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Babenko-Beckner constants of the Hausdorff-Young inequality
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Numbers
$n$
$p$ 
$A^{(n)}_p$
1
1:
1
1
5/4:
0.9308227833180347069360832026298387733457674177426944501315953616310948133295711549416945068055426600
1
4/3:
0.9366870743752481399168612675978212617167407847431873462803675101795218013976340438827420100688621739
1
3/2:
0.9531842929969365737574089953994271577729070946503468732759762069254178623976950413403019050653469463
1
5/3:
0.9704353834739619793970385516124885293950393557667909648611270048676492881731818366628063764627009862
1
7/4:
0.9785595827226036352573521421503122577071603032852364393298841826122783979919978268708499403103863461
1
2:
1
2
1:
1
2
5/4:
0.8664310539439329911028148143988392208501170625705362018638599631829684186172672026194994001518035271
2
4/3:
0.8773826753016616405461459345313327034400457268038814511546151057305778634648680887004954334977346210
2
3/2:
0.9085602964160698294456058781636302512141052315706098357406671489656548697296509328235708520632085361
2
5/3:
0.9417448334982556392318463897136199436939005715075258668215181333742969795450481387024818308951145136
2
7/4:
0.9575788569382361470603859629859686999947249622574445220490494378721024246539203781627657356965924914
2
2:
1
3
1:
1
3
5/4:
0.8064937651852699790652167498152237157807245851622205221056545841048966505270411912565882270052385616
3
4/3:
0.8218330112358417263234371834612269137747513984855596932066517073726450665128997879372323384650994321
3
3/2:
0.8660254037844386467637231707529361834714026269051903140279034897259665084544000185405730933786242878
3
5/3:
0.9139025086305021864616344174978315732306152241292544205292387316590148112555070835620659528769665318
3
7/4:
0.9370479666694681269538511971633489383554033685607988262845302281488849156557482510625282207833886368
3
2:
1
4
1:
1
4
5/4:
0.7507027712383945207763124334994783277575161947827020162300224115097478728826509303970820402205084684
4
4/3:
0.7698003589195010193455317073359432741968023350268358346914697686453035630705777942582871941143327003
4
3/2:
0.8254818122236566709686524881022712392117211610796233342433006342773189188947178839663131737912023629
4
5/3:
0.8868833314206572367676655351513477265659218061858556463407984205022650431256685290412029784069813924
4
7/4:
0.9169572672551389296001513743598426990788381116032959255149574149890821214733242891254441996054860481
4
2:
1
5
1:
1
5
5/4:
0.6987712429686842801278667714785238235751932373786017888346553891907877892618140310670045026183694609
5
4/3:
0.7210620460493233641426584311129342891248795630047591098660142904144619633303097858157014235201199212
5
3/2:
0.7868362975662361391456411172666972050897251746806689263260383726413360475011671235919050630629522854
5
5/3:
0.8606629658238704189287256443960888024073159345092424059083497258029962426526620074487305281908163373
5
7/4:
0.8972973208196476929956623359777780499613018062030280894206330237396111321836643192564687647964121173
5
2:
1
6
1:
1
6
5/4:
0.6504321932827133909498806393419115104562556980935398368285142174849407695663375559955332601801748344
6
4/3:
0.6754094983569711531832780005818180759082430656776091720989232477336197628187086858616983237022552506
6
3/2:
3/4
6
5/3:
0.8352177952811251234110756869936763336176350812114216894531176521652566708414494504294170986687280276
6
7/4:
0.8780588918393846497760596533442600086738197888586065750438276943972027252387319258796484485379914454
6
2:
1
7
1:
1
7
5/4:
0.6054371045110691963277964889846983675812457839126145180082443694161978734866284915411208200360553139
7
4/3:
0.6326473470212452749182842478134436150340770475301803934806340011885780444488144858709822699856420040
7
3/2:
0.7148882197477024303180567465495703683296803209877601549569821551940633967982712810052264287990102097
7
5/3:
0.8105249014479157312874220893076229503237059073476278426060980233240616814170937486682941786103617128
7
7/4:
0.8592329428042200012352588488240015916845856081439947317388578982325378297902070412383177924050209436
7
2:
1
8
1:
1
8
5/4:
0.5635546507450052957328215775157908759149683537996937531511233606660842380627806819209769783038776969
8
4/3:
16/27
8
3/2:
0.6814202223120523720842044086227226884105789236779573768055003617242411522972381996176781390474064020
8
5/3:
0.7865620435518033438844782766628575427198877716347100909817402922829379857899112880337875277507799918
8
7/4:
0.8408106299720122802480162874361014709387479505512945815041098504363571945571374198754298754655298378
8
2:
1
9
1:
1
9
5/4:
0.5245695085582887909159174877488544671959423424833807303661954369938863354156442446854667497561273628
9
4/3:
0.5550738218519988977285103807987088958321426872552221311291066726989758823097090630416248948556220290
9
3/2:
0.6495190528383289850727923780647021376035519701788927355209276172944748813408000139054298200339682159
9
5/3:
0.7633076383602574614436067405799721151592995228315397643775253109969747945623492475220736694563808074
9
7/4:
0.8227832992141418264391661133005694824193422908188407921292527691805615653652513287120630724119762421
9
2:
1
10
1:
1
10
5/4:
125/256
10
4/3:
0.5199304742528365277310494426852341946311382084763741932768089515440461413125144229336269235542131087
10
3/2:
0.6191113591677425032264893660767034294087908708097175006824754757079891891710384129747348803434017722
10
5/3:
20/27
10
7/4:
0.8051424819501177572102035120287111839525765999970821692116779906134567179045619459709980638106189227
10
2:
1
11
1:
1
11
5/4:
0.4545033121670091342461343762841009635477379969446750244783180476714330143210796654988742709011438770
11
4/3:
0.4870121548064247267101849976066529859405934213247761144928306414060118912669035780368784227941329968
11
3/2:
0.5901272231746771043592308379500229038172938810105016947445287794810020356258753426939287972972142141
11
5/3:
0.7188410247955273921459544826759174291815106339013266406378718554575179912393939530835602788612599898
11
7/4:
0.7878798911693486617036763573699700312572219727076400282883043746684562695465769189184745105088210891
11
2:
1
12
1:
1
12
5/4:
0.4230620380585610308119212960931832133057212219582696298163378726479337981529625012790524414803728160
12
4/3:
0.4561779904708154188714261969398182365610680503862730872245746777157354447825646188197257446603453039
12
3/2:
9/16
12
5/3:
0.6975887655542634364680343627508295879214078307463154569048282469439236885518875101499865414037885286
12
7/4:
0.7709874175382081911141472654102890899167044817177198629621325293438858602508881716367744825719799363
12
2:
1
13
1:
1
13
5/4:
0.3937957837818701069654378661207147049710569263487404893094016523949690676401568316682559702174016414
13
4/3:
0.4272960272884879194919457369558129120740027040028202132539343943196811634549983915944897324563673607
13
3/2:
0.5361661648107768227385425599121777762472602407408201162177366163955475475987034607539198215992576573
13
5/3:
0.6769648212077793973789884574058011653560112771327810522438805419696406009300052470830118169459011347
13
7/4:
0.7544571255905667871764233160865881684579877334363268472789336834175408178606941123671891433588161681
13
2:
1
14
1:
1
14
5/4:
0.3665540875187473245978088054196671522253497044837412188623156306199940785559818996079502149514201506
14
4/3:
0.4002426656930199426271277040484847857234032981793239538363989616199228224110866286587841918235586670
14
3/2:
0.5110651667340392790631533064670420163079341927584680326041252712931808642229286497132586042855548015
14
5/3:
0.6569506158671535087167892852972946122710531897673004787709617929646407726856803918823725765977639944
14
7/4:
189/256
14
2:
1
15
1:
1
15
5/4:
0.3411968959808028711561849470109979607300748229387704047044215767533143502254951323569357922941257133
15
4/3:
0.3749021315681453480997239987042628829831567689067735665070423710747129152289271027383598636951952616
15
3/2:
0.4871392896287467388045942835485266032026639776341695516406957129708561610056000104290723650254761619
15
5/3:
0.6375281228324966066138708477008065203017155070438832636358146117059231427056755610731337245857898795
15
7/4:
0.7224521919319222150923420111969102215103644426598034649740160566942211610175296456194946824947774196
15
2:
1
16
1:
1
16
5/4:
0.3175938443763248979247464059286677297149686025847362484514229577563187351398132597634439746973509933
16
4/3:
256/729
16
3/2:
0.4643335193758068774198670245575275720565931531072881255118566067809918918782788097310511602575513291
16
5/3:
0.6186798483563889803045005088842046915686185786751271773726797423446345709936662595773459990138726130
16
7/4:
0.7069625154739321554469255742357347042929805385416289635440247803040131182015271541904793908072499253
16
2:
1
17
1:
1
17
5/4:
0.2956235861870455060194318793870597497955301679260813076212130710040028679133928181353129004082301337
17
4/3:
0.3289326351715549023576357812140497160486771480030945962246558060438375598872350003209629006551834246
17
3/2:
0.4425954173810078282694231284625171778629704107578762710583965846107515267194065070204465979729106605
17
5/3:
0.6003888158873449861388311772649058891286710424797243278563689061659716158645138879024401323039716391
17
7/4:
0.6918049441426932656026479541557537083952001431952772584678758325005439416364703884797571630002361420
17
2:
1
18
1:
1
18
5/4:
0.2751731693090846170570417858963822636303556415037567154058219534502364443665921298442270401874402817
18
4/3:
0.3081069477053846090258070771468054486703041235415550774973682675816569726296381765532603991432373978
18
3/2:
27/64
18
5/3:
0.5826385507791135880625765012317463279406576086183037710975854016910651746591935466916944650005777717
18
7/4:
0.6769723574657080378688617568516026176792984495821208200090896540349082849939776978308943504899877464
18
2:
1
19
1:
1
19
5/4:
0.2561374553507269486894128358148703453105187219157132472491188657196710622146700413503255614043590639
19
4/3:
0.2885997954408442824949244430261647324092405459702376974772329726850440837137206388366686949891158500
19
3/2:
0.4021246236080826170539069199341333321854451805556150871633024622966606606990275955654398661994432430
19
5/3:
0.5654130654520425640323012893184978630809626095048442699092780081459072552313698129793138292269487462
19
7/4:
0.6624577876363805233444538339835939509479923491108293316425767245577597811824709232010648302598511335
19
2:
1
20
1:
1
20
5/4:
15625/65536
20
4/3:
0.2703276980567795074793636722606330290732255113400136813182664756833987820933716259672448857246490690
20
3/2:
0.3832988750505294592973649798502815122309506445688510244530939534698856481671964872849439532141661012
20
5/3:
400/729
20
7/4:
0.6482544162407956984674815409455669595287185159932681354815759150041848244926575546533342061471889968
20
2:
1
Definition
Let $n\geq1$ and $1\leq p\leq2$, and let $q$ be conjugate to $p$, with $q=\infty$ for $p=1$. This table gives the sharp Babenko-Beckner constant $A^{(n)}_p$ [4]: the least constant such that $\|\hat f\|_q\leq A^{(n)}_p\|f\|_p$ for all $f\in L^p(\mathbb{R}^n)$. [1], [2]
Parameters
$n$
—   dimension ($n\geq1$)
$p$
—   Lebesgue exponent ($1\leq p\leq2$)
Formulas
(1)
For $1<p\leq2$ and $q=p/(p-1)$, $A^{(n)}_p=(p^{1/p}/q^{1/q})^{n/2}$. At $p=1$ the same formula is read as its limit, $A^{(n)}_1=1$. [2]
Comments
(2)
The Fourier transform uses the $L^2$-unitary convention $\hat f(\xi)=\int_{\mathbb{R}^n}f(x)e^{-2\pi i x\cdot\xi}\,dx$. With this normalization the $p=2$ case is Plancherel's theorem.
(3)
The endpoint constants $A^{(n)}_1$ and $A^{(n)}_2$ are exactly $1$.
(4)
Centered Gaussians give equality in the sharp Hausdorff-Young inequality. [3]
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

R = RealBallField(numberdb.bits(100, losing=96))
n, p = 3, QQ(5) / QQ(4)
q = p / (p - 1)
print((R(p) ** (QQ(1) / p) / R(q) ** (QQ(1) / q)) ** (QQ(n) / QQ(2)))
References
[1]
K. I. Babenko, "An inequality in the theory of Fourier integrals", Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 531-542; English translation in Amer. Math. Soc. Transl. (2) 44 (1965), 115-128. (zbMATH) (MR)
[2]
William Beckner, "Inequalities in Fourier analysis", Ann. of Math. (2) 102 (1975), 159-182. (doi)
[3]
Elliott H. Lieb, "Gaussian kernels have only Gaussian maximizers", Invent. Math. 102 (1990), 179-208. (doi)
Links
Similar tables
Best Sobolev constant for $W^{1,p}(\mathbb{R}^n)$ —   that table stores the largest $S_{n,p}$ in $S_{n,p}\|u\|_{L^q}\leq\|\nabla u\|_{L^p}$, while this table stores the least $A^{(n)}_p$ on the right side of its Fourier-transform inequality
Sharp constants in the fractional Sobolev inequality —   another sharp Euclidean functional-inequality constant, for a Sobolev embedding rather than a Fourier-transform bound
Sharp constants in the Hardy-Littlewood-Sobolev inequality —   another sharp Euclidean functional-inequality constant, for the diagonal Hardy-Littlewood-Sobolev integral inequality rather than a Fourier-transform bound
Data properties
Entries are of type: real number
How well the digits are known: proven
How they were obtained:

Each non-exact value is computed in Sage's real ball field from (1), using exact rational values of $p$. The endpoint rows $p=1$ and $p=2$ are returned as the exact integer $1$.

more

For $p=a/b$ in lowest terms with $1<p<2$, the formula gives $A^{(n)}_p=B_p^{n/(2a)}$ for a rational $B_p$; for the exponents in this table, the rows with $p=5/4$ and $10\mid n$, $p=4/3$ and $8\mid n$, $p=3/2$ and $6\mid n$, $p=5/3$ and $10\mid n$, or $p=7/4$ and $14\mid n$ are stored as exact rationals.

Table is complete: no (it holds every dimension $n\leq20$ and every rational exponent $p$ with $1\leq p\leq2$ whose reduced denominator is at most $4$, a small grid chosen to include the endpoint, midpoint, and integer-conjugate cases while keeping the table to reference size)


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