\documentclass[12pt]{amsart}
\usepackage[a4paper]{geometry}
\geometry{verbose,tmargin=2.5cm,bmargin=2.5cm,lmargin=2.5cm,rmargin=2.5cm}

\begin{document}
%%%%%%%%%   Your contribution starts from here

\title{One-scale H-measures, variants and applications}

%Please underline name of the presenting author
\author{Nenad Antoni\' c $^1$}
\address{$^1$ University of Zagreb}
\email{nenad@math.hr}

\author{\underline{Marko Erceg} $^2$}
\address{$^2$ University of Zagreb}
\email{maerceg@math.hr}

\author{Martin Lazar $^3$}
\address{$^3$ University of Dubrovnik}
\email{mlazar@unidu.hr}

\keywords{ H-measures, localisation principle, semiclassical measures, compactness by compensation.}

\maketitle

Microlocal defect functionals (H-measures, H-distributions, semiclassical measures etc.) are 
objects which determine, in some sense, the lack of strong compactness for weakly convergent ${\rm L}^p$ 
sequences. In contrast to the semiclassical measures, H-measures are not suitable to treat problems 
with a characteristic length (e.g.~thickness of a plate). {\sc Luc Tartar}
in his recent book overcame the mentioned  restriction by introducing one-scale H-measures, a generalisation 
of H-measures with a characteristic length [2]. Moreover, these objects are also an extension of 
semiclassical measures, being functionals on continuous functions on a compactification of 
$\mathbb{R}^d\setminus\{ 0 \}$.

We improve and generalise Tartar's localisation principle for one-scale H-measures from which 
we are able to derive the known localisation principles for both H-measures and semiclassical 
measures. Moreover, we develop a variant of compactness by compensation suitable 
for equations with a characteristic length [1]. 

Since one-scale H-measures are adequate only for the ${\rm L}^2$ framework, we introduce
the generalisation, one-scale H-distributions, as a counterpart of H-distributions with a
characteristic length, and address some important features. 


%IF bibliography is needed
\begin{thebibliography}{9}

\bibitem{AEL} N. Antoni\'c, M. Erceg, M. Lazar, \textsl{Localisation principle for one-scale H-measures}, 
submitted 35 pp.

\bibitem{T} L. Tartar, \textsl{Multi-scale H-measures}, 
\textit{Discrete and Continuous Dynamical Systems}, S {\bf 8} (2015), 77--90.

\bibitem{H} Lars H\"{o}rmander, \textit{The Analysis of Linear Partial Differential Operators IV}, Springer, 1990.

\end{thebibliography}

\end{document}
