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Colloquium Mathematicum

2000 | 83 | 2 | 183-200
Tytuł artykułu

Fundamental solutions for translation and rotation invariant differential operators on the Heisenberg group

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
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, y<0}. 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 $H_1$.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
183-200
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-05-04
poprawiono
1999-06-14
Twórcy
autor
• Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy
Bibliografia
• [1] C. Benson, A. H. Dooley and G. Ratcliff, Fundamental solutions for powers of the Heisenberg sub-laplacian}, Illinois J. Math. 37 (1993), 455-476.
• [2] C. Benson, J. Jenkins, G. Ratcliff and T. Worku, Spectra for Gelfand pairs associated with the Heisenberg group, Colloq. Math. 71 (1996), 305-328.
• [3] J. Bochnak, M. Coste et M.-F. Roy, Géométrie Algébrique Réelle, Ergeb. Math. Grenzgeb. 12, Springer, Berlin, 1987.
• [4] W. Chang, Invariant differential operators and P-convexity of solvable Lie groups, Adv. Math. 46 (1982), 284-304.
• [5] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, 1953.
• [6] G. B. Folland and E. M. Stein, Estimates for the $\οverline 𝛛_b$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522.
• [7] L. Hörmander, Analysis of Linear Partial Differential Operators II, Springer, Berlin, 1983.
• [8] L. Schwartz, z Théorie des distributions. Tome I, Hermann, Paris, 1957.
• [9] E. M. Stein, An example on the Heisenberg group related to the Lewy operator, Invent. Math. 69 (1982), 209-216.
• [10] G. Szegő, Orthogonal Polynomials, Colloq.Publ. 23, Amer. Math. Soc., 1975.
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