Extreme invariant positive operators

Translation-invariant bilinear operators with positive kernels

We study L (or Lr,∞) boundedness for bilinear translation-invariant operators with nonnegative kernels acting on functions on R. We prove that if such operators are bounded on some products of Lebesgue spaces, then their kernels must necessarily be integrable functions on R, while via a counterexample we show that the converse statement is not valid. We provide certain necessary and some suffic...

Positive Operators and Hausdorff Dimension of Invariant Sets

In this paper we obtain theorems which give the Hausdorff dimension of the invariant set for a finite family of contraction mappings which are “infinitesimal similitudes” on a complete, perfect metric space. Our work generalizes the graph-directed construction of Mauldin and Williams (1988) and is related in its general setting to results of Schief (1996), but differs crucially in that the mapp...

On Extreme Averaging Operators

Scale invariant operators

Several types of operators acting on a continuous scale (mostly on the unit interval [O, I ] ) should be discretized before their real use in intelligent computing (computers work on discrete scales only). First of all, the operators allowing the discretization related to any discrete (finite) scale should be examined. However, different discrete (finite) scales may induce different restricted ...

Invariant differential operators

on the complex upper half-plane H, provably invariant under the linear fractional action of SL2(R), but it is oppressive to verify this directly. Worse, the goal is not merely to verify an expression presented as a deus ex machina, but, rather to systematically generate suitable expressions. An important part of this intention is understanding reasons for the existence of invariant operators, a...