<p>H-mere i H-distribucije su mikrolokalni objekti koji se koriste za ispitivanje jake konvergencije slabo konvergentnog niza u prostorima Lebega i prostorima Soboljeva. H-mere su uveli Tartar i Zerar (koji ih zove mikrolokalne mere defekta), u radovima [34] i [19]. H-mere su Radonove mere koje daju informacije o mogu ´ cim oblastima jake konvergencije slabo konvergentnog<em> L</em><sup>2</sup> niza. Da bismo mogli da posmatramo i slabo konvergentne<em> L</em><sup>p</sup> nizove za 1 < p < ∞, Antonić i Mitrović u radu [11] uvode H-distribucije.</p><p>U disertaciji dajemo konstrukciju H-distribucija za slabo konvergentne nizove u <em>W</em><sup>-k,p</sup> prostorima, kad je 1 < p < ∞, k ∈ ℕ i pokazujemo da kada je H-distribucija pridružena slabo konvergetnim nizovima jednaka nuli za sve test funkcije, onda imamo lokalno jaku konverenciju datog niza.</p><p>Takođe je pokazan i lokalizacijski princip, koji nam daje oblast u kojoj imamo lokalno jaku konvergenciju slabo konvergentnog niza. H-mere i H-distribucije deluju na test funkcije φ i ψ (odgovarajuće regularnosti) koje su definisane na ℝ<sup>d</sup> i S<sup>d-1</sup> (jedinična sfera u ℝ<sup>d</sup>), pri čemu je funkcija ψ, koju zovemo množilac, ograničena. U disertaciji uvodimo i H-distribucije sa neograničenim simbolom, pri čemu posmatramo slabo konvergentne nizove u Beselovim H<sup>p</sup><sub>-s</sub> prostorima, gde je 1 < p < ∞; s ∈ ℝ. U ovom delu koristimo teoriju pseudo-diferencijalnih operatora i dokazujemo kompaktnost komutatora [<i>A</i><sub>ψ</sub>, T<sub>φ</sub>] za razne klase množioca ψ, što je potrebno za dokaz postojanja H-distribucija. Takođe pokazujemo odgovarajuću verziju lokalizacijskog principa.</p> / <p>H-measures and H-distributions are microlocal tools that can be used to investigate strong conver-gence of weakly convergent sequences in the Lebesgue and Sobolev spaces.</p><p>H-measures are introduced by Tartar and Gérard (as microlocal defect measures) in papers [34] and [19]. H-measures are Radon measures and they provide information about the set of points where given weakly convergent sequence in <em>L</em><sup>2</sup> converges strongly. In paper [11], Antonić and Mitrović introduced H-distributions in order to work with weakly convergent <em>L</em><sup>p</sup> sequences.</p><p>In this thesis we give construction of H-distributions for weakly convergent <em>W<sup>-</sup></em><sup>k,p</sup> sequences, where 1 < p < ∞; k ∈ N. We show that if the H-distribution corresponding to given weakly convergent sequence is equal to zero, then we have locally strong convergence of the sequence. We also prove localization principle.</p><p>H-measures and H-distributions act on test functions φ and ψ (regular enough) which are defined on ℝ<sup>d</sup> and <sup>d-1</sup> (unit sphere in ℝ<sup>d</sup> ) and the function ψ, which is called multiplier, is bounded. We also introduce H-distributions with unboundedmultipliers and in this case we assume that weakly convergent sequences are in Bessel potential spaces H<sup>p</sup><sub>-s</sub> , where 1 < p < ∞, s ∈ ℝ. Theory of pseudo-differential operators is used in construction of H-distributions with unbounded multipliers. We prove compactness of the commutator [<em>A</em><sub><em>ψ</em></sub>,T<sub>φ</sub> ] for different classes of multipliers y and appropriate version of localization principle.</p>
| Identifer | oai:union.ndltd.org:uns.ac.rs/oai:CRISUNS:(BISIS)104554 |
| Date | 01 July 2017 |
| Creators | Vojnović Ivana |
| Contributors | Aleksić Jelena, Pilipović Stevan, Teofanov Nenad, Prangoski Bojan |
| Publisher | Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, University of Novi Sad, Faculty of Sciences at Novi Sad |
| Source Sets | University of Novi Sad |
| Language | Serbian |
| Detected Language | English |
| Type | PhD thesis |
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