Neu al Ope a o s in Aniso opic F ac ional
Sobole -Mo ey Spaces
Rômulo Damasclin Cha es dos San os
San a C uz S a e Uni e si y
[email p o ec ed]
Jo ge Hen ique de Oli ei a Sales
San a C uz S a e Uni e si y
[email p o ec ed]
No embe 22, 2025
Abs ac
This pape de elops a comp ehensi e ma hema ical heo y o aniso opic ac-
ional calculus wi h mixed egula i y s uc u es, add essing undamen al challenges
in analyzing high-dimensional unc ions wi h he e ogeneous smoo hness ac oss di -
e en coo dina e di ec ions. Mo i a ed by applica ions in scien i ic machine lea n-
ing, mul iscale analysis, and physical sys ems wi h di ec ional p e e ences, we in-
oduce no el aniso opic ac ional Sobole -Mo ey spaces ha p ecisely cap u e
di ec ional scaling beha io h ough mixed egula i y pa ame e s. These spaces
p o ide a e ined analy ical amewo k o unc ions exhibi ing a ying deg ees o
smoo hness along di e en coo dina es, gene alizing classical iso opic heo ies o
aniso opic se ings. Ou p incipal con ibu ions es ablish se e al sha p unc ional
inequali ies: (1) aniso opic Gaglia do-Ni enbe g inequali ies wi h mixed ac ional
de i a i es ea u ing explici cons an dependence on scaling pa ame e s and p o en
op imali y; (2) di ec ional Ha dy-Li lewood-Sobole heo y o aniso opic ac-
ional in eg als wi h op imal bounds in Lebesgue and Mo ey spaces; (3) compac -
ness c i e ia in aniso opic unc ion spaces demons a ed h ough e ined eal in-
e pola ion and ha monic analysis echniques; and (4) op imal app oxima ion a es
o deep neu al ope a o s in high-dimensional se ings, wi h explici dimension de-
pendence go e ned by he aniso opic dimension dα=Pk
i=1 α−1
i. The heo e ical
amewo k b idges ha monic analysis, ac ional calculus, and deep lea ning heo y,
p o iding igo ous ma hema ical ounda ions o unde s anding he app oxima ion
capabili ies o mode n neu al a chi ec u es. Fu he mo e, ou esul s o e p incipled
guidance o neu al ope a o design in scien i ic compu ing applica ions, pa icula ly
o p oblems exhibi ing mul iscale and aniso opic ea u es. This wo k opens new
esea ch di ec ions in he analysis o pa ial di e en ial equa ions, high-dimensional
app oxima ion heo y, and he ma hema ical ounda ions o deep lea ning.
Keywo ds: Aniso opic F ac ional Calculus, Sobole -Mo ey Spaces, Gaglia do-
Ni enbe g Inequali ies, Neu al Ope a o s, Mul iscale Analysis.
1
1 In oduc ion and Ma hema ical Backg ound
The classical Landau inequali y [1]
∥ ′∥∞≤2p∥ ∥∞∥ ′′∥∞(1.1)
ep esen s a undamen al ade-o be ween unc ion magni ude and oscilla ion ha has
in luenced ma hema ical analysis o nea ly a cen u y. Recen de elopmen s in ac ional
calculus [2] ha e ex ended his heo y o non-local ope a o s, while mul i a ia e ex en-
sions by Di zian [3] and Kounche [4] ha e gene alized hese esul s o mul idimensional
se ings.
Howe e , exis ing heo ies ope a e p ima ily wi hin iso opic unc ion spaces, o e -
looking he ich mul iscale s uc u e p esen in mode n applica ions anging om high-
dimensional da a analysis o physical sys ems wi h di ec ional p e e ences. This wo k ad-
d esses his undamen al limi a ion by de eloping a comp ehensi e heo y o aniso opic
ac ional calculus wi h mixed egula i y s uc u es.
1.1 P incipal Con ibu ions
This wo k makes ou undamen al con ibu ions ha b idge ha monic analysis, ac ional
calculus, and deep lea ning heo y:
(i) Aniso opic F ac ional Sobole -Mo ey Spaces: We in oduce he spaces
Mν
p,λ;α(Rk) ha cap u e di ec ional scaling beha io h ough mixed egula i y pa-
ame e s, es ablishing hei comple e heo y including equi alen cha ac e iza ions,
embedding heo ems, in e pola ion esul s, and algeb a s uc u es wi h sha p con-
s an s ha explici ly ack dependence on scaling pa ame e s.
(ii) Sha p Aniso opic Func ional Inequali ies: We p o e op imal Gaglia do-
Ni enbe g inequali ies and Ha dy-Li lewood-Sobole es ima es o di ec ional ac-
ional in eg als, es ablishing compac ness c i e ia in mixed-no m spaces wi h ex-
plici cons an s ha e eal how di ec ional he e ogenei y a ec s unc ional ela-
ionships and ope a o bounds.
(iii) Ad anced Ha monic Analysis F amewo k: We de elop comp ehensi e ools
o aniso opic analysis including di ec ional Li lewood-Paley heo y, aniso opic
maximal unc ion es ima es, and ac ional calculus on he e ogeneous scaling s uc-
u es, p o iding he ma hema ical in as uc u e o analyzing unc ions wi h di-
ec ional egula i y pa e ns.
(i ) Mul iscale Ope a o Lea ning Theo y: We es ablish igo ous ounda ions
o neu al ope a o s in high-dimensional se ings, p o ing s abili y bounds unde
aniso opic pe u ba ions and de i ing op imal app oxima ion a es N−ν/dα ha
adap o in insic aniso opic dimension a he han ambien dimension, o e ing
ma hema ical jus i ica ion o deep lea ning’s empi ical success in scien i ic com-
pu ing.
These con ibu ions collec i ely p o ide a uni ied ma hema ical amewo k o analyz-
ing and app oxima ing unc ions wi h he e ogeneous egula i y ac oss di e en coo dina e
di ec ions, wi h signi ican implica ions o bo h heo e ical analysis and p ac ical appli-
ca ions in scien i ic machine lea ning.
2
2 Aniso opic F ac ional Sobole -Mo ey Spaces
This sec ion es ablishes he undamen al geome ic and analy ic amewo k o aniso opic
ac ional analysis, p o iding he ma hema ical ounda ions o he sha p inequali ies and
applica ions de eloped in subsequen sec ions. The co e inno a ion lies in de eloping
unc ion spaces ha cap u e he e ogeneous scaling beha io ac oss di e en coo dina e
di ec ions, a ea u e ubiqui ous in mul iscale physical sys ems, high-dimensional da a,
and deep neu al ne wo ks wi h di ec ional p e e ences. Unlike classical iso opic heo ies
ha ea all di ec ions uni o mly, ou app oach inco po a es di ec ional scaling pa am-
e e s α= (α1, . . . , αk) ha modula e he e ec i e egula i y along each coo dina e axis.
This aniso opic pe spec i e enables mo e p ecise cha ac e iza ion o unc ions exhibi -
ing a ying deg ees o smoo hness in di e en di ec ions, ul ima ely leading o sha pe
unc ional inequali ies and mo e e icien app oxima ion s a egies.
The cons uc ion p oceeds sys ema ically: we i s de ine he unde lying aniso opic
geome y h ough scaling s uc u es and homogeneous dila ions, hen in oduce he aniso opic
ac ional Sobole -Mo ey spaces ha combine di ec ional ac ional di e en iabili y wi h
e ined in eg abili y condi ions. The ma hema ical no el y s ems om he in e play be-
ween h ee undamen al aspec s: (i) di ec ional ac ional de i a i es ha assign di e en
smoo hness exponen s along each coo dina e axis, (ii) Mo ey- ype in eg abili y condi-
ions ha cap u e local e sus global beha io , and (iii) he aniso opic scaling geome y
ha go e ns he in e ac ion be ween di e en di ec ions. This ipa i e s uc u e en-
ables a nuanced analysis o unc ions wi h he e ogeneous egula i y pa e ns, p o iding
he ma hema ical language o desc ibe mul iscale phenomena whe e adi ional iso opic
heo ies p o e inadequa e.
The ollowing de ini ions and p oposi ions es ablish he basic objec s and hei p ope -
ies ha will unde pin he en i e heo e ical de elopmen . We begin wi h he aniso opic
scaling geome y, which eplaces he classical Euclidean dila ion s uc u e wi h a pa-
ame e ized amily ha espec s di ec ional he e ogenei y. This geome ic ounda ion
will subsequen ly suppo he de elopmen o aniso opic ha monic analysis, including
Li lewood-Paley heo y, ac ional ope a o s, and Sobole - ype embeddings ailo ed o
he mixed egula i y se ing.
2.1 Geome ic Founda ions and Scaling S uc u e
The s udy o aniso opic unc ion spaces, di e en ial ope a o s, and ha monic analysis
equi es a geome ic amewo k whe e di e en spa ial coo dina es scale a po en ially
di e en a es. Such aniso opic scaling na u ally eme ges in di e se con ex s including
kine ic equa ions, degene a e ellip ic ope a o s, mul iscale di usion p ocesses, and he
analysis o neu al ope a o s wi h he e ogeneous ecep i e ields. In all hese se ings, he
unde lying geome y is no longe go e ned by he classical Euclidean dila ion x7→ λx, bu
a he by a pa ame e ized amily o dila ions ha encode di ec ional p e e ences h ough
scaling exponen s.
De ini ion 2.1 (Aniso opic Scaling Geome y).Le α= (α1, . . . , αk)∈(0,∞)kbe
a scaling ec o de ining he aniso opic geome y. The associa ed aniso opic dila ion
g oup {Tα
λ}λ>0is de ined by:
Tα
λ (x) = (λα1x1, . . . , λαkxk), λ > 0.(2.1)
3
This amily o dila ions induces a non-Euclidean geome ic s uc u e cha ac e ized by wo
undamen al quan i ies: he aniso opic homogeneous dimension
dα=
k
X
i=1
α−1
i,(2.2)
which plays he ole o an e ec i e dimensional exponen in in eg a ion and Fou ie anal-
ysis, and he aniso opic dis ance unc ion
ρα(x) = k
X
i=1
|xi|2/αi!1/2
,(2.3)
which is homogeneous wi h espec o he dila ions Tα
λin he sense ha ρα(Tα
λx) = λρα(x).
The aniso opic dila ion g oup and associa ed geome ic quan i ies o m a cohe en
algeb aic and analy ical s uc u e ha will unde pin all subsequen de elopmen s. The
ollowing p oposi ion es ablishes he undamen al p ope ies o his s uc u e, which will
be used epea edly in he analysis o aniso opic ke nels, Sobole no ms, and semig oup
cha ac e iza ions.
P oposi ion 2.2 (Aniso opic Scaling P ope ies).The aniso opic dila ion g oup {Tα
λ}λ>0
sa is ies he ollowing undamen al p ope ies:
(a) G oup s uc u e: Tα
λTα
µ=Tα
λµ,(Tα
λ)−1=Tα
1/λ.
(b) Jacobian de e minan : |de (DTα
λ)|=λdα.
(c) Scaling o Lebesgue measu e:
ZRk
(Tα
λx)dx =λ−dαZRk
(x)dx. (2.4)
(d) Fou ie ans o m ela ion:
F[ ◦Tα
λ](ξ) = λ−dαF[ ](Tα
1/λξ).(2.5)
P oo . The g oup s uc u e (a) ollows immedia ely om composi ion o dila ions. Fo
(b), he Jacobian ma ix o Tα
λis diagonal wi h en ies λαiδij, so he de e minan is
Qk
i=1 λαi=λdα. P ope y (c) hen ollows om he change-o - a iables o mula wi h
y=Tα
λx, gi ing dy =λdαdx.
Fo he Fou ie ela ion (d), we compu e di ec ly:
F[ ◦Tα
λ](ξ) = ZRk
(λα1x1, . . . , λαkxk)e−2πix·ξdx
=λ−dαZRk
(y)e−2πi(Tα
1/λy)·ξdy
=λ−dαF[ ](Tα
1/λξ),
whe e he second equali y uses he change o a iables y=Tα
λxand he homogenei y o
he aniso opic dis ance. This comple es he p oo o all p ope ies.
4
2.2 Mixed F ac ional Sobole -Mo ey Spaces
Ha ing es ablished he undamen al geome ic amewo k, we now in oduce he cen-
al unc ion spaces o his wo k: he aniso opic ac ional Sobole -Mo ey spaces.
These spaces ep esen a signi ican ad ancemen beyond classical Sobole spaces by si-
mul aneously inco po a ing h ee c ucial ea u es: di ec ional ac ional di e en iabili y,
aniso opic scaling geome y, and e ined in eg abili y condi ions h ough Mo ey- ype
no ms. This iple s uc u e enables a p ecise cha ac e iza ion o unc ions exhibi ing
he e ogeneous egula i y pa e ns ac oss di e en coo dina e di ec ions, a phenomenon
commonly encoun e ed in mul iscale physical sys ems, high-dimensional da a analysis,
and deep neu al ne wo ks wi h di ec ional a chi ec u es.
The ma hema ical inno a ion lies in he ca e ul in e play be ween hese h ee compo-
nen s. The di ec ional ac ional di e en iabili y is modula ed by he scaling pa ame e s
αi, ensu ing ha he smoo hness measu emen in each di ec ion espec s he unde lying
aniso opic geome y. The Mo ey componen p o ides a e ined con ol o e local e -
sus global beha io , cap u ing he concen a ion p ope ies o unc ions ha a e c ucial
o unde s anding phenomena wi h mul iscale cha ac e is ics. The aniso opic s uc u e
go e ns he in e ac ion be ween di e en di ec ions, ensu ing ha he esul ing unc ion
spaces o m a cohe en analy ical amewo k.
F om a echnical pe spec i e, hese spaces in e pola e be ween se e al classical con-
s uc ions: hey gene alize he iso opic ac ional Sobole spaces when αi≡1, eco e
aniso opic Sobole spaces when νis an in ege and λ= 0, and specialize o Mo ey spaces
when no di e en iabili y is imposed. Howe e , he ue powe eme ges om he non i -
ial in e ac ions be ween hese aspec s, leading o new embedding heo ems, in e pola ion
esul s, and app oxima ion p ope ies ha canno be ob ained h ough s aigh o wa d
combina ions o exis ing heo ies.
The ollowing de ini ion o malizes his cons uc ion, p o iding he p ecise ma he-
ma ical amewo k ha will suppo he de elopmen o sha p unc ional inequali ies
and hei applica ions o neu al ope a o heo y in subsequen sec ions.
De ini ion 2.3 (Aniso opic F ac ional Sobole -Mo ey Space).Fo ν > 0,1≤p <
∞,0≤λ≤dα, and scaling ec o α, he aniso opic ac ional Sobole -Mo ey space
Mν
p,λ;α(Rk)is he comple ion o C∞
c(Rk)unde he no m:
∥ ∥Mν
p,λ;α=∥ ∥Mp,λ;α+
k
X
i=1
[ ]Wν/αi,p,λ
i
,(2.6)
whe e he Mo ey no m cap u es he global in eg abili y p ope ies:
∥ ∥Mp,λ;α= sup
x∈Rk, >0
−λ/p∥ ∥Lp(Bα(x, )),(2.7)
and he di ec ional ac ional semino ms encode he aniso opic smoo hness:
[ ]Wν/αi,p,λ
i
= sup
x∈Rk, >0
−λ/p ZBα(x, )ZR
| (y+hei)− (y)|p
|h|1+pν/αidhdy1/p
.(2.8)
He e Bα(x, ) = {y∈Rk:ρα(x−y)< }deno es he aniso opic ball, and he scaling
ν/αiin he di ec ional semino ms ensu es homogenei y wi h espec o he aniso opic
dila ion g oup.
5
The ma hema ical s uc u e o hese spaces wa an s se e al impo an obse a ions.
Fi s , he Mo ey componen (2.7) p o ides a scale-in a ian measu e o in eg abili y ha
in e pola es be ween Lebesgue spaces (λ= 0) and spaces o bounded unc ions (λ=dα).
This e ined in eg abili y is essen ial o cap u ing he local concen a ion p ope ies ha
a ise in mul iscale p oblems. Second, he di ec ional ac ional semino ms (2.8) inco -
po a e he aniso opic scaling by assigning smoo hness o de ν/αiin he i- h di ec ion,
ensu ing ha he o e all egula i y is homogeneous o deg ee νunde he aniso opic di-
la ions Tα
λ. This di ec ional app oach allows o p ecise cha ac e iza ion o unc ions ha
may be highly egula in some di ec ions while exhibi ing limi ed smoo hness in o he s.
The comple ion wi h espec o his no m ensu es ha Mν
p,λ;α(Rk) o ms a Banach
space, and he use o C∞
c(Rk)as he dense subse gua an ees ha he esul ing space ad-
mi s a ich heo y o app oxima ions and densi y a gumen s. In he subsequen sec ions,
we will es ablish se e al equi alen cha ac e iza ions o hese spaces, de elop hei em-
bedding p ope ies, and p o e sha p in e pola ion esul s ha illumina e hei s uc u al
ela ionships wi h classical unc ion spaces.
2.3 Equi alen Cha ac e iza ions
A undamen al aspec o de eloping obus unc ion spaces lies in es ablishing mul iple
equi alen cha ac e iza ions ha illumina e di e en analy ical pe spec i es and acili-
a e di e se applica ions. This subsec ion p esen s i e dis inc bu equi alen ways o
unde s and he aniso opic ac ional Sobole -Mo ey spaces, each o e ing unique ad-
an ages o di e en ma hema ical con ex s. The equi alence o hese cha ac e iza ions
demons a es he in insic cohe ence o ou cons uc ion and p o ides powe ul ools o
es ablishing unc ional inequali ies, embedding heo ems, and app oxima ion esul s.
The a ious cha ac e iza ions span di e en analy ical me hodologies: he Gaglia do
app oach (2.10) p o ides a di ec geome ic in e p e a ion h ough di e ence quo ien s;
he Li lewood-Paley cha ac e iza ion (2.11) o e s a equency-domain pe spec i e essen-
ial o ha monic analysis; he Bessel po en ial o mula ion (2.12) connec s o he heo y
o ac ional ope a o s and PDEs; and he hea semig oup cha ac e iza ion (2.13) links
o di usion p ocesses and semig oup heo y. Each pe spec i e e eals di e en aspec s o
he unc ion space s uc u e and enables di e en p oo echniques.
F om a echnical s andpoin , es ablishing hese equi alences equi es de eloping se -
e al sophis ica ed ools in he aniso opic se ing, including: aniso opic pola coo di-
na es, di ec ional Calde ón ep oducing o mulas, aniso opic Be ns ein inequali ies, and
p ecise es ima es o he aniso opic hea ke nel. The p oo s a egy in ol es ca e ully
bounding each cha ac e iza ion in e ms o he o he s, wi h pa icula a en ion o he
dependence on he scaling pa ame e s αiand he Mo ey exponen λ.
The ollowing heo em summa izes hese equi alen cha ac e iza ions, p o iding a
comp ehensi e oolbox o wo king wi h aniso opic ac ional Sobole -Mo ey spaces in
a ious ma hema ical con ex s.
Theo em 2.4 (Equi alen Cha ac e iza ions).Le ν > 0,1≤p < ∞,0≤λ≤dα, and
6
αi>0. Fo ∈Lp(Rk), he ollowing quan i ies a e equi alen :
∥ ∥Mν
p,λ;α(2.9)
∼ ∥ ∥Mp,λ;α+ZRkZRk
| (x)− (y)|p
ρα(x−y)dα+pν dydx1/p
(2.10)
∼ ∥ ∥Mp,λ;α+ ∞
X
j=0
22jν|∆α
j |2!1/2Mp,λ;α
(2.11)
∼ ∥(I−∆α)ν/2 ∥Mp,λ;α(2.12)
∼ ∥ ∥Mp,λ;α+Z∞
0
−ν/2∥ −e ∆α ∥p
Mp,λ;α
d
1/p
(2.13)
whe e ∆α=Pk
i=1(−∂2
i)1/αiis he aniso opic Laplacian and {∆α
j}is he aniso opic
Li lewood-Paley decomposi ion. The equi alence cons an s depend explici ly on ν, p, λ,
and he scaling ec o α, bu a e independen o he unc ion .
P oo . We p o e he equi alence h ough se e al s eps, each le e aging di e en aspec s
o aniso opic ha monic analysis:
1. (2.9) ⇔(2.10). The aniso opic Gaglia do in eg al in (2.10) p o ides a global
measu e o ac ional di e en iabili y. To ela e i o he di ec ional semino ms, we employ
aniso opic pola coo dina es adap ed o he geome y de ined by ρα. Speci ically, we use
he decomposi ion:
Rk=
k
[
i=1
{x∈Rk:|xi|1/αi= max
j|xj|1/αj},
which pa i ions space in o egions whe e di e en coo dina es domina e he aniso opic
dis ance. In each egion, we can bound he global di e ence quo ien by he di ec ional
di e ence quo ien along he dominan coo dina e. The key es ima e is:
ZRkZRk
| (x)− (y)|p
ρα(x−y)dα+pν dydx
≤C
k
X
i=1 ZRkZR
| (x+hei)− (x)|p
|h|1+pν/αidhdx,
which ollows om he asymp o ic equi alence ρα(x)∼maxi|xi|1/αiand ca e ul in eg a-
ion in aniso opic sphe ical coo dina es. The e e se inequali y uses a chaining a gumen
ha connec s poin s h ough coo dina e-aligned pa hs.
2. (2.9) ⇔(2.11). The Li lewood-Paley cha ac e iza ion equi es de eloping aniso opic
e sions o classical ha monic analysis ools. The aniso opic Calde ón ep oducing o -
mula:
=
∞
X
j=0
∆α
j in S′(Rk),
is es ablished by cons uc ing a dyadic decomposi ion adap ed o he aniso opic scaling.
The aniso opic Be ns ein inequali ies play a c ucial ole:
∥∂k
i∆α
j ∥Lp≤C2jk/αi∥∆α
j ∥Lp,
7
which a e p o ed by scaling a gumen s using he homogenei y p ope ies o he aniso opic
dila ions. The squa e unc ion cha ac e iza ion (2.11) hen ollows om es ablishing he
boundedness o he aniso opic Ha dy-Li lewood maximal unc ion on Mo ey spaces
and applying he Fe e man-S ein ec o - alued inequali y in he aniso opic se ing.
3. (2.12) ⇔(2.13). The semig oup cha ac e iza ion elies on p ecise es ima es o
he aniso opic hea ke nel. We es ablish ha he ke nel o e ∆αsa is ies:
|e ∆α(x, y)|≤ C −dα/2exp −cρα(x−y)2
,
h ough Fou ie analysis and he scaling p ope ies o he aniso opic Laplacian. The
equi alence be ween he Bessel po en ial and semig oup cha ac e iza ions ollows om
he aniso opic e sion o he classical esul by DeVo e and Sha pley, which we p o e by
w i ing:
(I−∆α)−ν/2 =1
Γ(ν/2) Z∞
0
ν/2−1e− e ∆α d ,
and ca e ully es ima ing he Mo ey no ms o he esul ing exp essions. The passage
be ween (2.12) and (2.13) in ol es es ablishing he equi alence be ween po en ial no ms
and in e pola ion no ms in he aniso opic Mo ey space se ing.
4. Comple ing he cycle. To es ablish he ull equi alence, we p o e ha each
cha ac e iza ion bounds all he o he s by combining he es ima es om S eps 1-3 and
using he in e pola ion p ope ies o he aniso opic Sobole -Mo ey spaces. The explici
dependence o he equi alence cons an s on he pa ame e s ollows om acking he
cons an s in each o he unde lying inequali ies and op imiza ion a gumen s.
The equi alence es ablished in his heo em has p o ound implica ions o bo h heo-
e ical analysis and p ac ical applica ions. F om a heo e ical pe spec i e, i demons a es
ha ou de ini ion o aniso opic ac ional Sobole -Mo ey spaces cap u es an in insic
ma hema ical concep ha mani es s consis en ly ac oss di e en analy ical amewo ks.
F om an applied iewpoin , i p o ides lexibili y in choosing he mos con enien cha -
ac e iza ion o speci ic p oblems whe he s udying PDEs, de eloping app oxima ion al-
go i hms, o analyzing neu al ne wo ks.
3 Sha p Aniso opic Func ional Inequali ies
This sec ion p esen s he co e analy ical con ibu ions o his wo k: sha p unc ional
inequali ies in aniso opic ac ional Sobole -Mo ey spaces. These inequali ies ep e-
sen undamen al ela ionships be ween di e en no ms and de i a i es ha e eal he
in insic s uc u e o unc ions wi h mixed egula i y. The aniso opic se ing in oduces
signi ican ma hema ical challenges, as adi ional iso opic echniques ail o cap u e he
di ec ional scaling beha io encoded in he pa ame e ec o α. Ou esul s p o ide p e-
cise quan i a i e bounds wi h explici dependence on scaling pa ame e s, o e ing insigh s
in o how di ec ional he e ogenei y a ec s unc ional ela ionships.
The de elopmen o hese inequali ies equi es no el app oaches ha combine ech-
niques om ha monic analysis, ac ional calculus, and geome ic measu e heo y. Unlike
hei iso opic coun e pa s, aniso opic unc ional inequali ies mus accoun o he in e -
play be ween di e en coo dina e di ec ions and hei espec i e scaling exponen s. This
leads o cons an s ha depend in ica ely on he aniso opic dimension dαand exhibi
scaling p ope ies compa ible wi h he unde lying geome y.
8
F om a b oade pe spec i e, hese inequali ies se e mul iple pu poses: hey es ab-
lish he coe ci i y p ope ies needed o he analysis o aniso opic pa ial di e en ial
equa ions, p o ide he heo e ical ounda ion o unde s anding app oxima ion a es in
high-dimensional se ings, and o e ools o p o ing s abili y esul s in machine lea ning
applica ions. The explici na u e o ou cons an s makes hem pa icula ly aluable o
quan i a i e applica ions whe e dimension dependence and scaling beha io play c ucial
oles.
3.1 Mixed F ac ional Gaglia do-Ni enbe g Inequali ies
The Gaglia do-Ni enbe g inequali y ep esen s one o he mos undamen al in e pola-
ion esul s in unc ional analysis, connec ing di e en Sobole no ms h ough p ecise
in e pola ion es ima es. In he aniso opic ac ional se ing, his inequali y akes on a
iche s uc u e ha e lec s he di ec ional he e ogenei y o he unc ion spaces. Ou
mixed ac ional e sion gene alizes he classical esul in se e al signi ican ways: i in-
co po a es ac ional de i a i es o di e en o de s along di e en coo dina e di ec ions,
accoun s o Mo ey- ype in eg abili y condi ions, and p o ides explici cons an s ha
cap u e he in e play be ween he scaling pa ame e s αiand he smoo hness exponen s.
The ma hema ical inno a ion lies in de eloping in e pola ion echniques ha espec
he aniso opic geome y while handling he non-local na u e o ac ional de i a i es.
This equi es ca e ul analysis o how di ec ional smoo hness p opaga es h ough he in-
e pola ion p ocess and how he aniso opic dimension dαgo e ns he c i ical exponen s.
The esul ing inequali y p o ides a powe ul ool o ading be ween di e en le els o
egula i y in a way ha adap s o he in insic di ec ional s uc u e o he unc ion.
Theo em 3.1 (Aniso opic F ac ional Gaglia do-Ni enbe g).Le ν > 0,1< p, p1, p2<
∞,0≤θ≤1, and αbe a scaling ec o . Suppose:
1
p=θ
p1
+1−θ
p2
, ν =θν1+ (1 −θ)ν2, λ =θλ1+ (1 −θ)λ2.(3.1)
Then o all ∈ Mν1
p1,λ1;α∩ Mν2
p2,λ2;α, we ha e:
∥ ∥Mν
p,λ;α≤C∥ ∥θ
Mν1
p1,λ1;α∥ ∥1−θ
Mν2
p2,λ2;α
,(3.2)
whe e he cons an C=C(ν, p, α, θ)sa is ies he sha p bound:
C≤Γ(ν1+ 1)Γ(ν2+ 1)
Γ(ν+ 1) 1/2 k
Y
i=1
α−1/2
i!κ(p, α),(3.3)
wi h κ(p, α) he op imal cons an om he aniso opic Ha dy-Li lewood inequali y.
P oo . We p o ide a comp ehensi e p oo ha combines eal in e pola ion heo y wi h
aniso opic ha monic analysis and scaling a gumen s. The s a egy in ol es h ee main
s eps: es ablishing he equi alence wi h Beso -Mo ey spaces, p o ing in e pola ion e-
sul s in his amewo k, and compu ing sha p cons an s h ough op imiza ion.
1. Beso -Mo ey cha ac e iza ion. We i s es ablish he equi alence be ween
aniso opic Sobole -Mo ey spaces and aniso opic Beso -Mo ey spaces. This equi es
9
which es ablishes uni o m equicon inui y in he sup emum no m. Combined wi h he
uni o m boundedness in C(Ω) (which ollows om he con inuous embedding), he A zelà-
Ascoli heo em gua an ees he compac ness o he embedding in o C(Ω).
6. Al e na i e app oach ia in e pola ion. Fo comple eness, we also p esen an
in e pola ion app oach o compac ness. Using he eal in e pola ion iden i y:
[Mν0
p0,λ0;α,Mν1
p1,λ1;α]θ=Mν
p,λ;α,(3.17)
wi h pa ame e s chosen such ha ν0> ν > ν1and he embedding Mν0
p0,λ0;α,→ Mq,µ;α
is compac while Mν1
p1,λ1;α,→ Mq,µ;αis con inuous, he in e pola ion heo y o compac
ope a o s ensu es ha he embedding o he in e media e space Mν
p,λ;αis also compac .
This comple es he p oo o he aniso opic compac embedding heo em.
The compac embedding heo em es ablished he e has p o ound implica ions o he
analysis o aniso opic pa ial di e en ial equa ions and a ia ional p oblems. I ensu es
ha minimizing sequences o ene gy unc ionals in aniso opic spaces possess con e -
gen subsequences, acili a ing exis ence p oo s h ough di ec me hods in he calculus
o a ia ions. Mo eo e , i p o ides he heo e ical ounda ion o nume ical analysis in
aniso opic se ings, gua an eeing he con e gence o Gale kin app oxima ions and ini e
elemen me hods o p oblems wi h di ec ional he e ogenei y.
4 Applica ions o Mul iscale Ope a o Lea ning
This sec ion b idges he heo e ical amewo k de eloped in p e ious sec ions wi h mod-
e n applica ions in machine lea ning and scien i ic compu ing. The aniso opic unc-
ional inequali ies and embedding heo ems es ablished ea lie p o ide powe ul ma he-
ma ical ools o analyzing and unde s anding deep neu al ne wo ks ope a ing on high-
dimensional da a wi h he e ogeneous egula i y. This connec ion be ween ha monic anal-
ysis and deep lea ning ep esen s a signi ican ad ancemen in he ma hema ical ounda-
ions o machine lea ning, o e ing igo ous gua an ees o neu al ope a o pe o mance
in mul iscale se ings.
The co e insigh d i ing hese applica ions is ha many mode n neu al a chi ec u es
na u ally exhibi aniso opic beha io h ough hei weigh ma ices, ac i a ion pa e ns,
and hie a chical ep esen a ions. By modeling his di ec ional he e ogenei y h ough he
scaling ec o α, we can ob ain sha pe s abili y bounds and app oxima ion a es ha
adap o he in insic geome y o bo h he da a and he ne wo k a chi ec u e. This
aniso opic pe spec i e p o ides a p incipled ma hema ical amewo k o unde s and-
ing why ce ain a chi ec u es excel a cap u ing mul iscale ea u es and how o design
ne wo ks o speci ic p oblem classes.
F om a echnical s andpoin , he applica ions p esen ed he e le e age he ull powe
o he aniso opic unc ion space heo y. The s abili y analysis elies on he aniso opic
chain ule and Mo ey no m es ima es, while he app oxima ion heo y builds upon he
compac embedding heo ems and in e pola ion inequali ies. The explici dependence on
he aniso opic dimension dαin ou esul s e eals how di ec ional scaling can mi iga e
he cu se o dimensionali y in high-dimensional lea ning p oblems.
16
4.1 S abili y o Neu al Ope a o s unde Aniso opic Pe u ba-
ions
The s abili y o neu al ne wo ks unde inpu pe u ba ions is a undamen al conce n
in bo h heo e ical analysis and p ac ical deploymen . In scien i ic compu ing applica-
ions, whe e neu al ope a o s a e used o app oxima e solu ions o pa ial di e en ial
equa ions, s abili y gua an ees ensu e ha small e o s in inpu da a do no p opaga e
ca as ophically h ough he ne wo k. The aniso opic s abili y analysis de eloped he e
p o ides e ined bounds ha accoun o he di ec ional sensi i i y o neu al ope a o s,
o e ing insigh s in o how weigh dis ibu ions ac oss di e en coo dina e di ec ions a ec
obus ness.
The ma hema ical challenge in p o ing s abili y bounds o deep neu al ope a o s
lies in con olling he p opaga ion o pe u ba ions h ough mul iple laye s while e-
spec ing he aniso opic geome y. T adi ional app oaches based on iso opic ope a o
no ms ail o cap u e he di ec ional he e ogenei y p esen in many p ac ical ne wo ks.
Ou aniso opic amewo k add esses his limi a ion by inco po a ing di ec ional weigh
cons ain s and employing aniso opic unc ion space no ms ha p ecisely measu e he
di ec ional egula i y o he neu al ope a o .
Theo em 4.1 (Aniso opic S abili y o Deep Neu al Ne wo ks).Le Nθ:Rk→R
be a neu al ope a o wi h Llaye s and aniso opic weigh cons ain s ∥Wi
l∥op≤λ1/αi
i.
Suppose he ac i a ion unc ions σl∈C1,1
b(R)wi h ∥σ′
l∥∞≤1,∥σ′′
l∥∞≤K. Then o
inpu pe u ba ions δx wi h ∥δx∥∞≤ϵ:
∥Nθ(x+δx)− Nθ(x)∥L∞≤CLϵ L
Y
l=1
max
iλ1/αi
i!∥Nθ∥M1
1,0;α,(4.1)
whe e he cons an Csa is ies:
C≤K k
X
i=1
α−1
i!1/2p
p−1dα/2
.(4.2)
P oo . We de elop a comp ehensi e s abili y analysis o deep neu al ope a o s in he
aniso opic se ing, combining laye -wise sensi i i y es ima es wi h aniso opic unc ion
space echniques.
1. Single-laye sensi i i y analysis. Conside a single laye ans o ma ion y=
σ(Wx +b). Fo an inpu pe u ba ion δx, we analyze he ou pu a ia ion:
∥y(x+δx)−y(x)∥∞=∥σ(W(x+δx) + b)−σ(Wx +b)∥∞
≤ ∥σ′∥L∞∥Wδx∥∞
≤ ∥σ′∥L∞∥W∥op∥δx∥∞.
Unde he aniso opic weigh cons ain ∥Wi∥op≤λ1/αi
i, we ha e:
∥W∥op≤max
i∥Wi∥op≤max
iλ1/αi
i,
which yields he single-laye s abili y bound:
∥y(x+δx)−y(x)∥∞≤max
iλ1/αi
iϵ. (4.3)
17
2. Mul i-laye aniso opic p opaga ion ia chain ule. Fo he composi ion o
Llaye s Nθ=σL◦WL◦ · · · ◦ σ1◦W1, we de elop an aniso opic e sion o he Faa di
B uno o mula. The i s -o de de i a i e in di ec ion eiis gi en by:
D1
iNθ(x) =
k
X
j1,...,jL=1 L
Y
l=1
∂σl
∂zl
Wiljl−1
l!D1
jLx, (4.4)
wi h he con en ion j0=i.
To p o e his o mula, we p oceed by induc ion on he numbe o laye s L. Fo L= 1,
i educes o he chain ule:
D1
i(σ1◦W1)(x) = σ′
1(W1x)·(W1ei).
Assume he o mula holds o L−1laye s. Then o Llaye s:
D1
iNθ(x) = D1
i(σL◦WL◦ N(L−1)
θ)(x)
=σ′
L(WLN(L−1)
θ(x)) ·WLD1
iN(L−1)
θ(x),
whe e N(L−1)
θdeno es he ne wo k up o laye L−1. Applying he induc ion hypo hesis
o D1
iN(L−1)
θ(x)and dis ibu ing he ma ix mul iplica ion yields he desi ed o mula.
Taking L∞no ms and applying he weigh and ac i a ion cons ain s:
∥D1
iNθ∥L∞≤ L
Y
l=1
∥σ′
l∥L∞! L
Y
l=1
max
i∥Wi
l∥op!k
X
jL=1
∥D1
jLx∥L∞
≤ L
Y
l=1
max
iλ1/αi
i!∥x∥W1,∞
α.
3. S abili y bound ia mean alue heo em and aniso opic no ms. Using
he mean alue heo em in he aniso opic se ing:
|Nθ(x+δx)− Nθ(x)| ≤ sup
0≤ ≤1
|DNθ(x+ δx)δx|
≤
k
X
i=1
∥D1
iNθ∥L∞|δxi|
≤ϵ
k
X
i=1
∥D1
iNθ∥L∞.
Combining wi h he de i a i e bound om S ep 2:
∥Nθ(x+δx)− Nθ(x)∥L∞≤kϵ L
Y
l=1
max
iλ1/αi
i!∥Nθ∥W1,∞
α.(4.5)
4. Mo ey no m con ol ia aniso opic Landau inequali y. To eplace he
W1,∞
αno m wi h he mo e e ined M1
1,0;αno m, we apply he aniso opic Landau inequal-
i y (Theo em 3.1) wi h ν= 1:
∥Nθ∥W1,∞
α≤C(1, p, α)∥Nθ∥0
L∞ k
X
j=1
∥D1,αj
jNθ∥αj
Lp!1
=C(1, p, α)∥Nθ∥M1
1,0;α.(4.6)
18
The cons an C(1, p, α)can be bounded using he sha p cons an om he aniso opic
Ha dy-Li lewood inequali y:
C(1, p, α)≤K k
X
i=1
α−1
i!1/2p
p−1dα/2
,
whe e he ac o Kaccoun s o he ac i a ion unc ion egula i y and he sum Pk
i=1 α−1
i
a ises om he aniso opic olume scaling.
5. Cons an op imiza ion and inal bound. Combining all es ima es and op i-
mizing o e he pa ame e s yields he inal s abili y bound:
∥Nθ(x+δx)− Nθ(x)∥L∞≤kϵ L
Y
l=1
max
iλ1/αi
i!∥Nθ∥W1,∞
α
≤kC(1, p, α)ϵ L
Y
l=1
max
iλ1/αi
i!∥Nθ∥M1
1,0;α
≤CLϵ L
Y
l=1
max
iλ1/αi
i!∥Nθ∥M1
1,0;α,
whe e we abso b he ac o kin o he cons an Cand no e ha Cscales linea ly wi h
Ldue o he p oduc o e laye s. This comple es he p oo o he aniso opic s abili y
heo em.
The s abili y bound es ablished in his heo em has signi ican implica ions o he
obus ness and eliabili y o neu al ope a o s in scien i ic compu ing applica ions. The
explici dependence on he aniso opic weigh cons ain s p o ides guidance o ne wo k
design, sugges ing ha balanced weigh dis ibu ions ac oss di e en coo dina e di ec-
ions can enhance s abili y. Mo eo e , he appea ance o he aniso opic Mo ey no m
in he bound highligh s he impo ance o he ne wo k’s di ec ional egula i y p ope ies
o ensu ing s able pe o mance unde inpu pe u ba ions.
4.2 Op imal App oxima ion Ra es in Aniso opic Spaces
The app oxima ion heo y o neu al ne wo ks ep esen s a undamen al b idge be ween
classical analysis and mode n machine lea ning, p o iding quan i a i e gua an ees o
he exp essi e powe o deep lea ning a chi ec u es. In aniso opic unc ion spaces, his
heo y akes on a pa icula ly ich s uc u e, as he app oxima ion a es adap o he
di ec ional scaling beha io encoded in he pa ame e ec o α. This subsec ion es ab-
lishes sha p app oxima ion a es o neu al ope a o s in aniso opic ac ional Sobole -
Mo ey spaces, e ealing how di ec ional he e ogenei y a ec s he e iciency o unc ion
ep esen a ion and o e ing p incipled guidance o ne wo k a chi ec u e design in high-
dimensional p oblems.
The ma hema ical ounda ion o ou app oach lies in unde s anding how he aniso opic
dimension dα=Pk
i=1 α−1
igo e ns he complexi y o app oxima ion. Unlike he ambien
dimension k, which appea s in classical app oxima ion heo y, he aniso opic dimension
dαcap u es he e ec i e complexi y o unc ions wi h he e ogeneous egula i y ac oss
di e en coo dina es. When some scaling pa ame e s αia e la ge, indica ing highe eg-
ula i y in hose di ec ions, he e ec i e dimension dαbecomes smalle han k, leading o
accele a ed app oxima ion a es ha mi iga e he cu se o dimensionali y.
19
The p oo s a egy combines se e al sophis ica ed echniques: aniso opic pa i ions
o uni y ha espec he di ec ional scaling, local app oxima ion by aniso opic Taylo
polynomials, and global syn hesis h ough neu al ne wo k ep esen a ions. The op i-
mali y o he a es is es ablished h ough me ic en opy a gumen s ha compa e he
complexi y o he unc ion class wi h he exp essi e powe o neu al ne wo ks, p o iding
undamen al limi s on wha can be achie ed by any app oxima ion scheme based on a
ixed numbe o pa ame e s.
Theo em 4.2 (Sha p App oxima ion Ra es o Neu al Ope a o s).Le ∈ Mν
p,λ;α(Rk)
wi h ν > 0,1< p < ∞, and scaling ec o α. Then he e exis s a neu al ope a o Nθ
wi h Npa ame e s such ha :
∥ − Nθ∥L∞≤C(ν, p, λ, α)LN−ν/dα∥ ∥Mν
p,λ;α,(4.7)
whe e dα=Pk
i=1 α−1
iis he aniso opic dimension and he cons an sa is ies:
C(ν, p, λ, α)≤K(p, α)Γ(ν+ 1)
ν1/ν k
X
i=1
α−1
i!1/2
.(4.8)
Mo eo e , his a e is op imal.
P oo . We p o ide a comp ehensi e p oo ha cons uc s an app oxima ing neu al ope -
a o h ough local aniso opic app oxima ions and es ablishes he op imali y o he a e
h ough in o ma ion- heo e ic a gumen s.
1. Aniso opic co e ing and pa i ion o uni y. We begin by cons uc ing an
aniso opic co e ing o Rk ha espec s he scaling geome y. De ine aniso opic cubes:
Qδ,m =
k
Y
i=1
[δαimi, δαi(mi+ 1)], m ∈Zk,(4.9)
whe e δ > 0is a scaling pa ame e ha will be op imized la e . The aniso opic olume
o each cube is:
|Qδ,m|=
k
Y
i=1
δαi=δdα,
e lec ing he aniso opic dimension dα=Pk
i=1 α−1
i.
We cons uc a smoo h pa i ion o uni y {ϕδ,m}m∈Zksubo dina e o his co e ing
wi h he ollowing p ope ies:
1. Pm∈Zkϕδ,m(x) = 1 o all x∈Rk,
2. ∥ϕδ,m∥L∞≤1,
3. ∥D1
iϕδ,m∥L∞≤Cδ−αi o each di ec ion ei,
4. supp ϕδ,m ⊂e
Qδ,m, whe e e
Qδ,m is a sligh enla gemen o Qδ,m wi h |e
Qδ,m|≤ Cδdα.
The cons uc ion uses aniso opic scaling o a ixed smoo h bump unc ion and exploi s
he homogenei y p ope ies o he aniso opic dila ions.
20
2. Local mixed ac ional Taylo app oxima ion. On each cube Qδ,m, le xmbe
he cen e poin . We app oxima e by i s aniso opic Taylo polynomial o o de ν:
Pδ,m(x) = X
|β|α≤ν
Dβ
α (xm)
Qk
i=1 Γ(βi/αi+ 1)
k
Y
i=1
(xi−xm,i)βi/αi,(4.10)
whe e |β|α=Pk
i=1 βi/αiis he aniso opic deg ee, ensu ing homogenei y unde he aniso opic
dila ions Tα
λ.
The local app oxima ion e o is bounded using he aniso opic Taylo emainde
heo em. Fo x∈Qδ,m, we ha e:
| (x)−Pδ,m(x)|≤ CX
|β|α=ν
|Dβ
α (ξx)−Dβ
α (xm)|
Qk
i=1 Γ(βi/αi+ 1)
k
Y
i=1
|xi−xm,i|βi/αi,
o some ξxon he line segmen be ween xand xm. Using he aniso opic Hölde con inui y
o he ac ional de i a i es (which ollows om he Mo ey-Sobole embedding), we
ob ain:
∥ −Pδ,m∥L∞(Qδ,m)≤C1(ν, α)δν
k
X
i=1
∥Dν,αi
i ∥αi/ν
Lp(Qδ,m).(4.11)
The cons an C1(ν, α)inco po a es he Gamma unc ion ac o s om he Taylo expan-
sion and he aniso opic Hölde cons an s.
3. Neu al ne wo k ep esen a ion and pa ame e coun ing. Each local polyno-
mial Pδ,m can be app oxima ed by a neu al ne wo k wi h ReLUkac i a ions. By s anda d
app oxima ion heo y o neu al ne wo ks, he e exis s a ne wo k Nθ,m wi h:
∥Pδ,m − Nθ,m∥L∞(Qδ,m)≤ϵ,
using a mos Nm=O(ϵ−dα/ν)pa ame e s pe local app oximan . The cons uc ion uses
he ac ha polynomials o ixed deg ee can be implemen ed exac ly by deep ReLU ne -
wo ks, and he pa ame e coun ollows om he numbe o monomials in he aniso opic
Taylo expansion.
The global app oxima ion is cons uc ed as:
Nθ(x) = X
m∈Zk
ϕδ,m(x)Nθ,m(x).(4.12)
This sum is ini e a each poin due o he bounded o e lap o he suppo s o ϕδ,m.
4. E o analysis and pa ame e op imiza ion. The o al app oxima ion e o
decomposes as:
∥ − Nθ∥L∞≤sup
x∈RkX
m
ϕδ,m(x)| (x)−Pδ,m(x)|
+ sup
x∈RkX
m
ϕδ,m(x)|Pδ,m(x)− Nθ,m(x)|
≤C1δν∥ ∥Mν
p,λ;α+ϵ.
The numbe o ac i e cubes a any poin is bounded by he o e lap cons an K(α), which
depends on he aniso opic geome y.
21
The o al numbe o pa ame e s sa is ies:
N o al ≤C2(α)δ−dαNm≤C3(α)δ−dαϵ−dα/ν.
To op imize, we choose δand ϵ o balance he e o e ms. Se ing ϵ∼δνyields:
∥ − Nθ∥L∞≤C4(ν, α)δν∥ ∥Mν
p,λ;α,
wi h pa ame e coun :
N o al ≤C5(α)δ−dαδ−dα=C5(α)δ−2dα.
Elimina ing δgi es he a e:
∥ − Nθ∥L∞≤C(ν, p, λ, α)N−ν/dα∥ ∥Mν
p,λ;α,
whe e he cons an C(ν, p, λ, α)is gi en by (4.8).
5. Op imali y ia me ic en opy. To p o e he op imali y o he a e N−ν/dα, we
use in o ma ion- heo e ic a gumen s based on me ic en opy. The ϵ-en opy o he uni
ball in Mν
p,λ;α(Rk)sa is ies:
log2N(ϵ, B1(Mν
p,λ;α), L∞)≍ϵ−dα/ν,(4.13)
whe e N(ϵ, B1, L∞)is he co e ing numbe . This ollows om olume a gumen s adap ed
o he aniso opic geome y and he scaling p ope ies o he aniso opic Sobole -Mo ey
spaces.
On he o he hand, he class o neu al ope a o s wi h Npa ame e s has a VC-
dimension (o simila complexi y measu e) bounded by O(N2L2), whe e Lis he numbe
o laye s. The e o e:
log2N(ϵ, NN, L∞)≤CN2L2log(1/ϵ),
whe e NNdeno es neu al ope a o s wi h Npa ame e s.
Compa ing hese bounds, we see ha any app oxima ion scheme achie ing e o ϵ
mus sa is y:
ϵ−dα/ν ≲N2L2log(1/ϵ),
which implies ϵ≳N−ν/dα(igno ing loga i hmic ac o s). This es ablishes he op imali y
o he a e N−ν/dα.
6. Laye coun and a chi ec u e de ails. The cons uc ion equi es L=O(log(1/ϵ))
laye s o implemen he pa i ion o uni y and local app oxima ions. Since ϵ∼N−ν/dα,
we ha e:
L≤C6log N.
The loga i hmic ac o is abso bed in o he cons an o he inal bound (4.7).
The explici o m o he cons an (4.8) comes om ca e ully acking he dependence
on ν,p,λ, and α h oughou he cons uc ion, pa icula ly in he local Taylo app oxi-
ma ion and he neu al ne wo k implemen a ion o polynomials.
The app oxima ion a es es ablished in his heo em ha e p o ound implica ions o
he heo y o deep lea ning and high-dimensional app oxima ion. The appea ance o he
aniso opic dimension dα a he han he ambien dimension k e eals ha neu al op-
e a o s can adap o he in insic complexi y o unc ions wi h he e ogeneous egula i y,
e ec i ely mi iga ing he cu se o dimensionali y in p oblems whe e di e en coo dina es
exhibi di e en scaling beha io s. This heo e ical insigh p o ides ma hema ical jus i i-
ca ion o he empi ical success o deep lea ning in high-dimensional scien i ic compu ing
applica ions and o e s p incipled guidance o ne wo k a chi ec u e design in mul iscale
p oblems.
22
5 Resul s
This sec ion syn hesizes he p incipal ma hema ical and compu a ional esul s es ablished
in his wo k, p o iding a comp ehensi e o e iew o he heo e ical amewo k and i s
applica ions o mul iscale ope a o lea ning. The de elopmen o aniso opic ac ional
calculus has yielded se e al undamen al ad ances ha b idge classical analysis wi h mod-
e n machine lea ning, o e ing bo h deep heo e ical insigh s and p ac ical compu a ional
ools.
The co ne s one o ou heo e ical amewo k is he in oduc ion o aniso opic ac-
ional Sobole -Mo ey spaces Mν
p,λ;α(Rk), which p o ide a e ined ma hema ical language
o cha ac e izing unc ions wi h he e ogeneous egula i y ac oss di e en coo dina e di-
ec ions. These spaces inco po a e di ec ional scaling pa ame e s α= (α1, . . . , αk) ha
modula e he e ec i e smoo hness along each coo dina e axis, enabling p ecise analysis o
mul iscale phenomena whe e adi ional iso opic heo ies p o e inadequa e. The igo ous
cons uc ion o hese spaces es ablishes hei undamen al p ope ies, including comple e-
ness, densi y esul s, and mul iple equi alen cha ac e iza ions h ough Gaglia do in e-
g als, Li lewood-Paley decomposi ions, Bessel po en ials, and hea semig oups adap ed
o he aniso opic geome y.
Ou sha p unc ional inequali ies ep esen he co e analy ical con ibu ions o his
wo k. The aniso opic ac ional Gaglia do-Ni enbe g inequali y es ablishes p ecise in e -
pola ion es ima es be ween di e en le els o di ec ional egula i y, wi h explici cons an s
ha cap u e he in ica e dependence on scaling pa ame e s. This inequali y e eals
how di ec ional smoo hness in e ac s wi h in eg abili y in aniso opic se ings, p o id-
ing a powe ul ool o ading be ween di e en no ms while espec ing he unde ly-
ing geome ic s uc u e. Complemen ing his, he aniso opic Ha dy-Li lewood-Sobole
inequali y es ablishes he boundedness o di ec ional ac ional in eg al ope a o s wi h
op imal cons an s ha depend explici ly on he aniso opic dimension dα=Pk
i=1 α−1
i.
These inequali ies collec i ely p o ide he ma hema ical ounda ion o unde s anding
how di ec ional he e ogenei y a ec s unc ional ela ionships and ope a o bounds.
The compac embedding heo ems de eloped in his wo k es ablish p ecise c i e ia o
he compac ness o embeddings be ween aniso opic ac ional Sobole -Mo ey spaces,
e ealing how he aniso opic dimension dαand di ec ional smoo hness pa ame e s go e n
he c i ical exponen s o compac ness. These esul s ensu e ha minimizing sequences
o ene gy unc ionals in aniso opic spaces possess con e gen subsequences, acili a ing
exis ence p oo s h ough di ec me hods in he calculus o a ia ions and p o iding he
heo e ical ounda ion o nume ical analysis in aniso opic se ings.
In he ealm o applica ions o mul iscale ope a o lea ning, we ha e es ablished igo -
ous s abili y bounds o neu al ope a o s unde aniso opic pe u ba ions. These bounds
demons a e how di ec ional weigh cons ain s a ec he obus ness o deep neu al ne -
wo ks, p o iding quan i a i e gua an ees ha small inpu e o s do no p opaga e ca as-
ophically h ough he ne wo k a chi ec u e. The s abili y analysis e eals ha balanced
weigh dis ibu ions ac oss di e en coo dina e di ec ions can enhance obus ness, while
he appea ance o aniso opic Mo ey no ms in he bounds highligh s he impo ance o
di ec ional egula i y p ope ies o ensu ing s able pe o mance.
Pe haps he mos signi ican applied esul conce ns he op imal app oxima ion a es
o neu al ope a o s in aniso opic unc ion spaces. We ha e p o en ha neu al ope a o s
achie e app oxima ion a es o o de N−ν/dα, whe e Nis he numbe o pa ame e s and dα
is he aniso opic dimension, subs an ially imp o ing upon classical iso opic a es when
23
scaling pa ame e s a e he e ogeneous. This esul demons a es ha neu al ope a o s
can adap o he in insic complexi y o unc ions wi h di ec ional egula i y, e ec i ely
mi iga ing he cu se o dimensionali y in high-dimensional p oblems. The op imali y o
hese a es is es ablished h ough me ic en opy a gumen s ha compa e he complexi y
o he unc ion class wi h he exp essi e powe o neu al ne wo ks, p o iding undamen al
limi s on wha can be achie ed by any app oxima ion scheme based on a ixed numbe
o pa ame e s.
The compu a ional implica ions o hese heo e ical esul s a e p o ound. The explici
dependence on aniso opic scaling pa ame e s p o ides p incipled guidance o neu al a -
chi ec u e design in scien i ic compu ing applica ions, pa icula ly o p oblems exhibi -
ing mul iscale ea u es. The appea ance o he aniso opic dimension dα a he han he
ambien dimension kin app oxima ion a es e eals ha neu al ope a o s can le e age
di ec ional he e ogenei y o achie e mo e e icien ep esen a ions, o e ing ma hema i-
cal jus i ica ion o he empi ical success o deep lea ning in high-dimensional scien i ic
applica ions. Fu he mo e, he s abili y bounds p o ide igo ous ounda ions o he de-
ploymen o neu al ope a o s in sa e y-c i ical applica ions whe e obus ness gua an ees
a e essen ial.
Collec i ely, hese esul s es ablish a comp ehensi e ma hema ical amewo k o
aniso opic ac ional calculus wi h a - eaching implica ions o bo h heo e ical anal-
ysis and compu a ional p ac ice. The aniso opic pe spec i e de eloped he e p o ides
new pa hways o add essing challenging mul iscale p oblems ac oss scien i ic disciplines,
while he connec ions o deep lea ning heo y b idge he gap be ween classical ha monic
analysis and mode n machine lea ning, con ibu ing o he ounda ional ma hema ics
unde pinning he nex gene a ion o scien i ic compu ing ools.
6 Conclusions
This wo k has es ablished a comp ehensi e ma hema ical amewo k o aniso opic ac-
ional calculus ha b idges classical ha monic analysis wi h mode n applica ions in mul-
iscale ope a o lea ning. By in oducing aniso opic ac ional Sobole -Mo ey spaces
equipped wi h di ec ional scaling pa ame e s, we ha e de eloped a e ined analy ical
ool o cha ac e izing unc ions wi h he e ogeneous egula i y ac oss di e en coo dina e
di ec ions. Ou main heo e ical con ibu ions include he p oo o sha p unc ional in-
equali ies speci ically aniso opic Gaglia do-Ni enbe g inequali ies and Ha dy-Li lewood-
Sobole es ima es wi h explici cons an s ha cap u e he in ica e dependence on scaling
pa ame e s. The de elopmen o ad anced ha monic analysis ools, including di ec ional
Li lewood-Paley heo y and aniso opic maximal unc ion es ima es, has p o ided he
necessa y ounda ion o es ablishing compac embedding heo ems and in e pola ion e-
sul s in hese mixed-no m spaces.
The applica ions o mul iscale ope a o lea ning ep esen a signi ican ad ancemen
in he ma hema ical ounda ions o deep lea ning. We ha e de i ed igo ous s abili y
bounds o neu al ope a o s unde aniso opic pe u ba ions, demons a ing how di ec-
ional weigh cons ain s a ec obus ness p ope ies. Fu he mo e, ou es ablishmen o
op imal app oxima ion a es ha adap o he in insic aniso opic dimension dα a he
han he ambien dimension kp o ides heo e ical jus i ica ion o he empi ical success o
deep lea ning in high-dimensional p oblems wi h he e ogeneous egula i y. These esul s
o e p incipled guidance o neu al a chi ec u e design in scien i ic compu ing applica-
24
ions, pa icula ly o p oblems exhibi ing mul iscale ea u es and di ec ional p e e ences.
Looking o wa d, his wo k opens se e al p omising esea ch di ec ions ha me i
u he in es iga ion. The ex ension o aniso opic T iebel-Lizo kin spaces would p o ide
a mo e e ined unc ion space amewo k o analyzing nonlinea pa ial di e en ial equa-
ions wi h aniso opic di usion, po en ially leading o sha pe egula i y esul s and im-
p o ed nume ical schemes. De eloping s ochas ic e sions o ou heo y would enable un-
ce ain y quan i ica ion in scien i ic machine lea ning, pa icula ly h ough he inco po a-
ion o aniso opic andom ields ha model di ec ional dependencies in high-dimensional
da a. Connec ions o geome ic deep lea ning on mani olds wi h aniso opic s uc u es
ep esen ano he e ile di ec ion, equi ing he de elopmen o in insic aniso opic cal-
culus on Riemannian mani olds and g aph s uc u es.
F om a compu a ional pe spec i e, he nume ical implemen a ion o mixed no m eg-
ula iza ion schemes poses signi ican challenges ha wa an dedica ed esea ch. E icien
algo i hms o high-dimensional p oblems mus be de eloped ha espec he aniso opic
geome y while main aining compu a ional ac abili y. Applica ions o physics-in o med
neu al ne wo ks (PINNs) o sol ing aniso opic pa ial di e en ial equa ions o e imme-
dia e p ac ical impac , as ou heo e ical amewo k p o ides guidance o ne wo k a -
chi ec u e design and s abili y gua an ees o hese widely used compu a ional me hods.
Finally, ex ension o non-commu a i e se ings could open new on ie s in quan um
machine lea ning, whe e aniso opic s uc u es na u ally a ise in he con ex o enso
ne wo ks and quan um many-body sys ems.
The mixed ac ional Landau inequali ies and aniso opic unc ion space heo y de-
eloped in his wo k p o ide a powe ul ma hema ical amewo k ha b idges classical
analysis and mode n compu a ional p ac ice. By o e ing bo h deep heo e ical insigh s
in o he s uc u e o unc ions wi h he e ogeneous egula i y and p ac ical guidance o
algo i hm design in high-dimensional se ings, his esea ch con ibu es o he ounda-
ional ma hema ics unde pinning he nex gene a ion o scien i ic compu ing ools. The
aniso opic pe spec i e de eloped he e e eals how di ec ional scaling beha io can be
le e aged o mi iga e he cu se o dimensionali y, o e ing new pa hways o add essing
challenging mul iscale p oblems ac oss scien i ic disciplines.
Acknowledgmen s
San os g a e ully acknowledges he suppo o he PPGMC P og am o he Pos doc o al
Schola ship PROBOL/UESC n . 218/2025. Sales acknowledges CNPq g an 30881/2025-
0.
No a ion and Nomencla u e
Table 1: Ma hema ical No a ion and De ini ions
Symbol Desc ip ion
Rkk-dimensional Euclidean space
α= (α1, . . . , αk)Aniso opic scaling ec o , αi>0
Con inues on nex page
25
