Mikrolokalne distribucije defekta i primene
Microlocal defect distributions and applications
Author
Vojnović, Ivana
Mentor
Aleksić, Jelena
Committee members
Pilipović, Stevan
Aleksić, Jelena

Teofanov, Nenad

Prangoski, Bojan
Metadata
Show full item recordAbstract
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.