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W. Wong, The heat equation for the Hermite operator on the Heisenberg group, Hokkaido Math. J. 34 (2005), 393–404. [17] K. Yosida, Functional Analysis, Reprint of Sixth Edition, Springer-Verlag, 1995. W. ca Operator Theory: Advances and Applications, Vol. 189, 43–65 c 2008 Birkh¨ auser Verlag Basel/Switzerland Construction of the Fundamental Solution and Curvature of Manifolds with Boundary Chisato Iwasaki Abstract. The method of construction of the fundamental solution for the heat equations of the initial boundary value problem on manifolds with boundary, which is applicable to calculate traces of operators, is discussed.

Ir ) ∈ I (r = 2m) tr(βI (−R)m ) = 2n−r−m sign(π) sign(σ) π,σ∈Sr ×Riπ(1) iπ(2) iσ(1) iσ(2) · · · Riπ(r−1) iπ(r) iσ(r−1) iσ(r) . 4. (1-i) If (1-ii) If is odd, C (x, M ) = 0 is even ( = 2m), C (x, M ) = CI (x, M ), I∈I, (I)= where CI (x, M ) = n 1 √ 2 π 1 1 m! 2 m sign(π) sign(σ) π,σ∈S × Riπ(1) iπ(2) iσ(1) iσ(2) · · · · · · Riπ( −1) iπ( ) iσ( −1) iσ( ) for I = (i1 , i2 , . . , i ) ∈ I. 3, we have the following equation for (I) = r √ n r n r 2n−r t− 2 + 2 CI (x, M ) detg + 0(t− 2 + 2 +1 ), if r = 2m ; tr (βI u˜0 (t, x, x)) = n r 1 if r is odd.

Wong, Global solutions of semilinear heat equations in Hilbert spaces, Abstr. Appl. Anal. 1 (1996), 263–276. [8] J. W. Wong, Positive deﬁnite temperature functions on the Heisenberg group, Appl. Anal. 85 (2006), 987–1000. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993. [10] S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Birkh¨ auser, 1998. [11] S. Thangavelu, An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups, Birkh¨ auser, 2004.