Lusin area functions on local fields

On the littlewood-paley and lusin functions in higher dimensions.

Local Self-concordance of Barrier Functions Based on Kernel-functions

 Many efficient interior-point methods (IPMs) are based on the use of a self-concordant barrier function for the domain of the problem that has to be solved. Recently, a wide class of new barrier functions has been introduced in which the functions are not self-concordant, but despite this fact give rise to efficient IPMs. Here, we introduce the notion of locally self-concordant barrier functio...

Higher dimensional local fields and L - functions

Example. Let X be an algebraic projective irreducible surface over a field k and let P be a closed point of X , C ⊂ X be an irreducible curve such that P ∈ C . If X and C are smooth at P , then we let t ∈ OX,P be a local equation of C at P and u ∈ OX,P be such that u|C ∈ OC,P is a local parameter at P . Denote by C the ideal defining the curve C near P . Now we can introduce a two-dimensional l...

A Quantitative Lusin Theorem for Functions in BV

We extend to the BV case a measure theoretic lemma previously proved by DiBenedetto, Gianazza and Vespri ([1]) in W 1,1 loc . It states that if the set where u is positive occupies a sizable portion of a open set E then the set where u is positive clusters about at least one point of E. In this note we follow the proof given in the Appendix of [3] so we are able to use only a 1−dimensional Poin...

Lecture Notes on Local Fields

In this case, I have corrected the Texnical errors. Also I have corrected some mathematical errors that I found – which were mercifully slight. Overall my 23 year-old mathematical self bears a closer resemblance to the current model than I might have expected (good news or bad?). If I would say something different today, I make a brief comment about it, but I have made no attempt to systematica...