Download New Developments in Pseudo-Differential Operators: ISAAC by Maurice de Gosson (auth.), Luigi Rodino, M. W. Wong (eds.) PDF

By Maurice de Gosson (auth.), Luigi Rodino, M. W. Wong (eds.)

This quantity contains seventeen peer-reviewed papers regarding lectures on pseudo-differential operators offered on the assembly of the ISAAC crew in Pseudo-Differential Operators (IGPDO) held on the center East Technical college in Ankara,Turkey on August 13-18, 2007, and invited papers by means of specialists within the box. incorporated during this quantity are such issues as research and partial differential equations concerning the Heisenberg team; international research and pseudo-differential research on non-compact manifolds and manifolds with singularities; Fourier critical operators and Colombeau algebras with functions to partial differential equations; unique pseudo-differential operators and regularity effects on quasi-elliptic operators and hypoelliptic operators; Stockwell transforms in time-frequency research; and pseudo-differential operators on Lie teams with similar effects to sampling. This quantity is an invaluable supplement to the volumes “Advances in Pseudo-Differential Operators”, “Pseudo-Differential Operators and similar subject matters” and “Modern developments in Pseudo-Differential Operators” released within the related sequence in, respectively, 2004, 2006 and 2007.

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Extra info for New Developments in Pseudo-Differential Operators: ISAAC Group in Pseudo-Differential Operators (IGPDO), Middle East Technical University, Ankara, Turkey, August 2007

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W. P. E. W. Wong, World Scientific, 2005, 39–70. [13] J. Tie, The non-isotropic twisted Laplacian on Cn and the sub-Laplacian on Hn , Comm. Partial Differential Equations 31 (2006), 1047–1069. W. Wong, Weyl Transforms, Springer-Verlag, 1998. W. Wong, Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator, Ann. Global Anal. Geom. 28 (2005), 271–283. W. Wong, The heat equation for the Hermite operator on the Heisenberg group, Hokkaido Math. J. 34 (2005), 393–404. [17] K.

A formula for e−uLτ f, u > 0, for every function f in L2 (H) is given in Section 4. 2). In Section 5, we use the family L2s (H), s ∈ R, of Sobolev spaces with respect to the center of the Heisenberg group in [1] to obtain Sobolev estimates for e−uL f, u > 0, in terms of f L2(H) , and estimates for e−uL f L2 (H) , u > 0, in terms of the Sobolev norms f L2s(H) of f in L2s (H). 36 A. W. Wong 2. τ -Weyl transforms Let f and g be functions in L2 (R). 1) for all q and p in R. In fact, Vτ (f, g)(q, p) = |τ |1/2 V (f, g)(τ q, p), q, p ∈ R, where V (f, g) is the classical Fourier–Wigner transform of f and g.

2) and the fact that eτk L2 (R) = 1, k = 0, 1, 2, . . , we get e−uLτ f L2 (C) ≤ (2π)1/2 e−|τ |u ∞ k=0 e−2k|τ |u Wfτˆ eτk L2 (R) , u > 0. 7) 40 A. W. 7), we get e−uLτ f L2 (C) ≤ (2π)−(1/p)+(1/2) |τ |−(1/2)+(1/p) e−|τ |u ∞ e−2k|τ |u f Lp (C) k=0 = (2π)−(1/p)+(1/2) |τ |−(1/2)+(1/p) 1 f 2 sinh(|τ |u) Lp (C) , as asserted. 5. Sobolev estimates for e−uL , u > 0 Let s ∈ R. Then as in [1], we define L2s (H) to be the set of all functions f in L2 (H) such that ∞ C |τ |2s |f τ (z)|2 dτ dz < ∞. −∞ For every f in L2s (H), we define the norm f f 2 L2s (H) ∞ = C −∞ L2s (H) by |τ |2s |f τ (z)|2 dτ dz.

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