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## Studia Mathematica

1997 | 122 | 1 | 15-37
Tytuł artykułu

### Initial value problem for the time dependent Schrödinger equation on the Heisenberg group

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Treść / Zawartość
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EN
Abstrakty
EN
Let L be the full laplacian on the Heisenberg group $ℍ^{n}$ of arbitrary dimension n. Then for $f ∈ L^{2}(ℍ^{n})$ such that $(I-L)^{s/2}f ∈ L^{2}(ℍ^{n})$, s > 3/4, for a $ϕ ∈ C_{c}(ℍ^{n})$ we have $ʃ_{ℍ^{n}} |ϕ(x)| sup_{0 < t≤1} |e^{(√-1)tL}f(x)|^{2} dx ≤ C_{ϕ} ∥f∥_{W^{s}}^{2}$. On the other hand, the above maximal estimate fails for s < 1/4. If Δ is the sublaplacian on the Heisenberg group $ℍ^{n}$, then for every s < 1 there exists a sequence $f_{n} ∈ L^{2}(ℍ^{n})$ and $C_{n} > 0$ such that $(I-L)^{s/2} f_{n} ∈ L^{2}(ℍ^{n})$ and for a $ϕ ∈ C_{c}(ℍ^{n})$ we have $ʃ_{ℍ^{n}} |ϕ(x)| sup_{0 < t≤1} |e^{(√-1)tΔ} f_{n}(x)|^{2} dx ≥ C_{n} ∥f_{n}∥_{W^{s}}^{2}, lim_{n→∞}C_{n} = +∞$.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
15-37
Opis fizyczny
Daty
wydano
1997
otrzymano
1995-09-20
Twórcy
autor
• Institute of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, zenek@math.uni.wroc.pl
Bibliografia
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• [DK] B. E. J. Dahlberg and C. E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation, in: Harmonic Analysis, Lecture Notes in Math. 908, Springer, 1982, 205-209.
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• [E2] A. Erdélyi, W. Magnus, F. Oberhettinger and G. F. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953.
• [HR] A. Hulanicki and F. Ricci, A Tauberian theorem and tangential convergence of bounded harmonic functions on balls in $ℂ^n$, Invent. Math. 62 (1980), 325-331.
• [KPV1] C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33-69.
• [KPV2] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc. 4 (1991), 323-347.
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• [SS] P. Sjögren and P. Sjölin, Convergence properties for the time dependent Schrödinger equation, Ann. Acad. Sci. Fenn. Ser. AI Math. 14 (1989), 13-25.
• [S1] P. Sjölin, Regularity of solutions to the Schrödinger equations, Duke Math. J. 55 (1987), 699-715.
• [S2] P. Sjölin, Global maximal estimates for solutions to the Schrödinger equation, Studia Math. 110 (1994), 105-114.
• [S3] P. Sjölin, Radial functions and maximal estimates for solutions to the Schrödinger equation, J. Austral. Math. Soc. 59 (1995), 134-142.
• [Sz] G. Szegő, Orthogonal Polynomials, Colloq. Publ. 23, Amer. Math. Soc., 1939.
• [V] L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874-878.
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