×

On \(L^p\)-solutions of the Laplace equation and zeros of holomorphic functions. (English) Zbl 0893.31003

The authors consider the following problem for a given smooth domain \(\Omega \subset \mathbb{R}^n\) and \(p\in [1,\infty]\): To characterize those positive measures \(\mu\) on \(\Omega\) for which there exists a subharmonic function \(u\in L^p(\Omega)\) with \(\Delta u= \mu\). For \(\Omega\) either bounded or the whole \(\mathbb{R}^n\), \(n>2\), a nice answer is given in terms of the maximal fractional function \[ \mu^*_\lambda (x)=\sup_{\mu <\lambda \delta(x)} {\mu\bigl(B(x,t)\bigr) \over t^{n-2}}, \] where \(0<\lambda<1\), \(\delta(x)= \text{dist} (x,\partial \Omega)\) (in case of \(\Omega =\mathbb{R}^n\), \(\delta(x)= +\infty)\), and \(B(x,t)= \{y:| y-x| <t\}\):
\(\exists u\in L^p(\Omega)\), \(1\leq p<\infty\): \(\Delta u=\mu \Leftrightarrow \mu^*_\lambda \in L^p (\Omega)\) for some \(\lambda\in (0,1)\). A characterization for \(p=\infty\) is given, too.
For \(\Omega= \mathbb{D}\), the unit disc in \(\mathbb{C}\), a linear solution operator \(u=K [\mu]\) is found for \(1\leq p<\infty\). It is related with Toeplitz operators, Carleson measures for the Bergman spaces, and the bilaplacian.
The results are applied to zeros of holomorphic functions in \(\mathbb{D}\).

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
PDF BibTeX XML Cite
Full Text: Numdam EuDML

References:

[1] M. Andersson , Solution formulas for the \partial \partial -equation and weighted Nevanlinna classes in the polydisc , Bull. Soc. Math. France 109 ( 1985 ), 135 - 154 . Zbl 0598.32007 · Zbl 0598.32007
[2] E. Beller , Zeros of AP functions and related classes of analytic functions , Israel J. Math. 22 ( 1975 ) 68 - 80 . MR 385103 | Zbl 0322.30028 · Zbl 0322.30028
[3] E. Beller , Factorization for non-Nevanlinna classes of analytic functions , Israel J. Math. 27 ( 1977 ), 320 - 330 . MR 442234 | Zbl 0361.30012 · Zbl 0361.30012
[4] S. Bell , A duality theorem for harmonic functions , Michigan Math. J . 29 ( 1982 ), 123 - 128 . Article | MR 646379 | Zbl 0482.31004 · Zbl 0482.31004
[5] B. Berndtsson , \partial b and Carleson type inequalities , Lecture Notes in Math. 1277 , Springer-Verlag , pp. 42 - 55 . · Zbl 0627.32014
[6] E. Beller - C. Horowitz , Zero sets and random zero sets in certain function spaces, J . Analyse Math. 64 ( 1994 ), 203 - 217 . MR 1303512 | Zbl 0819.30003 · Zbl 0819.30003
[7] G. Bomash , A Blaschke-type product and random zero sets for Bergman spaces , Ark. Mat. 30 ( 1992 ), 45 - 60 . MR 1171094 | Zbl 0764.30029 · Zbl 0764.30029
[8] R.R. Coiffman - Y. Meyer - E.M. Stein , Some new function spaces and their application to harmonic analysis , J. Funct. Anal. 62 ( 1985 ), 304 - 335 . MR 791851 | Zbl 0569.42016 · Zbl 0569.42016
[9] C. Cochran , Random Blaschke products , Trans. Amer. Math. Soc. 322 ( 1990 ), 731 - 755 . MR 1022163 | Zbl 0709.30031 · Zbl 0709.30031
[10] S.A. Dautov - G.M. Henkin , Zeros of holomorphic functions of finite order and weighted estimates for solutions of \partial -problem , Math. Sbornik 107 ( 1978 ), 163 - 174 . Zbl 0392.32001 · Zbl 0392.32001
[11] H. Federer , Geometric measure theory , Springer-Verlag , New York, N.Y. , 1969 . MR 257325 | Zbl 0176.00801 · Zbl 0176.00801
[12] J.B. Garnett , Bounded analytic functions , Academic Press , New York, N.Y , 1981 . MR 628971 | Zbl 0469.30024 · Zbl 0469.30024
[13] P.R. Garabedian , Partial differential equations, second edition , Chelsea Publishing Company , New York, N.Y. , 1986 . MR 943117 · Zbl 0607.76124
[14] A. Heilper , The zeros offunctions in Nevanlinna’s area class , Israel J. Math . 34 ( 1979 ), 1 - 11 . MR 571391 | Zbl 0498.30040 · Zbl 0498.30040
[15] W.K. Hayman - P.B. Kennedy , Subharmonic functions , Academic Press , London , 1976 . Zbl 0419.31001 · Zbl 0419.31001
[16] C. Horowitz , Some conditions on Bergman spaces zero sets , J. Analyse Math. 62 ( 1994 ), 323 - 348 . MR 1269212 | Zbl 0795.30006 · Zbl 0795.30006
[17] E. Leblanc , A probabilistic zero condition for the Bergman space , Michigan Math. J . 37 ( 1990 ), 427 - 436 . Article | MR 1077326 | Zbl 0717.30008 · Zbl 0717.30008
[18] E. Ligocka , On the reproducing kernel for harmonic functions and the space of Bloch harmonic functions on the unit ball of Rn , Studia Math . 87 ( 1987 ), 23 - 32 . MR 924758 | Zbl 0658.31006 · Zbl 0658.31006
[19] D. Luecking , Multipliers of Bergman spaces into Lebesgue spaces , Proc. Edinburgh Math. Soc . ( 2 ) 29 ( 1986 ), 125 - 131 . MR 829188 | Zbl 0587.30048 · Zbl 0587.30048
[20] D. Luecking , Trace ideal criteria for Toeplitz operators , J. Funct. Anal. 73 ( 1987 ), 345 - 368 . MR 899655 | Zbl 0618.47018 · Zbl 0618.47018
[21] D. Luecking , Zero sequences for Bergman spaces , to appear in Complex Variables . MR 1413164 | Zbl 0871.30004 · Zbl 0871.30004
[22] X. Massaneda , Random sequences with prescribed radii in the unit ball , Complex Variables 31 ( 1996 ), 193 - 211 . MR 1423251 | Zbl 0865.41007 · Zbl 0865.41007
[23] B. Muckenhoupt - R.L. Wheeden , Weighted norm inequalities for fractional integrals , Trans. Amer. Math. Soc . 192 ( 1974 ), 261 - 274 . MR 340523 | Zbl 0289.26010 · Zbl 0289.26010
[24] J. Ortega-Cerdà , Zero sets of holomorphic functions in the bidisc , to appear in Ark. Mat . MR 1611181 | Zbl 0911.32003 · Zbl 0911.32003
[25] D. Pascuas , Zeros i interpolació en espais de funcions holomorfes del disc unitat , PhD. Thesis, Universitat Autònoma de Barcelona (March, 1988 ).
[26] W. Rudin , Function theory in the unit ball of Cn , Springer-Verlag , New York , 1980 . MR 601594 | Zbl 0495.32001 · Zbl 0495.32001
[27] R. Rudowicz , Random interpolating sequences with probability one , Bull. London Math. Soc . 26 ( 1994 ), 160 - 164 . MR 1272302 | Zbl 0831.30020 · Zbl 0831.30020
[28] H. Shapiro , Quasi-balayage and apriori inequalities , in preparation.
[29] E.M. Stein , Singular integrals and differentiability properties of functions , Princeton University Press , Princeton, New Jersey , 1970 . MR 290095 | Zbl 0207.13501 · Zbl 0207.13501
[30] G. Sweers , Positivity for a strongly coupled elliptic system , J. Geom. Anal. 4 ( 1994 ), 121 - 142 . MR 1274141 | Zbl 0792.35048 · Zbl 0792.35048
[31] K. Zhu , Operator theory in the function spaces , Marcel-Dekker , New York , 1990 . MR 1074007 | Zbl 0706.47019 · Zbl 0706.47019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.