##
**Analytic potential theory over the \(p\)-adics.**
*(English)*
Zbl 0847.31006

In these notes we develop the potential theory of the \(p\)-adic analogue of the symmetric stable distributions. We do this purely analytically and in an explicit manner. In \(\S 1\) we recall Weil’s formulation of (the local part of) Tate’s thesis in terms of \(\alpha\)-homogeneous distributions, and relate these in \(\S 2\) to probability. We prove that the function \(|x |^\alpha\) is negative definite over \(\mathbb{Q}_p\) for \(\alpha \in (0, \infty)\), generating a semigroup of probability measures \(\mu^\alpha_t\), explicitly given by
\[
\mu^\alpha_t (dx) = \sum_{n \geq 1} {(-t)^n \over n!} {\zeta_p (1 + n \alpha) \over \zeta_p (-n \alpha)} |x |^{-(1 + n \alpha)} dx
\]
(this formula being an analogue of a formula of Feller for the \(\alpha\)-symmetric stable distribution over the reals \(\mathbb{R})\). For \(\alpha \in (0,1)\), \(\mu^\alpha_t\) is transient, and its potential is the Riesz potential
\[
k^\alpha_t (x) = \int^\infty_0 \mu^\alpha_t (x)dt = {\zeta_p(1 - \alpha) \over \zeta_p (\alpha)} |x |^{\alpha - 1}.
\]
When we approach the boundary \( \alpha \to \infty\) (in analogy with the real case \(\alpha \to 2)\) we obtain the “normal law”, a very simple process which degenerates and “lives” on \(\mathbb{Q}_p/ \mathbb{Z}_p\). In \(\S 3\) we recall the analytic properties of the Riesz potentials, their distributional meromorphic continuation, and the Riesz reproduction formula.

In \(\S 4\) we begin to develop the potential theory of \(R ^\alpha_p\), \(\alpha \in (0,1)\). In view of \(\S 2\) we can identify our potentials with probabilistic potentials and deduce the various potential theoretic principles in one blow; we prefer, nevertheless, to develop these principles purely analytically, and to grasp things along the way in a most explicit manner. We note, for example, that the Harnack inequalities become in our \(p\)-adic situation trivial equalities. We prove the principles of descent, dichotomy, maximum, regularization, and uniqueness. We follow mostly the analogous real case as developed in N. S. Landkof, Foundations of modern potential theory, Springer (1972; Zbl 0253.31001), the \(p\)-adic setting offering many simplifications.

In \(\S 5\) we consider the finite energy measures leading to the proof that they form a complete positive cone in the Hilbert space of finite energy distributions. In \(\S 6\) we prove the existence and various characterizations of the equilibrium measure, and show that the \(\alpha\)-capacity satisfies the usual properties and, moreover, is given explicitly via the \(\alpha\)-diameter: \[ \text{cap}_\alpha (K) = {\zeta_p (\alpha) \over \zeta_p (1 - \alpha)} \lim_{N \to \infty} \biggl[ \min_{x_1, \dots, x_N \in K} {1 \over n(n - 1)} \sum_{i < j} |x_i - x_j |^{\alpha - 1}\biggr]^{-1}. \] In \(\S 7\), we approach balayage and the Green measure using the Keldysh transform; i.e. instead of using Cartan’s method of projections in the Hilbert space of finite energy distributions (which equally works well), we use the more geometrical situation of the analysis of the \(\text{PGL}_2 (\mathbb{Q}_p)\)-action on \(\mathbb{P}^1 (\mathbb{Q}_p)\) and on equilibrium measures. We calculate explicitly the Green measures of balls and their complements. Throughout \(\S 4\) to \(\S 7\) there are various strengthenings of the uniqueness principle which demonstrate the increasing grasp we have over our potentials.

In \(\S 8\), we develop the concepts of \(\alpha\)-(super)-harmonic functions. Explicitly, a function \(f : \mathbb{Q}_p \to \mathbb{C}\) is \(\alpha\)-harmonic at \(x \in \mathbb{Q}_p\) if for all \(N\) sufficiently large: \[ f(x) = {1 \over \zeta_p (\alpha)} \int_{|y |\geq 1} f(x + p^Ny) {d^*y \over |y |^\alpha}. \] We give an explicit solution to Dirichlet problems, prove the Riesz representation theorem, and prove our last principles for potentials: domination and harmonic minorant, concluding with a convexity property of the \(\alpha\)-capacity. We note that with slight modifications one can carry over the whole discussion to the case of an arbitrary finite dimensional vector space over an arbitrary non-archimedean local field.

In \(\S 4\) we begin to develop the potential theory of \(R ^\alpha_p\), \(\alpha \in (0,1)\). In view of \(\S 2\) we can identify our potentials with probabilistic potentials and deduce the various potential theoretic principles in one blow; we prefer, nevertheless, to develop these principles purely analytically, and to grasp things along the way in a most explicit manner. We note, for example, that the Harnack inequalities become in our \(p\)-adic situation trivial equalities. We prove the principles of descent, dichotomy, maximum, regularization, and uniqueness. We follow mostly the analogous real case as developed in N. S. Landkof, Foundations of modern potential theory, Springer (1972; Zbl 0253.31001), the \(p\)-adic setting offering many simplifications.

In \(\S 5\) we consider the finite energy measures leading to the proof that they form a complete positive cone in the Hilbert space of finite energy distributions. In \(\S 6\) we prove the existence and various characterizations of the equilibrium measure, and show that the \(\alpha\)-capacity satisfies the usual properties and, moreover, is given explicitly via the \(\alpha\)-diameter: \[ \text{cap}_\alpha (K) = {\zeta_p (\alpha) \over \zeta_p (1 - \alpha)} \lim_{N \to \infty} \biggl[ \min_{x_1, \dots, x_N \in K} {1 \over n(n - 1)} \sum_{i < j} |x_i - x_j |^{\alpha - 1}\biggr]^{-1}. \] In \(\S 7\), we approach balayage and the Green measure using the Keldysh transform; i.e. instead of using Cartan’s method of projections in the Hilbert space of finite energy distributions (which equally works well), we use the more geometrical situation of the analysis of the \(\text{PGL}_2 (\mathbb{Q}_p)\)-action on \(\mathbb{P}^1 (\mathbb{Q}_p)\) and on equilibrium measures. We calculate explicitly the Green measures of balls and their complements. Throughout \(\S 4\) to \(\S 7\) there are various strengthenings of the uniqueness principle which demonstrate the increasing grasp we have over our potentials.

In \(\S 8\), we develop the concepts of \(\alpha\)-(super)-harmonic functions. Explicitly, a function \(f : \mathbb{Q}_p \to \mathbb{C}\) is \(\alpha\)-harmonic at \(x \in \mathbb{Q}_p\) if for all \(N\) sufficiently large: \[ f(x) = {1 \over \zeta_p (\alpha)} \int_{|y |\geq 1} f(x + p^Ny) {d^*y \over |y |^\alpha}. \] We give an explicit solution to Dirichlet problems, prove the Riesz representation theorem, and prove our last principles for potentials: domination and harmonic minorant, concluding with a convexity property of the \(\alpha\)-capacity. We note that with slight modifications one can carry over the whole discussion to the case of an arbitrary finite dimensional vector space over an arbitrary non-archimedean local field.

### MSC:

31C15 | Potentials and capacities on other spaces |

31C45 | Other generalizations (nonlinear potential theory, etc.) |

11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |

60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |

43A05 | Measures on groups and semigroups, etc. |

30G06 | Non-Archimedean function theory |

### Keywords:

analytic potential theory; \(\alpha\)-homogeneous distributions; \(\alpha\)-capacity; \(\alpha\)-superharmonic functions; \(\alpha\)-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### Citations:

Zbl 0253.31001
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\textit{S. Haran}, Ann. Inst. Fourier 43, No. 4, 905--944 (1993; Zbl 0847.31006)

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