By Marco Bramanti
Hörmander's operators are an immense category of linear elliptic-parabolic degenerate partial differential operators with tender coefficients, which were intensively studied because the past due Sixties and are nonetheless an lively box of analysis. this article presents the reader with a common evaluation of the sphere, with its motivations and difficulties, a few of its basic effects, and a few contemporary strains of development.
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Additional info for An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields
3 28 2 Hörmander’s Operators: Why they are Studied since n n ξ f = ξ ξu = ξ ξz k u dz k k=1 ξz2h z k u dz h ∧ dz k = h,k=1 ⎠ ξz2h z k u − ξz2k z h u dz h ∧ dz k = 0. n = h>k=1 Generalizing, one considers the space C(∞p,q) (D) of forms ∈ f = f I,J dz I ∧ dz J I,J where I = j1 , j2 , . . , j p , J = j1 , j2 , . . , jq are multiindices, dz I = dz j1 ∧ dz j2 ∧ . . ∧ dz j p ; dz J = dz j1 ∧ dz j2 ∧ . . ∧ dz jq ; the sum ∈ is made over increasing multiindices and the f I,J ∗ C ∞ (D) are defined for arbitrary I, J so that they are antisymmetric.
A duality argument, and the symmetry property of the kernel, imply L p continuity also in the range 2 < p < ∗. All these continuity estimates on Tβ hold with constants independent of β. Finally, one has to prove the actual convergence of Tβ to some T , which as a consequence will satisfy the same L p bound. This can be a delicate point in the abstract setting, but in applications to PDEs usually is not a problem, since the existence of the limit as β → 0 is a starting point (representation formulas).
Remark 25 For α = 0, Lα is a sum of squares of Hörmander’s vector fields, therefore both hypoelliptic and locally solvable. It is usually called the sublaplacian on the Heisenberg group. This is by far the most studied among Hörmander’s operators, and we will say more about it in the following. Even though the Lα ’s involved in b are only those for α = n, n − 2, n − 4, . . , −n, these operators have been studied more generally for any α ∗ C, encountering interesting phenomena: Theorem 26 There exists a sequence of forbidden values α = ±n, ± (n + 2) , ± (n + 4) , .