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Mikrolokalne distribucije defekta i primene

Microlocal defect distributions and applications

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2017
Disertacija11047.pdf (710.4Kb)
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Author
Vojnović, Ivana
Mentor
Aleksić, Jelena
Committee members
Pilipović, Stevan
Aleksić, Jelena
Teofanov, Nenad
Prangoski, Bojan
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Abstract
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 L2 niza. Da bismo mogli da posmatramo i slabo konvergentne Lp nizove za 1 < p < ∞, Antonić  i Mitrović u radu [11] uvode H-distribucije. U disertaciji dajemo konstrukciju H-distribucija za slabo konvergentne nizove u W-k,p 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. 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 regul...arnosti) koje su definisane na ℝd i Sd-1 (jedinična sfera u ℝd), 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 Hp-s prostorima, gde je 1 < p < ∞; s ∈ ℝ. U ovom delu koristimo teoriju pseudo-diferencijalnih operatora i dokazujemo kompaktnost komutatora [Aψ, Tφ] za razne klase množioca ψ,  što je potrebno za dokaz postojanja H-distribucija. Takođe pokazujemo odgovarajuću verziju lokalizacijskog principa.

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. 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 L2 converges strongly. In paper [11], Antonić and Mitrović introduced  H-distributions in order to work with weakly convergent Lp sequences. In this thesis we give construction of H-distributions for weakly convergent W-k,p 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. H-measures and H-distributions act on test functions φ and ψ (regular enough) which are defined on ℝd and d-1 (unit sphere in ℝd ) and the func...tion 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 Hp-s , 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 [Aψ,Tφ ] for different classes of multipliers y and appropriate version of localization principle.

Faculty:
Универзитет у Новом Саду, Природно-математички факултет
Date:
01-07-2017
Keywords:
H-mere / H-measures / H-distributions / weak convergence / Bessel spaces / commutationlemma / H-distribucije / slaba konvergencija / Beselovi prostori / komutacijska lema
[ Google Scholar ]
Handle
https://hdl.handle.net/21.15107/rcub_nardus_8363
URI
https://nardus.mpn.gov.rs/handle/123456789/8363
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