Short Note on Inf-Convolution Preserving the Fatou Property

Positivity preserving forms have the Fatou property

If (un)n∈IN is a sequence in L2(E; m) converging m-almost everywhere to u, then Fatou’s lemma says that (u, u)L2 ≤ lim infn(un, un)L2 , where we set (u, u)L2 = ∞ if u 6∈ L2(E; m). The corresponding result, where a Dirichlet form replaces the inner product, was used by Silverstein [5; Lemma 1.7] and by Fukushima, Oshima, and Takeda [2; Theorem 1.5.2] to define extended Dirichlet space and study ...

A Theorem À La Fatou for the Square Root of Poisson Kernel in Δ-hyperbolic Spaces and Decay of Matrix Coefficients of Boundary Representations

In this note, we prove a theorem à la Fatou for the square root of Poisson Kernel in the context of quasi-convex cocompact discrete groups of isometries of δ-hyperbolic spaces. As a corollary we show that some matrix coefficients of boundary representations cannot satisfy the weak inequality of Harish-Chandra. Nevertheless, such matrix coefficients satisfy an inequality which can be viewed as a...

Law Invariant Risk Measures Have the Fatou Property

S. Kusuoka [K 01, Theorem 4] gave an interesting dual characterization of law invariant coherent risk measures, satisfying the Fatou property. The latter property was introduced by F. Delbaen [D 02]. In the present note we extend Kusuoka’s characterization in two directions, the first one being rather standard, while the second one is somewhat surprising. Firstly we generalize — similarly as M....

Subordination and Superordination Properties for Convolution Operator

In present paper a certain convolution operator of analytic functions is defined. Moreover, subordination and superordination- preserving properties for a class of analytic operators defined on the space of normalized analytic functions in the open unit disk is obtained. We also apply this to obtain sandwich results and generalizations of some known results.

A note on the comparison between Laplace and Sumudu transforms