Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

The boundary double layer potential, or the Neumann-Poincaré operator, is studied on the Sobolev space of order 1/2 along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the whole space. Poincaré’s program of studying the spectrum of the boundary double layer potential is developed in complete generality on closed Lipschitz hypersurfaces in euclidean space. Furthermore, the Neumann-Poincaré operator is realized as a singular integral transform bearing similarities to the Beurling-Ahlfors transform in 2 dimensions. As an application, in the case of planar curves with corners, bounds for the spectrum of the Neumann-Poincaré operator are derived from recent results in quasi-conformal mapping theory

Duality and distance formulas in spaces defined by means of oscillation

For the classical space of functions with bounded mean oscillation, it is well known that VMO∗∗=BMOVMO∗∗=BMO and there are many characterizations of the distance from a function f in BMOBMO to VMOVMO. When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this paper such duality results and distance formulas are obtained by pure functional analysis. Applications include general Möbius invariant spaces such as QK-spaces, weighted spaces, Lipschitz–Hölder spaces and rectangular BMOBMO of several variables

Atomic decompositions, two stars theorems, and distances for the Bourgain–Brezis–Mironescu space and other big spaces

Given a Banach space E with a supremum-type norm induced by a collection of operators, we prove that E is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space B introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual B⁎, the biduality result that B0⁎=B⁎ and B⁎⁎=B, and a formula for the distance from an element f∈B to B0

Decreasing Late Mortality Among Five-Year Survivors of Cancer in Childhood and Adolescence: A Population_Based Study in the Nordic Countries

PURPOSE: To assess the risk of death in patients who survive more than 5 years after diagnosis of childhood cancer and to evaluate causes of death in fatal cases. PATIENTS AND METHODS: This was a population-based study in the five Nordic countries (Denmark, Finland, Iceland, Norway, and Sweden) using data of the nationwide cancer registries and the cause-of-death registries. The study cohort included 13,711 patients who were diagnosed with cancer before the age of 20 years between 1960 and 1989 and who survived at least 5 years from diagnosis. By December 31, 1995, 1,422 patients had died, and death certificates were assessed in 1,402. Standardized mortality ratios (SMRs) for validated causes of death were calculated based on 156,046 patient-years at risk. RESULTS: The overall SMR was 10.8 (95% confidence interval [CI], 10.3 to 11.5), mainly due to high excess mortality from the primary cancer. SMR for second cancer was 4.9 (95% CI, 3.9 to 5.9) and was 3.1 (95% CI, 2.8 to 3.5) for noncancer death. The pattern of causes of death varied markedly between different groups of primary cancer diagnoses and was highly dependent on time passed since diagnosis. Overall late mortality was significantly lower in patients treated during the most recent period of time, 1980 to 1989, compared with those treated from 1960 to 1979 (hazard ratio, 0.61; 95% CI, 0.54 to 0.70), and there was no increase in rates of death due to cancer treatment. CONCLUSION: Long-term survivors of childhood cancer had an increased mortality rate, mainly dying from primary cancers. However, modern treatments have reduced late cancer mortality without increasing the rate of therapy-related deaths