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 P series c z converges in the unit disk and its sum has modulus less than P 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 n in C Let us review some of the recent generalisations of Bohrs theorem to functions of several complex variables n Given a complete Reinhardt domain D in C we denote by RD the P c z largest nonnegativenumber r with the prop ertythatifapower series P 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 n of nonnegativeintegers z z z is the tuple of complex variables and n n z z z In BKh the following result is proved in case D is the unit n p olydisk n n C jz j jng U fz j Theorem For n one has p log n n p p RU n n n We see from Theorem that RU whenn If D is the n hyp ercone C fz C jz j jz j g the situation is quite dierent n cf Aiz Theorem The fol lowing estimates are true p RC e P 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 z Bohr Phenomenon n p p For further estimates of RD in domains fz C jz j jz j g n 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 P ries c z converges in D and its sum has mo dulus less than then P sup jc z j r D b eing the homothetyof D The following result r D is contained in Aiz n Theorem For any boundedcomplete Reinhardt domain D in C the inequality holds r n rD 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 P Theorem If a function f z c z has positive real part in the P 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 f 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 P for al l f HolM with f c f X 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 D 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 n 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 K 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 f n Z in H all the elements b eing smo oth up to the b oundary Then given any x D the series X y K x y f x f n Z 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 n Z 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 P c f is a solution to Af in D satisfying Lemma If f n Z f B on the boundary then jc j s D sup jf j B f x D for al l where s D is the surfaceareaof D n Pro of Pick Z dierent from zero Letting ds denote the area elementof D wehave Z f f ds c D Since f is an orthogonal system we get n Z Z f ds D whence Z jc j f B f ds D Z sup jf j B f ds D D c sup jf j s D B f D 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 P Theorem Thereisanr with the property that if f c f n Z P full ls jf j in D then jc f j in the bal l B x r n Z 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 n Z X f x s D jf xj sup jf j f x D the last inequality b eing a consequence of Lemma Wenowinvoke Prop erty of the basis f to obtain n Z X X jj jj jj f xj sup jf j C a distx D jx x j j D X jj A C a distx D jx x j n Z n C 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 q r a distx D n C s D Bohr Phenomenon Thus for x B x r we get X jc f xj f x f x n Z 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 n DR satisfying some relations there and let Supp ose f is a basis in suchthatf and all the functions f