Distributions and ultradistributions on R+d through Laguerre expansionswith applications to pseudo-dierential operators with radial symbols

Abstract

Proučavamo razvoje elemenata iz S(ℝ+d) i S'(ℝ+d) preko Lagerove ortonormirane baze. Kao posledicu dobijamo Švarcovu teoremu o jezgru za preko Lagerove ortonormirane baze. Kao posledicu dobijamo Švarcovu teoremu o jezgru za S(ℝ+d) i S'(ℝ+d). Takođe, pokazujemo i Teoremu Vitnijevog tipa za S(ℝ+d) . Zatim, posmatramo prostore G-tipa i.e. prostore Gαα(ℝd), α ≥ 1 i njihove duale koji su analogni sa Geljfand-Šilovim prostorima i njihovim dualima. Zapravo, pokazujemo da postoji topološki izomorfizam između prostora G-tipa i potprostora Geljfand-Šilovih prostora Sα/2α/2(ℝd), α ≥ 1 koji sadrže "parne" funkcije. Dalje, dokazujemo da Furije Lagerovi koeficijenti elemenata iz prostora G-tipa i njihovih duala karakterišu ove prostore kroz eksponencijalni i sub-eksponencijalni rast tih koeficijenata. Opisujemo topološku strukturu ovih prostora i dajemo Švarcovu teoremu o jezgru. Takođe, dve strukturalne teoreme za duale prostora G-tipa su dobijene. Dalje, definišemo novu klasu Vejlovih pseudo-diferencijalnih operatora sa radijalnim simbolima koji se nalaze u prostorima G-tipa i njihovim dualima. Pokazana je neprekidnost ove klase Vejlovih pseudo-diferencijalnih operatora na prostorima Geljfand-Šilova i na njihovim dualima. Na ovaj način klasa Vejlovih pseudo-diferencijalnih operatora je proširena na radijalne simbole koji imaju eksponencijalni i sub-eksponencijalni rast.

We study the expansions of the elements in S(ℝ+d) and S'(ℝ+d) with respect to the Laguerre orthonormal basis. As a consequence, we obtain the Schwartz kernel theorem for S(ℝ+d) and S'(ℝ+d). Also we give the extension theorem of Whitney type for S(ℝ+d). Next, we consider the G-type spaces i.e. the spaces Gαα(ℝ+d), α≥1 and their dual spaces which can be described as analogous to the Gelfand-Shilov spaces and their dual spaces. Actually, we show the exist-ence of the topological isomorphism between the G-type spaces and the subspaces of the Gelfand-Shilov spaces Sα/2α/2(ℝd), α≥1 consisting of "even" functions. Next, we show that the Fourier Laguerre coecients of the elements in the G-type spaces and their dual spaces characterize these spaces through the exponential and sub-exponentia l growth of the coecients. We provide the full topological description and the kernel theorem is proved. Also two structural theorems for the dual spaces of G-type spaces are obtained. Furthemore, we dene the new class of the Weyl pseudo-dierential operators with radial symbols belonging to the G-type spaces and their dual spaces. The continuity properties of this class of pseudo-dierential operators over the Gelfand-Shilov type spaces and their duals are proved. In this way the class of the Weyl pseudo-dierential operators is extended to the one with the radial symbols with the exponential and sub-exponential growth rate.