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Sharp Gagliardo-Nirenberg constants of Del Pino and Dolbeault
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Numbers
$n$
$a$ 
$A_{n,a}$
3
9/8:
0.8995966967108490646446085925629613747285006284192151895470472968131909297641147595242045788989899712
3
8/7:
0.8878189897134250063140727452768211012323592254167108788218082085806927546167395459616050674457250048
3
7/6:
0.8729087404365831468602694489325448479647598950360724956902741791685748882322380052328254476332609984
3
6/5:
0.8534207661074015539348178027689885518996118352097614665884056820511920442576604706348439062374686182
3
5/4:
0.8268509412500422322876455740077079739783618084535982265732906073957762791443170386371977443514951201
3
9/7:
0.8095704698215068702711222349092912993871995865230250893911750080238324357415410348742767658909939666
3
4/3:
0.7884180569595735073434306662730713824423106247658753179971718635338670687740814011333449003918517942
3
11/8:
0.7714505908870285434532352007454763269220081520533026314168754069857869544867857866753732568439218023
3
7/5:
0.7618795853958058471546928128308917467719800808160191703224998140185474030302856570074246756705503289
3
10/7:
0.7514471386458670388622301817769863720950683283797799796414842493419413425613489486862529516741888730
3
3/2:
0.7274433720730808107064352621901312029348488179854605542073821821572421107945107195957208076772991588
3
11/7:
0.7059395015068666218376972205939297856260212667053119351793401845591748064964288160963076857377950595
3
8/5:
0.6979273962317105316267332060360261454546120946585212744884585250163903732214276342207572420504472045
3
13/8:
0.6911609874357331671672555277759723700825634799920343684793133859963630447765784460721994936997389683
3
5/3:
0.6803455590916884497370240800152191899828907980687585486031246875531598555944388017833034279970959849
3
12/7:
0.6686185396562551450792769421648426336103860173814025999612624428644567818290423000442212126498224781
3
7/4:
0.6602161701594570021066475922613053348768787851267229397426880566483919104858733912004049693121967606
3
9/5:
0.6489499064402296399934334930356401508942931657404065019393314564315751119431098674581510901105604238
3
11/6:
0.6417256670621424790190855649442527936793702378671239355241744959467600963046086069397524277475054925
3
13/7:
0.6366915795471067768741650414858057293816169393742101599974933934536792991825783429079338363352982582
3
15/8:
0.6329799066388991426984924516827073778287677275275088938718258714760857460226971424422547243818151424
3
2:
0.6082914467207952700548600461723281324184527254059116583864245728025187583680538395567532564011330302
3
17/8:
0.5852887928400294024277736785322322301438202652265953833697139312892624691121681904912597229486325646
3
15/7:
0.5820995752380966963279240145174107677090209550574202723869609036540686743219685708544583009393958902
3
13/6:
0.5778763852076071262906706148373697351969515010550051035628555922078380457096360977771489426989344112
3
11/5:
0.5720129945630128321018965572158220138599986714503821336221004608497560489953128668510987365819740382
3
9/4:
0.5633050322470127004519270851033186967190384934678103793511042542930246543251336928130233295656837149
3
16/7:
0.5571334049486547498194566456828453699075260362721347095295529249700908292324107779475297122751430037
3
7/3:
0.5489457688534923536190957224292893211866895815275217730283132173428077173525840632317776576944510104
3
19/8:
0.5418023637060759209291759760280434429972380461897592279899037094749296077358808077251298187387205825
3
12/5:
0.5375186307901861921231797502872386545497315468518532584884034622706121998231171356009031827200000192
3
17/7:
0.5326199041747221927674938074851819044378180468846407035547173830934345453194578623046711673079244423
3
5/2:
0.5203286166887543575204475386903117167710255076295162063943841896174232961796202929850217979850133794
3
18/7:
0.5079168133427904232212889645870797136162934326538670828166420968339354605342272146452489441847540978
3
13/5:
0.5029029756348327475574953150219016156312725089538139947605300195601711388210372871336888017529112960
3
21/8:
0.4984877548972453942052220279578583997267169945054233630306160806681003508264191792508153969463681598
3
8/3:
0.4910628850256609517181643849825367472414771996162214378289094829534453309881946763273400056613240436
3
19/7:
0.4824618420844568387207818928937336428127443056327303167060373420591968263852822399530731686007673966
3
11/4:
0.4759195199807840462234251052127253309181634313529640385976284705539801840850963134035156620237694153
3
14/5:
0.4666122572874441544772629049991317755217640096271967113840215862121486800962964718584167168997824763
3
17/6:
0.4603022258569950750681269853999152110946263863004213184810886430950373490739052080753269941290498432
3
20/7:
0.4557394965389648442428324118436780219117732907042112769626348311859199870803280756483871488449225012
3
23/8:
0.4522855267874260079154802672873645147507452763483156033256562367709732982570888680097589906910650963
3
3:
0.4272605428625266649876716112987146712543444475230011487534456211614623680303006737657006591538785342
comment: Sobolev endpoint: this value is $1/S_{3,2}$.
4
9/8:
0.8497110067248949453026113884437860717657015610268388495106089987317900104395294071134682748005213211
4
8/7:
0.8318025795548100477428416060232804007573590309977131806981215737297127103745186760001166341117365321
4
7/6:
0.8089980541036343261434281050056794494877248927521626676526029109344247828153888032860029938850724368
4
6/5:
0.7789258862624775663279486069468837720498656072620772062387239805039691258316018049910589692826423959
4
5/4:
0.7373162963077371223914132163457654706502255795130834801511943050332649832916752466762647197662639006
4
9/7:
0.7097792251966090020039903995672853402472154000342496001062630650450901763364800607799088377374294764
4
4/3:
0.6754270216403788278446128699368490075476841301263754529270667485586414495989887105358233483610156286
4
11/8:
0.6472430490998172774435708241147992109570407983680107830816634424444185661646890959235171458549413288
4
7/5:
0.6310517475675126334359240541642562737459675265665398151916013081138236578761486487067098508384079772
4
10/7:
0.6131273405898445288821823166943428497158293885577106184524420963345013529998320639529396164038357760
4
3/2:
0.5705984522739834996248955691764317021362594046041774187717205640351375966926632386205223656226443748
4
11/7:
0.5306253422964105864658616877347816156262138164192701584770903811962531199721738460014064595720439945
4
8/5:
0.5151840435736654052296029125741667155953277254051909294136672998100687204613262532035184761635830380
4
13/8:
0.5018806206792494486091754837222941373634538467062422631961247988752841329832816129684963448603128166
4
5/3:
0.4800704482934333095926295276470409294617826380461637341131219443140747384119774437445998245323401724
4
12/7:
0.4555850612374063921667195341670073594009336489325322402140098890352773811156903834989143481184486607
4
7/4:
0.4374509359869989984485262137682507728320135227969842048854644539683852218641951728248097086219113874
4
9/5:
0.4122877638910988885494316081211816231147516862082056761397463211077420011019150812712050226495104940
4
11/6:
0.3956033576345310688918491146000750496307110901717746588836523396026222485440855887981597756223424376
4
13/7:
0.3837090174874528760717020293289286056804494770601277201975688977319647842511822819833832520707484410
4
15/8:
0.3747934958728748978120620293434681726700456769744739587211128462140339656769880815785720590028659549
4
2:
0.3121892056977779516773160629259526518917585721758716167276486221255009949074751977532205797312876936
comment: Sobolev endpoint: this value is $1/S_{4,2}$.
5
9/8:
0.7963364511822358791799972127316993558375839198913342983450439386151962809584284616889038149671007047
5
8/7:
0.7718268460806866318137886571222184938174034988891208720399514581972453679939725851732427287041625301
5
7/6:
0.7404962561851331704345372596434983319555685528296617251570330621775658748694948641277159183273747836
5
6/5:
0.6989380387636935711832177463610610364432806053026854727761656496727312223008053271383260121048466166
5
5/4:
0.6408774146230721199295290817312542892690090455540706910440343376443228789731645776818312892652242287
5
9/7:
0.6020214682901318842722291707527404320155772751033204157982361106762345718765211053813330302101784767
5
4/3:
0.5529751647571221463980982760279942001055989233317109957513067532217399018400007954272277170941195836
5
11/8:
0.5121955226321076227483814661374423314583386082693040269226854924837641655708830071964736999884286027
5
7/5:
0.4885280580502206555384560541444230314161356532781350810069695920823498957548489126019381418579551571
5
10/7:
0.4621129821529798553842235428486388429495924058685175571521144989915869352363486873209592536384947269
5
3/2:
0.3985229965199785919371665367094966288301758609864304550854591483492604954280126709435944834887215148
5
11/7:
0.3376708836479384232758238801354782841897860253588005018185250889001092339609442540278738068823288500
5
8/5:
0.3139494746889123000233139283194494122301094072990811717478072721582292967090004745686826185823910619
5
13/8:
0.2934557710229257153474267249078458332885676450736192294405681251924728257247227857741616019499103541
5
5/3:
0.2598330806849343119379194929296201461415609819664724177789706540078270484563531030571028355419014629
comment: Sobolev endpoint: this value is $1/S_{5,2}$.
6
9/8:
0.7404794790805661475091456945181769741226966073350492422445383378937802722143879204662926614700059240
6
8/7:
0.7090943484480914102712964472330150841759659571803247292657627812677640845166057837520310306766089277
6
7/6:
0.6688983520529088321845881824233055141092791453580194418443983323805425853719609480098693527488761800
6
6/5:
0.6154333404849340786511373287838117516109824551598104533792450710068533214798426751870438846938920537
6
5/4:
0.5404361562439507747275263251187451749485189965346814442068545608442497801119981197340364735860628533
6
9/7:
0.4900597621411578734289029010544829785604174802862793386014786227894020706595455879495364525148895863
6
4/3:
0.4263281056146415380956872080928359005063644417638666385537088992420395519462633355630796517504684458
6
11/8:
0.3733388178482235148852824888495970507040191863270830741774677204132307819236553227189586148104003614
6
7/5:
0.3426627199772905227684252024568856517173619412542523345492575770485197584583050340670825550393548051
6
10/7:
0.3085753332199938952486113908717689203499739053760281237113794769507687944438244131707062019686640261
6
3/2:
0.2278651897947799410796353912138799426764417140039570847449266396512912405898666925214399748901270286
comment: Sobolev endpoint: this value is $1/S_{6,2}$.
7
9/8:
0.6829415502145542479520659851112128400900052393364386525656896283219825371910468980665872488940967679
7
8/7:
0.6445976980557087787481276390862258676103799303018690305125309264120582283936698986708844836273869681
7
7/6:
0.5955017026114767203955898578722233471105861577577785111856961383237477156735385599623054153627121476
7
6/5:
0.5302570581958142618591631230539754729026449853335624102839599637003230465456298061244099815071318529
7
5/4:
0.4390289482000034662413502683429438472202237178682966282975513175836471526574727350292883663328114212
7
9/7:
0.3781675529827849094731708084494423945587955449776539678239957617565885884814243303638674416724277375
7
4/3:
0.3021446872425684772482488789525712286494497128980220916587361866224481146744850394899633714407230785
7
11/8:
0.2404010316340795415608690121830466145847534391961132432037565842403476288481990986640947941026096413
7
7/5:
0.2056224413422255527043067376111386195989789677408363866085753209733829080187440090876528820490129230
comment: Sobolev endpoint: this value is $1/S_{7,2}$.
8
9/8:
0.6244204942669146598745521544104599712300703913436878198569995103791033606030987622006973695475704376
8
8/7:
0.5792334238247013440764194364548298942242000634347598355896920191524257432648187132902644461923462613
8
7/6:
0.5215372987858420514672404050466296977671290492584471513171228802151616207857794119967289715210574385
8
6/5:
0.4452837673620965641049084825418511594385439567095636530529803152776857498218841005562826963876670304
8
5/4:
0.3400545540595805497184071961764142482657944067999679077167125077948092000936564208330402673325798795
8
9/7:
0.2714378417002608770394447545688324752895245798845310293038234299751231704348312565513134086645559871
8
4/3:
0.1889467193439846325440693329447504026465053844165565876315197765591370340392575045340810072705567949
comment: Sobolev endpoint: this value is $1/S_{8,2}$.
9
9/8:
0.5655602786244717326725881650856565680464865063882444336455422178997840618310490045375398410088770661
9
8/7:
0.5138576961700154995301429574242262277859723198513423789113232853290759703159819804851337118690753919
9
7/6:
0.4482334763897866341500944161466357562553660447012927765712972725220935937522007764163104604665337146
9
6/5:
0.3624924726690705416514651081940443534926125554649384754367209305000447518065003843200908588135298540
9
5/4:
0.2473567293347315732767772666844697677414951322282902141077641195648371874637710285210226802544023204
9
9/7:
0.1758224690742126630904374222779734867993643099260659630294156644004689751260435527842588985183680292
comment: Sobolev endpoint: this value is $1/S_{9,2}$.
10
9/8:
0.5069776149609241565785316867097109372003181703676185491627994845100321051029020461409111690629866662
10
8/7:
0.4493151176312113568337035974492997197594394911931974306573995488267467412392712920912440359349030307
10
7/6:
0.3768456019563392196672155489826940306147997634683147832716655555143690448488508445460671390458692648
10
6/5:
0.2839869725744144524629734969117692900936432679106061893972472222995170173101316445101371403307338568
10
5/4:
0.1651335241130897770507885310764143181330961654052492073640988467028102710712107314263981151713489947
comment: Sobolev endpoint: this value is $1/S_{10,2}$.
11
9/8:
0.4492762745562832200714861209793199825279055874667336991410149631219563055401807318176102465579257203
11
8/7:
0.3864520852047833006913011758814113287959615233898379694893012522997536802558645020182508339725075759
11
7/6:
0.3086623541554715283313092650021934350547946325818113083292039871448423742212945247002298278448907904
11
6/5:
0.2119611219180173122739454932989692574267999809935326307376790725031261656849946308625916694614342840
11
11/9:
0.1562031739436406776992714609082912878530411727637013765340123548812362265954110392969492154628873827
comment: Sobolev endpoint: this value is $1/S_{11,2}$.
12
9/8:
0.3930537260131540461819550267242869522059756198113638229187073368428961012530363671692369879270930245
12
8/7:
0.3261190813915599709830992975132284722347754343579108411325554278770363473929385049422851848686160604
12
7/6:
0.2449903175833459912971226812014845414252090876114376116321833996525557832478787933265557281699024502
12
6/5:
0.1485932152235923569270491813240724959428176407991520251012559309256205132353268093466810771846321480
comment: Sobolev endpoint: this value is $1/S_{12,2}$.
13
9/8:
0.3389020929006721063011224877911642501053292264087432918662936102724310408737893436551981126288021206
13
8/7:
0.2691631422842842062016416443118457435721105186785303680580932520073315624070697895499492865566627223
13
7/6:
0.1871142418170863102478657600188293234867425293374216043311220527473801763250229221866050551935167769
13
13/11:
0.1420052973437669267371313297784243772841568470046026823456147129543548882129140829614940005485536512
comment: Sobolev endpoint: this value is $1/S_{13,2}$.
14
9/8:
0.2874041901714157095073444733894261288627992113093302208092718805154033852402108824149323414970169989
14
8/7:
0.2164102037756193419397925338135548020146140280670269409215701643666389675175908049012871394374181975
14
7/6:
0.1362280987990931098987695025805902784106428148610147693125661875925027935708579991563644544444053415
comment: Sobolev endpoint: this value is $1/S_{14,2}$.
15
9/8:
0.2391247611241755555638764825275556308253674608313431700553050723193701975284203381139996767615777990
15
8/7:
0.1686362641603887042736429440323172939813018183474047312036289424108034310513965815385128668033466154
15
15/13:
0.1311070746621206464130357939705841566543046067870172915493683228603628493193113827917642121156264336
comment: Sobolev endpoint: this value is $1/S_{15,2}$.
16
9/8:
0.1945967186502434710260533374147372076338516230060876811699288958524844750895565586616571748187879600
16
8/7:
0.1265261728230072355629061653940822618862378858226439571580626851590728568474733641613824416140691877
comment: Sobolev endpoint: this value is $1/S_{16,2}$.
17
9/8:
0.1543021096167892566508950406086590699588379570937525046897328023510377146232490764942579986168988895
17
17/15:
0.1223962858397266476469815407995742678357065492145964380092438412415934507639646731224906879042672169
comment: Sobolev endpoint: this value is $1/S_{17,2}$.
18
9/8:
0.1186476836010625797143878929225527585124442650596338631511086794857292328846497257286226258565134432
comment: Sobolev endpoint: this value is $1/S_{18,2}$.
19
19/17:
0.1152248973175112840151190362521999493304224894940563276947214771461740498499376352994139159398419845
comment: Sobolev endpoint: this value is $1/S_{19,2}$.
20
10/9:
0.1120831667148931756117731460924436100601936306928741103149126237644733304296692232496981359804507791
comment: Sobolev endpoint: this value is $1/S_{20,2}$.
Definition
Let $n\geq3$, $1<a\leq n/(n-2)$, and $\theta=n(a-1)/(a(n+2-(n-2)a))$. The number $A_{n,a}$ is the least constant for which the inequality [1] [2] $\|u\|_{L^{2a}}\leq A_{n,a}\|\nabla u\|_{L^2}^{\theta}\|u\|_{L^{a+1}}^{1-\theta}$ holds on $\mathbb{R}^n$ for all $u\in D_a$.
Parameters
$n$
—   dimension ($n\geq 3$)
$a$
—   interpolation exponent ($1<a\leq n/(n-2)$)
Formulas
(1)
With $y=(a+1)/(a-1)$, $A_{n,a}=\left(\frac{y(a-1)^2}{2\pi n}\right)^{\theta/2} \left(\frac{2y-n}{2y}\right)^{1/(2a)} \left(\frac{\Gamma(y)}{\Gamma(y-n/2)}\right)^{\theta/n}$.
(2)
Equality is attained, up to the usual translations and dilations, by $u(x)=(1+|x|^2)^{-1/(a-1)}$.
Comments
(3)
The table holds the $a>1$ Del Pino-Dolbeault family. The companion family with $a<1$ has a different closed form and is not held here.
(4)
The class $D_a(\mathbb{R}^n)$ consists of the functions $u$ with $u\in L^{a+1}(\mathbb{R}^n)$, $u\in L^{2a}(\mathbb{R}^n)$, and weak gradient $\nabla u\in L^2(\mathbb{R}^n)$; this is Del Pino and Dolbeault's $D_p(\mathbb{R}^d)$ with $p=a$ and $d=n$.
(5)
At $a=n/(n-2)$, the inequality becomes the Sobolev inequality and $A_{n,a}=1/S_{n,2}$, where $S_{n,2}$ is the largest constant with $S_{n,2}\|u\|_{L^{2n/(n-2)}(\mathbb{R}^n)}\leq\|\nabla u\|_{L^2(\mathbb{R}^n)}$, as the Sobolev constant table stores it.
(6)
Some sources write the exponent $a$ as $p$, and write the left-hand norm's exponent $2a$ as $2\sigma+2$. This table's $a$ is the exponent parameter in the defining inequality.
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=160))
n = QQ(3)
a = QQ(2)
theta = n * (a - 1) / (a * (n + 2 - (n - 2) * a))
y = (a + 1) / (a - 1)
A = (R(y * (a - 1)**2) / (2 * R.pi() * R(n)))**(R(theta) / 2)
A *= ((2 * R(y) - R(n)) / (2 * R(y)))**(R(QQ(1) / (2 * a)))
A *= (R(y).gamma() / R(y - n / 2).gamma())**(R(theta / n))
print(A)
References
[1]
M. del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9) 81 (2002), no. 9, 847-875. (doi) (MR)
Links
Similar tables
Best Sobolev constant for $W^{1,p}(\mathbb{R}^n)$ —   the endpoint $a=n/(n-2)$ is the reciprocal of the $p=2$ row
Sharp constants in the fractional Sobolev inequality —   another sharp Sobolev-scale inequality, with the constant stored in the lower-bound direction
Sharp constants in the Hardy-Littlewood-Sobolev inequality —   another sharp conformal inequality whose constants are finite products of powers of $\pi$ and Gamma values
Babenko-Beckner constants of the Hausdorff-Young inequality —   another sharp Euclidean functional-inequality constant, for a Fourier-transform bound rather than a Gagliardo-Nirenberg interpolation inequality
Sharp constants in Young's convolution inequality —   another sharp Euclidean functional-inequality constant, for convolution rather than a Gagliardo-Nirenberg interpolation inequality
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds every dimension $3\leq n\leq20$, every rational exponent $a$ in the open admissible range whose reduced denominator is at most $8$, and the Sobolev endpoint $a=n/(n-2)$ in each dimension; the denominator bound is $8$ rather than $4$ because the admissible window closes as $n$ grows, and with denominator at most $4$ most dimensions above $n=9$ would hold only their Sobolev endpoint)
How they were obtained:

Each value is computed from (1) in Sage's real ball field, using exact rational values of $a$ and 160 guard bits beyond the 100 digits written.

more

The endpoint rows are checked against the reciprocal of the $s=1$ rows of the fractional Sobolev constant table, and also against the $p=2$ rows of the Sobolev constant table where that table carries them. The formula is checked independently by evaluating the three displayed norms on (2).



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