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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

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Ivana Vojnović (Department of Mathematics and Informatics, Faculty of Sciences, Novi Sad)

H-distributions with unbounded multipliers

H-measures were introduced independently by Tartar, [4] and Gérard, [3]. They are used to determine weather a weakly convergent sequence in $L^2({\mathbb R}^d)$ converges strongly. Antonić and Mitrović in [2] introduced H-distributions, extension of H-measures to an $L^p - L^q$ setting for $1 < p < \infty$ and $ q=p/p-1$. In [1], H-distributions are constructed for sequences in dual Sobolev spaces, $W^{-k,p} - W^{k,q}$. Test functions for H-measures and H-distributions are bounded Fourier multipliers. Using theory of pseudo-differential operators we construct H-distributions for sequences in dual Bessel potential spaces, $H_q^k - H_{p}^{-k}, k \in \mathbb R$, $1 < p < \infty$. In this case we consider classes of unbounded test functions. Also, a necessary and sufficient condition is given so that the weak convergence of sequence in $H_{p}^{-k}$ implies the strong one. Results are applied on a weakly convergent sequence of solutions to a family of partial differential equations. This is a joint work with Jelena Aleksić and Stevan Pilipović.

References

[1] Aleksić, J.; Pilipović, S.; Vojnović, I. H - distributions via Sobolev spaces, Mediterranean Journal of Mathematics, 2016., pp 1-14
[2] Antonić, N.; Mitrović, D. H-distributions: an extension of H-measures to an L^p-L^q setting. Abstr. Appl. Anal. 2011, Art. ID 901084, 12 pp.
[3] Gérard, P. Microlocal defect measures. Comm. Partial Differential Equations 16 (1991), no. 11, 1761--1794.
[4] Tartar, L. H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 3-4, 193--230.

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