The aim of this paper is to introduce and study multilinear pseudo-differential operators on Z n and T n = (R n /2πZ n ) the n-torus. More precisely, we give sufficient conditions and sometimes necessary conditions for L p -boundedness of these classes of operators. L 2boundedness results for multilinear pseudo-differential operators on Z n and T n with L 2 -symbols are stated. The proofs of these results are based on elementary estimates on the multilinear Rihaczek transforms for functions in L 2 (Z n ) respectively L 2 (T n ) which are also introduced.We study the weak continuity of multilinear operators on the m-fold product of Lebesgue spaces L p j (Z n ), j = 1, . . . , m and the link with the continuity of multilinear pseudo-differential operators on Z n .Necessary and sufficient conditions for multilinear pseudo-differential operators on Z n or T n to be a Hilbert-Schmidt operators are also given. We give a necessary condition for a multilinear pseudo-differential operators on Z n to be compact. A sufficient condition for compactness is also given.
Following Wong's point of view, we construct the minimal and maximal extension in L p (R n ), 1 < p < 1 for M-hypoelliptic pseudo-differential operators, which have been introduced and studied by Garello and Morando. We give some facts about the domain of minimal and maximal extensions of M-hypoelliptic pseudo-differential operators. For M-hypoelliptic pseudo-differential operators with constant coefficients, the spectrum and essential spectrum are computed.
We give a formula for the one-parameter strongly continuous semigroups e −tL λ and e −tà , t > 0 generated by the generalized Hermite operator L λ , λ ∈ R\{0} respectively by the generalized Landau operator A. These formula are derived by means of pseudo-differential operators of the Weyl type, i.e. Weyl transforms, Fourier-Wigner transforms and Wigner transforms of some orthonormal basis for L 2 (R 2n ) which consist of the eigenfunctions of the generalized Hermite operator and of the generalized Landau operator. Applications to an L 2 estimate for the solutions of initial value problems for the heat equations governed by L λ respectivelyÃ, in terms of L p norm, 1 ≤ p ≤ ∞ of the initial data are given.
Mathematics Subject Classification (2010). Primary 47G10, 47G30; Secondary 35S10.Following Wong's point of view (see [6], by Wong), we give a formula for the one-parameter strongly continuous semigroup e −tL λ , t > 0, generated by the generalized Hermite operator L λ , for a fixed λ ∈ R \ {0}, in terms of the Weyl transforms. Then we use it to obtain an L 2 estimate for the solution of the initial value problem for the heat equation governed by L λ , in terms of the L p norm, 1 ≤ p ≤ ∞, of the initial data.Similar results have also been derived for the generalized Landau operatorà which was firstly introduced by M.A. de Gosson (see [3] by de Gosson) who has studied its spectral properties.This work was completed with the support of our T E X-pert.
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