Microlocal defect distributions and applications

Abstract

<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&nbsp; Zerar (koji ih zove mikrolokalne mere defekta), u radovima [34] i [19]. H-mere su Radonove mere koje daju informacije o mogu &acute; cim oblastima jake konvergencije slabo konvergentnog<em> L</em>² niza. Da bismo mogli da posmatramo i slabo konvergentne<em> L</em>^p nizove za 1 &lt; p &lt; &infin;, Antonić&nbsp; i Mitrović u radu [11] uvode H-distribucije.</p><p>U disertaciji dajemo konstrukciju H-distribucija za slabo konvergentne nizove u <em>W</em>^-k,p prostorima, kad je 1 &lt; p &lt; &infin;, k &isin; ℕ&nbsp;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&nbsp; konvergenciju slabo konvergentnog niza. H-mere i H-distribucije deluju na test funkcije &phi;&nbsp;i &psi;&nbsp;(odgovarajuće regularnosti) koje su definisane na ℝ^d i S^d-1 (jedinična sfera u ℝ^d), pri&nbsp; čemu je funkcija &psi;, koju zovemo množilac, ograničena. U disertaciji uvodimo i H-distribucije sa neograničenim simbolom, pri čemu posmatramo slabo&nbsp; konvergentne nizove u Beselovim H^p_-s prostorima, gde je 1 &lt; p &lt; &infin;; s &isin; ℝ. U ovom delu koristimo teoriju pseudo-diferencijalnih operatora i dokazujemo kompaktnost komutatora [<i>A</i>_&psi;, T_&phi;] za razne klase množioca &psi;,&nbsp; &scaron;to 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 G&eacute;rard (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>² converges strongly. In paper [11], Antonić and Mitrović introduced&nbsp; H-distributions in order to work with weakly convergent <em>L</em>^p sequences.</p><p>In this thesis we give construction of H-distributions for weakly convergent <em>W^-</em>^k,p sequences, where 1 &lt; p &lt; &infin;; k &isin;&nbsp;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 &phi; and &psi;&nbsp;(regular enough) which are defined on ℝ^d and ^d-1 (unit sphere in ℝ^d ) and the function &psi;, which is called multiplier, is bounded. We also introduce H-distributions with unboundedmultipliers and in this&nbsp; case we assume that weakly convergent sequences are in Bessel potential spaces H^p_-s , where 1 &lt; p &lt; &infin;, s &isin; 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>_{<em>&psi;</em>},T_&phi; ] for different classes of multipliers y and appropriate version of localization principle.</p>