By O. Liess, L. Rodino (auth.), Luigi Rodino (eds.)

The NATO complex examine Institute "Microlocal research and Spectral The­ ory" used to be held in Tuscany (Italy) at Castelvecchio Pascoli, within the district of Lucca, hosted through the foreign holiday heart "11 Ciocco" , from September 23 to October three, 1996. The Institute recorded the significant development learned lately within the box of Microlocal research. In a huge experience, Microlocal research is the fashionable model of the classical Fourier method in fixing partial differential equa­ tions, the place now the localization continuing occurs with recognize to the twin variables too. accurately, in the course of the instruments of pseudo-differential operators, wave-front units and Fourier vital operators, the overall thought of the lin­ ear partial differential equations is now attaining a mature shape, within the body of Schwartz distributions or different generalized features. even as, Microlocal research has grown up right into a convinced and self sufficient a part of Math­ ematical research, with different functions throughout arithmetic and Physics, one significant subject being Spectral concept for Schrodinger equation in Quantum Mechanics.

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But beyond the conclusion that subellipticity implies Gevrey regularity as in the above Theorems, most attention has gone to study real analytic regularity when there is a maximal estimate (usually requiring a lot of symmetry - cf. , [6], [7] with a series of papers [34], [13] proving local real analytic hypoellipticity for either the 8-Neumann problem or for Db whose Levi forms degenerate in well-controlled ways). Semi-global regularity is also an active area of research currently. In addition, some mixed results, where the characteristic manifold splits into an involutive portion and a symplectic portion were studied for pseudodifferential operators very recently by the author and Antonio Bove [4].

Thus whenever a derivative in y lands on the localizing function, we have no way to produce better than G 2 results unless we make the a priori assumption that where such a derivative is non zero, the solution is already smooth. Thus if there is no singularity in y derivatives in a band El ::; Iyl ::; E2, there will be none at y = either. ° ° ° 8. Breaking the GS - Barrier; Other Rational Exponents Recently there has been study of Gevrey hypoellipticity when s takes on values other than l/m. Christ [9] has very recently obtained sharp isotropic results in rational Gevrey classes that are better than those predicted by 52 the subellipticity index.

1 extends to the case of codimension n' > 1. 4 for quasihomogeneous symbols; this in turn would depend on a generalization to the quasihomogeneous case of the infinite order calculus in [25}. 3. 11) and r(K-I)/k is a classical analytic symbol of order (K - 1)/k. lp(y, ",)1(1 + 1",I)-KIf3I/k . 7 and we omit the details. 13) we deduce the existence of a s-microlocal inverse of P, acting on M8(r) for k/ K ~ s ~ 00. 3. 3 leave open the case when 1m h-I-K(P, X) vanishes at P = po, but h-I-K(p, X) is not real-valued nearby.

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