Petak, 02.03.2012. u 14h sala II MI
Professor Sandro Coriasco, Faculty of Mathematics, University of Torino
$L^P(R^n)$-BOUNDEDNESS OF ANISOTROPIC MULTIPLIERS AND OF $SG$ FOURIER INTEGRAL OPERATORS

Abstract: I will illustrate some recently obtained results about the continuity on $L^p(\mathbb{R}^n)$ of certain pseudodifferential and Fourier integral operators, defined through symbol classes satisfying global estimates on $\mathbb{R}^n$.

More precisely, I will first discuss necessary conditions for the $L^p(\mathbb{R}^n)$-continuity of multipliers $\sigma(D)$ associated with suitable, strictly positive, weight functions $\lambda,\psi=(\psi_1, \dots, \psi_n) \in\mathcal{C}(\mathbb{R}^n)$, $\lambda$ bounded. Namely, the derivatives of the symbol $\sigma$ satisfy, for all $\alpha\in\mathbb{Z}_+^n$ and suitable constants $C_{\alpha}\ge 0$, the `anisotropic estimates'' $$ |D^\alpha \sigma(\xi)|\le\lambda(\xi)\cdot \psi(\xi)^{-\alpha}, \quad \xi\in\mathbb{R}^n, $$ % where $\displaystyle\psi(\xi)^{-\alpha}=\prod_{j=1}^n \psi_j(\xi)^{-\alpha_j}$. This generalises a classical result by Beals, where no difference in the components of $\psi$ was allowed.

Subsequently, I will present an extension of a famous result by Seeger, Sogge and Stein, to the Fourier integral operators defined on $\mathbb{R}^n$ by means of the so-called $SG$-symbols (a class of symbols independently introduced in the '70s by Cordes and Parenti). The lack of compactness in the supports gives additional complications, which imply the need of controlling the behaviour at infinity of the involved amplitude and phase functions. An interesting aspect of the result is that this reflects in a corresponding `loss of decay at infinity'', completely analogous to the well-known `loss of smoothness''.

Petak, 23.03.2012. u 14h sala II, MI
Prof Miodrag Mateljević, Matematički fakultet, Beograd
PRIKAZ MONOGRAFIJA POD ZAJEDNIČKIM NASLOVOM "KONFORMNA, KVAZIKONFORMNA I HARMONIJSKA PRESLIKAVANJA"