Analytic potential theory over the p -adics

Shai Haran

Annales de l'institut Fourier (1993)

  • Volume: 43, Issue: 4, page 905-944
  • ISSN: 0373-0956

Abstract

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Over a non-archimedean local field the absolute value, raised to any positive power α > 0 , is a negative definite function and generates (the analogue of) the symmetric stable process. For α ∈ ( 0 , 1 ) , this process is transient with potential operator given by M. Riesz’ kernel. We develop this potential theory purely analytically and in an explicit manner, obtaining special features afforded by the non-archimedean setting ; e.g. Harnack’s inequality becomes an equality.

How to cite

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Haran, Shai. "Analytic potential theory over the $p$-adics." Annales de l'institut Fourier 43.4 (1993): 905-944. <http://eudml.org/doc/75030>.

@article{Haran1993,
abstract = {Over a non-archimedean local field the absolute value, raised to any positive power $\alpha &gt;0$, is a negative definite function and generates (the analogue of) the symmetric stable process. For $\alpha \in (0,1)$, this process is transient with potential operator given by M. Riesz’ kernel. We develop this potential theory purely analytically and in an explicit manner, obtaining special features afforded by the non-archimedean setting ; e.g. Harnack’s inequality becomes an equality.},
author = {Haran, Shai},
journal = {Annales de l'institut Fourier},
keywords = {analytic potential theory; -homogeneous distributions; -capacity; -superharmonic functions; -harmonic functions; Riesz potentials; descent; dichotomy; regularization; finite energy measures; equilibrium measure; balayage; Green measure; Keldish transform; uniqueness principle; Dirichlet problems; Riesz representation; domination; harmonic minorant; convexity; local field},
language = {eng},
number = {4},
pages = {905-944},
publisher = {Association des Annales de l'Institut Fourier},
title = {Analytic potential theory over the $p$-adics},
url = {http://eudml.org/doc/75030},
volume = {43},
year = {1993},
}

TY - JOUR
AU - Haran, Shai
TI - Analytic potential theory over the $p$-adics
JO - Annales de l'institut Fourier
PY - 1993
PB - Association des Annales de l'Institut Fourier
VL - 43
IS - 4
SP - 905
EP - 944
AB - Over a non-archimedean local field the absolute value, raised to any positive power $\alpha &gt;0$, is a negative definite function and generates (the analogue of) the symmetric stable process. For $\alpha \in (0,1)$, this process is transient with potential operator given by M. Riesz’ kernel. We develop this potential theory purely analytically and in an explicit manner, obtaining special features afforded by the non-archimedean setting ; e.g. Harnack’s inequality becomes an equality.
LA - eng
KW - analytic potential theory; -homogeneous distributions; -capacity; -superharmonic functions; -harmonic functions; Riesz potentials; descent; dichotomy; regularization; finite energy measures; equilibrium measure; balayage; Green measure; Keldish transform; uniqueness principle; Dirichlet problems; Riesz representation; domination; harmonic minorant; convexity; local field
UR - http://eudml.org/doc/75030
ER -

References

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  1. [1] C. BERG, G. FORST, Potential theory on locally compact abelian groups, Ergebn. d. Math., 87, Springer-Verlag (1975). Zbl0308.31001MR58 #1204
  2. [2] J. BLIEDTNER, W. HANSEN, Potential theory, Universitext, Springer-Verlag (1986). Zbl0706.31001
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  5. [5] S. HARAN, Riesz potentials and explicit sums in arithmetic, Inventiones Math., 101 (1990), 697-703. Zbl0788.11055MR91g:11132
  6. [6] S. HARAN, Index theory, potential theory and the Riemann hypothesis, in Proc. LMS Symp. on L-functions and Arithmetic, Durham, 1989. Zbl0744.11042
  7. [7] G.A. HUNT, Markoff processes and potentials I-III, Illinois J. Math., 1 (1957), 44-93 and 316-369 ; 2 (1958), 151-213. Zbl0100.13804
  8. [8] N.S. LANDKOF, Foundations of modern potential theory, Grundl. d. math. Wiss., 180, Springer-Verlag (1972). Zbl0253.31001MR50 #2520
  9. [9] S.C. PORT, C.J. STONE, Infinitely divisible processes and their potential theory I-II, Ann. Inst. Fourier, 21-2 (1971), 157-275 ; 21-4 (1971), 179-265. Zbl0195.47601MR49 #11640
  10. [10] M. TAIBLESON, Fourier analysis over local fields, Princeton Univ. Press, 1975. Zbl0319.42011MR58 #6943
  11. [11] A. WEIL, Fonction zêta et distributions, séminaire Bourbaki (1966), 312. Zbl0226.12008

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