Classical approximation of a linearized three waves kinetic equation

Jou nal o Func ional Analysis 282 (2022) 109390
Con en s lis s a ailable a ScienceDi ec
Jou nal o Func ional Analysis
www.else ie .com/loca e/j a
Classical app oxima ion o a linea ized h ee wa es
kine ic equa ion
M. Escobedo
Depa amen o de Ma emá icas, Uni e sidad del País Vasco (UPV/EHU), Apa ado
644, 48080 Bilbao, Spain
a i c l e i n o a b s a c
A icle his o y:
Recei ed 9 Oc obe 2020
Accep ed 25 Decembe 2021
A ailable online 21 Janua y 2022
Communica ed by Benjamin Schlein
Keywo ds:
Th ee wa es collisions
Classical app oxima ion
Cauchy p oblem
Bose gas
The pu pose o his wo k is o sol e he Cauchy p oblem o
he classical app oxima ion o an iso opic linea ized h ee
wa es kine ic equa ion ha appea s in he kine ic heo y o
a condensed gas o bosons nea he c i ical empe a u e. The
undamen al solu ion is ob ained, i is p o ed o be unique
in a sui able space o dis ibu ions, and some o i s egula i y
and in eg abili y p ope ies a e desc ibed. The ini ial alue
p oblem o in eg able and locally bounded ini ial da a is hen
sol ed. Classical solu ions a e ob ained as unc ions, whose
egula i y depends on ime and ha sa is y he expec ed
conse a ion o ene gy.
© 2022 The Au ho (s). Published by Else ie Inc. This is an
open access a icle unde he CC BY license
(h p://c ea i ecommons.o g/licenses/by/4.0/).
1. In oduc ion
Ou pu pose is o s udy he classical app oxima ion o he linea ized e sion o a h ee
wa e kine ic equa ion, a ound one o i s equilib ia,
∂u
∂τ (τ,x)=
∞
ˆ
0
(u(τ,y)−u(τ,x))K(x, y)dy, τ > 0,x>0 (1.1)
E-mail add ess: [email protected].
h ps://doi.o g/10.1016/j.j a.2022.109390
0022-1236/© 2022 The Au ho (s). Published by Else ie Inc. This is an open access a icle unde he CC
BY license (h p://c ea i ecommons.o g/licenses/by/4.0/).
2M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
K(x, y)=1
|x2−y2|−1
x2+y2y
x,∀x>0,∀y>0,x=y. (1.2)
In a condensed gas o quan um Bose pa icles ([17,27]), co ela ions a ise be ween he
supe fluid componen and he no mal fluid pa co esponding o he exci a ions. This
causes numbe -changing p ocesses, whe e an exci a ion spli s in o wo o he s in p esence
o he condensa e. A kine ic equa ion which includes hese p ocesses in a uni o m Bose gas
was fi s deduced in a se ies o pape s by Ki kpa ick and Do man [22]. Mo e ecen ly,
Za emba & al. ex ended he ea men o a apped Bose gas by including Ha ee–Fock
co ec ions o he ene gy o he exci a ions, and de i ed coupled kine ic equa ions o he
dis ibu ion unc ions o he no mal and supe fluid componen s, some imes called ZNG
sys em (see [33]). Kine ic equa ions o quan um pa icles al hough simila in many
aspec s wi h he classical Bo zmann equa ion, p esen new and in e es ing p ope ies
and ha e al eady been conside ed in he ma hema ical li e a u e (c . [14,23,29,30]and
e e ences he ein).
Only solu ions ha do no depend on he space a iable a e conside ed in his pape .
Fi s because ou in e es is mainly cen e ed on he p ope ies o he collision ope a o ,
bu also because he homogenei y hypo hesis simplifies e y much he difficul ies. These
solu ions a e called spa ially homogenous, o simply homogeneous. As no iced in [31],
§5.2, hey na u ally a ise in nume ical analysis whe e all nume ical schemes achie e a
spli ing o he anspo ope a o and he collision ope a o . I is also expec ed ha
spa ial homogenei y is a s able p ope y, in he sense ha a weakly inhomogeneous ini ial
da um leads o a weakly inhomogeneous solu ion o he Bol zmann equa ion, as i has
been ma hema ically jus ified in [2] unde some ad hoc smallness assump ions.
Unde he condi ions o spa ial homogenei y, in he limi o empe a u e below bu
close o he c i ical empe a u e, he ollowing sys em was fi s deduced in [10,22],
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
∂n
∂ ( , p)=C1,2(nc( ),n( ))(p) >0,p∈R3,(a)
n
c( )=−ˆ
R3
C1,2(nc( ),n( ))(p)dp > 0,(b) (1.3)
whe e C1,2(nc, n)is he h ee wa es collision in eg al,
C1,2(nc( ),n( )) = nc( )I3(n( ))(p) (1.4)
I3(n( ))(p)=¨
(R3)2R(p, p1,p
2)−R(p1,p,p
2)−R(p2,p
1,p)dp1dp2,(1.5)
R(p, p1,p
2)=
δ(|p|2−|p1|2−|p2|2)δ(p−p1−p2)×
×[n1n2(1 + n)−(1 + n1)(1 + n2)n].(1.6)
In hese no a ions n=n( , p), n( , p) deno es he densi y o pa icles in he no mal gas
ha a ime >0ha e momen um pand ene gy ω(p) =|p|2, and nc( ) he densi y o
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 3
he condensa e a ime . The e m (1.4) desc ibes he 1 ↔2 spli ing o an exci a ion
in o wo o he s in he p esence o he condensa e. Fo example (1.6)is o he spli ing
o he pa icle wi h momen um pin pa icles o momen um p1and p2, and simila ly o
R(p1, p, p2)and R(p2, p1, p). The specific o m o such a e m depends on he dispe sion
ela ion ω(|p|) o he ene gy o quasipa icles and on he ma ix elemen Mo he
effec i e Hamil onian desc ibing he in e ac ion be ween hem. The exp ession |p|2 o
he dispe sion ela ion is deduced om he well es ablished Bogoliubo app oxima ion
([6], [17])
ω(|p|)=gn|p|2
m+|p|2
2m21/2
whe e mis he mass o he pa icles, g=4πam−1is he in e ac ion coupling cons an and
ais he s-wa e sca e ing leng h, nis he o al pa icle densi y. When he empe a u e T
o he gas is low bu s ill such ha kBT>>gn
c(whe e kBis he Bol zmann cons an )
he app oxima ions ω(|p|) ∼|p|2
2m+gncand |M|2=g2nc
2π2a e used. In o de o simpli y
he no a ions i is assumed in (1.4)–(1.6) wi hou loss o gene ali y, ha he mass o he
pa icles is m =2and he in e ac ion coupling cons an is g=1.
O he heo e ical models do exis o desc ibe Bose gases in p esence o a condensa e
(c . [27]), bu ZNG sys em, and (1.3a), (1.3b) in pa icula , a e specially well sui ed o
apply analy ical PDE’s me hods and ob ain quan i a i e es ima es o some impo an
p ope ies.
I is well known ha he equa ion (1.3a) has a amily o non i ial equilib ia n0,
n0(p)=ν0(|p|2) (1.7)
ν0(ω)=eβω −1−1,∀ω>0.(1.8)
The pa ame e βmay be any posi i e cons an and is ela ed o he empe a u e T>0o
he gas a equilib ium n0 h ough he o mula, β=1/(kBT) whe e kBis he Bol zmann’s
cons an . I is easily checked ha R(p, pk, p) ≡0i n =n0.
I is known (c . [8]) ha o all cons an s ρ >0and all non nega i e measu es nin
wi h a fini e fi s momen , he sys em (1.3a)–(1.6)has a weak solu ion (n( ), nc( )) wi h
ini ial da a (nin, ρ). Fo all >0, n( )is a non nega i e measu e wi h fini e fi s momen
ha does no cha ge he o igin, and nc( ) >0. Sys em (1.3a)-(1.6)was also ea ed in
[3].
One basic aspec o he non equilib ium beha io o he sys em condensa e–no mal
fluid is he g ow h o he condensa e a e i s o ma ion (c . [33,5,27], and e e ences
he ein). Al hough he ela ion o ncwi h he condensa e ampli ude is no s aigh o -
wa d (c . [33,19,27]), i seems ne e heless e y closely ela ed o he o al numbe o
pa icles o he sys em ha ing an ene gy less han an a bi a ily small, bu fixed, alue
(c . [19]). I u ns ou ha he e olu ion o nc( ) c ucially depends on he beha io o
4M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
n( , p)as |p| →0as indica ed o example by P oposi ion 2 in [30]. When he measu e
n( )is w i en as n( , p) =|p|−1g( , |p|2), i is p o ed in [8] ha , i g( )has no a omic
pa and has an algeb aic beha io as |p| →0 hen,
n( , p)=
|p|→0a( )|p|−2,(1.9)
o some unc ion a( ), (c . [8]). These esul s o [8]and [30]make use o some egula i y
hypo hesis on he solu ion n( , p). Bu none o hese p ope ies ha e been p o ed o hold
o he solu ions no he sys em (1.3a)–(1.6) ob ained up o now.
1.1. Small iso opic pe u ba ion o a Planck dis ibu ion
Only adially symme ic pe u ba ions o he equilib ium n0(p)a e conside ed in
wha ollows. Unde such condi ion all he angula in eg a ions can be pe o med in
he collision in eg al (1.3a) and ob ain an equa ion wi h only wo eal, non nega i e
independen a iables and |p|. Fo non iso opic pe u ba ions Ω( , p), i expanded
in sphe ical ha monics as Ω( , p) =,m Ω,mY,m(p), simila , al hough sligh ly mo e
in ol ed equa ions a e ob ained o he e olu ion o he diffe en angula momen um
eigens a es Ω,m( , |p|)(c . equa ions (21), (22) in [16]). I would be o cou se o in e es
o know he possible effec s o non adial pe u ba ions, bu his is ou o he scope o
his a icle and le o u u e wo k.
In o de o p o e he exis ence o iso opic, egula classical solu ions o (1.3a)-(1.6)
sa is ying (1.9), we fi s conside he linea iza ion o (1.3a) a ound an equilib ium n0.
The linea ized equa ion was essen ially ob ained in [16]as b iefly desc ibed in §5.3 o
he Appendix: conside fi s he new iso opic dependen a iable Ω,
n( , p)=n0(p)+n0(p)(1 + n0(p))Ω( , |p|)=n0(p)+ Ω( , |p|)
4sinh
2β|p|2
2.(1.10)
When (1.10)is plugged in (1.3a), and only he linea e ms in Ωa e kep , hen a e he
change o a iables
x=√β
2|p|,τ=
ˆ
0
mc0(s)π
32
β3
2
ds, u(τ,x)=Ω( , |p|)
|p|2,(1.11)
he linea ized equa ion o u eads (c . [16]and §5.3 o he Appendix)
∂u
∂τ =pc(τ)
∞
ˆ
0
(u(τ,y)−u(τ,x))M(x, y)dy (1.12)
M(x, y)=1
sinh |x2−y2|−1
sinh(x2+y2)y3sinh x2
x3sinh y2,(1.13)
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 5
whe e pc(τ) =nc( ). When coupled wi h he equa ion
p
c(τ)=−pc(τ)
∞
ˆ
0
∞
ˆ
0
(u(τ,y)−u(τ,x))M(x, y)x2dy dx, (1.14)
i is easily checked ha , i Fubini’s Theo em may be applied when he collision in eg al
in (1.12)is mul iplied by n0(x)(1 +n0(x))x2and n0(x)(1 +n0(x))x4and in eg a ed o e
(0, ∞),
p
c(τ)+ d
dτ
∞
ˆ
0
n0(x)(1 + n0(x))u(τ,x)x4dx =0,
d
dτ
∞
ˆ
0
n0(x)(1 + n0(x))u( , x)x6dx =0.
These iden i ies eflec he conse a ion o he o al numbe o pa icles and ene gy and
he e o e, sys em (1.12), (1.14) seems a easonable linea iza ion o (1.3a), (1.3b). The
ac o pc(τ)may now be scaled in equa ion (1.12)wi h a new change o ime a iable,
deno ed again wi h some abuse o no a ion,
=
τ
ˆ
0
nc(s)ds
o ob ain he equa ion,
∂u
∂ ( , x)=
∞
ˆ
0
(u( , y)−u( , x))M(x, y)dy (1.15)
M(x, y)=1
sinh |x2−y2|−1
sinh(x2+y2)y3sinh x2
x3sinh y2.(1.16)
The ke nel Min (1.16) di ec ly ollows om he linea iza ion o C1,2(nc( ), n( )) in he
igh hand side o (1.3a) and he exp ession o n0(p)(1 +n0(p)) =eβ|p|2
22sinhβ|p|2
2−1
in he le hand side as explained in §5.3 o he Appendix.
The Cauchy p oblem o equa ion (1.15)is s ill delica e and as a fi s s ep in ha
di ec ion we conside in his wo k he simplified equa ion (1.1), (1.2), ob ained only
keeping in M he leading e ms o he hype bolic sine unc ions o small alues o
hei a gumen s. This eminds somewha he classical field limi we e la ge occupa ion
numbe s o diffe en modes a e assumed ([27], Chap e 10).

6M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
Fo Kgi en in (1.2), ou pu pose is hen o sol e he ollowing p oblem,
∂u
∂ ( , x)=
∞
ˆ
0
(u( , y)−u( , x))K(x, y)dy, > 0,x>0,(1.17)
L(u( ))(x)=
∞
ˆ
0
(u( , y)−u( , x))K(x, y)dy (1.18)
u(0,x)= 0(x) (1.19)
Again, o gene al, non necessa ily iso opic, pe u ba ions, simila simplified app oxi-
ma ed equa ions may be ob ained o he non adial componen s Ω,m o Ω(c . equa ions
(36), (38)–(43) in [16], o  =1and  =2).
The Cauchy p oblem o equa ion (1.15)is conside ed [11]as a pe u ba ion o (1.1).
In eg o diffe en ial equa ions o ha o m, in se e al dimensions bu wi h in eg als o e
all RN, ha e been much s udied, unde condi ions on he ke nels K, Mensu ing ha
he in eg o diffe en ial ope a o sa isfies an ellip ici y p ope y o some o de s >0. The
bes known is he ke nel C|x −y|−1−s o s ∈(0, 2) ha , o some cons an C>0, gi es
he ope a o (−Δ)s/2. Bu weake condi ions on mo e gene al ke nels may be ound in
[18]and he many e e ences he ein. A case whe e s =0is conside ed in [20].
Fo ua egula unc ion, equa ion (1.1)may be w i en (c (5.48)in he Appendix),
∂u
∂ ( , x)=
∞
ˆ
0
Hx
y∂u
∂y( , y)dy
y(1.20)
H( )=10< <1
1
log 1+ 2
1− 2+1 >1
1
log 1−1
4.(1.21)
Equa ion (1.20)may be sol ed using he Mellin ans o m. Simila ques ions we e consid-
e ed wi h simila me hods in [12], and in [13] o “pos gela ion” solu ions o a coagula ion
equa ion. Some o he echnical esul s in he las Sec ion o [13] will be o some use in
his wo k. The equa ion (1.3a) may ac ually be w i en as a coagula ion- agmen a ion
equa ion, wi h nonlinea agmen a ion, in e ms o he ene gy ω=|p|2as indepen-
den a iable o a measu e gdefined as |p|n( , p) =g( , ω)(c . [15], and [4] o gene al
coagula ion-collisional agmen a ion equa ions).
Rema k 1.1. The linea equa ion (1.1)also ollows i , fi s only quad a ic e ms a e kep
in (1.5), (1.6), and hen linea iza ion is pe o med a ound he equilib ium ω−1(p) =|p|−2.
The fi s s ep yields a h ee wa e u bulence ype equa ion, conside ed by se e al au ho s
[9,15,21], and (1.20)is he linea iza ion o ha equa ion a ound he equilib ium ω−1(p).
Ou se ing is a bi na ow wi hin he h ee wa es a ea, since he specific o m o he
dispe sion ela ion ωand o he ma ix elemen Ma e s ongly used. O he examples
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 7
o h ee wa e kine ic equa ions may be ound in [32], o capilla y wa es, weak acous ic
wa es and o he s.
1.2. Main esul s
The use o he Mellin ans o m, ha is deno ed by M, makes he spaces E
p, q o
p <q, p esen ed o example in Chap e 11 o [24], e y sui able. They a e defined as
he dual o he spaces Ep, q o all he unc ions φ ∈C∞(0, ∞)such ha :
Np,q,k(φ)=sup
x>0kp,q(x)xk+1 φk(x)<∞
kp,q(x)=x−p,i 0 <x≤1
x−q,i x>1
wi h he opology defined by he se o semino ms {Np,q,k}k∈N. I ollows ha E
p, q a e
he subspaces o D([0, ∞)) o Mellin ans o mable dis ibu ions ([24]). We call
Sp,q ={s∈C;Re(s)∈(p, q)},∀p∈R,∀q∈R,p<q. (1.22)
We also deno e Hρ
loc he se o locally Hölde con inuous unc ions o o de ρ ha
sa is y,
∀K⊂(0,∞)compac se ,∃CK>0; | (x)− (y)|≤CK|x−y|ρ,∀x∈K, ∀y∈K.
Fo α∈(0, 1) and x >0we deno e Θα(x) =|x −1|−α(log x); and o θ∈(0, 1),
|| 0||1,θ =|| 0||1+sup
0<x<1
xθ| 0(x)|.
We deno e a g and log he p incipal alues o he a gumen and loga i hm unc ions.
The second momen o a unc ion (x), o M( )(3), is some imes called he ene gy o
, because i is equal, up o a cons an , o he o al ene gy o a sys em o pa icles wi h
ene gy |p|2, whose momen um densi y unc ion is n(p) = (|p|).
Theo em 1.2. The e exis s a unc ion Λ ∈C((0, ∞); L1(0, ∞)) sa is ying (1.20)) in
D((0, ∞) ×(0, ∞)) and such ha
lim
→0Λ( )=δ1,weakly in D(0,∞).(1.23)
(log x)Λ ∈C((0,∞)×[0,∞)) (1.24)
lim
→0 −1e−1/ Y
1−2 Λ , 1+e−1/ Y= 1 (1.25)
uni o mly o Yon bounded subse s o R. Fo all T>0, Λ( ) ∈E
0,2, M(Λ( )) is bounded
on S0,2 o all ∈(0, T). The unc ion Λis such ha ,
8M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
(log x)∂mΛ
∂ m∈C((0,∞)×(0,∞)) ∀m∈N {0},(1.26)
(log x)2∂1+mΛ
∂ m∂x ∈C((0,∞)×(0,∞)),∀m∈N,(1.27)
∀k∈N,Λ∈Cmk+1
2,∞;Ck(0,∞),∀m∈N.(1.28)
∀ ∈(0,1/2),∀α∈[0, ); ΘαΛ∈C(2 , 1) ; H −α
loc (0,∞),(1.29)
(whe e we ecall ha Θα(x) =|x −1|−αlog x), and sa isfies (1.1) o almos e e y >0
and x >0. The second momen o Λ( )is one o all >0.
Theo em 1.3. I o some T>0, Λj∈C((0, T); L1(R+)), j=1, 2a e supposed o be
wo solu ions o (1.20), ha sa is y (1.23), such ha , o all ∈(0, T), Λj( ) ∈E
0,2and
M(Λ1( ) −Λ2( )) is bounded in S0,2, hen Λ1( ) =Λ
2( )in E
0,2 o all ∈(0, T).
The undamen al solu ion Λo he linea ized equa ion inhe i s he conse a ion o he
ene gy p ope y ha holds o he nonlinea equa ion (1.3a). As shown by (1.24), he
Di ac measu e a x =1is ins an ly egula ized o a unc ion Λ( ), whose egula i y is
gi en by (1.26)-(1.29). P ope y (1.25)shows ha , o small alues o >0, Λ( ) beha es
a x =1, like |x −1|2 −1. The egula i y o Λ( )a x =1shown in Theo em 1.2 imp o es
as he alue o inc eases, as seen in (1.28). By (1.29), (3.54), o all ∈(0, 1/2) he
unc ion Λ( )is locally Hölde con inuous a ound x =1o o de −α o any <2
and α∈(0, ). Fo >1i ollows om (1.28) ha Λis C1. De ailed es ima es o Λ( , x)
and some o i s de i a i es a e gi en in he Sec ions below. This undamen al solu ion is
used o sol e he ini ial alue p oblem.
Theo em 1.4. Suppose ha 0∈L1(0, ∞)and define,
u( , x)=
∞
ˆ
0
0(y)Λ 
y,x
ydy
y,∀ >0,∀x>0.(1.30)
Then, u ∈L∞((0, ∞); L1(0, ∞)) ∩C((0, ∞); L1(0, ∞)) and i sa isfies (1.20)in
D((0, ∞) ×(0, ∞)),
∞
ˆ
0
u( , x)x2dx =
∞
ˆ
0
0(x)x2dx, ∀ >0,(1.31)
he e exis s a cons an C>0such ha
||u( )||1≤C|| 0||1,∀ >0 (1.32)
and u( )
→0 0,in D(0,∞).(1.33)
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 9
I 0∈L1(0, ∞) ∩L∞(0, ∞) hen u( ) ∈L∞(0, ∞) o all >0, he e exis s a cons an
C∞>0such ha ,
||u( )||∞≤C∞|| 0||∞,∀ >0.(1.34)
I 0∈L1(0, ∞) ∩L∞
loc(0, ∞),
L(u)∈L∞
loc((0,∞); L∞(0,∞)),(1.35)
he e exis s a cons an C>0such ha , o all >0and x >0,

∂u
∂ ( , x)≤⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
C 2x−4+ 3x−3+ε|| 0||1,a.e.x>3
C(1 + |log |x/2 −1||)
(1 −θ)2xsup
z∈(2 ,3x)| 0(z)|, a.e.x ∈(2 /3,3 )
C( −2+ −3x)|| 0||1,a.e.x∈(0,2 /3),
(1.36)
and usa isfies (1.1) o a.e. >0, x >0.
The solu ion ualso sa isfies he ollowing p ope ies,
P oposi ion 1.5. I 0∈L1(0, ∞)and uis gi en by (1.30), he ollowing holds.
1.- Fo e e y δ>0as small as desi ed, and o all >0,
u( , x)=
x→0( 0, )+⎛
⎝ −2+δ
ˆ
0| 0(y)|dy + 5+δ
∞
ˆ
| 0(y)|dy
y7⎞
⎠Oδx1−δ,(1.37)
( 0; )=A1 −3
ˆ
0
0(y)y2dy +A2 −4
ˆ
0
0(y)y3dy +
ˆ
0
0(y)b1
ydy
y(1.38)
o A1, A2cons an s gi en in (4.75)and b1( ) =
→∞
O( −8)gi en in (3.15).
2.- Fo all >0, he unc ion u( )is locally Hölde con inuous on (0, ∞). Mo e p ecisely,
(i) The e exis nume ical cons an s C>0and σ∗
0∈(−2, −1) such ha
|u( , x)−u( , x)|≤C|| 0||1 −2+x−1−σ∗
0 −1+σ∗
0|x−x|, o 0<x
<x< , (1.39)
(ii) Fo all c ∈(0, 2) he e exis s a cons an C>0such ha ,
|u( , x)−u( , x)|≤C|| 0||1|x−x|x−1−c −1+c+ x−4+
+C|| 0||1|x−x|1−α
x1−α|log(x/ )|1−α+|x−x| −α
x −α|log(x/ )|(1+α)( −α),
i 0<x
< <x, (1.40)
16 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
whe e log(z) =log(|z|) +iA g(z)and A g(z) ∈(−2π, 0]. The equa ion (2.1)on Vyields
he ollowing equa ion o H:
ze−slog(−z)B(s)H(z,s)=−ze−slog(z)W(s−1)B(s−1)H(z,s −1) + 1
B(s)H(z,s)=−W(s−1)B(s−1)H(z,s −1) + eslog(−z)
z
B(s)H(z,s)=B(s)H(z,s −1) + eslog(−z)
z
and hen, o all z∈Csuch ha Re(z) >0and s ∈C, Re(s) ∈(0, 2)
H(z,s)−H(z,s −1) = eslog(−z)
zB(s).(2.23)
We may use again he change o a iables (2.9)and define,
h(z,ζ)=H(z,s),
B(ζ)=B(s)
and deduce om (2.23) ha hhas o sa is y
h(z, −i0) = h(z, +i0) + e2iπβα(z) α(z)
z
B( ),∀ >0; α(z)=log(−z)
2iπ .(2.24)
I ollows ha
α(z)=log(−z)
2iπ =−ilog |z|
2π+A g(−z)
2π
and he choice o he log(z)is such ha −1 <(Re(α(z))) <0. By P oposi ion (2.5)i
ollows ha he in eg al
h(z,ζ)= 1
2iπ
e2iπβα(z)
z
∞
ˆ
0
α(z)

B( )
d
( −ζ)
is absolu ely con e gen and defines a unc ion hanaly ic on he domain
{(z,s); z∈C,Re(z)>0,s∈C [0,∞)}
ha sa isfies (2.24). Using he o iginal a iables we ob ain ha
H(z,s)=1
zˆ
Re(σ)=β
eσlog(−z)
B(σ)
dσ
(1 −e2iπ(s−σ))(2.25)
is well defined, analy ic on z∈C, Re(z) >0, s ∈C, Re(s) ∈(β, β+1) whe e i sa isfies

M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 17
H(z,s)−H(z,s −1) = eslog(−z)
zB(s).(2.26)
Since β∈(0, 2) is a bi a y, using a con ou de o ma ion a gumen in he in eg al o he
igh hand side o (2.26), His ex ended as an analy ic unc ion z∈C, Re(z) >0and
s ∈C, Re(s) ∈(0, 2).
Using now (2.22)we eco e he unc ion
V(z,s)=B(s)
2iπz ˆ
Re(σ)=β
e(σ−s)log(−z)
B(σ)
dσ
(1 −e2iπ(s−σ)).
Since Bis analy ic and non ze o on Re(s) ∈(0, 2) and β∈(0, 2) is a bi a y he unc ion
Vis analy ic on z∈C, Re(z) >0and s ∈C, Re(s) ∈(0, 2) and sa isfies he equa ion
(2.21) o Re(s) ∈(1, 2). 
Co olla y 2.9. The ollowing in e se Laplace ans o m o V
U( , s)= 1
2iπ
d+i∞
ˆ
d−i∞
ez V(z,s)dz, β −1<d<β,
is well defined o >0and Re(s) ∈(0, 2). Fo all >0, U( , ·)is analy ic on S0,2,
∀s∈S0,2,U( , s)=B(s)
2iπ ˆ
Re(σ)=β
−(σ−s)Γ(σ−s)
B(σ)dσ, ∀β∈(Re(s),2) (2.27)
may be ex ended o Cas a me omo phic unc ion such ha U( , 3) =1and sa isfies,
∀k∈N,U∈Ck((0,∞)×S0,2) (2.28)
∂U
∂ ( , s)=W(s−1)U( , s −1) ∀ >0,∀s∈S1,3.(2.29)
P oo . Fo all σand ssuch ha Re(s) <Re(σ), and c >0,
1
2iπ
c+i∞
ˆ
c−i∞
ez
ze(σ−s)log(−z)dz = −(σ−s)Γ(σ−s)e2iπ(σ−s)−1.
We use now ha S i ling’s o mula o Γ(z)is uni o mly alid o a gz∈(−π+ε0, π−ε0)
wi h ε0>0, o deduce ha , o all R>0and β∈(0, 2)
|Γ(σ−s)|≤CR
e−π|σ|
2
1+|σ|,∀s;|s|≤R. (2.30)
18 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
The igh hand side o (2.27)is hen absolu ely con e gen . The iden i y (2.27)and he
analy ici y o U( , ·)on S(0, 2) ollow o β−1 <Re(s) <β. We also deduce om (2.30)
ha he in eg als
ˆ
Re(σ)=β
d
d  −(σ−s)Γ(σ−s)
B(σ)dσ, k ∈N
a e absolu ely con e gen and analy ic unc ions o son he s ip Re(s) ∈(0, 2). The e-
o e,
∂k
∂ kU( , s)=−B(s)
2iπ ˆ
Re(σ)=β
d
d  −(σ−s)Γ(σ−s)
B(σ)dσ,
and (2.28) ollows. On he o he hand, since
1
2iπ
d+i∞
ˆ
d−i∞
ez e(σ−s)log(−z)dz = −(σ−s)−1Γ(1 + σ−s)e2iπ(σ−s)−1
he in e se Laplace ans o m o zV (z)is well defined o all >0and gi en by,
1
2iπ
d+i∞
ˆ
d−i∞
ez zV (z,s)dz =−B(s)
2iπ ˆ
Re(σ)=β
−(σ−s)−1Γ(1 + σ−s)
B(σ)dσ.
The exp ession (2.27) indica es ha U(·, s) ∈C((0, ∞)). I he in eg a ion con ou in
(2.27)is de o med owa ds lowe alues o βand he pole o he unc ion Γ(σ−s)a
σ−s =0is c ossed,
U( , s)=1−B(s)
2iπ ˆ
Re(σ)=β
−(σ−s)Γ(σ−s)
B(σ)dσ, β∈(0,Re(s)).(2.31)
Since now Re(σ−s) <0, i ollows ha U(·, s) ∈C([0, ∞)) and U(0, s) =1. By classical
de o ma ion o con ou a gumen s U( )is now ex ended as a me omo phic unc ion o
all o C, and U( , 3) =0by (2.31)and because B(3) =0(c . P oposi ion 2.3). Using
L(U (·,s)) (z)=zV (z,s)−U(0,s),
we deduce
∂U
∂ ( , s)= 1
2iπ
d+i∞
ˆ
d−i∞
ez (zV (z,s)−1) dz
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 19
We apply now he in e se Laplace ans o m o bo h sides o he equa ion (2.21)wi h
Re(s) ∈(1, 2), since U( )is analy ic on S0,2and so is Won S−2,4, (2.29) ollows. 
The ollowing p ope ies o Ua e impo an o wha ollows,
P oposi ion 2.10. Fo all T>0, he e exis s a posi i e cons an CTsuch ha o all
s ∈S, ∈(0, T),
|U( , s)|≤CTe−2 log |bs|,b=eγe
2
2,(2.32)
(1 + |s|)
∂U
∂s ( , s)+(1+|s|)2
∂2U
∂s2( , s)≤CT e−2 log(|bs|)(2.33)

∂U
∂ ( , s)≤CT e−2 log(|bs|)(1 + |log |s||) (2.34)
(1 + |s|)
∂
∂s ∂U
∂ ( , s)+(1+|s|)2
∂2
∂s2∂U
∂ ( , s)≤CT(1 + |log |s||)
e2 log(|bs|).(2.35)
The p oo o P oposi ion 2.10 is essen ially he same as ha o P oposi ion 8.1 in
[13], only diffe ing in small de ails, and is p esen ed in he Appendix. I is based on he
exp ession o U( , s)gi en in (2.27)and also
U( , s)=−B(s)
2iπ ˆ
Re(Y)=β−Re(s)
−YΓ(Y)
B(s+Y)dY =ˆ
Re(σ)=β
eψ(s,σ, )A(Y)dY (2.36)
whe e
Ψ(s, Y, )= ˆ
Re(ρ)=β
log (−W(ρ)) Θ(ρ−s, Y )dρ −Ylog −Y+Y−1
2log Y,
wi h Θ defined in (2.19), and
A(Y)= Γ(Y)
2iπe−YYY−1/2.
The es ima es o |s| ollow om con ou de o ma ion me hods on (2.27). The es ima es
o |s|la ge a e deduced using he s a iona y phase a gumen on (2.36).
As a Co olla y, he in e se Mellin ans o m o U( )is well defined.
Co olla y 2.11. Fo e e y >0 he e exis s a unique dis ibu ion Λ( ) := M−1(U( )) ∈
E
0,2, he in e se Mellin ans o m o U( )such ha :
M(Λ( )) (s)=U( , s),∀s∈S0,2(2.37)
Λ∈C((0,∞); E
0,2).(2.38)
20 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
Fo all >0i is gi en by he ollowing exp ession,
Λ( , x)=x∂
∂x2⎛
⎝1
2πi
c+i∞
ˆ
c−i∞
U( , s)s−2x−sds⎞
⎠,c∈(0,2).(2.39)
When >1/2,
Λ( , x)= 1
2πi
c+i∞
ˆ
c−i∞
U( , s)x−sds, c ∈(0,2).(2.40)
P oo . By Co olla y 2.9, o e e y >0, he unc ion U( )is analy ic on he s ip
Re(s) ∈(0, 2). By P oposi ion 2.10
|U( , s)|≤|bs|−2 ,∀ ∈(0,1).
I ollows ha , o all >0, he unc ion s−K+2U( , s)is analy ic and bounded on he
s ip Re(s) ∈(0, 2) as |s| →∞ o K=2. I ollows om Theo em 11.10.1 in [24] ha
he e exis s a unique empe ed dis ibu ion Λ( ) ∈E
0,2 ha sa isfies (2.37)and is gi en
by (2.39). As soon as >1/2, he in eg al in he igh hand side o (2.40)is absolu ely
con e gen and i s Mellin ans o m is U( ) om whe e i is equal o Λ( ). P ope y
(2.38) ollows om (2.28)and he con inui y o he in e se Mellin ans o m. 
I is now possible o apply he in e se Mellin ans o m o bo h sides o (2.29).
P oposi ion 2.12.
Λ( )∈C1(0,∞;E
1,3) (2.41)
∂Λ
∂ =∂Λ
∂x ∗Hin C((0,∞); E
1,3) (2.42)
whe e His he unc ion defined in (1.21).
P oo . By (2.38), ∂xΛ( ) ∈C(0, ∞; E
1,3)and o all s ∈S1,3,
M(∂xΛ( ))(s)=−(s−1)U(s−1),and
Since M(H)(s) =−W(s−1)
s−1, i hen ollows o all >0,
M−1(W(s−1)U( , s −1))(x)=∂Λ( )
∂x ∗H(x)inE
1,3
On he o he hand, by (2.29)and P oposi ion 2.10
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 21
M−1∂U( )
∂ (x)≡M−1(W(s−1)U( , s −1)) =
=x∂
∂x2⎛
⎝1
2πi
c+i∞
ˆ
c−i∞
W(s−1)U( , s −1)s−2x−sds⎞
⎠.(2.43)
By P oposi ion 2.10 again, o all >0and x >0,
d
d ⎛
⎝1
2πi
c+i∞
ˆ
c−i∞
U( , s)s−2x−sds⎞
⎠=1
2πi
c+i∞
ˆ
c−i∞
W(s−1)U( , s −1)s−2x−sds (2.44)
and he in eg al in he igh hand side o (2.44)is absolu ely con e gen , uni o mly o
xand in compac s subse s o (0, ∞) ×(0, ∞). I is hen a con inuous unc ion on
(0, ∞) ×(0, ∞). I is hen possible o apply he ope a o (x∂x)2 o bo h sides o (2.44)
in he sense o dis ibu ions o ob ain (2.42). 
The ollowing P oposi ion, shows some impo an p ope ies o Λ.
P oposi ion 2.13. The unc ion Λ( )defined in Co olla y 2.11 sa isfies (1.24), and
(1.26)–(1.29).
P oo . By i s defini ion, Λ( ) ∈E
0,2, and o all m ∈N,
M(((log x)∂m
Λ( ))(x)) = ∂s∂m
U( , s)
=∂sU( , s −m)
m

=1
W(s−)in Sm,2+m.(2.45)
P ope y (1.26) ollows om he decay o he unc ion a he igh hand side o (2.45)as
|Im(s)| →∞. Indeed, by P oposi ion 2.10, (2.4)and (2.5), o e e y m ≥1and >0
he e exis s a posi i e cons an C>0, depending on m, band , such ha ,
∂sU( , s −m)
m

=1
W(s−)≤C(1 + |s|)−1− ,∀s∈Sm,2+m.
I ollows ha , o c∈(m, 2 +m),
((log x)∂m
Λ( ))(x)= 1
2πi
c+i∞
ˆ
c−i∞
∂sU( , s −m)
m

=1
W(s−)x−sds (2.46)
whe e he in eg al in (2.46)is absolu ely con e gen . Since mo eo e he in eg al con-
e ges uni o mly o xand on compac subse s o (0, ∞) ×(0, ∞), p ope y (1.26)
ollows. No ice ha he same p oo shows (log x)Λ ∈C((0, ∞) ×(0, ∞)). Simila ly,

22 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
M(((log x)2∂x∂m
Λ( ))(x)) = ∂2
s(log(s−1) ∂m
U( , s))
=∂2
slog(s−1) U( , s −m)
m

=1
W(s−),in Sm,2+m.
(2.47)
Again by P oposi ion 2.10, (2.4)and (2.5), o e e y m ≥0, >0, he e exis s a posi i e
cons an C>0such ha ,
∂2
sU( , s −m)
m

=1
W(s−)≤C(1 + |s|)−2− |log s)|,∀s∈Sm,2+m.
Then,
(log x)2∂m
( , x)( , x)= 1
2πi
c+i∞
ˆ
c−i∞
∂2
sU( , s −m)
m

=1
W(s−)x−sds
and, since he in eg al is absolu ely and uni o mly con e gen o xand on compac
subse s o (0, ∞) ×(0, ∞), p ope y (1.27) ollows.
In o de o p o e (1.24)we fi s no ice ha o >1/2, o mula (2.40)may be used.
Using (3.4), i we de o m he in eg a ion con ou in (2.40) owa ds lowe alues o Res
and c oss he pole o B(s)a s =0, using Res(B(s), s =0)) =−B(1)/W (0) we ob ain
Λ( , x)= 1
4π2ˆ
Re(s)=c
x−sˆ
Re(σ)=β
B(s)
B(σ)Γ(σ−s) −(σ−s)dσds
=B(1)
2iπW (0) ˆ
Re(σ)=β
Γ(σ) −σ−1
B(σ)dσ+
+1
4π2ˆ
Re(s)=c
x−sˆ
Re(σ)=β
B(s)
B(σ)Γ(σ−s) −(σ−s)dσds, c ∈(−1,0)
I ollows ha Λ ∈C([1/2, ∞) ×[0, ∞)) since bo h in eg als con e ge uni o mly o x
and on compac subse s o [0, ∞) ×[1/2, ∞). Fo ∈(0, 1/2)
(log x)Λ( , x)= 1
2πi
c+i∞
ˆ
c−i∞
∂sU( , s)x−sds. (2.48)
I ollows om (2.27) ha U( )is me omo phic on he s ip S−1,2wi h a simple pole a
s =0. Then, ∂sU( , s)is also me omo phic on S−1,2and has a pole o o de 2 a s =0.
We deduce
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 23
(log x)Λ( , x)=−B(1)
2iπW (0) ˆ
Re(σ)=β
Γ(σ) −σ−1
B(σ)dσ +1
2πi
c+i∞
ˆ
c−i∞
∂sU( , s)x−sds,
o c ∈(−1, 0) and, a guing as be o e, (log x)Λ ∈C((0, 1/2) ×[0, ∞)) and (1.24) ollows.
In o de o p o e (1.28)we no ice ha , by P oposi ion 2.10, (2.4)and (2.5) again, o
all k∈Nand m ∈N he e exis s C>0such ha
(s−k)kU( , s −m)
m

=1
W(s−)≤C|s|k−2 |log |s||m, o |s|>> 1.
Then, o >1, k<2 −1, and c∈(m +k, 2 +m +k) he iden i y (2.40)may be used
o w i e
∂k+mΛ
∂xk∂ m=(−1)k
2πi
c+i∞
ˆ
c−i∞
(s−k)kU( , s −m)
m

=1
W(s−)x−s−kds, (2.49)
whe e he in eg al in (2.49)con e ges absolu ely. Since he con e gence is uni o m in
compac s o ((k+1)/2, ∞) ×(0, ∞), p ope y (1.28) ollows.
We p o e now p ope y (1.29). Fo all ∈(0, 1/2), ∈(0, 2 ), and |s|la ge,

∂
∂sU( , s − )Γ(1 −s+ )
Γ(1 −s)x1−s≤|x|−Re(s+ )|s|−2 −1+ ,(2.50)
he ac ional de i a i e o o de o (log x)Λ, is hen
∂ (log x)Λ( )
∂x =1
2πi
c+i∞
ˆ
c−i∞
Γ(1 −s+ )
Γ(1 −s)
∂
∂sU( , s − )x−sds (2.51)
c∈( , 2), (c . [26], §2.10), whe e he in eg al in he igh hand side o (2.51)con e ges
absolu ely o xand in compac subse s o (0, ∞) ×(0, ∞). Fo each >0 he unc ion
(log x)Λ( )has con inuous ac ional x-de i a i e o o de on e e y compac subse o
(0, ∞)and by (2.51), o all >0
∀ ∈( , 2),∃C >0,
∂ ((log x)Λ( , x))
∂x ≤C x− ,∀x>0.(2.52)
By Theo em 3.1 [28], (1.29) ollows o α=0and ∈(0, 2 ). Mo eo e , since
(log x)Λ( , x)= 1
2iπ ˆ
Re(s)=c
∂U( , s)
∂s x−sds,
24 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
by he con inui y p ope y (1.26), and an in eg a ion by pa s,
lim
x→1(log x)Λ( , x)= 1
2iπ ˆ
Re(s)=c
∂U( , s)
∂s ds =0.
Then p ope y (1.29)is deduced o α∈(0, )using he esul in [25], p. 14. 
Co olla y 2.14. The unc ion Λsa isfies
lim
→0Λ( )=δ1,in D(0,∞).(2.53)
P oo . Conside any es unc ion ϕ ∈D(0, ∞)and suppose ha supp(ϕ) ⊂(a, b) o
some 0 <a <b <∞. Then
Λ( ),ϕ−ϕ(1) =
∞
ˆ
0
M−1(U( )−1) (x)ϕ(x)dx
=1
2iπ
∞
ˆ
0
c+i∞
ˆ
c−i∞
(U( , s)−1) x−sdsϕ(x)dx
=1
2iπ
c+i∞
ˆ
c−i∞
∞
ˆ
0
x−sϕ(x)dx (U( , s)−1) ds
=1
2iπ
c+i∞
ˆ
c−i∞
M(ϕ)(1 −s)(U( , s)−1) ds.
By defini ion, o s =c +i , ∈R, Re(s) ∈(β, 2),
M(ϕ)(1 −s)=
∞
ˆ
0
ϕ(x)x−sdx =1
(1 −s)(2 −s)
∞
ˆ
0
ϕ(x)x2−sdx ≤C
1+|s|2.
As we ha e seen abo e (c . (2.31)), o Re(s) ∈(β, 2),
|U( , s)−1|=|B(s)|ˆ
Re(σ)=β
−(σ−s)Γ(σ−s)
B(σ)dσ
,β
∈(0,Re(s))
≤|B(s)| Re(s−β)ˆ
Re(σ)=β
−(iIm(σ−s))Γ(σ−s)
B(σ)dσ
≤CeRe(s−β)log log |s|
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 25
Then, o Re(s) =c >β
:
|Λ( ),ϕ−ϕ(1)|=1
2iπ 
c+i∞
ˆ
c−i∞
M(ϕ)(1 −s)(U( , s)−1) ds
≤Ce(c−β)log
c+i∞
ˆ
c−i∞
|M(ϕ)(1 −s)|log |s||ds|≤Ce(c−β)log ˆ
R
log | |d
1+| |2→
→00.
3. Fu he p ope ies o Λ
In his Sec ion we fi s gi e he main e ms in he asymp o ic beha io o he un-
damen al solu ion Λ( , x)in diffe en egions o he , x)plane. Mo e de ailed esul s
a e also gi en on he con inui y and de i abili y p ope ies o he unc ion Λ, in pa -
icula a ound he poin x =1, whe e he Di ac’s del a o ma ion is desc ibed. These
esul s a e used la e , fi s o sol e he Cauchy p oblem associa ed o equa ion (1.1) o
a la ge se o ini ial da a, and hen o ge he p ecise beha io o he solu ions. This will
be mainly done wi h he ep esen a ion o Λas a con ou in eg al, using he classical
con ou de o ma ion a gumen and Cauchy’s esidue Theo em.
ρ(σ)=Res1
B(s),s=σ, (σ)=Res(B(s),s=σ) (3.1)
˜ (σ)=Res(s−2B(s),s=σ),˜ρ(σ)=Res1
W(s),s=σ(3.2)
P(n)=ResΓ(ω)
B(ω),ω=−n,Q(n)=ResΓ(ω+1)
B(ω),ω=−n=−nP(n).(3.3)
No ice ha −nis a simple pole o
Γ(ω)
B(ω) o n ∈{0, ···5}and a double pole o n ≥6.
3.1. Beha io o Λ o >1
The unc ion Λsa isfies he ollowing es ima es when >1
P oposi ion 3.1. Fo all >1,
Λ( , x)= −3Q1(θ)+Q2( , θ),θ=x
(3.4)
Q1(θ)= c1
2iπ ˆ
Re(s)=c
θ−sB(s)Γ(3 −s)ds (3.5)
Q2( , θ)=−1
4π2ˆ
Re(s)=c
θ−sˆ
Re(σ)=β2
B(s)
B(σ)Γ(σ−s) −σdσds (3.6)
32 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
since he fi s pole o Γ(σ−σ∗
n)/B(σ)wi h nega i e eal pa is loca ed a σ=−6, (3.28)
ollows.
The same me hod gi es es ima e (3.29). S a ing om (2.39)and (2.29), we deduce
∂
∂ Λ( , x)=x∂
∂x2⎛
⎝1
2πi
c+i∞
ˆ
c−i∞
U( , s −1)W(s−1)s−2x−sds⎞
⎠
=−x∂
∂x2⎛
⎜
⎝x−1
4π2
c+i∞
ˆ
c−i∞
ˆ
Re(σ)=β
B(s)Γ(σ−s+1)
B(σ)s−2 −σx
−(s−1) dσds⎞
⎟
⎠.
Wi h he same a gumen as be o e we deduce,
∂
∂ Λ( , x)=x−1μ( )
∞

k=1 x
−k−β
1+1 Ak(k+β
1)2+x−1
∞

n=1 x
−4n(4n+1)
2νn( )

∂
∂ Λ( , x)≤C1x−1x
−β
1 6+C2x−1x
−4 2,x
>1,0< <1.
Fo es ima e (3.30), whe e x ∈(0, /2) he sin eg a ion con ou is mo ed owa ds smalle
alues o Re(s). The sequence o poles o B(s), wi h Re(s) ≤0is hen c ossed. These
a e loca ed a s =0, −1and poin s σ∗
no P oposi ion (2.1). We deduce, a guing as be o e
∂ Λ( , x)=x∂
∂x21
˜μ1( )+˜μ2( )x
2+
∞

n=0 x
−σ∗
n˜νn( ),
=˜μ2( )x
2+
∞

n=0
(σ∗
n)2x
−σ∗
n˜νn( )
˜νn( )= ˜ σ∗
n
2iπ ˆ
Re(ω)=β
Γ(ω−σ∗
n)
B(ω) −ωdω;˜μ2( )= ˜ −1
2iπ ˆ
Re(σ)=β
Γ(σ+2)
B(σ) −σdσ
The unc ions ˜νnand ˜μ2a e now de e mined by he sequence o ze os o B(σ)such ha
Re(σ) ≤0. Since he fi s one is a s = 6 es ima e (3.30) ollows.
F om (2.39)and basic p ope ies o he Mellin ans o m,
∂Λ
∂x( , x)=x∂
∂x3
(J( , x)) whe e, o c∈(1,2),
J( , x)=−1
4π2
c+i∞
ˆ
c−i∞
ˆ
Re(σ)=β
−σ−1B(s−1)Γ(σ−s)
B(σ)(s−1)s−3x
−sdσds.

M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 33
Fo ∈(0, 1) and x ∈(0, ρ ) he beha io o J( , x)is ob ained by de o ming he con ou
in eg als o lowe alues, fi s o Re(s)and hen o Re(σ). In he fi s s ep we c oss fi s
he pole s =0 hen, he poles σ∗
n+1o B(s −1) o (s −1)−2B(s), and ob ain,
J( , x)=ResB(s−1)
s3;s=0
R∗
0( )+
2

j=1 x
−1−σ∗
j
R∗
j( )+Ox−1−σ∗
3+ε −σ∗
3−ε
R∗
0( )= ˆ
Re(σ)=β
−σ−1Γ(σ)
B(σ)dσ, 
R∗
j( )= (1 + σ∗
j)σ∗
j
(1 + σ∗
j)3− −σ∗
j
B(1 + σ∗
j)+O 1−σ∗
j
and, (3.33) ollows, wi h he same a gumen as in he p oo o (3.27), (3.28).
The es ima e (3.34) whe e x/ >x >1 equi es o de o m fi s he scon ou in eg als
in J owa ds la ge alues o Re(s). Since by cons uc ion c <βwe fi s he pole o
Γ(σ−s)a s =σ, om whe e, o c∈(β, 2),
J( , x)= −1
2iπ ˆ
Re(σ)=β
(σ−1)
W(σ−1)σ−3x−σdσ +J1( , x)
J1( , x)=−1
4π2ˆ
Re(σ)=β
c+i∞
ˆ
c−i∞
−σ−1(s−1)B(s−1)Γ(σ−s)
B(σ)s3x
−sdσds
J( , x)=2˜ρ(2)
27 −1x−3+O −1x−4+ε−J1( , x), o a bi a ily small ε>0
The sin eg a ion con ou in J1is mo ed o la ge alues. The nex pole o B(s −1) is
a s =6. Since σ=3is a ze o o B(σ), ha we do no wan o c oss, he condi ion
σ−s ∈(−1, 0) can no be main ained. The singula i ies o Γ(σ−s)a σ−s =−1and
σ−s =−2a e c ossed. Since
1
2iπ ˆ
Re(σ)=β
x−σ−1σ
(σ+1)
3dσ =1
2(log x)21(0,1)(x)=0,∀x>1,
his gi es, using (2.15), o d ∈(5, 6),
J1( , x)=−
2iπ ˆ
Re(σ)=β
x−σ−2(σ+1)
(σ+2)
3W(σ)dσ +J2( , x)
J2( , x)= 1
4π2ˆ
Re(σ)=β
d+i∞
ˆ
d−i∞
−σ−1(s−1)B(s−1)Γ(σ−s)
B(σ)s3x
−sdσds
hen J1( , x)=O( x−6)+J2( , x),J
2( , x)=O( 1+εx−5−ε)
and (3.34) ollows om he loca ion o he ze os and poles o Wand B.
34 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
3.2.2. P ope ies o Λ o ∈(0, 1) and 0 <|x −1| <1
P oposi ion 3.6. The e exis s a cons an Csuch ha
|Λ( , x)|≤ C
|x−1|,∀x;0<|1−x|<1,∀ ∈(0,1) (3.47)

∂
∂ Λ( , x)≤C(1 + |log |x−1||)
|x−1|,∀x;0<|1−x|<1,∀ ∈(0,1).(3.48)
P oo . We define he new a iables
X=logx, ˜
Λ( , X)=Λ( , x),∀ >0,x>0.(3.49)
Then, ∀X∈R,˜
Λ( , X)= 1
2iπ ˆ
Re(s)=c
e−sXU( , s)ds. (3.50)
A e wo in eg a ions by pa s:
˜
Λ( , X)= 1
X2ˆ
Re(s)=ce−sX −1∂2U
∂s2( , s)ds. (3.51)
When |s| <1, we use |e−sX −1| ≤|sX|and deduce om (3.51)and P oposi ion 2.10
˜
Λ( , X)≤
|X|ˆ
Re(s)=c
|s|<1
|s||ds|
1+|s|2+
1
X2ˆ
Re(s)=c
|s|>1e−sX −1∂2U
∂s2( , s)ds
.
Bu ,
ˆ
Re(s)=c
|s|>1e−sX −1∂2U
∂s2( , s)ds =1
Xˆ
Re(u)=cX
|u|>|X|
e−u−1∂2U
∂s2 , u
Xdu
and by P oposi ion 2.10

ˆ
Re(s)=c
|s|>1
e−sX −1
1+|s|2ds
≤
|X|ˆ
Re(u)=cX
|u|>|X|
|e−u−1|
1+|u/X|2|du|
= |X|ˆ
Re(u)=cX
|u|>|X|
|e−u−1|
|X|2+|u|2|du|< |X|ˆ
Re(u)=cX
|u|>|X|
|e−u−1|
|u|2|du|.
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 35
I s =c +i , hen e−u=e−cXe−i , and o Xbounded,
e−u−12=e−2cX (cos2( X)−1) + sin2( X)≤C
and, i u =cX +iw,
ˆ
Re(u)=cX
|u|>|X|
|e−u−1|
|u|2|du|≤Cˆ
Re(u)=cX
c2|X|2+w2>|X|2
dw
c2X2+w2≤Cˆ
R
dw
c2X2+w2=C
X
This shows (3.47)and a simila calcula ion gi es (3.48)using ha ,
∀X∈R,∂
∂ ˜
Λ( , X)= 1
2iπ ˆ
Re(s)=c
e−sX ∂U
∂ ( , s)ds. 
Lemma 3.7. Fo all ε >0as small as desi ed, he e exis s a cons an Cε>0such ha
o all ∈(0, 1), α∈(0, 2 ), and all x ∈(0, 2),
|(log x)1−αΛ( , x)|≤ Cε
xε|log x|α.(3.52)
P oo . I ollows om (2.33) ha , he e exis s C>0, independen o ε, such ha o
∈(0, 1) and x ∈[0, 2),
|(log x)Λ( , x)|≤C ˆ
Re(s)=ε
(1 + |s|)−1−2 x−sd|s|=Cε
xε,
and hen, o all α∈(0, 1),
|(log x)1−αΛ( , x)|=|(log x)Λ( , x)|
|log x|α≤Cεx−ε
|log x|α.
Ou nex goal is an es ima e o he Hölde p ope y (1.29) o Λ( ). We s a wi h,
Lemma 3.8. Fo all ∈(0, 1/2), all ε >0a bi a ily small, he e exis s a cons an
C( , ε) >0such ha , o α∈[0, )and a, bsa is ying 0 <a <b <∞,
∀ ∈(0,1),∀(x, y)∈(a, b)×(a, b),
|Θα(x)Λ( , x)−Θα(y)Λ( , y)|≤(2 + α)C( , ε)|x−y| −α
aε(3.53)
whe e, Θα=|x −1|−α(log x)
36 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
P oo . We deduce om (1.24)and (3.52)(c . Theo em 3.1 [28] o example) ha o all
ε >0 a bi a ily small, he e exis s a cons an C( , ε) >0such ha , o a, bsa is ying
0 <a <b <∞,
∀ ∈(0,1),∀(x, y)∈(a, b)×(a, b),
|(log x)Λ( , x)−(log y)Λ( , y)|≤C( , ε)|x−y|
aε
This is (3.53) o α=0. P ope y (3.53) o all α∈[0, ) ollows by simple s aigh o wa d
calcula ion (c . o example 5oin [25], p. 14). 
Co olla y 3.9. Fo all ∈(0, 1/2), o all ε >0a bi a ily small, he e exis s a cons an
C=C( , α) >0such ha , o α∈[0, )and a ∈(0, 2),
∀ ∈(0,1),∀(x, y)∈(a, 2) ×(a, 2),
|log x|1−αΛ( , x)−|log y|1−αΛ( , y)≤C|x−y| −α
a −α|log a|(1+α)( −α)(3.54)
P oo . Le us w i e,
|log x|1−αΛ( , x)=ϕ( , x)w(x)
ϕ( , x)=(log x)Λ( , x)
|x−1|α,w(x)=|x−1|α
log x|log x|1−α.
By Lemma 3.8 and he mean alue Theo em,
ϕ( , x)w(x)−ϕ( , y)w(y)=(ϕ( , x)−ϕ( , y))w(y)+ϕ(x)(w( , x)−w( , y))
|ϕ( , x)−ϕ( , y)||w(y)|≤(2 + α)C( , T, ε)|x−y| −α
aεsup
z∈(a,2) |w(z)|
|w(x)−w(y)|≤ sup
z∈(0,2) |w(z)||x−y|≤ C
a|log a|1+α|x−y|,
because,
w(x)=α|x−1|α−2(x−1)
log x|log x|1−α−|x−1|α
x|log x|2|log x|1−α+
+|x−1|α
log x|log x|−αH(x−1)
x,H= Hea iside’s unc ion
|w(x)|≤ C
a|log a|1+α,∀x∈(a, 2).
Bu since, |w(x)| ≤w(2) o all x ∈(0, 2) we deduce by in e pola ion, o all θ∈(0, 1),
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 37
|w(x)−w(y)|≤ C|x−y|θ
aθ|log a|(1+α)θ
and, o θ= −α ha may be supposed o be la ge ha ε,
|ϕ( , x)w(x)−ϕ( , y)w(y)|≤(4 + 2α)C( , ε)|x−y| −α
aε+C|x−y| −α
a −α|log a|(1+α)( −α)
≤C|x−y| −α
a −α|log a|(1+α)( −α)
o some cons an C>0 ha depends on , αbu no on a.
P oposi ion 3.10. Fo all ∈(0, 1/2), α∈[0, )and ∈(2 , 1), i m(x, y) =min(x, y)
|Λ( , y)−Λ( , x)|≤ 2|Λ( , x)||x−y|1−α
m(x, y)1−α|log y|1−α+C|x−y| −α
m(x, y) −α|log m(x, y)|(1+α)( −α)(3.55)
and he unc ion Λsa isfies p ope y (1.29)
P oo . Λ( , y)−Λ( , x)=|log x|1−αΛ( , x)A1(x, y)+A2( , x, y) (3.56)
A1(x, y)=1
|log x|1−α−1
|log y|1−α(3.57)
A2( , x, y)=|log x|1−αΛ( , x)−log y)1−αΛ( , y)
|log y|1−α(3.58)
|A1(x, y)|=|log x|1−α−|log y|1−α
|log x|1−α|log y|1−α≤||log x|−|log y||1−α
|log x|1−α|log y|1−α
=1
|log x|1−α|log y|1−αd
dz |log z|(ξ)|x−y|1−α
=1(1,∞)(ξ)|x−y|1−α
ξ1−α|log x|1−α|log y|1−α
o some ξbe ween xand y, and hen,
|A1(x, y)|≤ 2|x−y|1−α
min(x, y)1−α|log x|1−α|log y|1−α.(3.59)
Using Co olla y 3.9 o es ima e A2, he esul ollows. 
3.3. Beha io o Λas x →1
In he ollowing P oposi ion he beha io o Λis gi en in he neighbo hood o x =1.
I s p oo , somewha echnical is gi en in he Appendix.

38 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
P oposi ion 3.11. Fo all bounded subse A ⊂R, he e exis s a cons an Ca>0such
ha ,
sup
X∈A, ∈(0,1)
−1|X|1−2 |˜
Λ( , X)|≤CA(3.60)
sup
X∈A, ∈(0,1)
|X|1−2
(1 + 2 log |X|)
∂˜
Λ
∂ ( , X)≤CA,(3.61)
and, uni o mly on A,
lim
→0 −1|X|1−2 ˜
Λ( , X)=1,(3.62)
lim
→0|X|1−2
(1 + 2 log |X|)
∂˜
Λ
∂ ( , X)=1.(3.63)
Rema k 3.12. Fo any ϕ ∈CC(R),
lim
→0 ˆ
R
|X|−1+2 ϕ(X)dX =ϕ(0).
Co olla y 3.13.
lim
→0 −1e−1/ Y
1−2 Λ , 1+e−1/ Y= 1 (3.64)
uni o mly on bounded subse s o R.
P oo . Fo >0 sufficien ly small, depending on he bounded se Ko Rwhe e Y a ies,
1 +e−1/ Y>0. Then we define 1 +e−1/ Y=eXand by defini ion Λ( , 1 +e−1/ Y) =
˜
Λ( , X). By (3.62), uni o mly o Xin bounded subse s o R,
lim
→0 −1|X|2 −1˜
Λ( , X) = 1 (3.65)
lim
→0 −1|X|2 −1Λ( , 1+e−1/ Y) = 1 (3.66)
Bu , since
lim
→0e−1/ Y=0,uni o mly o Yon K,
i ollows ha
lim
→0eX=1,uni o mly o Yon K.
Then
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 39
lim
→0
e−1/ Y
X= lim
→0
eX−1
X=1
om whe e
lim
→0 −1|X|2 −1Λ( , 1+e−1/ Y) = lim
→0 −1|e−1/ Y|2 −1Λ( , 1+e−1/ Y) = 1 (3.67)
uni o mly o Y∈K.
Co olla y 3.14. Fo all R∈(0, 1) he e exis s CR>0such ha
|Λ( , x)|≤ CR
|x−1|1−2 ,∀x;|x−1|e1/ ≤R, ∈(0,1),(3.68)

∂
∂ Λ( , x)≤CR(1 + 2 log |x−1|)
|x−1|1−2 ,∀x;|x−1|e1/ ≤R, ∈(0,1).(3.69)
P oo . By (3.60), o all ∈(0, 1),
|Λ , 1+e−1/ Y|≤ 2
e−1/ Y1−2 ,∀Y∈(−R, R).(3.70)
In e ms o x =1 +e−1/ Y, (3.68) ollows. Simila ly, (3.69) ollows om (3.61). 
Co olla y 3.15. The unc ion Λsa isfies,
Λ∈C((0,∞),L
1(0,∞)),(3.71)
and he e exis s C>0such ha ,
||Λ( )||1≤C
1+ 2,∀ >0 (3.72)
P oo . We p o e (3.72) fi s . Fo ∈(0, 1) we use he es ima es in Sec ion 3.2
∞
ˆ
0|Λ( , x)|dx =
1/2
ˆ
0|Λ( , x)|dx +ˆ
|x−1|<1/2
|Λ( , x)|dx +
∞
ˆ
3/2
|Λ( , x)|dx. (3.73)
The fi s and hi d in eg als in he igh hand side o (3.73)a e es ima ed as,
1/2
ˆ
0|Λ( , x)|dx ≤
1/2
ˆ
0
dx
|x−1|≤ .
∞
ˆ
3/2
|Λ( , x)|dx ≤C1 7+β
1
∞
ˆ
3/2
x−1−β
1dx +C2 7
∞
ˆ
3/2
x−6dx.
40 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
Fo he second in eg al in he igh hand side o (3.73)we w i e,
ˆ
|x−1|<1/2
|Λ( , x)|dx =ˆ
0<|x−1|<e−1/
|Λ( , x)|dx +ˆ
e−1/ <|x−1|<1/2
|Λ( , x)|
≤C ˆ
0<|x−1|<e−1/
dx
|x−1|1− +C ˆ
e−1/ <|x−1|<1/2
dx
|x−1|
=2C
e−1/
ˆ
0
dz
z1− +2C
1/2
ˆ
e−1/
dz
z=2C
e−2C log 2 + 2C.
Fo >1, by P oposi ion (3.1)
∞
ˆ
0|Λ( , x)|dx = −3
∞
ˆ
0|Q1(θ)|dx +
∞
ˆ
0|Q2( , θ)|dx,
= −2
∞
ˆ
0|Q1(θ)|dθ +
∞
ˆ
0|Q2( , θ)|dθ,
whe e we used he change o a iable θ=x
. Then (3.72) ollows since, by P oposi ion
(3.2), Q1∈L1(0, ∞) and, by P oposi ion (3.3),
∞
ˆ
0|Q2( , θ)|dθ ≤C −4(3.74)
On he o he hand i 1>0and | − 1| <
1/4, o any ε >0small fixed and Rla ge o
be fixed,
∞
ˆ
0|Λ( 1,x)−Λ( , x)|=I1+I2+I3+I4
I1=
1−ε
ˆ
0|Λ( 1,x)−Λ( , x)|dx ≤sup
x∈[0,1−ε)|Λ( 1,x)−Λ( , x)|
I2=
1+ε
ˆ
1−ε|Λ( 1,x)−Λ( 2,x)|dx ≤2εsup
x∈[1−ε,1+ε)
∈3 1
4
5 1
4|Λ( , x)|
I3=
R
ˆ
1+ε|Λ( 1,x)−Λ( 2,x)|dx ≤sup
x∈[1+ε,R)|Λ( 1,x)−Λ( , x)|
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 41
I4=
∞
ˆ
R
|Λ( 1,x)−Λ( 2,x)|dx ≤
∞
ˆ
R
|Λ( 1,x)|dx +
∞
ˆ
R
|Λ( 2,x)|dx
The e ms I1, I2and I3 end o ze o as → 1by he con inui y o (log x)Λ( , x) o
>0and x ∈R+ {1}. I 0 <
1<1, we deduce I4≤CR−β
1 om an es ima e simila
o (3.71) w i en o Rins ead o 3/2. Fo >1, i ollows om (3.4)and (2.30) ha
I4≤CR1−cwhe e cmay by chosen in he in e al (0, 2). The choice c ∈(1, 2) ensu es
ha o all >0, I4→0when R→∞. This p o es (3.71). 
In o de o check ha Λsa isfies (1.1)le us show fi s ha L(Λ( )) is well defined.
When >1 his ollows om he C1 egula i y o he unc ion Λ( ).
P oposi ion 3.16. L(Λ) ∈C((1, ∞) ×(0, ∞)). Fo all >1, he e exis s a nume ical
cons an C>0such ha
L(Λ( ))(x)<C
x 2min 1
,1
x,∀x>0.
P oo . Fo >1, Λ( ) ∈C1(0, ∞)and by P oposi ions 3.1–3.3
|Λ( , x)|≤min( −3,x
−3).
The e o e, o e e y x >0, and y∈(0, x/2)
|Λ( , y)−Λ( , x)|K(x, y)≤Cx−2min( −3,x
−3)+min( −3,y−3)
Then, i x ∈(x0−ε, x0+ε) o some x0>2ε >0,
|Λ( , y)−Λ( , x)|K(x, y)10<y<x/2≤C(x0−ε)210<y<(x0+ε)/2
(min( −3,(x0−ε)−3)+min( −3,y−3))
and since he igh hand side belongs o L1(0, ∞)i ollows ha
x/2
ˆ
0
(Λ( , y)−Λ( , x))K(x, y)dy ∈C(0,∞).
Mo eo e :
x/2
ˆ
0|Λ( , y)−Λ( , x)|K(x, y)dy ≤Cmin( −3,x
−3)x−1+
+Cx−2
x/2
ˆ
0
min( −3,y−3)dy ≤Cmin( −3,x
−3)x−1+C
x 2min( −1,x
−1).
48 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
∂U
∂ ( , s)=W(s−1)U( , s −1) + ( , s) (3.87)
( , s)=M(Λ( ))(s)( ) (3.88)
and he unc ion is bounded on (0, T) ×S0,2, ( ) ≡0i 0 ≤ ≤T/2. We may hen
Laplace ans o m bo h sides o (3.87)and ob ain, o some cons an C>0,
z˜
V(z,s)=−W(s−1) ˜
V(z,s −1) + ˜ (z,s),Re(z)>0,Re(s)∈(1,2) (3.89)
|˜ (z,s)|≤Ce−T
2Re(z),∀s∈S,Re(z)>0.(3.90)
The unc ion
˜
Vmay be spli as
˜
V=˜
Vp+˜
Vhwhe e
˜
Vpis he pa icula solu ion o (3.89),
˜
Vp(z,s)= 1
2iπ
B(s)
zˆ
Re(σ)=β
e(σ−s)log(−z)
B(σ)
˜ (z,σ)dσ
(1 −e2iπ(s−σ))
and
˜
Vhmus sa is y
∂˜
Vh
∂ ( , s)=−W(s−1) ˜
Vh( , s −1),Re(z)>0,Re(s)∈(1,2) (3.91)
The unc ion
˜
Vp(z, s)is analy ic on s ∈S o all Re(z) >0, analy ic on Re(z) >0and
o all s ∈S. By (3.90), and ou choice o he b anch o he log unc ion in (2.20), o
all z∈C, Re(z) ≥z0>0
˜
Vp(z,s)≤Ce−T
2Re(z)1
|z|ˆ
Re(σ)=βe(σ−s)log(−z)
B(σ)|dσ|
1−e2iπ(s−σ)≤Cz0e−T
2Re(z).
(3.92)
On he o he hand, using he unc ion
˜
Vhwe define, ollowing he same a ionale as in
he defini ion o (2.22), in he P oo o P oposi ion (2.8)
˜
H(z,s)= ˜
Vh(z,s)eslog(−z)
B(s)
˜
h(z,ζ)= ˜
H(z,s),ζ=e2iπ(s−β).
Fo e e y zsuch ha Re(z) >0, he unc ion h(z, ·)is hen analy ic on C R+and, by
(3.91),
˜
h(z,ζ +i0) = ˜
h(z,ζ −i0),∀ζ∈R+.
I ollows ha o all Re(z) >0,
˜
h(z, ·)is analy ic on C {0}. Bu since, by P oposi ion 2.4
and (3.92), we also ha e

M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 49
|˜
h(z,ζ)|≤Ceslog(−z)=Ceclog zei(s−β)A g(−z)=Ceclog z|ζ|A g(−z)
2π=Ceclog z|ζ|1/2,
by Liou ille’s Theo em
˜
h(z) ≡0. The e o e
˜
H(z) =˜
Vh(z) =0and
˜
V=˜
Vp. By he
in e se Laplace o mula
U( , s)= 1
2iπ
a+i∞
ˆ
a−i∞
˜
V(z,s)ez dz,
and by (3.90)we ha e hen U( , s) =M(Λ( , )(s) =0 o all s ∈Sand 0 ≤ ≤T/2
om whe e he esul ollows. 
P oo o Theo em 1.2.All he p ope ies o Λ, up o (1.25), ha e al eady been p o ed
in P oposi ion 2.13, Co olla y 2.14, Co olla y 3.13, Co olla y 3.15 and P oposi ion 3.18.
Since W(2) =0, U( , 3) =U(0, 3) o all >0by (2.29), which is he conse a ion o
he second momen o Λ( ). 
4. Solu ion o he Cauchy p oblem o (1.1)
This Sec ion is de o ed o he p oo o he exis ence o solu ions o he Cauchy p ob-
lem o equa ion (1.1) o ini ial da a 0∈L∞(0, ∞)o L1(0, ∞), and he p oo s o
Theo em 1.4 and P oposi ion 1.5.
Fo all y>0we define,
G( , x;y)=y−1Λ
y,x
y,∀ >0,∀x>0.(4.1)
By (3.71), G ∈C((0, ∞) ×(0, ∞); L1(0, ∞)), o y>0fixed i is a weak solu ion o
(1.20)and
lim
→0G( , ·,y)=δy,in he weak sense o D(0,∞).(4.2)
The unc ion Galso sa isfies he ollowing impo an p ope y,
P oposi ion 4.1. The e exis s a posi i e cons an CG>0such ha , o all >0, x >0,
I( , x)=
∞
ˆ
0|G( , x;y)|dy < CG.(4.3)
The p oo o P oposi ion 4.1 has se e al auxilia y Lemmas and wo diffe en cases:
50 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
•I 0 < <x,
I( , x)=
ˆ
0
(······)
 
/y > 1,x/y >1
dy
y+
x
ˆ
(······)
 
/y < 1,x/y >1
dy
y+
∞
ˆ
x
(······)
 
/y < 1,x/y <1
dy
y.(4.4)
•Fo 0 <x < ,
I( , x)=
x
ˆ
0
(······)
 
/y > 1,x/y >1
dy
y+
ˆ
x
(······)
 
/y > 1,x/y <1
dy
y+
∞
ˆ
(······)
 
/y < 1,x/y <1
dy
y.(4.5)
Lemma 4.2. The e exis s C>0such ha , o all >0and x >0,
ˆ
0Λ
y,x
y
dy
y≤C.
P oo o Lemma 4.2.Since y∈(0, ), /y > 1and by P oposi ion 3.1 and P oposi-
ion 3.2,
Λ
y,x
y≤Cmax 
y,x
y−3
.
Then,
∀x>0,∀ ∈(0,x),
ˆ
0Λ
y,x
y
dy
y≤
ˆ
0x
y−3dy
y= 3
3x3≤1/3.
∀ >0,∀x∈(0, ),
ˆ
0Λ
y,x
y
dy
y≤
ˆ
0
y−3dy
y=1
3
I emains now o es ima e he wo las in eg als a he igh hand side o (4.4), and
he las one a he igh hand side o (4.5). To his end we will be using a unc ion δ(z),
defined and con inuous on z≥0such ha ,
δis dec easing,δ(u)<1 o all u>0,δ(1) = 1
2,δ(u)=e1−u
2,∀u≥1
2.(4.6)
4.1. The domain 0 < <x
Conside fi s he domain whe e 0 < <y<xwhe e 0 <
y<1 <x
y. In o de o use
he es ima e on Λ, his domain is s ill subdi ided.
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 51
Lemma 4.3. Define
H2(z)=z(1 + δ(z)) and H1(z)=z(1 −δ(z)),∀z>0
These wo unc ions a e mono one inc easing. Mo eo e
∀z>0,H
1(z)<z (4.7)
∀z>3/2,H
−1
2(z)>1 (4.8)
∀z>0,H−1
2(z)<z (4.9)
∀x>0,∀ ∈(0,2x/3),2x
3< H
−1
2x
.(4.10)
P oo . Since he unc ion H2is s ic ly inc easing, i s in e se H−1
2is well defined. The
choice δ(1) =1/2makes H2(1) =3/2 hen H−1
2(3/2) =1. By mono onici y i ollows
ha H−1
2(z) >H
−1
2(3/2) =1 o all z>3/2and his p o es (4.8). Since H2(z) >zi
ollows ha z>H
−1
2(z)and his shows (4.10).
Since δ(1) =1/2, we ha e
2
3(1 + δ(1)) = 1 and he unc ion δ(z)is s ic ly dec easing
because so is ρ(z). The e o e δ(z) <1/2 o all z>1, and, o all ∈(0, 2x/3)
H22x
3 =2x
3 1+δ2x
3 <2x
3 (1 + δ(1)) = x
.
Since H2is s ic ly inc easing, so is H−1
2,
2x
3 ≤H−1
2x
and his p o es (4.10). 
Lemma 4.4. Fo all >0, x >0such ha <x,
x
ˆ
Λ
y,x
y
dy
y≤C1+ +Φ
1+Ψ
1+˜
Φ2,(4.11)
∞
ˆ
xΛ
y,x
y
dy
y≤C(1 + Φ3+Ψ
3),(4.12)
whe e: Φ1(x, )=
H−1
2x

ˆ
2x
3
1
y
x
y−1
−1dy
y,∀ ∈(0,2x/3),(4.13)
Ψ1(x, )=
x
ˆ
H−1
2x

y
x
y−1
−1+ 2
ydy
y,∀ ∈(0,2x/3),(4.14)
˜
Φ2(x, )=
x
ˆ
y
x
y−1
−1+ 2
ydy
y,∀ ∈(2x/3,x),(4.15)
52 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
Ψ3(x, )=
H−1
1x

ˆ
x
y
x
y−1
−1+ 2
ydy
y,∀ ∈(0,x),(4.16)
Φ3(x, )=
2x
ˆ
H−1
1x

y1−x
y
−1dy
y,∀ ∈(0,x).(4.17)
P oo o Lemma 4.4.We show (4.11) fi s and s a assuming ∈(0, 2x/3). By (4.10),
x
ˆ
Λ
y,x
y;1

dy
y=
2x
3
ˆ
(···)dy +
H−1
2x

ˆ
2x
3
(···)dy +
x
ˆ
H−1
2x

(···)dy. (4.18)
In he fi s in eg al o he igh hand side o (4.18), since y<2x/3, by P oposi ion 3.5
Λ
y,x
y≤C1x
−1−β
1
y6
+C2x
−6
y2
,
and hen,
2x
3
ˆ
Λ
y,x
y
dy
y≤C1 6
2x
3
ˆ
y−6dy +C2 2
2x
3
ˆ
y−2dy ≤C . (4.19)
In he second in eg al o he igh hand side o (4.18), simple compu a ions yield,
y∈2x
3, H−1
2x
=⇒
yH2(y/ )<x
y<3
2=⇒δy
<x
y−1<1
2.
Since x >3 /2we ha e y/ >1. On he o he hand, x/ may ake alues a bi a ily la ge,
and hen H−1
2x
and y/ oo. We deduce ha δ(y/ ) ∈(0, 1/2) and by P oposi ion 3.6,
H−1
2x

ˆ
2x
3Λ
y,x
y
dy
y≤CΦ1(x, ).(4.20)
In he hi d in eg al o he igh hand side o (4.18), since H−1
2x
<y, i ollows
ha H−1
2x
<y, om whe e
x
<H
2y
=y
1+δy
. Then
x
y<1 +δy
and,
since x/y > 1also,
0<x
y−1<δy
.(4.21)
We no ice now ha since x/ >3/2and
3
2<x
=u(1 +δ(u)) ≤2u, we also ha e
u =H−1
2(x/ ) >3/4. Then y/ a ies on he hal line (3/4, ∞)and δ(y/ ) a ies on
(0, δ(3/4)). We deduce om (4.21), using Co olla y 3.13, ha o some cons an C>0,
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 53
Λ
y,x
y≤C
y
x
y−1
−1+ 2
y
.(4.22)
I ollows om (4.19), (4.20)and (4.22) ha o 0 < <2x/3,
x
ˆ
Λ
y,x
y
dy
y≤C( +Φ
1(x, )+Ψ
1(x, )) .(4.23)
Suppose now ha ∈(2x/3, x). We fi s deduce ha since x/ <3/2and H−1
2is
inc easing, H−1
2(x/ ) <H
−1
2(3/2) =1and hen H−1
2(x/ ) < . Since y∈( , x)i ollows
ha y> H
−1
2(x/ )and he e o e,
H2(y/ )≡y
(1 + δ(y/ )) >x
=⇒1+δ(y/ )>x
y⇐⇒ x
y−1<δ(y/ ).
Then, o all 0 < <y<x, we ha e x/y > 1 and, 0 <x
y−1 <δ(y/ ). By Co olla y 3.14,
and (4.15)we deduce, when ∈(2x/3, x),
x
ˆ
Λ
y,x
y
dy
y≤˜
CΦ2(x, ),(4.24)
and (4.11) ollows om (4.23)and (4.24).
We p o e now (4.12). To his end we w i e, he le hand side as
∞
ˆ
x
(···)dy
y=
H−1
1x

ˆ
x
(···)dy
y+
2x
ˆ
H−1
1x

(···)dy
y+
∞
ˆ
2x
(···)dy
y(4.25)
In he fi s e m a he igh hand side o (4.25)x <y< H
−1
1x
, hen 0 <1 −x
y<δy

om whe e, by Co olla y 3.13 and (4.16)
H−1
1x

ˆ
xΛ
y,x
y;1

dy
y≤CΨ3(x, ),0< <x. (4.26)
In he second in eg al a he igh hand side o (4.25), H−1
1x
<y<2xand so
δy
<1 −x
y<1
2and by (4.17)and P oposi ion 3.6,
2x
ˆ
H−1
1x
Λ
y,x
y
dy
y≤CΦ3(x, )0< <x. (4.27)
In he las in eg al a he igh hand side o (4.25), since y>2x, by P oposi ion 3.6,

54 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
∞
ˆ
2xΛ
y,x
y
dy
y≤C
∞
ˆ
2x
1
|1−x/y|
dy
y2≤C
∞
ˆ
2x
dz
y2≤C. (4.28)
The es ima e (4.12) ollows now by (4.26)–(4.28). 
4.2. The domain 0 <x <
We es ima e now he las in eg al a he igh hand side o (4.5)
Lemma 4.5. Fo all >0and x ∈(0, ),
∀ >2x,
∞
ˆ
Λ
y,x
y
dy
y≤C(4.29)
∀ ∈(x, 2x),
∞
ˆ
Λ
y,x
y
dy
y≤C(1 + Φ3+Ψ
4) (4.30)
whe e Ψ4=
H−1
1x

ˆ
y
x
y−1
−1+ 2
ydy
y,∀ ∈(x, 2x).(4.31)
P oo o Lemma 4.5.I >2x hen, x/y < 1/2and P oposi ion 3.6 gi es (4.29).
Fo ∈(x, 2x),
x
>1
2≡H1(1) and < H
−1
1x
by he mono onici y o H1. On he
o he hand,
H12x
=2x
1−δ2x
≥2x
(1 −δ(1)) = x
(whe e use has been made o 2x/ ≥1), and hen, H−1
1x
<2x. The e o e,
∞
ˆ
(···)dy =
H−1
1x

ˆ
(···)dy +
2x
ˆ
H−1
1x

(···)dy +
∞
ˆ
2x
(···)dy. (4.32)
In he fi s e m a he igh hand side o (4.32)0 <1 −x
y<δ
y
because y∈
 , H−1
1x
,
H−1
1x

ˆ
Λ
y,x
y
dy
y≤CΨ4(x, ),(4.33)
by (4.31)and Co olla y 3.14. In he second in eg al o he igh hand side o (4.32)
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 55
y∈ H−1
1x
,2x=⇒δy
<1−x
y<1
2.
By P oposi ion 3.6 and (4.17)
2x
ˆ
H−1
1x
Λ
y,x
y
dy
y≤CΦ3(x, ).(4.34)
In he hi d in eg al o he igh hand side o (4.32)y>2x hen by P oposi ion 3.6,
∞
ˆ
2xΛ
y,x
y
dy
y≤C(4.35)
and (4.30) ollows om (4.32)–(4.35) o ∈(x, 2x). 
4.3. Es ima es o he unc ions Φand Ψ
In his sub Sec ion some use ul p ope ies o he unc ions Φand Ψdefined in
(4.13)–(4.17)a e ob ained.
Lemma 4.6. The e exis s a cons an C>0such ha ,
Φ1+Ψ
1+˜
Φ2+Φ
3+Φ
4+Ψ
4≤C(4.36)
P oo o Lemma 4.6.(i) Es ima e o Φ1. By defini ion, o x >0and ∈(0, 2x/3),
Φ1(x, )=C
x
xH−1
2x

ˆ
2
3
|1− |−1d =−
xlog 1−
xH−1
2x
+log3
.(4.37)
Then, o all ε >0, Φ1(x, )is bounded o all ( , x)such ha 0 < <xand
xH−1
2x
∈
[0, 1 −ε]. Assume now ha
xH−1
2x
→1, and deno e u =H−1
2(x/ ). Since,
xH−1
2x
=u
H(u)=1
1+δ(u)(4.38)
i
xH−1
2x
→1i ollows ha δ(u) →0. This implies ha u →∞, and by elemen a y
calculus,
xH−1
2x
=1
1+e1−u
2
=1−e1−u
2+Oe−2u,as u→∞
and
xH−1
2x
=
u→∞ 1+Oe−u,u=H−1
2x
=
u→∞
x
1+Oe−u.(4.39)
56 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
Using (4.38), (4.39)and he defini ion o δ, o ρ >0as small as desi ed and u →∞,
xH−1
2x
=1
1+e−x
(1 + Oe−(1−ρ)u)=1
1+e−x
1+Oe−(2−ρ)u
and i ollows ha
log 1−
xH−1
2x
=
u→∞ −x
+Oe−x
.
We deduce he exis ence o a cons an C>0such ha o all 0 < <2x/3,
Φ1(x, )≤C. (4.40)
(ii) Es ima e o Ψ1. Since ∈(0, 2x/3) and y> H
−1
2x
 hen x/ <H
2(y/ ) <2y/ .
Using ha y<x, also we deduce 0 <x
y−1<1. Since 1/y > 1/x,
Ψ1(x, )≤
x
ˆ
H−1
2x
x
y−1−1+ 2
xdy
y2= x−1
1
ˆ
xH−1
2x

(1 −ρ)−1+ 2
xρ−1−2
xdρ
By (4.10), 2H−1
2x
>4x
3 , hen
xH−1
2x
>1
2and,
Ψ1(x, )≤ x−1
1
ˆ
1
2
(1 −ρ)−1+ 2
xρ−1−2
xdρ =C. (4.41)
(iii) Es ima e o
˜
Φ2. When ∈(2x/3, x)and y∈( , x), 0 <x
y<1and hen, by (4.15)
˜
Φ2(x, )≤
x
ˆ
x
y−1−1+ 2
xdy
y2=
x
1
ˆ
x
(1 − )−1+ 2
x −1−2
xd
≤
x
1
ˆ
2
3
(1 − )−1+ 2
x −1−2
xd =2
−2
x−1≤2−4/3.(4.42)
(i ) Es ima e o Φ3. By defini ion, o 0 < <x,
Φ3(x, )=
x
2
ˆ
xH−1
1x

( −1)−1d
=−
xlog 
xH−1
1x
−1(4.43)
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 57
because, i =H−1
1x
 hen
x
=H1( ) = (1 −δ( )), and
xH−1
1x
=1
1−δ( )>1.
The same a gumen s as in he es ima e o he igh hand side o (4.37), show he exis ence
o a cons an C>0such ha o all 0 < <x,
Φ3(x, )≤C. (4.44)
( ) Es ima e o Ψ3. Fo all yin he domain o in eg a ion o Ψ3, y< H
−1
1x
, and hen
2
y>2
H−1
1x
. Since y>xalso, we ha e 1−x
y∈(0, 1) and we deduce om (4.16),
Ψ3(x, )≤
H−1
1x

ˆ
x1−x
y−1+ 2
H−1
1x
dy
y2=
x
xH−1
1x

ˆ
1
( −1)−1+ 2
H−1
1x

1+ 2
H−1
1x
d
We use now ha , because δ(x/ ) <1/2, z<H
1(2z)and so
xH−1
1x
<2, o ob ain,
Ψ3(x, )≤
x
2
ˆ
1
( −1)−1+ 2
H−1
1x

1+ 2
H−1
1x
d =
xH−1
1(x/ )2−1−2
H−1
1(x/ )≤C. (4.45)
( i) Es ima e o Ψ4. By defini ion, x < <y< H
−1
1x
<2x, o all yin he domain
o in eg a ion. The e o e, as o Ψ3, we ha e
2
y>2
H−1
1x
and 1−x
y∈(0, 1). A guing
as o Ψ3, we deduce om (4.31), o all ∈(x, 2x),
Ψ4(x, )≤
H−1
1x

ˆ
1−x
y−1+ 2
H−1
1x
dy
y2≤
2
ˆ
1
( −1)−1+ 2
H−1
1x

1+ 2
H−1
1x
d
= x−1H−1
1(x/ )2−1−2
H−1
1(x/ )≤C. (4.46)
Lemma 4.6 ollows om (4.40)–(4.46)
P oo o P oposi ion 4.1.P oposi ion 4.1 ollows om Lemma a 4.2–4.6 
I is now possible o define he solu ion uo he Cauchy p oblem.
4.4. P oo s o Theo em 1.4 and P oposi ion 1.5
Theo em 4.7. (i) Fo any 0∈L1(0, ∞),
∞
ˆ
0
∞
ˆ
0Λ
y,x
y 0(y)
dy
ydx < ∞,∀ >0.(4.47)
64 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
and since 1/2 < x/z < x/ 1<3/2 o all ∈I, o a posi i e cons an Cindependen
on xand 1,

∂Λ
∂ 
z,x
z| 0(z)|
z2≤C| 0(z)|
z2.(4.69)
A simila es ima e holds o z∈Dc
1( , x)wi h a simila a gumen .
We a e hen le wi h he domain D( , x), whe e, by (3.69)in Co olla y 3.14

∂Λ
∂ 
z,x
z| 0(z)|
z2≤C(1 + 2( /z)|log |(x/z)−1||)| 0(z)|
|(x/z)−1|1−2( /z)z2
≤C(1 + |log |(x/z)−1||)| 0(z)|
|(x/z)−1|1−2( /z)z2
Since z∈D( , x), E+(x/ ) ≤z/ ≤E−(x/ )and |(x/z) −1| <1. Then, o ∈( 1, 2),
1
E−(x/ 1)≤
z≤1
E+(x/ 2)
=⇒|(x/z)−1|1−2
z≥|(x/z)−1|1−ρ(x, 1)
ρ(x, 1)= 2
E−(x/ 1)>0

∂Λ
∂ 
z,x
z| 0(z)|
z2≤C(1 + |log |(x/z)−1||)| 0(z)|
|(x/z)−1|1−ρ( 1,x)z2.
No ice ha o z∈Dc, whe e (4.69)holds,
x
z−1≤1+x
z≤1+ x
2 1
and, om (4.69), o z∈Dc( , x) oo,

∂Λ
∂ 
z,x
z| 0(z)|
z2≤C(1 + |log |(x/z)−1||)| 0(z)|
|(x/z)−1|1−ρ( 1,x)z2.(4.70)
The e o e, when x/3 <
0<3x
2 he unc ion H(x, z)may hen be aken as ollows,
H(x, z)=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Cz2| 0(z)|
max( 4
1,x
4),∀z∈(0,2 1)
(1 + |log |(x/z)−1||)| 0(z)|
|(x/z)−1|1−ρ( 1,x)z2,∀z∈(2 1,3x)
C| 0(z)|
z2,∀z>3x,
(4.71)

M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 65
whe e he cons an Cmay depend on xand 1bu no on o z. We ha e hen, shown
ha o all 0>0, and almos e e y x >0 he e exis s a neighbo hood I=( 1, 2)such
ha , unde he hypo hesis... on 0,

∂Λ
∂ 
z,x
z| 0(z)|
z2≤H(x, z)
∞
ˆ
0
H(x, z)dz < ∞.
I ollows om classical p ope ies o Lebesgue’s in eg al, ha o all >0and a.e.
x >0,
∂
∂
∞
ˆ
0
0(z)Λ 
z,x
zdz
z=
∞
ˆ
0
0(z)∂ Λ
z,x
zdz
z2,(4.72)
and usa isfies (1.1) o all >0and a.e. x >0.
I is now possible, o ob ain poin wise es ima es o ∂ u( , x), using essen ially he same
igh hand side e ms ha in (4.57), (4.59), (4.71), excep ha needs no be eplaced
by 1o 2now. I easily ollows,

∂u
∂ ( , x)≤C 2x−4+ 3x−3+ε|| 0||1,a.e.x>3
C( −2+ −3x)|| 0||1,a.e.x∈(0,2 /3).(4.73)
I x ∈(2 /3, 3 ) deno e,
θ( , x)=1−2
E−(x/ )
and hen,
ˆ
D( ,x)
(1 + |log |(x/z)−1||)| 0(z)|
|(x/z)−1|1−ρ( 1,x)z2≤sup
z∈(2 ,3x)| 0(z)|3x
ˆ
2
C(1 + |log |(x/z)−1||)
|(x/z)−1|1−ρ( 1,x)z2
and
3x
ˆ
2
(1 + |log |(x/z)−1||)
|(x/z)−1|θz2=1
x
x/2
ˆ
1/3
(1 + |log |y−1||)
|y−1|θ
≤1
(1 −θ)x2
31−θ
+x
2 −1
1−θ+(2/3)1−θ(1 + (1 −θ)|log(2/3)|)
(1 −θ)2x+
66 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
+|x/2 −1|1−θ(1 + (1 −θ)|log |x/2 −1||)
(1 −θ)2x.
Since x ∈(2 /3, 3 )i ollows ha x/2 ∈(−2/3, 1/2) and |x/2 −1| ≤2/3. Then,
3x
ˆ
2
(1 + |log |(x/z)−1||)
|(x/z)−1|θz2≤2
(1 −θ)x+(1 + |log(2/3)|
(1 −θ)2x+(1 + |log |x/2 −1||)
(1 −θ)2x
≤C(1 + |log |x/2 −1||)
(1 −θ)2x
and,

∂u
∂ ( , x)≤C(1 + |log |x/2 −1||)
(1 −θ)2xsup
z∈(2 ,3x)| 0(z)|, a.e.x ∈(2 /3,3 ).(4.74)
We deduce om (4.73), (4.74) ha L(u)sa isfies (1.36). 
P oo o P oposi ion 1.5.When >0is fixed and x →0we a e in he egion whe e
2x < and we w i e, using he defini ion (1.30)o u,
u( , x)=I1+I2+I3,I
1=
x
ˆ
0
Λ
y,x
y 0(y)dy
y,
I2=
ˆ
x
Λ
y,x
y 0(y)dy
y,I
3=
∞
ˆ
Λ
y,x
y 0(y)dy
y.
In he wo fi s in eg als o he igh hand side /y > 1, and hen by (3.4), (3.10)and
(3.12), o all δ>0as small as desi ed,
Λ
y,x
y=
y−3
Q1x
+Q2
y,x

Q1x
=2c1B(1)
W(0) +Ox
,x
→0
Q2
y,x
=c2
y−4
+b1
y+Oδ
y−4x

1−δ,x
→0,
y>1,
=c2
y−4
+b1
y+ −4y4Ox

1−δ,x
→0,
y>1.
The e o e,
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 67
I1+I2=
ˆ
0
Λ
y,x
y 0(y)dy
y=2c1B(1)
W(0) −3
ˆ
0
0(y)y2dy 1+Ox
+
+c2 −4
ˆ
0
0(y)y3dy +
ˆ
0
0(y)b1
ydy
y+ −4Oδx

1−δ
ˆ
0
0(y)y3dy.
Since 2x < <yin I3, i ollows ha x/y < /(2y), and by (3.27), (3.28)
I3≤Cx 5
∞
ˆ
| 0(y)|dy
y7≤Cx1−δ 5+δ
∞
ˆ
| 0(y)|dy
y7.
This concludes he p oo o (1.37), (1.38), whe e b1is he unc ion gi en in (3.15)and
A1=−2
W(1)W(2)W(0),A
2=6˜ρ(2)
W(0)W(3)W(1).(4.75)
In o de o p o e (1.39)–(1.41), conside now 0 <x
<xand w i e,
u( , x)−u( , x)=I1+I2(4.76)
I1=
ˆ
0Λ
y,x
y−Λ
y,x
y 0(y)dy
y,(4.77)
I2=
∞
ˆ
Λ
y,x
y−Λ
y,x
y 0(y)dy
y(4.78)
Th ee cases may now be conside ed, depending on whe he x<x < , <x
<xo
x< <x.
Suppose fi s ha x<x < . Since /y > 1in I1, by P oposi ion 3.4 and he mean
alue Theo em,
∃ξ=ξ(x, xy)∈x
y,x
y;|I1|≤|x−x|
ˆ
0
∂Λ
∂x 
y,ξ| 0(y)|dy
y2
≤|x−x| −4
ˆ
0| 0(y)|y2dy. (4.79)
In I2, y> >ρx >x
 om whe e, by P oposi ion 3.5, and again by he mean alue
Theo em,
∃ξ=ξ(x, xy)∈x
y,x
y;|I2|≤|x−x|
∞
ˆ

∂Λ
∂x 
y,ξ| 0(y)|dy
y2
68 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
We use now P oposi ion 3.5 o ob ain,
|I2|≤C|x−x|
∞
ˆ
ξ−1−σ∗
0| 0(y)|y2dy ≤C|x−x|x−1−σ∗
0
∞
ˆ
| 0(y)|y−2+σ∗
0dy, (4.80)
whe e σ∗
0∈(−2, −1) is defined in P oposi ion 2.1. Then, o x<x < :
|u( , x)−u( , x)|≤C|x−x| −4
ˆ
0| 0(y)|y2dy +C|x−x|x−1−σ∗
0
∞
ˆ
| 0(y)|y−2+σ∗
0dy,
(4.81)
and his shows (1.39).
Suppose now ha x >x
> . By a simila a gumen as be o e, using now P oposi ion 3.4,
|I1|≤C|x−x| 2
x4
ˆ
0| 0(y)|dy. (4.82)
The e m I2mus be decomposed in h ee in eg als. Two o hem a e es ima ed as in
he p e ious case using P oposi ion 3.5
x
ˆ
Λ
y,x
y−Λ
y,x
y 0(y)dy
y≤|x−x|
x
ˆ

∂Λ
∂x 
y,ξ(y)| 0(y)|dy
y2
≤C|x−x| −1x−1
x
ˆ
| 0(y)|dy (4.83)
and
∞
ˆ
xΛ
y,x
y−Λ
y,x
y 0(y)dy
y≤|x−x|
∞
ˆ
x
∂Λ
∂x 
y,ξ(y)| 0(y)|dy
y2
≤C|x−x| x−4
∞
ˆ
x| 0(y)|dy. (4.84)
The las in eg al is,
J( , x, x)=
x
ˆ
xΛ
y,x
y−Λ
y,x
y 0(y)dy
y.
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 69
The in eg a ion in e al goes om x o x, and hen /y < 1and x/y < 1 <x/y. We
mus hen use he P oposi ion 3.10:
Λ
y,x
y−Λ
y,x
y≤2|x−x|1−α
x1−α|log(x/y)|1−αΛ
y,x
y
+C|x−x| −α
x −α|log(x/y)|(1+α)( −α)(4.85)
=
2x,α=(M−2)
2Mx ∈(0, ).M>3=⇒(1 + α)( −α)≤3
2M<1
2(4.86)
om whe e,
|J( , x, x)|≤2|x−x|1−α
x1−α
x
ˆ
xΛ
y,x
y| 0(y)|dy
y|log(x/y)|1−α+
+C|x−x| −α
x −α
x
ˆ
x
| 0(y)|dy
y|log(x/y)|(1+α)( −α)(4.87)
Since (1 +α)( −α) ∈(0, 1), i 0∈L∞
loc(0, ∞),
∞
ˆ
0
| 0(y)|1(x,x)(y)dy
y|log(x/y)|(1+α)( −α)≤|| 0||L∞(x,x)
x/x
ˆ
1
dz
z|log(z)|(1+α)( −α)
=|| 0||L∞(x,x)
1−(1 + α)( −α)(log(x/x))1−(1+α)( −α)≤|| 0||L∞(x,x)(log(x/x))1−(1+α)( −α)
≤|| 0||L∞(x,x)(1 + log(x/x)).(4.88)
By simila a gumen s and (4.49)in Theo em 4.7
%
%
%
%| 0(y)1(x,x)|dy
y|log(x/y)|1−α%
%
%
%1≤
∞
ˆ
x
| 0(y)|dy
y|log(x/y)|1−α≤|| 0||L∞(x,∞)
3x
ˆ
x
dy
y|log(x/y)|1−α+
+1
3|log 3|1−αx
∞
ˆ
3x| 0(y)|dy ≤|| 0||L∞(x,3x)
2x
(log 3) /2x+1
3|log 3|1−αx
∞
ˆ
3x| 0(y)|dy
(4.89)
and hen,
x
ˆ
xΛ
y,x
y| 0(y)|dy
y|log(x/y)|1−α

70 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
≤C|| 0||L∞(x,3x)
x
(log 3) /2x+1
|log 3|1−αx
∞
ˆ
3x| 0(y)|dy.(4.90)
I ollows om, (4.83), (4.84), (4.87))–(4.90)
|I2|≤C|x−x| −1x−1
x
ˆ
| 0(y)|dy +C|x−x| x−4
∞
ˆ
x| 0(y)|dy+
+2|x−x|1−α
x1−α|| 0||L∞(x,3x)
x
(log 3) /2x+1
|log 3|1−αx
∞
ˆ
3x| 0(y)|dy+
+C|x−x| −α
x −α|| 0||L∞(x,x)(1 + log(x/x)) (4.91)
Then, since 1 −α∈(0, 1/2) and /2x <1/2
|u( , x)−u( , x)|≤C|x−x| 2
x4
ˆ
0| 0(y)|dy +1
x
x
ˆ
| 0(y)|dy +
x4
∞
ˆ
x| 0(y)|dy+
+2|x−x|1−α
x1−α|| 0||L∞(x,3x)
x
+1
x
∞
ˆ
3x| 0(y)|dy+
+C|x−x| −α
x −α|| 0||L∞(x,x)(1 + log(x/x)),∀x>x
> >0,(4.92)
and his shows (1.41).
Assume now ha x< <x. Then, in he fi s e m I1we use ha o all τ>1and
z>0,
∂Λ
∂x(τ,z)=−1
4π2ˆ
Re(s)=c
sx−s−1U(τ,s)ds
(c . (2.27)and (3.20)), and by P oposi ion 2.10, o all c ∈(0, 2) he e exis s a nume ical
cons an C=C(c) >0such ha ,

∂Λ
∂x(τ,z)≤Cx−1−cˆ
R
|s|(1 + |s|)−2τds ≤Cx−1−c(1 + τ2)−1.
We ha e hen, using he same no a ion ξ∈(x/y, x/y),
|I1|≤C|x−x|
ˆ
0
ξ−1−c| 0(y)|dy
y(1 + ( /y)2)≤C|x−x|
x1+c 1−c
ˆ
0| 0(y)|dy. (4.93)
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 71
On he o he hand, we spli he I2 e m as
I2=
x
ˆ
···dy +
∞
ˆ
x···dy =I2,1+I2,2
The e m I2,2is es ima ed exac ly as in (4.84). In he e m I2,1es ima es (4.85)and
(4.86)a e used again o ob ain,
|I2,1|≤ 2|x−x|1−α
x1−α|log(x/ )|1−α
x
ˆ
Λ
y,x
y| 0(y)|dy
y
+C|x−x| −α
x −αlog(x/ )|(1+α)( −α)
x
ˆ
| 0(y)|dy
≤C|| 0||1|x−x|1−α
x1−α|log(x/ )|1−α+|x−x| −α
x −αlog(x/ )|(1+α)( −α).(4.94)
Then, i x< <x, o all c ∈(0, 2) he e exis s a cons an Csuch ha ,
|u( , x)−u( , x)|≤C|x−x|1
x1+c 1−c
ˆ
0| 0(y)|dy +
x4
∞
ˆ
x| 0(y)|dy+
+C|| 0||1|x−x|1−α
x1−α|log(x/ )|1−α+|x−x| −α
x −αlog(x/ )|(1+α)( −α).(4.95)
And his p o es (1.40). On he o he hand, by (2.37)and (4.48), o all >0and sfixed,
M(u( ))(s)=
∞
ˆ
0
U
y,s
 0(y)ys−1.
Since B(3) =0(c . P oposi ion (2.3)), p ope ies (1.42)and (1.43) ollow om (2.31). 
5. Appendix
5.1. The p oo o P oposi ion 2.10
P oo . Based on he exp ession (2.27)o U( )
U( , s)=B(s)
2iπ ˆ
Re(σ)=β
−(σ−s)Γ(σ−s)
B(σ)dσ, β ∈(0,2)
72 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
he p oo closely ollows ha o P oposi ion 8.1 in [13] (simila o (5.1) in [13]). As in
(8.34) o [13], his may be w i en,
U( , s)=B(s)
2iπ ˆ
Re(Y)=β−Re(s)
−YΓ(Y)
B(s+Y)dY =ˆ
Re(σ)=β
eψ(s,σ, )A(Y)dY (5.1)
whe e
Ψ(s, Y, )= ˆ
Re(ρ)=β
log (−W(ρ)) Θ(ρ−s, Y )dρ −Ylog −Y+Y−1
2log Y, (5.2)
wi h Θ defined in (2.19), and
A(Y)= Γ(Y)
2iπe−YYY−1/2.(5.3)
The unc ion Adefined in (5.3)is he same as in (8.5) o [13], up o he cons an ac o
−i(2π)−1/2. The unc ion Ψ defined in (5.2)is simila o (8.4) in [13], he only diffe ence
lies in he unc ion Wins ead o Φ.
The p oo o he es ima es (2.32), (2.33)o P oposi ion 2.10 ollows hen he same
a gumen s as in [13]wi h only mino diffe ences. Fo sin bounded se s, con ou de o -
ma ion and me hod o esidues in he in eg als (5.1), (5.2). Fo |s|la ge, hese a gumen s
a e combined wi h he s a iona y phase Theo em applied o Ψ(s, Y, )as a unc ion o
Y, whe e sand a e fixed. The a iable Yis scaled as Y=2Zlog |s|, acco ding o he
beha io o W(s)as Im(s) →∞, o Re(s)in a fixed bounded in e al and he esul
ollows om he ollowing. I we define,
˜
F(s, ζ)= ˆ
Re(ρ)=β
log (−W(ρ)) Θ(ρ−s, ζ)dρ (5.4)
F(s, Z)= ˆ
Re(ρ)=β
log (−W(ρ)) Θ(ρ−s, 2Zlog |s|)dρ, =˜
F(s, 2Zlog |s|) (5.5)
Es ima es (2.34)and (2.35) ollow now using (2.4)and (2.5). 
Le us define,
TL=S0,2∪{s∈C:Re(s)≤L|s|≥2L}(5.6)
whe e S0,2, defined as S0,2={s ∈C; Re(s) ∈(0, 2)}, is he egion o analy ici y o
U( ).
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 73
Lemma 5.1. Fo any cons an C>0, he e exis s a cons an L >0and s0∈C, bo h
depending on C, such ha , o all s ∈TL∩Bs0(0)c he unc ion Fmay be ex ended
analy ically o Z∈D(s, C) ∩B|log |s||
8(0) whe e
D(s, C)=&s∈C,Re(s)<0,|Re(s)|≤C|Im(s)+|log |s||
8|'
The e also exis s a cons an C>0, ha depends on C, such ha , o all Z∈D1(s, C) ∩
B|log |s||
8(0) and s ∈TL∩Bs0(0)c,
|F(s, Z)+Zlog(−W(s)) log |s|| ≤ CZ2+O1
log |s|.(5.7)
P oo . The P oo o (5.1) closely ollows ha o Lemma 14.1 in [13]. The unc ion Fis
ex ended as analy ical unc ion on D(s, C) ∩B|log |s||
8(0) by a modifica ion o he ep e-
sen a ion o mula (5.5)using con ou de o ma ion. The in eg al in he new in eg a ion
con ou Cis hen w i en as
ˆ
C
log (−W(ρ)) Θ(ρ−s, 2Zlog |s|)dρ =log(−W(s)) ˆ
C
Θ(ρ−s, 2Zlog |s|)dρ+
+ˆ
C
log W(ρ)
W(s)Θ(ρ−s, 2Zlog |s|)dρ.
The fi s in eg al may be explici ly calcula ed. The second is es ima ed using he cu off
p ope ies o he unc ion Θand elemen a y calculus a gumen s comple ely simila o
hose o Lemma 14.1 in [13]. 
Due o he slow decay o he unc ion U( , s)as |s| →∞, he ollowing is also needed
Lemma 5.2. The e exis s a cons an C>0such ha , o all s ∈TL∩Bs0(0)c, and ζ
such ha Z=ζ/|s|∈D1(s, C) ∩B|log |s||
8(0),

∂˜
F
∂s (s, ζ)≤C|ζ|2
|s|2log |s|+Ce−a|s|(5.8)
P oo . By (5.5)
∂˜
F
∂s (s, ζ)= ˆ
Re( )=β−Re(s)
∂
∂s (log (−W( +s))) Θ( , ζ)d
=−ˆ
Re( )=β−Re(s)
W( +s)
W( +s)Θ( , ζ)d =ˆ
Re(ρ)=β
W(ρ)
W(ρ)Θ(ρ−s, ζ)dρ
80 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
On he o he hand, by P oposi ion (2.5),
|B(s)|≥C|log |s||,∀s, |s|≥R
Then, o Re(s)on any compac subse o (0, 2), he unc ion |B(s)|is uni o mly bounded
om below by a posi i e cons an . I ollows, o |ζ| ≥δ0/ 2, and small,

∂˜
F
∂s (σ/ρ( ),ζ)e˜
F(σ/ρ( ),ζ)Γ(ζ) −ζ=
∂˜
F
∂s (σ/ρ( ),ζ)
Bσ
ρ( )
Bσ
ρ( )+ζΓ(ζ) −ζ
≤C(1 + |ζ|)ρ( )2 −4
|σ|2log |σ/ρ( )|+e−a|σ/ρ( )||log |σ/ρ( )||e−|π||ζ|
2e−(β1−α1)log
≤C(1 + |ζ|)ρ( )2 −4
ε2
0( −1+logε0)+e−a|σ/ρ( )|(log M+ −1)e−|π||ζ|
2e(β1−α1)|log |
≤C(1 + |ζ|)ρ( )2 −4+e−aε0/ρ( )(log M+ −1)e−|π|
4 2e(β1−α1)|log |e−|π||ζ|
4,
and
|J3|≤Cρ( )2 −4+e−aε0/ρ( )(log M+ −1)e−|π|
4 2ˆ
Re(ζ)=β1−α1
Im(ζ)≥δ0
2
(1 + |ζ|)e−|π||ζ|
4dζ
≤Cρ( )2 −4+e−aε0/ρ( )(log M+ −1)e−|π|
4 2
The e o e, uni o mly o |σ| ∈(ε0, M), log M∈(0, −θ),
lim
→0ρ( )−1|J3|=0.
P oceeding simila ly wi h ∂U/∂ , since o β∈(0, 2) such ha β−1 <c <β,
∂
∂ U( , s)= 1
2iπ ˆ
Re(ζ)=β−Re(s)
e˜
F(s,ζ) −ζ−1Γ(ζ+1)dζ (5.31)
i ollows,
∂
∂s ∂U
∂  , σ
ρ( )=1
2iπ ˆ
Re(ζ)=β1−α1
∂˜
F
∂s σ
ρ( ),ζe˜
Fσ
ρ( ),ζΓ(ζ+1) −ζ−1dζ,
om whe e (5.27)is deduced wi h he same a gumen s used o ob ain (5.25). 

M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 81
Lemma 5.5.
H , σ
ρ( )=− ρ( )
2σexp −2 log 
bσ
ρ( ).
H1 , σ
ρ( )=∂H
∂  , σ
ρ( )
P oo . The in eg al in (5.26)can be compu ed adding he esidues o he in eg and a
he poles ζ=−no he Gamma unc ion,
H , σ
ρ( )=ρ( )
2σlog |bσ/ρ( )
2|
∞

n=0
(−1)n n
n!nexp nlog 2log
bσ
ρ( )
=− ρ( )
2σexp −2 log 
bσ
ρ( ).
On he o he hand,
H1 , σ
ρ( )=ρ( )
2σlog |bσ/ρ( )
2|
∞

n=0
(−1)n n
n!(n+1)exp(n+1)log2log
bσ
ρ( )
=
exp −2 log bσ
ρ( )ρ( )
2σ−2log
bσ
ρ( )
exp −2 log bσ
ρ( )ρ( )
2σ
=∂H
∂  , σ
ρ( ).
P oposi ion 5.6.
M−1(H( ))(X)=−2
πΓ(−2 )sin(π )|X|2 sign(X).
P oo . I we call X=ρ( )Y,
M−1(H( ))(X)= 1
2iπ ˆ
Re(s)=α1
H( , s)e−sρ( )Yds
=1
2iπρ( )ˆ
Re(σ)=α1ρ( )
H , σ
ρ( )e−σY dσ
=
4iπ ˆ
Re(σ)=α1ρ( )
σ−1exp −2 log 
bσ
ρ( )e−σY dσ
we de o m he in eg a ion con ou o Re(σ) =0, and change a iables b → ,
82 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
4iπ ˆ
Re(σ)=0
σ−1e−2 logb
ρ( )e−i Y dσ =
4πˆ
R
−1e−2 log
ρ( )e−i Y
bd
Then, a e he change o a iables =ρ( )w, d =ρ( )dw,
4πˆ
R
−1exp −2 log 
ρ( )e−i Y
bd =
4πˆ
R
−1exp (−2 log |w|)e−iw ρ( )Y
bdw
=−2
πΓ(−2 )sin(π )|X|2 sign(X).
P oo o P oposi ion 3.11.We use (3.50) o w i e he le hand side o (3.65)as,
−1|X|1−2 ˜
Λ( , X)= −1|X|1−2 X−1X˜
Λ( , X)
=1
2iπ −1|X|1−2 X−1ˆ
R (s)=α1
∂U
∂s ( , s)e−sXds.
Fo X=ρ( )Y,
ˆ
R (s)=α1
∂U
∂s ( , s)e−sXds =1
2iπρ( )ˆ
Re(σ)=α1ρ( )
∂U
∂s  , σ
ρ( )e−σY dσ
(5.32)
ˆ
Re(σ)=α1ρ( )
∂U
∂s  , σ
ρ( )e−σY dσ =I1+I2+I3(5.33)
Ik=1
2iπ ˆ
Re(σ)=α1ρ( )
σ∈Dk
∂U
∂s  , σ
ρ( )e−σY dσ (5.34)
D1=Bε0(0),D
2=BM( )(0) Bε0(0),D
3=BM( )(0)c(5.35)
whe e log M( ) = −3/2. On D1and D3we use (2.33)o P oposi ion 2.10,

∂U
∂s  , σ
ρ( )≤CT e−2 log(|bσ/ρ( )|)1+
σ
ρ( )−1
≤C e−2 log |b |e2 log(ρ( )) 1+
σ
ρ( )−1
≤C ρ( )|σ|−2 −1,
om whe e,
|I1|≤C ρ( )ε0,|I3|≤Cρ( )M( )−2 .(5.36)
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 83
On D2,
I2=I2,1+I2,2
I2,1=1
2iπ ˆ
Re(σ)=α1ρ( )
σ∈D2
∂U
∂s  , σ
ρ( )−H , σ
ρ( )e−σY dσ
I2,2=1
2iπ ˆ
Re(σ)=α1ρ( )
σ∈D2
H , σ
ρ( )e−σY dσ
The fi s in eg al is es ima ed as
|I2,1|≤ 1
2iπ ˆ
Re(σ)=α1ρ( )
σ∈D2

∂U
∂s  , σ
ρ( )−H , σ
ρ( )|dσ|.
and by Lemma 5.4: lim
→0ρ( )−1|I2,1|=0.(5.37)
We w i e he second as

I2,2−1
2iπ ˆ
Re(σ)=α1ρ( )
H , σ
ρ( )e−σY dσ≤C
ˆ
Re(σ)=α1ρ( )
σ∈D1
H , σ
ρ( )e−σY dσ
+
+C
ˆ
Re(σ)=α1ρ( )
σ∈D3
H , σ
ρ( )e−σY dσ
(5.38)
and he exp ession o H( )gi es, by calcula ions simila o hose gi ing (5.36),

ˆ
Re(σ)=α1ρ( )
σ∈D1
H , σ
ρ( )e−σY dσ
≤C ρ( )ε0(5.39)

ˆ
Re(σ)=α1ρ( )
σ∈D3
H , σ
ρ( )e−σY dσ
≤C ρ( )M( )−2 (5.40)
I ollows om (5.33)and (5.36)–(5.40) ha o all ε0> he e exis s τsmall enough such
ha , o all ∈(0, τ)and all Y≥0,
84 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
−1ρ( )−1|I1|+|I3|+|I2,1|+I2,2−ρ( )(M−1(H( ))(ρ( )Y)≤
≤Cε0+ −1M( )−2 
and hen, uni o mly on Y∈R,
lim
→0 −1ρ( )−1|I1|+|I3|+|I2,1|+I2,2−ρ( )(M−1(H( ))(ρ( )Y)=0.(5.41)
The e o e, since o X=ρ( )Y
ˆ
Re(s)=α1
∂U
∂s ( , s)e−sXds =ρ( )−1(I1+I2+I3)
=ρ( )−1(I1+I3+I2,1+I2,2−ρ( )(M−1(H( ))(X))+(M−1(H( ))(X)
and,
−1X−1|X|1−2 ˆ
Re(s)=α1
∂U
∂s ( , s)e−sXds = −1X−1|X|1−2 ρ( )−1(I1+I3+I2,1+
+I2,2−ρ( )M−1(H( )(X)+ −1X−1|X|1−2 M−1(H( ))(X)
and by (5.41)we deduce,
lim
→0 −1X−1|X|1−2 ˆ
Re(s)=α1
∂U
∂s ( , s)e−sρ( )Yds = lim
→0|X|1−2 M−1(H( ))(X)
X =1
uni o mly o Xin bounded subse s o R. P ope y (3.60) ollows o sufficien ly small,
and hen o ∈(0, 1). The same a gumen s gi e (3.63)and hen (3.61). 
5.3. Linea iza ion: he equa ion (1.15)
When R(p, p1, p2) −R(p1, p, p2) −R(p2, p1, p)is w i en in e ms o he unc ion Ω
defined in (1.10)and only linea e ms in Ωa e kep , he esul is
n0(1 + n0)∂Ω( )
∂ =nc( )LI3(Ω( )) (5.42)
LI3(Ω( )) =
∞
ˆ
0
(U(k,k)Ω( , k)−V(k,k)Ω( , k)) k2dk,(5.43)
1
8nca2m−2U(k,k)=(mθ(k−k)
kk
×n0(ω(k))[1 + n0(ω(k))][1 + n0(ω(k)−ω(k))] + (k↔k))
M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390 85
−m
kkn0(ω(k)+ω(k))[1 + n0(ω(k))][1 + n0(ω(k))],(5.44)
1
8nca2m−2V(k,k)=mθ(k−k)
kkn0(ω(k))[1 + n0(ω(k))][1 + n0(ω(k)−ω(k))]
+mθ(k−k)
kkn0(ω(k))[1 + n0(ω(k))][1 + n0(ω(k)−ω(k))] (5.45)
whe e k=|p|and k=|p|. The unc ions U(k, k)and V(k, k)ha e a non in eg able
singula i y along he diagonal k=k. Howe e , hese singula i ies cancel each o he
when he wo e ms a e combined as in (5.43)as a as i is assumed ha , o all >0,
Ω( ) ∈Cα(0, ∞) o some α>0. Bu he in eg and (U(k, k)Ω( , k)−V(k,k)Ω( , k))
can no be spli as o he linea ized Bol zmann equa ions o classical pa icles ([7]).
Howe e , an explici calcula ion shows ha , o all k>0,
LI3(ω)(k)=
∞
ˆ
0U(k,k)k2−V(k,k)k2k2dk= 0 (5.46)
om whe e we deduce, o all k>0,
∞
ˆ
0U(k,k)k2
k2Ω( , k)−V(k,k)Ω( , k)k2dk=Ω( , k)
kLI3(ω)(k)=0.
We may hen w i e,
LI3(Ω( )) = nc( )
∞
ˆ
0
(U(k,k)Ω( , k)−V(k,k)Ω( , k)) k2dk
=nc( )
∞
ˆ
0
U(k,k)Ω( , k)
k2−Ω( , k)
k2k2k2dk
and he linea ized equa ion eads,
n0(1 + n0)∂Ω( )
∂ =nc( )
∞
ˆ
0
U(k,k)Ω( , k)
k2−Ω( , k)
k2k2k2dk.(5.47)
Use o he change o a iables (1.10)-(1.11)in (5.47) yields equa ion (1.12) o he unc-
ion u.

86 M. Escobedo / Jou nal o Func ional Analysis 282 (2022) 109390
5.4. F om (1.1) o (1.20)
I uis a egula unc ion, he igh hand side o he equa ion (1.1)may be w i en,
∞
ˆ
0
(u(y)−u(x))K(x, y)dy)=
∞
ˆ
0
y
ˆ
x
∂u
∂z(z)dzK(x, y)dy
=−
x
ˆ
0
∂u
∂z(z)
z
ˆ
0
K(x, y)dydz +
∞
ˆ
x
∂u
∂z(z)
∞
ˆ
z
K(x, y)dydz
=
∞
ˆ
0
∂u
∂z(z)Hx
zdz
z(5.48)
Hx
z=1z>x
∞
ˆ
z
K(x, y)dy −10<z<x
z
ˆ
0
K(x, y)dy (5.49)
whe e an explici in eg a ion o he wo in eg als in he igh hand side o (5.49)gi es
(1.21), and hen, he igh hand side o equa ion (1.20).
Acknowledgmen s
The esea ch o he au ho is suppo ed by g an s PID2020-112617GB-C21 o
MINECO and IT1247-19 o he Basque Go e nmen . The hospi ali y o IAM o he
Uni e si y o Bonn, and i s suppo h ough SFB 1060 a e g a e ully acknowledged. The
au ho hanks P . M. Valle a he Uni e sidad del País Vasco (UPV/EHU) o enligh -
ening discussions.
The au ho is also g a e ul o he e e ees o hei ca e ul eading o he manusc ip ,
hei commen s and sugges ions.
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