Abstract. In this paper we continue our investigations of square function inequalities in harmonic analysis. Here we investigate oscillation and variation inequalities for singular integral operators in dimensions d ≥ 1. Our estimates give quantitative information on the speed of convergence of truncations of a singular integral operator, including upcrossing and λ jump inequalities.
We prove L 2 variation inequalities for operators defined by the convolution powers of probability measures on locally compact Abelian groups. In some cases we also obtain L p results for 1 < p < ∞. These inequalities imply the pointwise convergence of these operators and give an estimate of the number of upcrossings. Bourgain [4]; and Jones et al [10].We can also define the continuous version of the v( ) norm as follows.Definition 2.4. Given a continuously indexed system, {x t } t ∈R , define its variation norm by: J = 1, 2, . . . ; t 0 ≤ t 1 ≤ t 2 ≤ · · · ≤ t J .
The Alaskan boreal forest is characterized by frequent extensive wildfires whose spatial extent has been mapped for the past 70 years. Simple predictions based on this record indicate that area burned will increase as a response to climate warming in Alaska. However, two additional factors have affected the area burned in this time record: the Pacific decadal oscillation (PDO) switched from cool and moist to warm and dry in the late 1970s and the Alaska Fire Service instituted a fire suppression policy in the late 1980s. In this paper a geographic information system (GIS) is used in combination with statistical analyses to reevaluate the changes in area burned through time in Alaska considering both the influence of the PDO and fire management. The authors found that the area burned has increased since the PDO switch and that fire management drastically decreased the area burned in highly suppressed zones. However, the temporal analysis of this study shows that the area burned is increasing more rapidly in suppressed zones than in the unsuppressed zone since the late 1980s. These results indicate that fire policies as well as regional climate patterns are important as large-scale controls on fires over time and across the Alaskan boreal forest.
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