Discrete Holomorphic Local Dynamical Systems

278 Tien-Cuong Dinh and Nessim Sibony
potentials of R. In general, dd c u ∧ S does not depend continuously on u and S. The
following proposition is a consequence of Hartogs’ lemma.
Proposition A.30. Let u (n) be p.s.h. functions on X which converge in the Hartogs’
sense to a p.s.h. function u. If u is locally integrable with respect to the trace
measure of S, then dd c u (n) ∧S are well-defined and converge to dd c u∧S. If u is continuous
and Sn are positive closed (1,1)-currents converging to S, then dd c u (n) ∧ Sn
converge to dd c u ∧ S.
If u1,...,uq, with q ≤ k − p, are p.s.h. functions, we can define by induction the
wedge-product
dd c u1 ∧ ...∧ dd c uq ∧ S
when some integrability conditions are satisfied, for example when the ui are locally
bounded. In particular, if u (n)
j ,1≤ j ≤ q, are continuous p.s.h. functions converging
locally uniformly to continuous p.s.h. functions u j and if Sn are positive closed
converging to S,then
dd c u (n)
1 ∧ ...∧ ddc u (n)
q ∧ Sn → dd c u1 ∧ ...∧ dd c uq ∧ S
The following version of the Chern-Levine-Nirenberg inequality is a very useful
result [CLN, DEM].
Theorem A.31. Let S be a positive closed (p, p)-current on X. Let u1,...,uq, q≤
k − p, be locally bounded p.s.h. functions on X and K a compact subset of X. Then
there is a constant c > 0 depending only on K and X such that if v is p.s.h. on X then
�vdd c u1 ∧ ...∧ dd c uq ∧ S�K ≤ c�v� L 1 (σS) �u1� L ∞ (X) ...�uq� L ∞ (X),
where σS denotes the trace measure of S.
This inequality implies that p.s.h. functions are locally integrable with respect to
the current dd c u1 ∧ ...∧ dd c uq. We deduce the following corollary.
Corollary A.32. Let u1,...,up, p≤ k, be locally bounded p.s.h. functions on X.
Then, the current dd c u1 ∧ ...∧ dd c up has no mass on locally pluripolar sets, in
particular on proper analytic sets of X.
We give now two other regularity properties of the wedge-product of currents
with Hölder continuous local potentials.
Proposition A.33. Let S be a positive closed (p, p)-current on X and q a positive
integer such that q ≤ k − p. Let ui be Hölder continuous p.s.h. functions of Hölder
exponents αi with 0 < αi ≤ 1 and 1 ≤ i ≤ q. Then, the current dd c u1 ∧...∧dd c uq ∧S
has no mass on Borel sets with Hausdorff dimension less than or equal to
2(k − p − q)+α1 + ···+ αq.