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Journals and Conferences
Complex powers of a class of hypoelliptic pseudodifferential operators in Rn, as well as their heat kernels are studied. An application to the Shatten-Von Neumann property of pseudodifferential operators is given. AMS Subject Classification: 35S05
For a Hörmander’s symbol class S(m, g), it is proved that the weight m is in L(R), with 1 6 p < ∞, if and only if all pseudo-differential operators with Weyl symbol in S(m, g) are in the Schatten-von Neumann class Sp(L ).
We investigate connections between certain dispersive estimates of a (pseudo)differential operator of real principal type and the number of nonvanishing curvatures of its characteristic manifold. More precisely, we obtain sharp thresholds for the range of Lebesgue exponents depending on the specific geometry.
Using microlocalization, the positive and the negative part of a class of second order formally self-adjoint pseudodifferential operators are constructed. AMS Subject Classification: 35S05
We study the local solvability problem for a class of semilinear equations whose linear part is the Kohn Laplacian, acting on top degree forms. We also study the validity of the Poincaré lemma, in top degree, for semilinear perturbations of the tangential Cauchy-Riemann complex.
We deal with the restriction phenomenon for the Fourier transform. We prove that each of the restriction conjectures for the sphere, the paraboloid, the elliptic hyperboloid in R implies that for the cone in R. We also prove a new restriction estimate for any surface in R locally isometric to the plane and of finite type.
A necessary condition is established for the optimal (L, L) restriction theorem to hold on a hypersurface S, in terms of its Gaussian curvature. For some classes of flat hypersurfaces we give sharp thresholds for the range of admissible exponents p, depending on the specific geometry.
We study the action on modulation spaces of Fourier multipliers with symbols eiμ(ξ), for real-valued functions μ having unbounded second derivatives. In a simplified form our result reads as follows: if μ satisfies the usual symbol estimates of order α ≥ 2, or if μ is a positively homogeneous function of degree α, the corresponding Fourier multiplier is… (More)
For a class of pseudodifferential operators with characteristics of even multiplicity k ≥ 2 we prove a lower bound with gain of k/2+1 derivatives. In the case of operators with double characteristics (k = 2), our result reduces exactly to Hörmander’s inequality. AMS Subject Classification: 35S05.