Fundamental solutions for translation and rotation invariant differential operators on the Heisenberg group
Colloquium Mathematicum, Tome 83 (2000) no. 2, pp. 183-200
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Let $H_1$ be the three-dimensional Heisenberg group. Consider the left invariant differential operators of the form D=P(-iT,-L), where P is a polynomial in two variables with complex coefficients, L is the sublaplacian on $H_1$ and T is the derivative with respect to the central direction. We find a fundamental solution of D, whose definition is related to the way the plane curve defined by P(x,y)=0 intersects the Heisenberg fan F = A ∪ B, A = {(x,y)∈ ℝ^2: y=(2m+1)|x|, m ∈ ℕ, B= {(x,y) ∈ ℝ^2: x=0, y0}. We can write an explicit expression of such a fundamental solution when the curve P(x,y)=0 intersects F at finitely many points, all belonging to A and, if one of them is the origin, the monomial $y^k$ has a nonzero coefficient, where k is the order of zero at the origin. As a consequence, such operators are globally solvable on $H_1$.
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author = {Priscilla Gorelli},
title = {Fundamental solutions for translation and rotation invariant differential operators on the {Heisenberg} group},
journal = {Colloquium Mathematicum},
pages = {183--200},
publisher = {mathdoc},
volume = {83},
number = {2},
year = {2000},
doi = {10.4064/cm-83-2-183-200},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm-83-2-183-200/}
}
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Priscilla Gorelli. Fundamental solutions for translation and rotation invariant differential operators on the Heisenberg group. Colloquium Mathematicum, Tome 83 (2000) no. 2, pp. 183-200. doi: 10.4064/cm-83-2-183-200
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