On L^p-solutions to the Laplace equation and zeros of holomorphic functions
Abstract
The problem we solve in this paper is to characterize, in a smooth domain $\Omega$ in $\Bbb R^n$ and for $1\le p\le\infty$, those positive Borel measures on $\Omega$ for which there exists a subharmonic function $u\in L^p(\Omega)$ such that $\Delta u=\mu$. The motivation for this question is mainly for $n=2$, in which case it is related with problems about distributions of zeros of holomorphic functions: If $\{a_n\}^{\infty}_{n=1}$ is a sequence in $\Omega\subset\Bbb C$ with no accumulation points in a simply connected domain $\Omega$, and $\mu=2\pi\sum_n\delta_{a_n}$, then all solutions $u$ of $\Delta u=\mu$ are of the form $u=\log |f|$, with $f$ holomorphic vanishing exactly on the poits $a_n$. Thus our results give the characterization of the zero sequences of holomorphic functions with $\log |f|\in L^p(\Omega)$. A related class had been considered by Beller.
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ON L p -SOLUTIONS OF THE LAPLACE EQUATION AND ZEROS OF HOLOMORPHIC FUNCTIONS Joaquim Bruna and Joaquim Ortega-Cerd a 1. Introduction The problem we solve in this pap er is to characterize, in a smo oth domain in R n and for 1 p 1 , those p ositive Borel measures on for which there exists a subharmonic function u 2 L p () such that u = . The motivation for this question is mainly for n =2, in whichcase it is related with problems ab out distributions of zeros of holomorphic functions: if f a n g 1 n =1 is a sequence in C with no accumulation points in a simply connected domain , and = 2 X n a n , then all solutions u of u = are of the form u = log j f j ,with f holomorphic vanishing exactly on the points a n .Thus our results give the characterization of the zero sequences of holomorphic functions with log j f j2 L p (). A related class had been considered by Beller (Bel1]). In section 2 we consider rst = R n . In this case, it is easily seen lo oking at the behavior at 1 that the problem only makes sense for n 3and p > n= ( n ; 2), and that it is equivalent to the description of the measures on R n having a Newtonian potential in L p ( R n ). This was achieved byMuckenhoupt and Wheeden, and we givean alternative pro of as well. The results and the metho ds of pro ofs serve as a guideline for the b ounded case. In section 3 we solve the problem for a b ounded smo oth domain in R n .Among other characterizations we nd that u = has a solution in L p () if and only if the fractional maximal function ( x )= sup r ( x ) r 2 ; n ( B ( x r )) belongs to L p (). Here ( x )is the distance from x to @ and is a xed constant, 0 < < 1. The case p = 1 was previously known (Hei], DH]) in dimension 2, and amounts to say that Z ( x ) 2 d ( x ) < + 1 : Supp orted by DGICYT grant PB92-0804-C02-02 and by CIRIT grantGRQ94-2014. 1
2 J. BRUNA AND J. ORTEGA-CERD A We also consider at the end the case p = 1 . In section 4 we lo ok closely to the case = D , the unit disc in C . We nd here a linear solution op erator u = K ] of u = with u 2 L p (), and relate the problem with others, suchasTo eplitz op erators, Carleson measures for Bergman spaces, and the bilaplacian. In section 5 we particularize to the motivating question, the zeros of holomorphic functions. It turns out that the condition characterizing the zero sequences f a n g of functions with log j f j2 L p ( D ) dep ends b oth on the radii and the angular disp osition of the points a n , as in many other classes of functions. We nd adyadic expression of it and also we nd the best condition on the sequence of radii r n = j a n j . A probabilistic version is also considered. As a nal remark, we point out that we have b ound ourselves to the situation describ ed |subharmonic functions and p ositive measures| in view of the applications we had in mind, zeros of holomorphic functions. It is quite probable that our results extend to measures or distributions, and also replacing the laplacian by a general strongly elliptic op erator, but we have not tried to do so. Another generalization, replacing the Laplace equation u = by the equation i@ @u = in C n and with applications to zero varieties of holomorphic functions of several variables, app ears in Ort]. 2. Subharmonic functions in L p ( R n ) In this section we shall characterize the laplacians of subharmonic functions in L p ( R n ). For general facts on subharmonic functions that we will use, see for instance Hay]. First note that if h is harmonic in L p ( R n ), the mean-value prop erty implies that h 0. Hence all subharmonic functions u in L p ( R n ) are completely determined by its laplacian = u , a p ositive Borel measure in R n . The sub-mean value prop erty implies that any such u is non-p ositive, hence we can consider in each ball B (0 R )the Green decomp osition u ( x )= h R ( x )+ Z j y j R G R ( x y ) d ( y ) j x j <R: Here h R is the least harmonic ma jorant of u in B (0 R ) and G R ( x y ) is the Green function for the ball B (0 R ), given by G R ( x y )= 8 > > < > > : c n j x ; y j 2 ; n ; R n ; 2 j y j 2 ; n j x ; y 0 j 2 ; n n> 2 c 2 log j x ; y 0 jj y j R j x ; y j n =2 with y 0 = yR 2 j y j ; 2 . Clearly u ( x ) h R ( x ) h R 0 ( x ) 0 if R R 0 .The limit h ( x ) = lim R !1 h R ( x ), the least harmonic ma jorant of u in R n , is then in L p ( R n ), hence
ON L p SOLUTIONS TO THE LAPLACE EQUATION 3 is identically zero. Thus we conclude that u ( x )= lim R !1 Z j y j R G R ( x y ) d ( y ) : It is immediate to check that for n> 2, lim R !1 G R ( x y )= c n j x ; y j 2 ; n , while this limit is innite for n =2. By the monotone convergence theorem, this means that there are no non-zero subharmonic functions in L p ( R 2 ) and that for n> 2 they are exactly those Newtonian p otentials u ( x )= c n Z R n j x ; y j 2 ; n d ( y ) b elonging to L p ( R n ). Now note that unless is identically zero, j u ( x ) j dominates j x j 2 ; n for large j x j and so p must b e greater than n= ( n ; 2). In conclusion, wehave seen that the question only makes sense for n> 2 and p>n= ( n ; 2) and that it is equivalent to the description of the p ositive measures for which the Riesz p otential of order 2 is in L p ( R n ). We use the notation T ( x )= Z R n j x ; y j ; n d ( y ) for the Riesz potential of order , 0 < < n . We consider the maximal fractional function of , dened by ( x )=sup t> 0 ( B ( x t )) j B ( x t ) j 1 ; n = sup t> 0 t ; n ( B ( x t )) : Note that is pointwise dominated by T .Next theorem was proved by Muckenhoupt and Wheeden in MW] for absolutely continuous measures. We will give an alternative pro of for it contains some concepts that will be used later. Theorem 2.1. Let be a positive measure in R n , n> 2 , 1 <p< 1 and such that 0 <<n . Then T 2 L p ( R n ) if and only if 2 L p ( R n ) . Proof. Write T in the form T ( x )=( n ; ) Z 1 0 t ; n ; 1 ( B ( x t )) dt: Consider in R n +1 + , with co ordinates ( x t ), x 2 R n , t> 0, the pro duct measure d ~ ( x t )= d ( x ) t ; 1 dt . Then we can write T ( x )=( n ; ) Z ;( x ) t ; n d ~ ( y t ) def = A (~ )( x )
4 J. BRUNA AND J. ORTEGA-CERD A where ;( x )is the cone f ( y t ) : j x ; y j < t g .By CMS, x 6], A (~ ) b elongs to L p ( R n ), 1 <p , simultaneously with the function C (~ )( x ) = sup ~ ( Q ( B )) j B j x 2 B : Here Q ( B ) denotes the Carleson window over the ball B .(We p oint out that this denition of A , C do es not coincide with that of CMS]). In our case ~ ( Q ( B )) ' ( B ) j B j =n , whence C (~ )( x ) ( x ). 3. Subharmonic functions in L p () In this section we will characterize, for asmo oth b ounded domain in R n , n 2, the p ositive (Borel) measures on such that the Poisson equation = has a (subharmonic) solution u 2 L p (), 1 p 1 . 3.1. For x 2 , we denote by ( x ) the distance from x to the b oundary of . For a p ositive measure on and a xed , 0 < < 1, we consider lo cal versions of the fractional maximal function and Newtonian potentials in the previous section. These are ( x )= sup r ( x ) ( B ( x r )) r n ; 2 and T ( x )= Z B ( x ( x )) j x ; y j 2 ; n d ( y ) B ( x r ) b eing the ball of radius r centered at x . Theorem 3.1. The fol lowing are equivalent for a positive measure on and 1 p< 1 : (a) There is a subharmonic function u 2 L p () such that u = . (b) T 2 L p () for some 0 << 1 . (c) 2 L p () for some 0 << 1 . In case p =1 ,an equivalent condition is (d) R ( x ) 2 d ( x ) < + 1 . In particular, conditions (b), (c) do not dep end on . The theorem should be compared with Theorem 2.1. 3.2. It is trivial that (b) implies (c), b ecause T . An application of Fubini's theorem shows that up to constants the integral in (d) is the L 1 -norm of T ,using that ( x ) ' ( y ) whenever j x ; y j ( x ). Obviously ( x ) & ( B ( x ( x )) ( x ) 2 ; n : and so 2 L 1 () implies again byFubini's theorem that (d) holds. Therefore (b), (c) and (d) are all equivalent for p =1. We will now prove that 2 L p (), 1 <p< 1 implies T 2 L p () for some . The rst step is alo cal version of Theorem 2.1.
ON L p SOLUTIONS TO THE LAPLACE EQUATION 5 Lemma 3.2. For a positive measure on B (0 8 " ) and 1 <p< 1 (1) Z B (0 " ) Z j x ; y j " j x ; y j 2 ; n d ( y ) ! p dA ( x ) C p Z B (0 3 " ) sup r 5 " ( B ( x r )) r n ; 2 p dA ( x ) : Proof. We use the same methodasinTheorem 2.1. The inner integral of the left-hand side equals ( n ; 2) Z " 0 t 1 ; n ( B ( x t )) dt: We consider in the half-space R n +1 + with co ordinates ( x t ) the pro duct measure d ~ ( x t )= 1I B (0 2 " ) ( x ) d ( x ) 1I (0 " ) ( t ) tdt , so that the previous integral equals Z ;( x ) t ; n d ~ ( x t )= A (~ )( x ) x 2 B (0 " ) : By the result of CMS] already used in Theorem 2.1, the left-hand side of (1) is b ounded by a constant times the L p -norm of C (~ )in R n . Now, for j x j 3 " C (~ )( x ) ' sup r 5 " ( B ( x r )) r n ; 2 hence the integral of C (~ ) p over j x j 3 " is already b ounded by the right hand side of (1). For j x j > 3 " , C (~ )( x ) . ( B (0 2 " )) " 2 j x j n and hence Z j x j 3 " ( C (~ )( x )) p dA ( x ) . " n ; p ( n ; 2) ( B (0 2 " )) p whichis also b ounded by the right hand side. Assume < 1 = 8 and consider covered by the family of balls f B ( x ( x )) x 2 g . By Besicovitch covering lemma (see Fed 2.8.14] for instance), there is a nite set of families B 1 ::: B N of balls in the covering whose total union is also and such that each family consists of disjoint balls. Then, writing B x for B ( x ( x )), B 0 x for B ( x 3 ( x )), Z T p ( x ) dA ( x ) N X i =1 X B x 2 B i Z B x T p ( x ) dA ( x ) which by Lemma 3.2 is dominated by N X i =1 X B x 2 B i Z B 0 x p 5 ( x ) dA ( x ) :
6 J. BRUNA AND J. ORTEGA-CERD A We claim now that, for each family B i , the balls B 0 x cover each p oint of at most a nite numb er (dep ending only on n ) of times. Indeed, if y 2 B 0 x then m ; 1 ( y ) ( x ) m ( y ) for some constant m = m ( ), whence the balls B x 2 B i , such that y 2 B 0 x have volume ' ( y ) n ,are disjoint and are all of them in the ball B ( y (3 + m ) ( y )), and the claim follows. It is then clear that the last expression is b ounded by C ( p ) Z p 5 ( x ) dA ( x ) : Therefore, 2 L p () implies T 2 L p () for small enough. 3.3. We will prove now that (a) implies (b). For r < ( x ) consider the Green decomp osition of u in B ( x r ) 1 n r n ; 1 Z j x ; y j = r u ( y ) d ( y ) ; u ( x )= Z j x ; y j <r G r ( x y ) d ( y ) : Here n is the surface measure of the unit sphere and G r ( x y )is Green's function for the ball B ( x r ). Multiply the ab ove by nr n ; 1 , integrate between 0 and R ( x ), and divide by R n . The result is, for n 3 n n R n Z j x ; y j R u ( y ) dA ( y ) ; u ( x )= = 1 ( n ; 2) n Z j x ; y j R j x ; y j 2 ; n 1 ; j x ; y j n R n 1 ; n 2 ; n 2 j x ; y j n ; 2 R n ; 2 d ( y ) and for n =2 1 R 2 Z j x ; y j R u ( y ) dA ( y ) ; u ( x )= = 1 2 Z j x ; y j R log R j x ; y j + 1 2 j x ; y j 2 R 2 ; 1 d ( y ) : The functions 1+ ; n 2 ; 1 x n ; n 2 x n ; 2 and log 1 x + 1 2 ( x 2 ; 1) are positivedecreasing functions in (0 1). Therefore for each < 1 there is c ( )such that 1 j B ( x R ) j Z B ( xR ) u ( y ) dA ( y ) ; u ( x ) c ( ) Z j x ; y j R j x ; y j 2 ; n d ( y ) : Writing ( x )= R with R = p ( x ), it follows that for (a) ) (b) it is enough to prove that for a xed < 1, u 2 L p () implies that the average of u over B x = B ( x ( x )) belongs to L p () as well. To see this we use Holder's inequality Z 1 ( x ) n Z B x u ( y ) dA ( y ) p dA ( x ) Z 1 ( x ) n Z B x j u ( y ) j p dA ( y ) dA ( x ) : If y 2 B x , then ( x ) ' ( y ) and Fubini theorem nishes the pro of that (a) implies (b).
ON L p SOLUTIONS TO THE LAPLACE EQUATION 7 3.4. Next weprove that (d) implies (a) in case p =1. More precisely wewanttoshow that whenever is a p ositive measure such that Z ( x ) 2 d ( x )= M < + 1 then there is u 2 L 1 () such that u = and k u k 1 CM , where C is a constant dep ending only on . The dual statement of this is contained in next theorem Theorem 3.3. For any test function in , j ( x ) j C ( x ) 2 k k 1 x 2 : These are indeed equivalent statements. Assume Theorem 3.3 holds and let be as ab ove. Dene a linear functional as follows () = Z ( x ) d ( x ) is well dened, and by Theorem 3.3 j () j CM k k 1 : By Hahn-Banach theorem, extends to a continuous functional on the space C c () of continuous functions with compact supp ort, with norm CM . By Riesz's representation theorem, we conclude that there is a measure on , with total variation k k CM such that () = Z ( x ) d ( x ) : This means that = in the weak sense, and then must be in fact absolutely continuous d = udx , with k u k 1 CM and u = . It is immediate to check that this argument can be reversed. Proof of Theorem 3.3. Let G ( x y ) denote the fundamental solution for in R n , G ( x y )= c n j x ; y j 2 ; n if n 3, G ( x y )= c 2 log j x ; y j in R 2 , so that ( x )= Z G ( x y )( y ) dA ( y ) : Since j ( x ) j k k 1 Z j G ( x y ) j dA ( y ) C k k 1 it is enough to prove the inequality for points x close to @ . Let U be a tubular neighborhood of @ , that is, there is awell-dened pro jection p : U ! @ of class C 1 , p ( x )is the unique closest p oint to x , and the line between x and p ( x )is orthogonal to @ at p ( x ). Let z =2 p ( x ) ; x be the symmetric point of x with resp ect @ .
8 J. BRUNA AND J. ORTEGA-CERD A The idea is that in the integral representation of ab ove, one can subtract to G ( x y ) any harmonic function h in b ecause these annihilate ( y ), by Green's theorem. Fixed x , we take as h ( y ) the second-order Taylor p olynomial of G ( y ) at z , evaluated at x , h ( y )= G ( z y )+2 ( x ) @ @n G ( z y )+ 2 ( x ) 2 @ 2 @n 2 G ( z y ) : Here n is the unit direction from z to x .Thus we have j ( x ) jk k 1 Z j G ( x y ) ; h ( y ) j dA ( y ) If j x ; y j 3 ( x ), by Taylor's formula, j G ( x y ) ; h ( y ) j . ( x ) 3 j x ; y j n +1 and hence the contribution of n B ( x 3 ( x )) in the last integral is dominated by ( x ) 2 . If j x ; y j 3 ( x ), we write j G ( x y ) ; h ( y ) j j G ( x y ) ; G ( z y ) j +3 ( x ) jr z G ( z y ) j +5 ( x ) 2 jr 2 z G ( z y ) j : This is b ounded by c j x ; y j 2 ; n if n> 2, and by c +log 3 ( x ) j x ; y j if n =2. In b oth cases the integral over B ( x 3 ( x )) is estimated by ( x ) 2 . The same pro of will go through if for any p oint x close to the b oundary,we can nd apoint z outside the op en set , such that the distance of z to the b oundary of is comparable to j z ; x j .This holds, for instance, if satises a uniform exterior cone condition. H. Shapiro Sha], has let us known that it is possible to get the same result under very mild regularity assumptions on the b oundary of . 3.5. To complete the pro of of Theorem 3.1, we nally prove that (b) implies (a) when p > 1. Let be a test function supp orted in j y j < 1, equal to one on j y j < 1 = 2, 0 1 with 0 < < 1 = 3, 0 < < small enough to be dened later, we dene an \approximate solution" by K ( )( x )= Z R n x ; y ( x ) K ( x y ) d ( y ) : Here K ( x y ) is the fundamental solution for in R n , K ( x y )= G ( x y )= c n j x ; y j 2 ; n if n> 2 if n =2 we need to mo dify it by a function H ( x y ) K ( x y )= G ( x y ) ; H ( x y )= c 2 log ( x ; y ) ; H ( x y ) where H ( x y )is harmonic in x , equal to G ( x y 0 )for y close to @ , y 0 being the symmetric of y with resp ect to @ , as in the previous paragraph. Another choice, in all
ON L p SOLUTIONS TO THE LAPLACE EQUATION 9 dimensions, is K ( x y ) = G ( x y ) the Green's function for , if is smo oth enough. In any event, K ( )= + R ( ) with R ( )( x )= Z R n 2 r x x ; y ( x ) r x K ( x y )+ x x ; y ( x ) K ( x y ) d ( y ) : We claim now that under the assumption (b), both K ( )and 2 R ( )are in L p (). This is clear if n> 2, b ecause both are pointwise estimated by T .If n =2, b oth are pointwise estimated by ( B ( x ( x ))) + Z B ( x ( x )) log ( x ) j x ; y j d ( y ) and so it is enough to prove that the function dened by this last integral, call it ( x ), is in L p (). By Holder's inequality and Fubini's theorem Z ( x ) p dA ( x ) Z Z E log ( x ) j x ; y j p ( B ( x ( x ))) p ; 1 dA ( x ) d ( y ) : The inner integral is over the set E = f x : y 2 B ( x ( x )) g .With " = 2 1 ; one has B ( x ( x )) B ( y " ( y )) if x 2 E ,and a computation in p olar co ordinates as in the previous section gives Z B ( y" ( y )) log " ( y ) j x ; y j p dA ( x ) ' ( y ) 2 : Therefore Z ( x ) p dA ( x ) . Z ( B ( y " ( y ))) p ; 1 ( y ) 2 d ( y ) : Now we can just rep eat the same argument: this last integral is comparable to Z ( B ( y " ( y ))) p ; 1 ( Z B ( y " ( y )) dA ( x ) ) d ( y )= = Z Z F ( B ( y " ( y )) p ; 1 d ( y ) dA ( x ) the set F b eing now f y : x 2 B ( y " ( y )) g .Cho osing " suchthat 2 " 1 ; " = , B ( y " ( y )) B ( x ( x )) for y 2 F and hence the ab ove is b ounded by Z ( B ( x ( x ))) p dA ( x )= Z T ( x ) p dA ( x ) < + 1 : It is then enough to prove that given a function f (= K ( )) with 2 f 2 L p () there is u 2 L p () such that u = f in the distribution sense, with Z j u j p dA M Z 2 p j f j p dA the constant M dep ending only on . In the same way as b efore it is enough to prove the dual statement that follows:
16 J. BRUNA AND J. ORTEGA-CERD A part of a subharmonic function is controlled by the p ositive part in L 1 .By the same reason log + j f j can be replaced by j log j f jj in the denition of W p ( D ). The connections with the problem u = is well known and classic: if Z = f a n g is a sequence with no accumulation p oints in D and = 2 X n a n ,the fact that an holomorphic function f vanishes exactly at Z is equivalent to say that log j f j = . 5.1. For a sequence Z = f a j g in D with no accumulation p oints in D , we call n jk the number of p oints of Z in the dyadic region T kj ,and N k = 2 k X j =1 n jk the numb er of p oints of Z in the annulus 1 ; 2 ; k < j z j 1 ; 2 ; k ; 1 ,counting multiplicities. Theorem 5.1. Z is the zero sequence of a function f 2A p ( D ) if and only if (6) 1 X k =0 2 ; 2 k 2 k X j =1 n p kj < + 1 and it is the zero sequence of f 2W p ( D ) if and only if (7) 1 X k =0 2 ; ( p +1) k N p k < + 1 : Proof. The rst part is the restatement of Theorem 3.1 in subsection 4.4. We prove now the necessity of condition (7). Let f 2 W p ( D ) and assume without loss of generality that f (0) = 1. Set = 2 X n a n , N ( r )= fj z j r g . By Jensen's formula, for r close to 1 N ( r )(1 ; r ) ' Z r 0 log r j j d ( )= Z 2 0 log j f ( re i ) j d and hence, Z 1 0 N ( r ) p (1 ; r ) p dr < + 1 whichis easily seen to be equivalent to (7). For the su#ciency, we will prove that the solution u = K ] of u = , considered in the previous section, satises Z 1 0 Z 2 0 u + ( re it ) dt p dr < + 1 :
ON L p SOLUTIONS TO THE LAPLACE EQUATION 17 Note that the estimate (5) of the kernel K ( z )of K implies that K ]is a subharmonic function in L 1 lo c ( D ) whenever satises Z D (1 ;j j ) 2 d ( ) < + 1 or X k 2 ; 2 k N k < + 1 and that (7) implies this (as corresp onds to the inclusion W p ( D ) W 1 ( D ) = A 1 ( D )). Moreover (4) implies u + ( z ) . Z D (1 ;j j 2 ) 2 j 1 ; z j 2 d ( ) b ecause log s +1 ; s 0 for s 1. Then Z 2 0 u + ( re it ) dt . Z D (1 ;j j 2 ) 2 (1 ; r j j ) d ( ) : The right-hand side is comparable to Z j j r (1 ;j j ) d ( )+ 1 1 ; r Z r< j j < 1 (1 ;j j ) 2 d ( ) : In terms of the distribution function N ( r ), we nally obtain Z 2 0 u + ( re it ) dt . Z r 0 N ( t ) dt + N ( r )(1 ; r )+ 1 1 ; r Z 1 r N ( t )(1 ; t ) dt and the result follows using Hardy's and Holder's inequalities. The problem of describing the zero sequences of B p ( D ) remains op en (Bel1], Bel2], BH]). This is a harder problem than the ones we deal with, mainly b ecause it is a nonlinear problem. Note the di$erentcharacter of conditions (6) and (7). While (7) only dep ends on the radii of the p oints of Z , (6) also dep ends on their angular distribution. We can see directly that (6) is stronger than (7), because by Jensen's inequality for convex functions, N p k (2 ; k ) p ; 1 =2 k 0 @ 2 ; k 2 k X j =1 n kj 1 A p 2 k X j =1 n p kj : The only way to have equality ab ove is that all n kj are equal, that is, the p oints in Z are uniformly distributed in the angle. Only in this case conditions (6), (7) are the same. In particular, (7) is the necessary and su#cient condition on a sequence of radii r n for the existence of f 2A p ( D ) (or f 2B p ( D )) with zeros a n , r n = j a n j .Next we will make this even more precise.
18 J. BRUNA AND J. ORTEGA-CERD A 5.2. Let f r n g be a xed increasing sequence of radii. Let n ( w )be a sequence of indep endent random variables, all uniformly distributed in 0 2 ]. We consider the random sequence f a n g = Z ( w )with a n ( w ) = r n e i n ( w ) and ask ab out the probability that Z ( w )be a zero sequence of A p ( D ) or B p ( D ). Questions of this typ e have been considered in other classes of functions in Co c], Bom], Leb], Rud], Hor], Mas] and BH]. Theorem 5.2. If the sequence f r n g satises (7), then Z ( w ) is the sequenceofzeros of a function f 2A p ( D ) with probability 1. Proof. It is enough to prove that the random variable X ( w )= 1 X k =0 2 k X j =1 n kj ( w ) p 2 ; 2 k has nite mean, by Theorem 5.1. Here n kj is the random variable that counts the number of points a n = r n e i n ( w ) in T kj ,which has binomial law B ( N k 2 ; k ) for each j =1 ::: 2 k . Then E ( n p kj ) . p X l =1 (2 ; k N k ) l and E ( X ) . 1 X k =0 2 k X j =1 p X l =1 (2 ; k ) l +1 N l k ' ' p X l =1 Z 1 0 N ( r ) l (1 ; r ) l dr . Z 1 0 N ( r ) p (1 ; r ) p dr < + 1 : 5.3. Finally wemake some remarks ab out prop erties of the zero sequences of functions in A p ( D ) and B p ( D ). It is immediate from the characterization in Theorem 5.1 that those of A p ( D ) are hyperbolical ly stable ,in the sense that whenever f a n g is such a sequence, and a n ; b n 1 ; a n b n < 0 < < 1 then f b n g is also a zero sequence of A p ( D ). It is also clear that a subsequence of a zero sequence of A p ( D ) is also a zero sequence of A p ( D ). Even though the characterization of those of B p ( D ) is unknown, this later prop erty can still b e proved using an idea from Lue3]. Theorem 5.3. Every subsequence of a zero sequence of B p ( D ) is also a zero sequence of B p ( D ) . Proof. We will show that if is a positive Borel measure on D for which there is u , u + 2 L p ( D ), with u = , and 0 , then there is v suchthat v =
ON L p SOLUTIONS TO THE LAPLACE EQUATION 19 and v + 2 L p ( D ). Recall from subsection 4.1 the decomp osition v = P v ]+ K v ], in particular v (0) = 1 Z D v ( ) dA ( )+ 1 2 Z D f log j j 2 +1 ;j j 2 g v ( ) dA ( ) : Fixed z 2 D , we apply this to v = u ' z , where ' z is the automorphism of D permuting z with 0, and change variables in the integrals. The result is u ( z )= Q u ]( z )+ 1 2 Z D ( log ; z 1 ; z 2 + (1 ;j j 2 )(1 ;j z j 2 ) j 1 ; z j 2 ) d ( ) with Q u ]( z )= 1 Z D 1 ;j z j 2 j 1 ; z j 2 2 u ( ) dA ( ) : Subtracting from u = P u ]+ K u ], and using (4), wesee that Q u ]( z )= P u ]( z )+ Z D j z j 2 1 ;j j 2 j 1 ; z j 2 d ( ) : We takeas solution of v = the function v = K ]+ P u ]. From (4), v ( z ) . Z D j z j 2 1 ;j j 2 j 1 ; z j 2 d ( )+ P u ] and therefore v ( z ) . Q u ]( z ). Then v + ( z ) . Z D 1 ;j z j 2 j 1 ; z j 2 2 u + ( ) dA ( ) : But Schur's lemma can be applied as well to the kernel ab ove, and since u + 2 L p ( D ) we obtain that v + 2 L p ( D ), as required. References And] M. Andersson, Solution formulas for the @ @ -equation and weighted Nevanlinna classes in the polydisc , Bull. So c. Math. France 109 (1985), 135{154. Bel1] E. Beller, Zeros of A p functions and related classes of analytic functions , Israel J. Math. 22 (1975), 68{80. Bel2] E. Beller, Factorization for non-Nevanlinna classes of analytic functions , Israel J. Math. 27 (1977), 320{330. Bell] S. Bell, A duality theorem for harmonic functions ,Michigan Math. J. 29 (1982), 123{128. Ber] B. Berndtsson, @ b and Carleson type inequalities , Lecture Notes in Math. 1276 ,42{55. BH] E. Beller and C. Horowitz, Zero sets and random zero sets in certain function spaces ,J. Analyse Math. 64 (1994), 203{217.
20 J. BRUNA AND J. ORTEGA-CERD A Bom] G. Bomash, A Blaschke-typeproduct and random zero sets for Bergman spaces , Ark. Mat. 30 (1992), 45{60. CMS] R. R. Coiman, Y. Meyer and E. M. Stein, Some new function spaces and their application to harmonic analysis ,J. Funct. Anal. 62 (1985), 304{335. Co c] C. Co chran, Random Blaschke products ,Trans. Amer. Math. So c. 322 (1990), 731{755. DH] S. A. Dautov and G. M. Henkin, Zeros of holomorphic functions of nite order and weighted estimates for solutions of @ -problem , Math. Sbornik 107 (1978), 163{174. Fed] H. Federer, Geometric measuretheory , Springer-Verlag, New York, N.Y., 1969. Gar] J. B. Garnett, Bounded analytic functions , Academic Press, New York, N.Y:, 1981. Gara] P. R. Garab edian, Partial dierential equations , second edition, Chelsea Publishing Company, New York, N.Y., 1986. Hei] A. Heilp er, The zeros of functions in Nevanlinna's area class , Israel J. Math. 34 (1979), 1{11. HK] W. K. Hayman and P. B. Kennedy, Subharmonic functions , Academic Press, London, 1976. Hor] C. Horowitz, Some conditions on Bergman spaces zero sets ,J. Analyse Math. 62 (1994), 323{348. Leb] E. Leblanc, Aprobabilistic zerocondition for the Bergman space ,Michigan Math. J. 37 (1990), 427{436. Lig] E. Ligo cka, On the reproducing kernel for harmonic functions and the spaceof Bloch harmonic functions on the unit bal l of R n , Studia Math. 87 (1987), 23{32. Lue1] D. Luecking, Multipliers of Bergman spaces into Lebesgue spaces , Pro d. Edinburgh Math. So c. (2) 29 (1986), 125{131. Lue2] D. Luecking, Trace ideal criteria for Toeplitz operators ,J. Funct. Anal. 73 (1987), 345{368. Lue3] D. Luecking, Zerosequences for Bergman spaces ,To app ear in Complex Variables. Mas] X. Massaneda, Random sequences with prescribedradii in the unit bal l , Preprint (1995). MW] B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals , Trans. Amer. Math. So c. 192 (1974), 261{274. Ort] J. Ortega-Cerda, Zero sets of holomorphic functions in the bidisc , Preprint(1995). Pas] D. Pascuas, Zeros i interpolacio en espais de funcions holomorfes del disc unitat , PhD. Thesis, Universitat Autonoma de Barcelona (March, 1988). Rudin] W. Rudin, Function theory in the unit bal l of C n , Springer-Verlag, New York, 1980. Rud] R. Rudowicz, Random interpolating sequences with probability one , Bull. London Math. So c. 26 (1994), 160{164. Sha] H. Shapiro, Quasi-balayage and apriori inequalities , In preparation. Ste] E. M. Stein, Singular integrals and dierentiability properties of functions , Princeton University Press, Princeton, New Jersey, 1970. Swe] G. Sweers, Positivity for a strongly coupled el liptic system , J. Geom. Anal. 4 (1994), 121{142. Zhu] K. Zhu, Operator theory in the function spaces , Marcel-Dekker, New York, 1990. J. Bruna: Dept. Matem atiques, UniversitatAut onoma de Barcelona, 08193 Bellaterra (Barcelona), Spain E-mail address : [email protected] J. Ortega-Cerda: Dept. Matema tica Aplicada I, Universitat Polit ecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain E-mail address : [email protected]