p}
\end{equation}
and the constant is sharp, with equality if and only if $f=\lambda g$ satisfy \eqref{maxdef} with $\lambda, a, c\in\mathbb{C}$ and $b\in\mathbb{C}^d$ where $|\operatorname{Re}b|<-\operatorname{Re}a$.
\end{corollary}
Corollary \ref{cor4} extends Corollary \ref{c:radial} in two ways. Estimate \eqref{e:alpha} extends \eqref{e:radialall} to general initial data in $\dot{H}^{\frac{d-1}{4}+\beta}({\mathbb R}^d)$ for $\beta \in (\frac{3-d}{4},\frac{5-d}{4}]$ (and in the symmetric case $f=g$). The upper threshold $\frac{5-d}{4}$ is a consequence of our method and it is quite possible the sharp estimate can be extended beyond this threshold (however, it seems that a new key idea is necessary for this).

For $\beta \in (\beta_d,\frac{3-d}{4})$, the estimate \eqref{spec} provides a natural substitute for the failure of \eqref{e:alpha} which is valid for all initial data in $\dot{H}^{\frac{d-1}{4}+\beta}({\mathbb R}^d)$, thus extending \eqref{e:radialall} for such $\beta$. Indeed, for such $\beta$ we have $p=\frac{2(d-1)}{3d-5+4\beta} > 1$ and, by Plancherel's theorem and H\"older's inequality,
\begin{equation}\label{PH}
\|f\|_{\dot{H}^{\frac{d-1}{4}+\beta}({\mathbb R}^d)}^2\leq |\mathbb{S}^{d-1}|^{\frac{3-d-4\beta}{2(d-1)}}\|T_\beta f\|_{L^p(\mathbb{S}^{d-1})},
\end{equation}
with equality when $f$ is radially symmetric.

\emph{Organisation.} In Section \ref{s:cor} we prove Proposition \ref{p:counter} and Corollary \ref{cor4}, and in Section \ref{s:proofs} we prove Theorem \ref{thm1} for $(+-)$ waves. Finally, in Section 4, we present analogous results for $(++)$ waves.

\section{Proofs of Proposition \ref{p:counter} and Corollary \ref{cor4}}\label{s:cor}
We begin this section with the proof of Corollary \ref{cor4} following arguments in \cite{BezJ}. Before proceeding, we introduce notation
\[
H_\lambda(f,g):=\int_{\mathbb{S}^{d-1}} \int_{\mathbb{S}^{d-1}}f(\omega_1)\overline{g(\omega_2)}|\omega_1-\omega_2|^{-\lambda}\,\mathrm{d}\omega_1\mathrm{d}\omega_2
\]
where $|\cdot|$ in this context means chordal distance on $\mathbb{S}^{d-1}$ (that is, euclidean distance on ${\mathbb R}^d$) and we fix $\lambda=3-d-4\beta$. Of course, $\lambda \geq 0$ if and only if $\beta \leq \frac{3-d}{4}$, and it is reasonable to expect the behaviour of $H_\lambda$ to change according to the sign of $\lambda$.
\proof[Proof of Corollary \ref{cor4}.]
The proof rests on the observation
\begin{equation} \label{HLStrick}
\mathcal{I}_\beta(f,g)=\frac{(2\pi)^{2d}}{2^{\frac{d-3}{2}+2\beta}}H_\lambda(T_\beta f, T_\beta g)
\end{equation}
which is easily verified.

We begin with the case $\beta \in (\frac{3-d}{4},\frac{5-d}{4}]$ for each $d \geq 2$, which means $\lambda \in [-2,0)$. In this case, we use the following result from \cite{BezJ} to bound $H_\lambda$.
\begin{lemma}\label{claim1}
Let $d\geq 2$, $\lambda \in [-2,0)$, and let $g$ be any $L^1$ function on $\mathbb{S}^{d-1}$. Then,
\[
H_\lambda(g,g) \leq 2^{2d-5+4\beta} \pi^{-\frac{1}{2}} \frac{\Gamma(d-2+2\beta) \Gamma(\frac{d}{2})}{\Gamma(\frac{3d-5}{2} + 2\beta)}   \left|\int_{\mathbb{S}^{d-1}}g\right|^2,
\]
with equality if $g$ is constant. If, in addition $\lambda>-2$, then equality holds only if $g$ is constant.
\end{lemma}
\begin{remark}
Strictly speaking, Lemma \ref{claim1} was proved in \cite{BezJ} only for $d\geq 3$, however the same argument also proves it for $d=2$, using the standard convention $|\mathbb{S}^0|=2$.
\end{remark}
Since $T_\beta f\geq 0$ and we have that
\[
\|T_\beta f\|_{L^1(\mathbb{S}^{d-1})}=\|f\|_{\dot{H}^{\frac{d-1}{4}+\beta}({\mathbb R}^d)}^2
\]
by Plancherel's theorem. Thus, Lemma \ref{claim1} implies that the right-hand side of \eqref{b-eq} with $f=g$ is at most
\begin{equation*}
\mathbf{C}(\beta,d)  \|f\|_{\dot{H}^{\frac{d-1}{4} + \beta}({\mathbb R}^d)}^4,
\end{equation*}
with equality if $T_\beta f$ is constant on $\mathbb{S}^{d-1}$, which certainly happens when $|\widehat{f}|$ is radial. An example of such a function of the form \eqref{maxdef} is given by $\widehat{f}=\frac{e^{-|\cdot|}}{|\cdot|}$ and this establishes sharpness of the constant in \eqref{e:alpha}.

The characterisation of extremisers for inequality \eqref{e:alpha} follows as in \cite{BezJ} without new arguments, so we give only brief details here. The idea is that any extremiser $f$ for \eqref{e:alpha} must be of the form \eqref{maxdef} since we applied \eqref{b-eq} to obtain \eqref{e:alpha}, and for such $f$ we have
\begin{align*}
T_\beta f(\omega) = e^{2\op{Re}c}\int_0^\infty e^{r(2\op{Re}a+2\op{Re}b\cdot\omega)}r^{\frac{3d-7}{2}+2\beta}\,\mathrm{d}r.
\end{align*}
Hence
\begin{equation} \label{e:Tbeta}
T_\beta f(\omega) =\frac{C}{\left(1+\xi\cdot\omega\right)^{\frac{3d-5}{2}+2\beta}}
\end{equation}
for some positive constant $C$ and where $\xi :=\frac{\op{Re}b}{\op{Re}a}\in{\mathbb R}^d$ satisfies $|\xi|<1$ by the assumptions on $a$ and $b$. For $\beta \in (\frac{3-d}{4},\frac{5-d}{4})$ (so that $\lambda \in (-2,0)$), the characterisation of extremisers in Lemma \ref{claim1} means $T_\beta f$ is constant and so $\xi$ must be zero. Hence $\op{Re}b = 0$ and this gives the claimed characterisation of extremisers for \eqref{e:alpha}. The remaining case $\beta = \frac{5-d}{4}$ (so that $\lambda = -2$) is more subtle because there are non-constant extremisers for the estimate in Lemma \ref{claim1}. However, the argument in Section 3 of \cite{BezJ} yields the same conclusion\footnote{Strictly speaking the argument in Section 3 of \cite{BezJ} is for the case $d=5$ but it can easily be shown to generalise to all $d \geq 2$.} that $\op{Re}b = 0$, and we omit the details.

Now we move to the case $\beta \in (\beta_d,\frac{3-d}{4}]$. When $\beta = \frac{3-d}{4}$ (or $\lambda = 0$), \eqref{spec} and the characterisation of extremisers follows from Theorem \ref{thm1} and \eqref{e:threshold}.

In the case $\beta \in (\beta_d,\frac{3-d}{4})$, we have $\lambda \in (0,d-1)$ so we apply the sharp Hardy--Littlewood--Sobolev inequality on the sphere and characterisation of extremisers (due to Lieb \cite{Lieb}) to obtain
\begin{equation}\label{HLS-ineq}
H_\lambda(T_\beta f,T_\beta g)\leq \pi^{\frac{\lambda}{2}}\frac{\Gamma\left(\tfrac{d-1-\lambda}{2}\right)}{\Gamma\left(d-1-\tfrac{\lambda}{2}\right)}\left(\frac{\Gamma\left(d-1\right)}{\Gamma\left(\tfrac{d-1}{2}\right)}\right)^{1-\frac{\lambda}{d-1}}\|T_\beta f\|_{L^p}\|T_\beta g\|_{L^p},
\end{equation}
for
\[p:=\frac{2(d-1)}{2(d-1)-\lambda}=\frac{2(d-1)}{3d-5+4\beta}.\]
Equality holds in \eqref{HLS-ineq} if and only if there exist $C_0, C_1\in\mathbb{C}$ and $\xi\in{\mathbb R}^d$ with $|\xi|<1$ such that
\begin{equation}\label{HLS-ext}
T_\beta f(\omega)=\frac{C_0}{\left(1+\xi\cdot\omega\right)^{\frac{2(d-1)-\lambda}{2}}},\qquad T_\beta g(\omega)=\frac{C_1}{\left(1+\xi\cdot\omega\right)^{\frac{2(d-1)-\lambda}{2}}}.
\end{equation}
That inequality \eqref{spec} is true with the stated constant now follows immediately from estimate \eqref{b-eq} in Theorem \ref{thm1} and using \eqref{HLS-ineq}. The optimality of the constant in \eqref{spec} and characterisation of extremisers is deduced from the additional observation that the functions $f, g$ which exhaust the extremals for \eqref{b-eq}, given by \eqref{maxdef}, also satisfy \eqref{HLS-ext}; this follows immediately from \eqref{e:Tbeta}.\endproof

\proof[Proof of Proposition \ref{p:counter}] Fix $f=f_{\delta}$ such that $\widehat{f}$ is a function of the form \eqref{maxdef} with $c=0$, $a=-1$ and $b=(1-\delta)\mathbf{e}_1$ with $0 < \delta < \frac{1}{100}$; that is
\begin{equation*}
\widehat{f}(\xi)=\frac{e^{-|\xi|+(1-\delta)\mathbf{e}_1 \cdot\xi}}{|\xi|}.
\end{equation*}
As usual $\mathbf{e}_1 = (1,0,\ldots,0)$ denotes the first basis vector in ${\mathbb R}^d$. For the rest of this section we shall let $C$ denote an arbitrary positive constant which may depend on $d$ and $\beta$ but not on $\delta$, and we use $x\lesssim y$ (respectively, $x\gtrsim y$) to mean $x\leq C y$ ($x\geq C y$), where $C$ may be different even in a single chain of inequalities. We denote by $x \sim y$ if both $x \lesssim y$ and $x \gtrsim y$ hold.

From \eqref{e:Tbeta} we have that
\[
T_\beta f(\omega)=\frac{C}{\left(1-(1-\delta)\mathbf{e}_1\cdot\omega\right)^{\frac{3d-5}{2}+2\beta}} = \frac{C}{\left(1-(1-\delta)\mathbf{e}_1\cdot\omega\right)^{\frac{d-1}{p}}},
\]
where $p$ is as in \eqref{spec}; note that $\beta \in (\beta_d,\frac{3-d}{4})$ implies that $p \in (1,2)$. Since $f$ is extremal for \eqref{spec} it is enough to show that
\[
\frac{\|T_\beta f\|_{L^p(\mathbb{S}^{d-1})}}{\|f\|_{\dot{H}^{\frac{d-1}{4}+\beta}({\mathbb R}^d)}^2}=\frac{\|T_\beta f\|_{L^p(\mathbb{S}^{d-1})}}{\|T_\beta f\|_{{L^1(\mathbb{S}^{d-1})}}}\rightarrow\infty
\]
as $\delta \rightarrow 0$. By changing variables, we have that
\[
\|T_\beta f\|_{L^p(\mathbb{S}^{d-1})}=C\left(\int_{-1}^1\frac{(1-t^2)^{\frac{d-3}{2}}}{(1-(1-\delta)t)^{d-1}}\,\mathrm{d}t\right)^{\frac{1}{p}}=:I_1(\delta),
\]
and
\[
\|T_\beta f\|_{L^1(\mathbb{S}^{d-1})}=C\int_{-1}^1\frac{(1-t^2)^{\frac{d-3}{2}}}{(1-(1-\delta)t)^{\frac{d-1}{p}}}\,\mathrm{d}t=:I_2(\delta),
\]
so to complete the proof of Proposition \ref{p:counter} it therefore suffices to prove that
\begin{equation}\label{lb}
I_1(\delta) \gtrsim \delta^{\frac{1-d}{2p}},
\end{equation}
and
\begin{equation}\label{ub}
I_2(\delta) \lesssim \delta^{\frac{1-d}{p}+\frac{d-1}{2}}.
\end{equation}

For this, we let $\sigma > \frac{d-1}{2}$ and change variables ($s = \delta^{-1}(1 - (1-\delta)t$)) to obtain
\begin{align*}
\int_{0}^1\frac{(1-t^2)^{\frac{d-3}{2}}}{(1-(1-\delta)t)^{\sigma}}\,\mathrm{d}t \sim \delta^{\frac{d-1}{2} - \sigma} \int_1^{\frac{1}{\delta}} \frac{(s-1)^{\frac{d-3}{2}}}{s^\sigma} \, \mathrm{d}s  \sim \delta^{\frac{d-1}{2} - \sigma}.
\end{align*}
Applying this with $\sigma = d-1$ immediately yields \eqref{lb}. For \eqref{ub}, by a simple change of variables and the fact that if $t>0$ then $1+(1-\delta)t > 1-(1-\delta)t$, we have
\[
I_2(\delta) \lesssim \int_0^1 \frac{(1-t^2)^{\frac{d-3}{2}}}{(1-(1-\delta)t)^{\frac{d-1}{p}}}\,\mathrm{d}t,
\]
from which \eqref{ub} follows by taking $\sigma = \frac{d-1}{p}$. \endproof

\section{Proof of Theorem \ref{thm1}}\label{s:proofs}
\subsection{Proof of the sharp inequality \eqref{b-eq}}
We broadly follow the strategy in \cite{BBJP} (which in turn was based on the approach in \cite{BBI}) to reduce to evaluating a certain integral over a submanifold of ${\mathbb R}^{2d}$. By Plancherel's theorem and using the relabelling $(x_1,x_2,y_1,y_2)\mapsto(x_1,y_2,y_1,x_2)$,
\begin{align*}
& 2^{-2\beta}(2\pi)^{3d-1}\||\square|^\beta (e^{itD}f\overline{e^{itD}g})\|_{L^2({\mathbb R}^{d+1})}^2 \\
& = \int_{{\mathbb R}^{4d}}(|y_1||y_2|+y_1\cdot y_2)^{2\beta}\widehat{f}(x_1)\overline{\widehat{g}(-x_2)}\overline{\widehat{f}(y_1)}\widehat{g}(-y_2)\delta\binom{-|x_1|+|x_2|+|y_1|-|y_2|}{x_1+x_2-y_1-y_2}\,\mathrm{d}x\mathrm{d}y \\
& = \int_{{\mathbb R}^{4d}}(|y_1||x_2|+y_1\cdot x_2)^{2\beta}\widehat{f}(x_1)\overline{\widehat{g}(-y_2)}\overline{\widehat{f}(y_1)}\widehat{g}(-x_2)\delta\binom{-|x_1|+|y_2|+|y_1|-|x_2|}{x_1+y_2-y_1-x_2}\,\mathrm{d}x\mathrm{d}y \\
& = \int_{{\mathbb R}^{4d}}(|y_1||x_2|+y_1\cdot x_2)^{2\beta}\frac{\widehat{F}(x)\overline{\widehat{F}(y)}}{\left(|x_1||x_2||y_1||y_2|\right)^{\frac{1}{2}}}\delta\binom{|y_1|+|y_2|-|x_1|-|x_2|}{x_1+y_2-y_1-x_2}\,\mathrm{d}x\mathrm{d}y
\end{align*}
for $\widehat{F}(y):=|y_1|^{\frac{1}{2}}|y_2|^{\frac{1}{2}}\widehat{f}(y_1)\widehat{g}(-y_2)$, for $y=(y_1,y_2)\in{\mathbb R}^{2d}$, and where we have used that if $\tau=|y_1|-|y_2|$ and $\xi=y_1+y_2$ then
\[
\left|\tau^2-|\xi|^2\right|=2\left|-|y_1||y_2|-y_1\cdot y_2\right|=2(|y_1||y_2|+y_1\cdot y_2).
\]
We now define a non-negative function on ${\mathbb R}^{4d}$,
\[\Psi_{x,y}=\Psi(x,y)=\left(\frac{|x_1||x_2|}{|y_1||y_2|}\right)^{\frac{1}{2}}.\]
Taking real parts and then applying the arithmetic-geometric mean inequality to $\widehat{F}(x)\Psi(x,y)^{\frac{1}{2}}$ and $\widehat{F}(y)\Psi(x,y)^{-\frac{1}{2}}$, it follows that
\begin{align*}
& 2^{-2\beta}(2\pi)^{3d-1}\||\square|^\beta(e^{itD}f\overline{e^{itD}g})\|_{L^2({\mathbb R}^{d+1})}^2 \\
& \leq \frac{1}{2}\int_{{\mathbb R}^{4d}}(|y_1||x_2|+y_1\cdot x_2)^{2\beta}\frac{|\widehat{F}(y)|^2\Psi_{x,y}^{-1}+|\widehat{F}(x)|^2\Psi_{x,y}}{\left(|x_1||x_2||y_1||y_2|\right)^{\frac{1}{2}}}\delta\binom{|y_1|+|y_2|-|x_1|-|x_2|}{x_1+y_2-y_1-x_2}\,\mathrm{d}x\mathrm{d}y \\
& = \int_{{\mathbb R}^{4d}}(|y_1||x_2|+y_1\cdot x_2)^{2\beta}\frac{|\widehat{F}(y)|^2}{|x_1||x_2|}\delta\binom{|y_1|+|y_2|-|x_1|-|x_2|}{x_1+y_2-y_1-x_2}\,\mathrm{d}x\mathrm{d}y,
\end{align*}
and equality holds if and only if
\[\widehat{F}(x)\Psi(x,y)^{\frac{1}{2}}=\widehat{F}(y)\Psi(x,y)^{-\frac{1}{2}},\]
or equivalently
\begin{equation}\label{f-eq}
|x_1||x_2|\widehat{f}(x_1)\widehat{g}(-x_2)=|y_1||y_2|\widehat{f}(y_1)\widehat{g}(-y_2)
\end{equation}
almost everywhere on the support of the delta measures; for example this equation is satisfied by $f,g$ given by \eqref{maxdef}.

We now define
\begin{equation} \label{e:Idefn}
\mathrm{I}_\beta(y):=\int_{{\mathbb R}^d}\int_{{\mathbb R}^d}\frac{(|y_1||x_2|-y_1\cdot x_2)^{2\beta}}{|x_1||x_2|}\delta\binom{|x_1|+|x_2|-|y_1|-|y_2|}{x_1+x_2-y_1-y_2}\,\mathrm{d}x_1\mathrm{d}x_2,
\end{equation}
for $y=(y_1,y_2)\in{\mathbb R}^{2d}$; by changing variables $(x_2,y_2) \mapsto -(x_2,y_2)$, to complete the proof of the sharp inequality it suffices to prove the following key lemma.
\begin{lemma}\label{L1}
For $y\in{\mathbb R}^{2d}$ and $\beta>\frac{1-d}{4}$ we have that
\[
\mathrm{I}_\beta(y) = (2\pi)^{\frac{d-1}{2}} \frac{\Gamma(\frac{d-1}{2} + 2\beta)}{\Gamma(d-1+2\beta)}  (|y_1||y_2|-y_1\cdot y_2)^{\frac{d-3}{2}+2\beta}.
\]
\end{lemma}
\begin{remark} In the case $\beta=0$, Lemma \ref{L1} was proved for $d=3$ by Foschi in \cite{Foschi} and for general dimensions in \cite{BR}. In each case the proof proceeds by  applying an appropriate Lorentz transformation to reduce to the case $y_1=-y_2$; we shall see that in fact this line of argument allows us to obtain the result for more general values of $\beta$. We also remark that in the case $\beta=0$, Lemma \ref{L1} may be seen to hold via a direct calculation using the homogeneity of the delta measure, and is contained in Lemma 4.1 of the earlier paper \cite{FK}.
\end{remark}

\proof[Proof of Lemma \ref{L1}]
In order to shorten the formulas, in what follows we define $\tau=|y_1|+|y_2|$ and $\xi=y_1+y_2$. It is then easy to see that $\mathrm{I}_\beta(y)$ equals
\begin{align*}
& \int_{{\mathbb R}^{2d+2}}\frac{\delta(\sigma_1-|x_1|)}{|x_1|}\frac{\delta(\sigma_2-|x_2|)}{|x_2|}\delta\binom{\sigma_1+\sigma_2-\tau}{x_1+x_2-\xi}(|y_1||x_2|-y_1\cdot x_2)^{2\beta} \,\mathrm{d}\sigma_1\mathrm{d}\sigma_2\mathrm{d}x_1\mathrm{d}x_2 \\
& =\int_{{\mathbb R}^{d+1}}\int_{{\mathbb R}^{d+1}}\frac{\delta(\sigma_1-|x_1|)}{|x_1|}\frac{\delta(\sigma_2-|x_2|)}{|x_2|}\delta\left(\begin{pmatrix}\sigma_1 \\ x_1\end{pmatrix}+\binom{\sigma_2}{x_2}-\binom{\tau}{\xi}\right) \\
& \qquad\qquad\qquad\qquad\times(|y_1|\sigma_2-y_1\cdot x_2)^{2\beta} \,\mathrm{d}\binom{\sigma_1}{x_1}\mathrm{d}\binom{\sigma_2}{x_2}.
\end{align*}
We now introduce the Lorentz transformation $L$, given by
\begin{equation}\label{L-def}
L\binom{t}{x}=\binom{\gamma(t-v\cdot x)}{x+\big(\frac{\gamma-1}{|v|^2}v\cdot x-\gamma t\big)v}, \qquad (t,x) \in {\mathbb R} \times {\mathbb R}^d
\end{equation}
with $v:=-\frac{\xi}{\tau}$ and
\begin{equation}\label{gamma}
\gamma:=\frac{1}{(1-|v|^2)^{\frac{1}{2}}}=\frac{\tau}{(\tau^2-|\xi|^2)^{\frac{1}{2}}}.
\end{equation}
It is not hard to check that
\[L\binom{(\tau^2-|\xi|^2)^{\frac{1}{2}}}{0}=\binom{\tau}{\xi},\]
that $|\op{det}L|=1$ and that the measure $|x|^{-1}\delta(t-|x|)$ is invariant under the transformation $L$. Applying the change of variables $\binom{\widetilde{\sigma_j}}{\widetilde{x_j}}=L^{-1}\binom{\sigma_j}{x_j}$ for $j=1,2$ it follows that
\begin{align*}
\mathrm{I}_\beta(y) & =\int_{{\mathbb R}^{2(d+1)}}\frac{\delta(\widetilde{\sigma_1}-|\widetilde{x_1}|)}{|\widetilde{x_1}|}\frac{\delta(\widetilde{\sigma_2}-|\widetilde{x_2}|)}{|\widetilde{x_2}|}\delta\left(L\begin{pmatrix}\widetilde{\sigma_1} \\ \widetilde{x_1}\end{pmatrix}+L\binom{\widetilde{\sigma_2}}{\widetilde{x_2}}-L\binom{(\tau^2-|\xi|^2)^{\frac{1}{2}}}{0}\right)\\
& \qquad\qquad\qquad\times\left(\binom{|y_1|}{-y_1}\cdot L\binom{\widetilde{\sigma_2}}{\widetilde{x_2}}\right)^{2\beta} \,\mathrm{d}\binom{\widetilde{\sigma_1}}{\widetilde{x_1}}\mathrm{d}\binom{\widetilde{\sigma_2}}{\widetilde{x_2}} \\
& = \int_{{\mathbb R}^{2d+2}}\frac{\delta(\widetilde{\sigma_1}-|\widetilde{x_1}|)}{|\widetilde{x_1}|}\frac{\delta(\widetilde{\sigma_2}-|\widetilde{x_2}|)}{|\widetilde{x_2}|}\delta\binom{\widetilde{\sigma_1}+\widetilde{\sigma_2}-(\tau^2-|\xi|^2)^{\frac{1}{2}}}{\widetilde{x_1}+\widetilde{x_2}} \\
& \qquad\qquad\qquad\times \left(\binom{|y_1|}{-y_1}\cdot L\binom{\widetilde{\sigma_2}}{\widetilde{x_2}}\right)^{2\beta}\,\mathrm{d}\widetilde{\sigma_1}\mathrm{d}\widetilde{\sigma_2}\mathrm{d}\widetilde{x_1}\mathrm{d}\widetilde{x_2} \\
& =\int_{{\mathbb R}^d}\frac{1}{|x|^2}\delta\left(2|x|-(\tau^2-|\xi|^2)^{\frac{1}{2}}\right)\left(\binom{|y_1|}{-y_1}\cdot L\binom{|x|}{x}\right)^{2\beta}\,\mathrm{d}x,
\end{align*}
where the last line follows by evaluating the integrals in $\widetilde{\sigma_1},\widetilde{\sigma_2}$ and $\widetilde{x_1}$ and then relabeling $\widetilde{x_2}=x$. We are now required to compute the quantity
\[\binom{|y_1|}{-y_1}\cdot L\binom{|x|}{x}=|y_1|\gamma(|x|-v\cdot x)-y_1\cdot\left(x+\left(\frac{\gamma-1}{|v|^2}v\cdot x-\gamma |x|\right)v\right)\]
for $v=-\frac{\xi}{\tau}$, $\gamma$ given by \eqref{gamma} and $x$ on the support of the remaining delta measure; this is contained in the following.

\begin{lemma}\label{L2}
For each $y_1,y_2\in{\mathbb R}^d$ there exists $\omega_\ast\in\mathbb{S}^{d-1}$ such that
\[\binom{|y_1|}{-y_1}\cdot L\binom{|x|}{x}=\frac{|y_1||y_2|-y_1\cdot y_2}{2}\left(1+\frac{x}{|x|}\cdot \omega_\ast\right)\]
for any $x\in{\mathbb R}^d$ with $2|x|=(\tau^2-|\xi|^2)^{\frac{1}{2}}$.
\end{lemma}
Assuming Lemma \ref{L2} to be true for the moment, using polar co-ordinates, the definitions of $\tau$ and $\xi$, and then rotation invariance to take $\omega_\ast$ to $\mathbf{e}_1$,
\begin{align*}
\mathrm{I}_\beta(y)
 = \frac{\left(|y_1||y_2|-y_1\cdot y_2\right)^{\frac{d-3}{2} + 2\beta}}{2^{2\beta+\frac{d-1}{2}}}\int_{\mathbb{S}^{d-1}}(1+\omega \cdot \mathbf{e}_1)^{2\beta}\,\mathrm{d}\omega.
\end{align*}
By changing variables, it follows that if $\beta>\frac{1-d}{4}$ then the integral term is
\begin{align*}
 |\mathbb{S}^{d-2}| \int_{-1}^1 (1+t)^{\frac{d-3}{2} + 2\beta}(1-t)^{\frac{d-3}{2}}\,\mathrm{d}t & = |\mathbb{S}^{d-2}|2^{d-2 + 2\beta} \mathrm{B}\left(\frac{d-1}{2} + 2\beta,\frac{d-1}{2}\right) \\
 & = 2^{d-1 + 2\beta} \pi^{\frac{d-1}{2}} \frac{\Gamma(\frac{d-1}{2} + 2\beta)}{\Gamma(d-1+2\beta)},
\end{align*}
where $\mathrm{B}$ is the classical beta function, and we have used the identity $\mathrm{B}(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ for appropriate $x$ and $y$. The claimed expression for $\mathrm{I}_\beta$ in the statement of Lemma \ref{L1} follows.  \endproof

It only remains to prove Lemma \ref{L2}.
\proof[Proof of Lemma \ref{L2}] After some straightforward calculations and simplifications we deduce that
\begin{align*}
L\binom{|x|}{x} & =\frac{1}{(\tau^2-|\xi|^2)^{\frac{1}{2}}}\binom{\tau|x|+\xi\cdot x}{x(\tau^2-|\xi|^2)^{\frac{1}{2}}+\xi\big(|x|+\xi\cdot x(\tau+(\tau^2-|\xi|^2)^{\frac{1}{2}})^{-1}\big)} \\
& = \frac{|x|}{(\tau^2-|\xi|^2)^{\frac{1}{2}}}\binom{\tau+\xi\cdot x^\prime}{x^\prime(\tau^2-|\xi|^2)^{\frac{1}{2}}+\xi\big(1+\xi\cdot x^\prime(\tau+(\tau^2-|\xi|^2)^{\frac{1}{2}})^{-1}\big)},
\end{align*}
where we recall $x^\prime:=\frac{x}{|x|}$. By our assumption $|x|=\frac{1}{2}(\tau^2-|\xi|^2)^{\frac{1}{2}}$,
\[
\binom{|y_1|}{-y_1}\cdot L\binom{|x|}{x}=\frac{1}{2}\left(|y_1|(\tau+\xi\cdot x^\prime) -2y_1\cdot x-y_1\cdot\xi\bigg(1+\frac{\xi\cdot x^\prime}{\tau+2|x|}\bigg) \right).
\]
Moreover, using the definitions $\tau=|y_1|+|y_2|$ and $\xi=y_1+y_2$,
\[|y_1|\tau-y_1\cdot\xi=|y_1||y_2|-y_1\cdot y_2,\]
and
\begin{align*}
& |y_1|\xi\cdot x^\prime-  2y_1\cdot x-\xi\cdot x^\prime\frac{\xi\cdot y_1}{\tau+2|x|} \\
& =x^\prime\cdot\left(|y_1|\xi-2|x|y_1-\xi\frac{\xi\cdot y_1}{\tau+2|x|}\right)  \\
& =\frac{1}{\tau+2|x|} x^\prime\cdot\bigg(\xi\big(|y_1|(\tau+2|x|)-\xi\cdot y_1\big)-2|x|(\tau+2|x|)y_1\bigg) \\
& =x^\prime\cdot\left(y_2\left(\frac{|y_1||y_2|-y_1\cdot y_2+2|x||y_1|}{|y_1|+|y_2|+2|x|}\right)-y_1\left(\frac{|y_1||y_2|-y_1\cdot y_2+2|x||y_2|}{|y_1|+|y_2|+2|x|}\right)\right)\\
& = x^\prime \cdot \frac{2|x|\big(y_2(|x|+|y_1|)-y_1(|x|+|y_2|)\big)}{|y_1|+|y_2|+2|x|},
\end{align*}
we get
\[
\binom{|y_1|}{-y_1}\cdot L\binom{|x|}{x}=\frac{1}{2}\left(|y_1||y_2|-y_1\cdot y_2+x^\prime\cdot z\right),
\]
where
\[
z=z(y_1,y_2):=\frac{2|x|\big(y_2(|x|+|y_1|)-y_1(|x|+|y_2|)\big)}{|y_1|+|y_2|+2|x|}.
\]
For $x$ such that
\begin{equation} \label{e:xcondition}
2|x|^2 = \frac{1}{2}(\tau^2-|\xi|^2) =  |y_1||y_2|-y_1\cdot y_2,
\end{equation}
we claim that
\begin{equation} \label{e:zlength}
|z| = |y_1||y_2|-y_1\cdot y_2
\end{equation}
which would complete the proof of Lemma \ref{L2} by taking $\omega_\ast = z'$.

To see \eqref{e:zlength}, first note that by expanding the square,
\begin{align*}
& \left|2|x|\big(y_2(|x|+|y_1|)-y_1(|x|+|y_2|)\big)\right|^2 \\
& = 4|x|^2\left( |x|^2(|y_1|^2 + |y_2|^2 - 2y_1 \cdot y_2) + 2(|y_1||y_2| - y_1 \cdot y_2)(|y_1||y_2| + |x||y_1| + |x||y_2|) \right)
\end{align*}
which may be rewritten as
\begin{align*}
& 2|x|^2\big(2|x|^2(|y_1|^2 + |y_2|^2 - 2y_1 \cdot y_2) \\
& \qquad + (|y_1||y_2| - y_1 \cdot y_2)(2|y_1||y_2| + 4|x||y_1| + 4|x||y_2|) + 4|x|^2|y_1||y_2|  \big).
\end{align*}
Using \eqref{e:xcondition} repeatedly, this can be expressed as
\begin{align*}
& 2|x|^2\big(2|x|^2(|y_1|^2 + |y_2|^2) + 4|x|^2( |y_1||y_2| - y_1 \cdot y_2) \\
& \qquad + (|y_1||y_2| - y_1 \cdot y_2)(2|y_1||y_2| + 4|x||y_1| + 4|x||y_2|)  \big)
\end{align*}
which is equal to
\begin{align*}
2|x|^2(|y_1||y_2| - y_1 \cdot y_2)(|y_1| + |y_2| + 2|x|)^2 = (|y_1||y_2| - y_1 \cdot y_2)^2(|y_1| + |y_2| + 2|x|)^2,
\end{align*}
and from this we obtain \eqref{e:zlength}.
\endproof

\subsection{Cases of equality in Theorem \ref{thm1}}
The remaining task in the proof of Theorem \ref{thm1} is to establish the claim regarding the characterisation of the extremisers.
As we have already observed, since the only place an inequality was used in the proof of \eqref{b-eq} was in the application of the arithmetic-geometric mean inequality, it follows that equality holds in \eqref{b-eq} if and only if (replacing $x_2$ and $y_2$ with $-x_2$ and $-y_2$ in \eqref{f-eq})
\[|x_1||x_2|\widehat{f}(x_1)\widehat{g}(x_2)=|y_1||y_2|\widehat{f}(y_1)\widehat{g}(y_2)\]
for almost every $(x_1,x_2,y_1,y_2)\in{\mathbb R}^{4d}$ satisfying
$$
x_1+x_2 = y_1+y_2 \qquad \mbox{and} \qquad |x_1|+|x_2|=|y_1|+|y_2|.
$$
Equivalently, equality holds if and only if
\begin{equation}\label{cone-eq}
h_1(x_1)h_2(x_2)=\Lambda(|x_1|+|x_2|,x_1+x_2)
\end{equation}
for almost every $x_1,x_2\in{\mathbb R}^d$ and some scalar function $\Lambda$, where $h_1:=|\cdot|\widehat{f}$ and $h_2:=|\cdot|\widehat{g}$, and, by symmetry and re-scaling, we can assume that $h_1=h_2=h$.

The functional equation \eqref{cone-eq} was solved by Foschi in \cite{Foschi} in the case $d=3$ under the assumption that $h$ is locally integrable, and this argument was generalised to obtain the solution for all $d \geq 2$ (under the same local integrability assumption) in \cite{BR}. In particular, the set of locally integrable functions $h$ satisfying \eqref{cone-eq} have the form
\[
h(x) = e^{a|x| + b \cdot x + c}
\]
for some $a, c \in \mathbb{C}$ and $b \in \mathbb{C}^d$.

Our goal will be to show that $|h|^\gamma$ is locally integrable for some $\gamma > 0$. Once established, it follows from \eqref{cone-eq} that
\[
|h(x)|^\gamma =  e^{a|x| + b \cdot x + c}
\]
for some $a, c \in \mathbb{R}$ and $b \in \mathbb{R}^d$, and taking $\gamma^{-1}$ powers, we obtain the desired form for $|h(x)|$. It then follows that $\frac{h}{|h|}$ is a bounded (hence locally integrable) solution of \eqref{cone-eq} and therefore
\[
\frac{h(x)}{|h(x)|} =  e^{ia'|x| + ib' \cdot x + ic'}
\]
for some $a', c' \in \mathbb{R}$ and $b' \in \mathbb{R}^d$. This gives the claimed characterisation of extremisers in Theorem \ref{thm1}.

To justify the remaining claim that $|h|^\gamma$ is locally integrable for some $\gamma > 0$, proceeding initially as in \cite{BR}, if $B$ is the euclidean ball centered at the origin of radius $N$, then by the Cauchy--Schwarz inequality,
\begin{align*}
\left(\int_B |h| \right)^4 & \leq \mathcal{I}_\beta(f,f) \int_B\int_B(|x_1||x_2|)^{\frac{5-d}{2}-2\beta}(1-x_1^\prime\cdot x_2^\prime)^{\frac{3-d}{2}-2\beta}\,\mathrm{d}x_1\mathrm{d}x_2 \\
& =  \mathcal{I}_\beta(f,f) \left(\int_{\mathbb{S}^{d-1}}\int_{\mathbb{S}^{d-1}}(1-\omega_1\cdot\omega_2)^{\frac{3-d}{2}-2\beta}\,\mathrm{d}\omega_1\mathrm{d}\omega_2\right)\left(\int_0^N r^{\frac{d+3}{2}-2\beta}\,\mathrm{d}r\right)^2.
\end{align*}
The integral in $r$ is finite when $\beta<\frac{d+5}{4}$, but the integral over the sphere is equal
\[
C_d\int_{-1}^1 (1-t)^{\frac{d-3}{2}}(1+t)^{-2\beta}\,\mathrm{d}t,
\]
which is finite for $\beta<\frac{1}{2}$.

Now suppose $\beta \geq \frac{1}{2}$, for which we make use of the reverse Hardy--Littlewood--Sobolev inequality on the sphere (for a proof see \cite{Beckner}), which states that if $g\geq 0$ then
\begin{equation*}
H_\lambda(g) \gtrsim \|g\|_{L^p(\mathbb{S}^{d-1})}^2,
\end{equation*}
where $\lambda<0$ and $p:=\frac{2(d-1)}{2(d-1)-\lambda}\in\left(0,1\right)$. Using \eqref{HLStrick}, this means
\[
\mathcal{I}_\beta(f,f) \geq C_{\beta,d} \|T_\beta f\|_{L^p(\mathbb{S}^{d-1})}^2
\]
where $p = \frac{2(d-1)}{3d-5+4\beta} \in (0,\frac{2}{3}]$, and thus it suffices to show
\begin{equation} \label{e:lastone}
\int_B |h|^{2p} \leq C_{\beta,d,N} \|T_\beta f\|_{L^p(\mathbb{S}^{d-1})}^p.
\end{equation}
Applying H\"older's inequality and Minkowski's inequality for integrals, using that $\frac{1}{p} > 1$, we obtain
\begin{align*}
\int_B |h(x)|^{2p} \,\mathrm{d}x
& = \int_0^N \left(\int_{\mathbb{S}^{d-1}}|\widehat{f}(r\omega)|^{2p} r^{2p+d-1} \,\mathrm{d}\omega\right)\,\mathrm{d}r \\
& \leq C_{\beta,d,N}\left(\int_0^N\left(\int_{\mathbb{S}^{d-1}}|\widehat{f}(r\omega)|^{2p}r^{2p+d-1}\,\mathrm{d}\omega\right)^{\frac{1}{p}} \,\mathrm{d}r\right)^{p}\\
& \leq C_{\beta,d,N} \int_{\mathbb{S}^{d-1}}\left(\int_0^N |\widehat{f}(r\omega)|^2 r^{2 + \frac{d-1}{p}}\,\mathrm{d}r\right)^{p}\mathrm{d}\omega.
\end{align*}
Since $2 + \frac{d-1}{p} > \frac{3}{2}(d-1) + 2\beta$, it follows that $r^{2 + \frac{d-1}{p}} \leq C_N r^{\frac{3}{2}(d-1) + 2\beta}$ for all $r \in [0,N]$, and \eqref{e:lastone} follows.

\endproof

\section{Sharp bilinear estimates for ($++$) waves}

In this last section, we present analogous results to Theorem \ref{thm1}, Corollary \ref{c:radial} and Corollary \ref{cor4} for $(++)$ waves instead of $(+-)$ waves.
\begin{theorem}\label{t:plusplus}
Let $d\geq 2$ and $\beta \in \mathbb{R}$. Then,
\begin{equation}\label{b-eq2}
\||\square|^\beta(e^{itD}fe^{itD}g)\|_{L^2({\mathbb R}^{d+1})}^2\leq \mathbf{W}^\prime(\beta,d)\mathcal{I}_\beta(f,g)
\end{equation}
holds with constant
\[
\mathbf{W}^\prime(\beta,d) = \frac{2^{\frac{5-7d + 4\beta}{2}}\pi^{\frac{2-5d}{2}}}{\Gamma(\frac{d}{2})}.
\]
For $\beta >\frac{2-d}{2}$ the constant is sharp and equality holds if and only if
$f=\lambda g$ satisfy \eqref{maxdef} for some $\lambda, a, c\in\mathbb{C}$ with $\operatorname{Re}a<0$, and $b\in\mathbb{C}^d$ with $|\operatorname{Re}b|<-\operatorname{Re}a$.
\end{theorem}

Note that the range of $\beta$ for the sharp estimates in Theorem \ref{t:plusplus} is the same as in Theorem \ref{thm1} when $d=2$, but is strictly larger otherwise. This is due to the presence of additional symmetry in the left-hand side of \eqref{b-eq2} which does not appear to occur in \eqref{b-eq}, permitting a simpler argument for which the restriction in Theorem \ref{thm1} does not arise. This is reminiscent of recent work in \cite{BBJP} where bilinear estimates (with and without complex conjugate) are studied for the Schr\"odinger evolution operator $e^{it\Delta}$. In that case, however, the presence of a complex conjugate causes a change in the shape of the multiplier at the level of sharp estimates and moreover the class of extremisers for the estimates analogous to \eqref{b-eq} and \eqref{b-eq2} are shown to be different.

Since the right-hand side of \eqref{b-eq2} is in terms of the same functional $\mathcal{I}_\beta(f,g)$, the arguments in Section \ref{s:cor} immediately give the following.
\begin{corollary}
Let $d \geq 2$, $\beta > \frac{2-d}{2}$ and let
\[
\mathbf{C}'(\beta,d) = 2^{4\beta - 1} \pi^{\frac{1-d}{2}} \frac{\Gamma(d-2 + 2\beta)}{\Gamma(\frac{3d-5}{2} + 2\beta)}.
\]
\emph{(i).} If $f$ and $g$ are radially symmetric, then
\begin{equation*}
\left\||\square|^\beta (e^{itD}f e^{itD}g)\right\|_{L^2({\mathbb R}^{d+1})}^2\leq \mathbf{C}'(\beta,d)\|f\|_{\dot{H}^{\frac{d-1}{4} + \beta}({\mathbb R}^d)}^2\|g\|_{\dot{H}^{\frac{d-1}{4} + \beta}({\mathbb R}^d)}^2
\end{equation*}
where the constant is sharp, with equality if and only if $f=\lambda g$ satisfy \eqref{maxdef} with $\lambda, c\in\mathbb{C}$, and $a \in \mathbb{C}$ such that $\operatorname{Re}a < 0$, and $b=0$.

\emph{(ii).} If $\beta \in (\frac{3-d}{4},\frac{5-d}{4}]$, then
\begin{equation*}
\left\||\square|^\beta(e^{itD}f)^2\right\|_{L^2({\mathbb R}^{d+1})}^2\leq \mathbf{C}'(\beta,d)\|f\|_{\dot{H}^{\frac{d-1}{4} + \beta}({\mathbb R}^d)}^4
\end{equation*}
and the constant is sharp, with equality if and only if $f$ satisfies \eqref{maxdef} with $c\in\mathbb{C}$, and $a \in \mathbb{C}, b \in \mathbb{C}^d$ are such that $\operatorname{Re}a < 0$ and $\operatorname{Re}b=0$.

\emph{(iii).} If $\beta \in (\frac{2-d}{4},\frac{3-d}{4}]$ and $p=\frac{2(d-1)}{3d-5+4\beta}$, then
\begin{equation*}
\left\||\square|^\beta (e^{itD}f e^{itD}g)\right\|_{L^2({\mathbb R}^{d+1})}^2\leq \mathbf{C}'(\beta,d)|\mathbb{S}^{d-1}|^{\frac{3-d-4\beta}{d-1}}\|T_\beta f\|_{L^p}\|T_\beta g\|_{L^p}
\end{equation*}
and the constant is sharp, with equality if and only if $f=\lambda g$ satisfy \eqref{maxdef} with $\lambda, a, c\in\mathbb{C}$ and $b\in\mathbb{C}^d$ where $|\operatorname{Re}b|<-\operatorname{Re}a$.
\end{corollary}

We end by briefly indicating how to modify the argument in Section \ref{s:proofs} to prove Theorem \ref{t:plusplus}. As noted earlier, it is also possible to follow the earlier arguments in \cite{Foschi} and \cite{BR}, but the advantage of our unified approach is that it highlights the additional difficulty in the case of $(+-)$ waves.

Firstly,
\begin{align*}
& 2^{-2\beta}(2\pi)^{3d-1}\||\square|^\beta (e^{itD}fe^{itD}g)\|_{L^2({\mathbb R}^{d+1})}^2 \\
& = \int_{{\mathbb R}^{4d}}(|y_1||y_2|-y_1\cdot y_2)^{2\beta}\widehat{f}(x_1)\widehat{g}(x_2)\overline{\widehat{f}(y_1)\widehat{g}(y_2)}\delta\binom{|x_1|+|x_2|-|y_1|-|y_2|}{x_1+x_2-y_1-y_2}\,\mathrm{d}x\mathrm{d}y \\
& = \int_{{\mathbb R}^{4d}}(|y_1||y_2|-y_1\cdot y_2)^{2\beta} \frac{\widehat{F}(x)\overline{\widehat{F}(y)}}{(|x_1||x_2||y_1||y_2|)^{\frac{1}{2}}}\delta\binom{|x_1|+|x_2|-|y_1|-|y_2|}{x_1+x_2-y_1-y_2}\,\mathrm{d}x\mathrm{d}y,
\end{align*}
but in this case we do not need to relabel any indices and we take $\widehat{F}:=(|\cdot|^{\frac{1}{2}}\widehat{f})\otimes(|\cdot|^{\frac{1}{2}}\widehat{g})$. Proceeding exactly as in the proof of \eqref{b-eq} we then obtain
\[
\frac{\||\square|^\beta (e^{itD}fe^{itD}g)\|_{L^2}^2}{2^{2\beta}(2\pi)^{1-3d}}\leq\int_{{\mathbb R}^{2d}}(|y_1||y_2|-y_1\cdot y_2)^{2\beta}|\widehat{F}(y)|^2 \mathrm{I}_0(y) \,\mathrm{d}y,
\]
where we have used the fact that $|x_1||x_2|-x_1\cdot x_2=|y_1||y_2|-y_1\cdot y_2$ on the support of the delta measures in this case, and $\mathrm{I}_0$ is defined in \eqref{e:Idefn}. Hence (for any admissible $\beta$) we need use only the $\beta=0$ case of Lemma \ref{L1} to obtain \eqref{b-eq2}. From a technical point of view, Lemma \ref{L1} is substantially easier in the case $\beta = 0$ and this explains why the $(++)$ case is easier.

Finally, the characterisation of extremisers follows in precisely the same manner, since the above argument shows that equality holds in $\eqref{b-eq2}$ if and only if
\[
|x_1||x_2|\widehat{f}(x_1)\widehat{g}(x_2)=|y_1||y_2|\widehat{f}(y_1)\widehat{g}(y_2)
\]
almost everywhere on the support of the delta measures, and this is the same functional equation which arose in the $(+-)$ case.

\begin{acknowledgement}
The first author was supported by JSPS Kakenhi grant number 26887008. The second author acknowledges support from the Universitas 21 PhD Scholarship 2014 (University of Birmingham). He also wishes to thank Waseda University, where part of this research was conducted, for their kind hospitality.
\end{acknowledgement}

redpajama_set_name: RedPajamaArXiv


There's something I just love about Andy Pye's work. He paints with a style that is sometimes unusual, and often highly accomplished – delivering beautiful and striking work in one moment, and work that challenges in another. I, like many others, am drawn to his Australian landscapes showcasing imperial-like powerful colours combined with wild composition – unrestricted and gutsy.
Encompassing all the chaos, drama and loneliness that the Australian bush can present, I feel like I can walk into some paintings; and that I've always known them. But in other cases I'm totally lost – the feeling is subdued, even bordering on conservative. Am I meant to be here? Am I meant to even admire this art?
But that's how how it goes in the creative field. It's virtually impossible to craft perfected magic on demand, and win every time – however much one might strive for greatness. And how the hell can you please everyone all of the time? Regardless, I love Andy's ability to press ahead and so very often derive great wins.
Unlike Boyd, Andy rarely paints people – other than one recent (and brilliant) self portrait; he concentrates almost exclusively on the Australian bush and manages to discover mesmerising scenes and objects of both natural composition and those that display man's interference within the landscape – it's keen observation at its best.
One example, the painting Brick Mine Study II, 2018, is one of my favourites. It has all the colour, composition and energy of a Fred Williams Pilbara scene, however for me, it takes this to a new and very daring level – displaying a large cutaway mountainside, scraped, pillaged and destroyed, yet also elevated with all the colour brilliance to light up a dull mind. It has an edginess too – like a fresh bleeding open wound; the cut is very apparent. For Andy, I feel it's a heartfelt statement piece – this is his (micro) Sistine Chapel ceiling, and it also manages to hold the same beauty and peace.
This painting is at the heart of who I feel Andy really is as a person. A humble, but quietly-assured bloke, with a voice that's dying to speak to the world in an articulate, clever and loving manner. He does this currently ever so cautiously which I must admit sometimes confuses me; but within his soul I also understand the great reason for his slower 'stepping stone' approach – Andy is no doubt taking his time in finding new ways to express himself through his art. This might very well be a wise move.
If you think I'm talking in riddles then perhaps I should cut to the chase. I'm simply suggesting that I admire the energy in Andy's brush strokes and choice in colour and feel so much, that I know he can do no wrong – therefore I'm wanting him to strive further much faster – but am I asking for too much? Somehow I feel I can predict the future and the result is a hero of Australian art right in front of me.
Andy's kaleidoscopic-coloured tree paintings make up an ongoing theme that are unquestionably strong. A string of exhibitions at Geelong's Boom Gallery have netted many sales, including a list of notable private collectors snapping up large works. His painting Lookout Near Jingellic, 2017 (shown top of page), is one of the most powerful in the series: depicting a sandy foreground gully that's sparse, dry and softly undulating.
This landscape steps back in layers to reveal greener distant hillsides and a polarised deep, deep blue sky above. The gemstone-like reflected blue-hued colours are highlighted in foreground boulders – quirky and cool – and other lighter cyan touches flowing across the canvas to create dried twigs, seem to appear as driftwood on a beach-like sandfly loam soil. Scratches of grass, wildflower stems and other branches are strewn across the scene. It is a totally honest and 'messy' display of the Australian landscape that's exciting to absorb with its uninhibited sense of power and pleasure. Not an inch of historic European landscape glamour makes its way into this painting; and its alarmingly beautiful colour palette of green, golds and aquamarines amplify the hidden colours of the Aussie bush; similar in approach to the way Australian master painter Fred Williams explored colour in landscape.
For me, I ache to see Andy step ahead more quickly and speak just a little louder, a little faster. But in oh-so-many cases his work seems to fall upon perfection – what a gift. I've now told myself to let go and appreciate who he his and what he can achieve. As a new Australian artist on the scene, he certainly makes me feel, think and dream – and isn't that the beauty of art right there? If it ain't got that, it's got nothin'.


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Our spacious tree-lined grounds are quiet and peaceful with lots of room for campers and motor homes alike. From the park you can admire the picturesque views of the surrounding mountain landscape. Our grounds are impeccably maintained and well established with lots of trees and grass to camp out on.
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text:
Nietzsche et la Philosophie est un ouvrage du philosophe français Gilles Deleuze paru en 1962 aux Presses universitaires de France.

Cet essai témoigne du regain d'intérêt pour l'œuvre de Nietzsche à une époque où, d'une part, la phénoménologie (Husserl, Heidegger) et, de l'autre, le marxisme dominent la scène intellectuelle française. Parmi les nombreux thèmes et concepts nietzschéens, Deleuze s'intéresse particulièrement au ressentiment et à la volonté de puissance. Il présente Nietzsche comme l'ennemi de la dialectique hégélienne, opposant à cette dynamique du dépassement, une philosophie vitaliste de l'affirmation.

Comme le remarque notamment Pierre-André Taguieff, et comme le montrent aussi les références en notes de bas de page, Deleuze s'est beaucoup appuyé dans cet essai sur La Volonté de puissance de Nietzsche, mais sans en faire un examen critique, alors que ce texte posthume est une recomposition a posteriori édité de façon malhonnête par sa sœur. Toute l'interprétation de la philosophie de Nietzsche par Deleuze doit donc être comprise à la lumière de cet important biais.

Voir aussi de Gilles Deleuze, son essai Nietzsche (PUF, coll. « Philosophies », 1965).

Table des matières
 I -- Le tragique
 II -- Actif et réactif
 III -- La critique
 IV -- Du ressentiment à la mauvaise conscience
 V -- Le surhomme, contre la dialectique

Œuvre de Gilles Deleuze
Bibliographie nietzschéenne
1962 en philosophie

redpajama_set_name: RedPajamaWikipedia


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In November of 1997, the program moved into its own brand new, state-of-the-art facility, which was named Fieldstone Farm Therapeutic Riding Center. This expanded facility is one of the largest facilities of its kind in North America and it includes a therapy room, classroom, indoor and outdoor arenas and ample workspace for an array of programming.


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"We know that maintaining our green spaces and keeping our roads and pavements clear of weeds and obstructions is important to our residents.
"As part of our weed control programme, we do sometimes use products that contain glyphosate, a product which is approved for use in the UK & EU. We are also guided by the Chemical Regulations Directorate (CRD), part of the HSE, which closely monitors and approves all herbicides in the UK which still approves glyphosate, should this advice change we could act accordingly. I can reassure the people of Dudley that all products that we use are fully approved as safe for use for weed control, and all of our operative undertake regular and comprehensive training, to ensure that it is applied safely and legally."