A Bohr Phenomenon for Elliptic Equations

A Bohr Phenomenon

for Elliptic Equations

Lev Aizenb erg

Department of Mathematics and Computer Science

BarIlan University

RamatGan

Israel

Nikolai Tarkhanov

Institut fur Mathematik

Universitat Potsdam

Postfach

Potsdam

Germany

August

L Aizenb erg and N Tarkhanov

Abstract

In Bohr proved that there is an r such that if a p ower

series converges in the unit disk and its sum has mo dulus less than

then for jz j r the sum of absolute values of its terms is again less

than Recently analogous results were obtained for functions of several

variables The aim of this pap er is to comprehend the theorem of Bohr

in the context of solutions to second order elliptic equations meeting

the maximum principle

Bohr Phenomenon

Contents

Preliminaries

Spaces with repro ducing kernels

Harmonic functions

Separately harmonic functions

Pluriharmonic functions

Polyharmonic functions

Solutions of elliptic equations

References

L Aizenb erg and N Tarkhanov

Preliminaries

It was in the spirit of function theory of the b eginning of this century that

Bohr Boh published the following theorem

Theorem There exists r with the property that if a power

series c z converges in the unit disk and its sum has modulus less than

jc z j for al l jz j r then

We dont knowany motivation of Bohrs result but very classical sub jects

and co ecient estimates much more precise than the Cauchy inequalities Per

haps this called on such mathematicians as M Riesz I Shur and F Wiener

who put Theorem in nal form by showing that one can take r and

this constant cannot b e improved

We call the b est constant r in Theorem ie the Bohr radiusIf

regarded as a homothety co ecient this concept extends easily to domains

in C Let us review some of the recent generalisations of Bohrs theorem to

functions of several complex variables

Given a complete Reinhardt domain D in C we denote by RD the

c z largest nonnegativenumber r with the prop ertythatifapower series

jc z j in converges in D and its sum has mo dulus less than then

the homothety r D Here the sums are over all multiindices

of nonnegativeintegers z z z is the tuple of complex variables and

z z z In BKh the following result is proved in case D is the unit

p olydisk

n n

C jz j jng U fz

Theorem For n one has

log n

p p

n n

We see from Theorem that RU whenn If D is the

hyp ercone C fz C jz j jz j g the situation is quite dierent

cf Aiz

Theorem The fol lowing estimates are true

Moreover if z C then there exists a series converging in C c z

P P

and such that j j is valid there but j fails at the point c z jc z

Bohr Phenomenon

n p p

For further estimates of RD in domains fz C jz j jz j g

p we refer the reader to Boa for other generalisations of Bohrs

theorem cf DR

Wenow discuss yet another natural multidimensional analogue of Bohrs

theorem Let r D stand for the largest r suchthatifapower se

ries c z converges in D and its sum has mo dulus less than then

sup jc z j r D b eing the homothetyof D The following result

r D

is contained in Aiz

Theorem For any boundedcomplete Reinhardt domain D in C the

inequality holds

It is worth p ointing out that this constant is near to the b est one for the

hyp ercone C

For holomorphic functions with p ositive real part there is another result

equivalent to Bohrs theorem cf AADb

Theorem If a function f z c z has positive real part in the

jc z j f for al l jz j and the unit disk and f then

constant cannot be improved

The coincidence of the constants in Theorems and is not acciden

tal as is shown in AADb These constants also coincide for holomorphic

functions on complex manifolds

In AADa the existence of a Bohr phenomenon is proved in HolM the

space of holomorphic functions on a complex manifold M

Theorem If f is a basis in HolM satisfying

al l the functions f vanish at a point z M

then there exist a neighbourhood U of z and a compact set K M such that

for al l f HolM with f c f

sup jc f jsup jf j

U K

In other words in this case we meet a Bohr phenomenon in M with Bohr

radius r M the more so with Bohr radius RM

Though the pro ofs make essen tial use of the algebra structure of holomor

phic functions the underlying fact seems to b e nothing but the maximum

L Aizenb erg and N Tarkhanov

principle More preciselywe need a substitute of Harnacks inequalityeg

Eva which reads that given a continuous function f in a domain

n k

DR with values in R such that f x for some x D the supremum

of jf j over any set D is dominated by the supremum of f itself over D

This amounts to saying that for every set D there is a constant c

such that

sup jf f x jc sup f f x

the inequality b eing uniform in f b elonging to an appropriate function class

and x

In this pap er we show that for bases in nuclear function spaces the esti

mate always implies a Bohr phenomenon

The pap er is organised as follows In Section we briey discuss Bohrs

phenomenon in Hilb ert spaces with repro ducing kernels the functions ob eying

Many function classes of mathematical physics meet among them are

harmonic separately harmonic and pluriharmonic functions In Sections

we will b e concerned with evaluating the Bohr radii for these classes of

functions The imp ortantpoint to note here is explicit formulas for the Bohr

radii for harmonic and separately harmonic functions

Polyharmonic functions fail to full the maximum principle in Section it

is shown that no Bohr phenomenon exists for p ositive p olyharmonic functions

Finally in Section weprove that for second order elliptic equations the

estimate reduces to classical Harnacks inequalitythus implying a Bohr

phenomenon

Spaces with repro ducing kernels

It seems that an eectivewayofproving Bohrs phenomenon in concrete cases

is trough establishing some co ecient estimates for the relevant classes of func

tions In this section we show co ecient estimates for expansions in Hilb ert

spaces with repro ducing kernels

Let D b e a relatively compact domain with C b oundary in R and A a

hyp o elliptic op erator in D This means that any distribution f in D satisfying

Af is actually a C function

We assume moreover that for each compact set K D there is a constant

h that csuc

sup jf jc kf k

L D

D satisfying Af in D for all f C

Denote by H the Hilb ert space obtained by completing the space of all

f C D satisfying Af in D with resp ect to the norm f kf k

L D

Bohr Phenomenon

In this waywe obtain what is known as a Hardy space of solutions to Af

in D

From we deduce that H is a Hilb ert space with repro ducing kernel

Namely x an orthonormal basis

in H all the elements b eing smo oth up to the b oundary Then given any

x D the series

y K x y f x f

converges in the norm of L D where f y f y Moreover it converges

uniformly in x on compact subsets of D so that K x y is actually a C

function in the pro duct DD This function is called the Szego kernel of D

relativeto A

has the following natural prop er Wenow supp ose that the basis f

ties

f is a constant

jj jj n

the constant C b eing jf xjCa jx x j for all x D and Z

indep endentof x and

c f is a solution to Af in D satisfying Lemma If f

f B on the boundary then

jc j s D sup jf j B f x

for al l where s D is the surfaceareaof D

Pro of Pick Z dierent from zero Letting ds denote the area

elementof D wehave

f f ds c

Since f is an orthogonal system we get

f ds

whence

jc j f B f ds

sup jf j B f ds

c sup jf j s D B

L Aizenb erg and N Tarkhanov

the second inequality b eing due to the fact that B f on D Taking into

accountthat

c f x

f f

s D f x

we arrived at the desired inequality

Co ecient estimates of the typ e are of indep endentinterest esp ecially

for the expansion of solutions to homogeneous elliptic equations with constant

co ecients cf Tar

Theorem Thereisanr with the property that if f c f

full ls jf j in D then jc f j in the bal l B x r

Pro of Indeed we can assume without loss of generalitythat f x

for if not we replace f by f Then

X X

jc f xj f x jc jjf xj

f x s D jf xj sup jf j f x

the last inequality b eing a consequence of Lemma Wenowinvoke Prop

erty of the basis f to obtain

X X

jj jj

f xj sup jf j C a distx D jx x j j

C a distx D jx x j

a distx D jx x j

If x x then the righthand side tends monotonically to zero Hence

there is an rsuch that it is less than s D when jx x j r In fact

wehave

a distx D

C s D

Bohr Phenomenon

Thus for x B x r we get

jc f xj f x f x

as desired

Lo osely sp eaking the theorem says that Bohrs phenomenon extends to

bases in Hardy spaces of solutions to elliptic equations of order The

question arises on evaluating the Bohr radius for some concrete bases We

discuss this problem in the next sections

Harmonic functions

To justify our formulation of Bohrs phenomenon for harmonic functions we

need an analogue of Theorem In the context of realvalued functions it is

fairly simple

Let b e a vector space of b ounded realvalued functions in a domain

DR satisfying some relations there and let Supp ose f is

a basis in suchthatf and all the functions f vanish at

t x D a p oin

Lemma Given any neighbourhood U D of x the fol lowing asser

tions areequivalent

P P

jc f j in U c f and jf j in D then If f

P P

jc f j f x in U c f and f in D then If f

Pro of Supp ose holds and f c f is a p ositive b ounded func

tion in D Cho ose a constant k with the prop erty that kfinD

and set F kf Then F satises jF j inD Since

F kc kc f

it follows bythat

j kc j jkc f j

L Aizenb erg and N Tarkhanov

in U Hence

X X

j kc j jc f j jkc f jjkc jj kc j

kc j kc j

in U as desired

Conversely let b e valid and let f c f satisfy jf j inD

We can certainly assume that c for if not we replace f by f Set

F f then F fullls F inD As

F c c f

we deduce by that

j c j jc f j c

in U Hence it follows that

X X

jc f j j c j jc f jjc jj c j

c c c

in U which completes the pro of

n n

Let B fx R jxj g b e the unit ball in R On the sphere B we

have an orthonormal basis of spherical harmonics Y where j j the

degree of Y is equal to k and given any k Z the index j varies from

to n k cf SW For example

n k n k

n k

k k

if k

Any spherical harmonics Y is the restriction to B of a homogeneous

harmonic p olynomial Y xin R

Y xjxj Y

kj kj

jxj

Bohr Phenomenon

These p olynomials form a basis in HarB the space of harmonic functions in

B Each function f HarB expands as a series

X X

f x c Y x

kj kj

in B whichconverges uniformly on compact subsets of B

By a Bohr radius for harmonic functions in B we will mean the largest

number r with the prop ertythatiff HarB expands as and

jf j inB then

X X

jc Y xj

kj kj

jxj rgWe denote this radius by R B in the ball B fx R

Har r

Theorem As denedabove R B is equal to the root of the equa

Har

tion

lying in the interval

Pro of According to Lemma R B is equal to the largest num

Har

ber r suchthatiff is a p ositive b ounded harmonic function in B with

expansion then

X X

jc Y xj c Y

kj kj

for all x B

Note that

where ds is the area form of B and the area of the unit sphere in R

Hence

and so implies

Further wehave

f x Y x ds c lim

kj r kj

B r

L Aizenb erg and N Tarkhanov

where ds is the area elementof B Using the mean value theorem for

r r

harmonic functions weget

nk nk

X X

jc j lim f x Y x ds

kj kj r

j j

Z Z

lim f x ds f xY x ds

r kj r

B B

r r

Z Z

lim f x ds lim f x jY xj ds

r kj r

r r

B B

r r

Z Z

f x r Z ds f x ds lim lim

k r r

r r

B B

r r

Z lim f x ds

k r

Z c

k n

Z b eing surface zonal harmonics cf SW

For further estimates we need an expansion of the Poisson kernel x y

by the zonal harmonics

Y Y Z

kj kj k

where j j andj j In this notation wehave Z Z for all

k k

k Z Moreover

jr j

r Z

cf SW Thus

nk nk

X X X X

jc Y r j r jc jjY j

kj kj kj kj

j j

k k

Bohr Phenomenon

v v

u u

nk nk

u u

X X X

t t

jc j jY j r

kj kj

j j

c r Z

n k

p r c

n n n

c r

Hence it follows that R B is not less than the ro ot of the equation

Har

lying in the interval It is a simple matter to check that has a

unique ro ot in

ConverselythePoisson kernel

jxj

jx j

p p

n n gives us an extremal function Indeed it is where

harmonic and p ositiveinB After the homothety x x the

function x is b ounded in B and Moreover

X X

x Y x Y

kj kj

X X

Y Y x

kj kj

ie in this case c Y Setting x r yields

kj kj

nk nk

X X X X

j Y jjY r j Y r

kj kj kj

j j

k k

Hence we conclude that if r is the ro ot of in the interval then

the inequality fails To nish the pro of it remains to pass to the limit

when

L Aizenb erg and N Tarkhanov

Note that if n then R B which agrees with the classical

Har

Bohr radius Furthermore

R B if n

Har

R B if n

Har

One can still write down R B in radicals for n however the

Har

formula is cumb ersome We just mention an asymptotic formula of R B

Har

when n namely

log

R B O

Har

n n

Separately harmonic functions

By a separately harmonic function in a domain DC is meantany function

f harmonic in every variable z x ix j n ie fz z

j j nj j j

in D

Let D U b e the unit p olydisk Each function f separately harmonic in

n n

U can b e represented in any smaller p olydisk rU r bythemultiple

Poisson integral where the integration is over the ndimensional skeleton of

n n

Expanding every Poisson rU ie s fz C jz j rjng

r j

kernel as a p ower series in z andz we obtain easily a basis in the space of

j j

functions separately harmonic in U endowed with the top ology of uniform

convergence on compact subsets of U This is w where I varies over

all ntuples consisting of there are such tuples and varies over all

multiindicesin Z Write I i i then w w w where

n I I In

w z ifi and w z ifi Under this notation we

Ij j j Ij j j

have w w w and any function f z z separately harmonic in U

I In

expands as a series

X X

f z zc c w

where

f z zw c lim

By a Bohr radius for functions separately harmonic in U we mean the

largest number r with the prop ertythatiff expands as and jf j

in U then

X X

jc j jc w j

n n

We denote this radius by R U in the p olydisk rU SHar

Bohr Phenomenon

Theorem

R U

SHar

Pro of By Lemma R U is equal to the largest number r

SHar

suchthatiff is a p ositive b ounded separately harmonic function in U with

expansion then

X X

c jc w j c

for all z rU

If f z z thenwe get using the mean value theorem for the skeleton

j f z z jw jc j lim

lim r f z z

for all and I Furthermore if z rU then

X X X X

c jc w j c c r

I I

c r

whence

R U

SHar

ConverselythePoisson kernel in the complex variable z can b e written in

the form

jz j

j z j

the second p ointbeing Given any xed we consider the

function

jz j

f z z

j z j

L Aizenb erg and N Tarkhanov

which is separately harmonic p ositive and b ounded in U For this function

wehave

for all and I Ifz s then

X X

jc w j c

Letting we deduce that

R U

SHar

which completes the pro of

Note that R U which agrees with the harmonic Bohr radius of

SHar

It is easy to show an asymptotic formula for R U whenn

SHar

namely

log

R U O

SHar

n n

cf One must takeinto account that the real dimension of U is n

hence the asymptotic formulas for the harmonic Bohr radius of the ball B in

n n

R and the separately harmonic Bohr radius of the p olydisk U coincide

Let D b e a complete Reinhardt domain in C We dene the Bohr radius

R D to b e the largest number r suchthatiff z z expands as

SHar

and jf z zj for all z D then is fullled in the homothety r D

Since D is the union of p olydisks we arrive at the following consequence of

Theorem

Corollary For any complete Reinhardt domain DC itfollows

that

R D

SHar

Theorem and gain in interest if we realise that the corresp onding

Bohr radii in these theorems are computed explicitly Till now mere estimates

of various Bohr radii were known in all cases except for the classical Bohr

theorem and the equivalent assertion for holomorphic functions with p ositive real part in the unit disk

Bohr Phenomenon

The pro ofs of these theorems give more namely explicit inequalities for

p ositive harmonic functions Since Z n k cf SW the pro of

k n

of Theorem contains an estimate of the ro ot mean square of the co ecients

in the expansion over the basis by homogeneous harmonic p olynomials of the

same degree

jc j

n k

for all k Z On the other hand the pro of of Theorem makes use of

the inequality jc jc for each and I It is the inequalities that are

the key to ol of evaluating the Bohr radii for various function classes They are

analogues of the classical Caratheo dory inequality Car which states that if

f z expands as a series c z in the unit disk and f z f

then jc jc for all

Pluriharmonic functions

By pluriharmonic functions are meant real parts of holomorphic functions of

several complex variables The coincidence of R U with the classical Bohr

Har

radius is not accidental Let f b e a basis in the space of holo

morphic functions in a domain DC with the natural top ology of uniform

convergence on compact subsets of D We assume that f and all the other

functions f vanish at a xed p oint z DThen f f form a basis

in the space of pluriharmonic functions in D Anysuch function expands as a

series

F z C C f C f

which converges uniformly on compact sets in D the co ecients C and C

b eing conjugate to each other

Theorem Given any neighbourhood U D of z the fol lowing asser

tions areequivalent

If F is a positive pluriharmonic function in D with expansion then

C jC f j F z in U

c f is a holomorphic function with positive real part in D If f

jc f j f z in U and f z then

Pro of The pro of is straightforward Each pluriharmonic function F has

the form f f where f is holomorphic Under the ab ove assumption we

L Aizenb erg and N Tarkhanov

have

F f

whence the desired conclusion follows

Theorem implies that all estimates of Bohr radii for holomorphic func

tions of BKh Aiz Aiz AADa AADb Boa DR dis

cussed in Section remain valid for Bohr radii of pluriharmonic functions in

the same domains

Polyharmonic functions

Recall that bya polyharmonic function in a domain DR is meantany

function f satisfying f in D where N For such functions there

is no Bohr phenomenon Indeed consider the function family c jxj

in the unit ball B parametrised by a p ositive constant c All the functions

are p olyharmonic and p ositive Moreover and jxj enter into a simplest

basis in the space of p olyharmonic functions in B Were the Bohr phenomenon

available one had c jxj in a ball rB for each c This amounts

to saying that

jxj

for all jxj rwhichcontradicts to the fact that the righthandsideisin

nitesimal when c

Solutions of elliptic equations

The preceding section shows that the Bohr phenomenon do es not extend to

solutions of elliptic equations of order larger than Moreover it is not valid

even for second order elliptic equations that do not meet the maximum prin

ciple Indeed c jz j c is a family of p ositive solutions to the elliptic

equation z f in the unit disk U C the solutions are known as

bianalytic functions Since and jz j enter into a simplest basis in the space

of bianalytic functions in U wemay rep eat the same reasoning as that in

Section

Thus we restrict ourselves to second order elliptic equation Af in a

domain DR whose solutions verify the maximum principle As is known

Bohr Phenomenon

eg Eva A should b e of the form

n n

X X

ax a x A a x

i ij

x x x

i j i

i ij

the co ecients b eing C realvalued functions in D satisfying a a for

ij ji

each i j nand a inD To ensure that SolD we actually

require a inD

Wecontinue to write SolD for the space of all functions f satisfying

Af inD By Weils lemma any reasonable solution of Af is

actually a C function in D Moreover any reasonable top ology in the

space SolD coincides with that induced bytheemb edding into C D cf

Tar

For a set K D weput

kf k sup jf xj

where f SolD The system of seminorms kf k K D denes the

top ology of uniform convergence on compact subsets of D When regarded

with this top ologySolD isanuclear Frechet space eg MV

A sequence f of functions in SolD issaidtobeabasis in this

space if for each solution f SolD there is a unique sequence of numb ers c

c f where the series converges uniformly on any compact such that f

subset of D

Lemma If f isabasis in the space SolD then for each

U D there exist K D and C such that given any f c f

in SolD we have

jc jkf k C kf k

U K

Pro of This result is a straightforward consequence of the theorem on ab

soluteness of bases in nuclear spaces proved in DM cf also Theorem

in MV

Certainly the generic situation is when U K which can b e assumed

without loss of generality

Lemma For each point x D and set D thereisaconstant

c such that whenever f SolD and f x we have

sup jf jc sup f

L Aizenb erg and N Tarkhanov

Pro of We can assume by enlarging if necessary that is connected and

contains x Cho ose a domain U D suchthat U Supp ose f SolD

vanishes at x Set

inf f m

sup f M

and

inf f m

sup f M

so that all the m M and m M are nonnegative Then

U U

m f x M for all x

m f x M for all x U

U U

Consider the function

ux M f x

in U It is a simple matter to see that u is a nonnegative solution to Au

in U and

inf u M M

sup u M m

According to Harnacks inequality eg Eva there exists a con

stant c dep ending only on and the co ecients of Asuchthat

M m c M M

U U

Hence

sup jf j M m

M m

c M M

c sup f

as desired

As we will shortly learn Lemmas and already imply a Bohr phe

Af While the rst of the two extends easily nomenon for solutions of

to solutions of general hyp o elliptic equations the second one is a ne result

on the geometry of graph hyp ersurfaces of solutions to second order elliptic

Bohr Phenomenon

equations Indeed an equivalentformulation of Lemma is that for eachset

D there is a constant c dep ending only on such that whenever

f SolD wehave

f x inf f c sup f f x

for all x To see this it is sucient to apply Lemma to the functions

f x f x Setting t c yields

f x t inf f t sup f

SolD The inequality for all x where t is indep endentof f

sp ecies the geometry of the hyp ersurface y f xinR through the

rigid constant t

Example Take D B R R and B x For the

family of harmonic functions

f x log x a x

in D a R the inequality reduces to

a R

a a

for all a R

Having disp osed of these preliminary steps wecannow extend Theorem

to solutions of Af

Theorem If f is a basis in SolD satisfying

al l the functions f vanish at a point x D

then there exist a neighbourhood U of x and a compact set K D such that

c f for any f SolD with f

sup jc f j sup jf j

U K

L Aizenb erg and N Tarkhanov

Pro of According to the GrothendieckPietsch criterion for nuclearityeg

Theorem of MV for every set U D there is a set D such

that

kf k

Set B fx R jx x j rgforr Since f x for all

wehave kf k whenr Thus from it follows that

kf k

sup

kf k

as r corresp onding to B

On the other hand Lemma implies that for any D there are

D and C with the prop erty that given any f c f in

SolD wehave

jc jkf k C kf k

Fix r suchthat B D cho ose D such that holds

and nd D so that holds Finallycho ose a compact set K D

whose interior contains By Lemma there exists a constant c with

the prop ertythat

kf f x k c sup f f x

for all f SolD

Nowsuppose f SolD Wemay assume without loss of generalitythat

c f x since otherwise wemultiply f by Then it follows from

and that

jc jkf k C kf f x k

Cc sup f f x

By wecancho ose rr such that

kf k

sup

kf k Cc

then

X X

kf k

sup jc jkf k jc f j f x

kf k

Bohr Phenomenon

kf k

f x Cc sup f f x sup

kf k

sup f

sup jf j

Hence we obtain the statementwith U B

In AADa Remark it is shown that if the space SolD has a basis

then it has also a basis satisfying the hyp otheses of Theorem While the

pro of is given for holomorphic functions the same arguments still work for the

space SolD

L Aizenb erg and N Tarkhanov

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