A NOTE ON THE HEAT KERNEL ON THE HEISENBERG GROUP

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1 A NOTE ON THE HEAT KERNEL ON THE HEISENBERG GROUP ADAM SIKORA AND JACEK ZIENKIEWICZ Abstract. We describe the analytic continuation of the heat ernel on the Heisenberg group H n (R. As a consequence, we show that the convolution ernel corresponding to the Schrödinger operator e isl is a smooth function on H n (R \ S s, where S s {(,, ±s H n (R : n, n +, n + 4,... }. At every point of S s the convolution ernel of e isl has a singularity of Calderón Zygmund type. Let H n (R denote the (n+-dimensional real Heisenberg group, that is, R n R n R with the group law (x, y, t(x, y, t ( x + x, y + y, t + t n Im (x r + iy r (x r iy r for all x, x, y, y in R n and t and t in R. For s in R, we define the set S s by r S s {(,, ±s H n (R : n, n +, n + 4,... }. We define H n (C lie H n (R. For (x, y, t H n (C, we write x for n r x r, and A and B for it (x + y /4 and (x + y / respectively. The vector fields X r, Y r (where r,..., n and T, given by X r y r x r t, Y r + x r y r t and T t, form a basis for the Lie algebra of left-invariant vector fields on H n (R. The Heisenberg Laplacian L is defined by n L Xr + Yr. r The subelliptic operator L admits a spectral resolution L λ de(λ, and therefore when Re s, one can define the operator e sl, bounded on L (H n (R, by the spectral theorem: e sl e sλ de(λ. Let p s be the convolution ernel of the operator e sl (see [5, (., (.]. When s >, e sl is the solution operator for the Heisenberg heat equation s u Lu and p s is called the heat ernel (see [6, (7.3, p. 7]. The goal of this note is to study the analytic continuation of the heat ernel p s. This is interesting from the point of view of the theory of analytic hypoellipticity (see [, ]. Another reason to study the analytic continuation of p s is to investigate the operator L α, 99 Mathematics Subject Classification. E3. Key words and phrases. Heisenberg group, Schrödinger equation. The research for this paper was supported by the Australian National University, Wroc law University, and the Polish Research Council KBN (KBN P3A 58 4.

2 ADAM SIKORA AND JACEK ZIENKIEWICZ equal to L + iαt, where α C (see Remar below and [6, (7.53 p. 73] (see also [3] for a detailed study of the operators L α and an explanation of the significance of L α. However, we are also motivated by the possibility of explicitly computing the ernel of the Schrödinger propagator e isl, where s R, using the analytic continuation of the heat ernel. Indeed, e isl is the solution operator for the Schrödinger equation ( s u(x, y, t, s ilu(x, y, t, s. In [5, p ], Strichartz noticed that In principle we could attempt to solve ( by analytic continuation from the solution of the heat equation. However, he abandoned this idea as this analytic continuation is delicate, so we approach the problem directly. Strichartz proved that, when s R, the convolution ernel of the Schrödinger operator p is is a smooth function on the open set {(x, y, t H n (R : t < n s }. In this note we propose a simple computation which allows us to handle the analytic continuation of the heat ernel in a straightforward manner. Then, using the analytic continuation of the heat ernel we show that p is is smooth on H n (R \ S s when s R, and at points in S s, the ernel p is has singularities of Calderón Zygmund type. This is in contrast to the Euclidean case where, for the standard Laplace operator, the ernel of the operator e is is bounded and smooth. In this context it is interesting to note that the convolution ernel of the operator e is(l T is smooth. The smoothness of the convolution ernel of the operator e is(l T is the last result proved in this note. A comprehensive discussion of harmonic analysis on the Heisenberg group can be found in [5] or [6, Chapter ]. The following theorem is the main result of this note. Theorem. The function p extends to an analytic function on H n (C \ S, where S {(x, y, t H n (C : ±it n x +y 4 N}. For all (x, y, t H n (C \ S, p (x, y, t (P (x, y, t + P (x, y, t, (4π n+ where ( P (x, y, t n n! h j ( ( n + j + h n + j h j B j (n + (j + h A n+j+. The sum ( is absolutely uniformly convergent on compact subsets of H n (C \ S. Moreover, (3 p s (x, y, t s p ( x y,, t n+ s s s for all s such that Re s.

3 A NOTE ON THE HEAT KERNEL ON THE HEISENBERG GROUP 3 Proof. By virtue of the well nown formula for the heat ernel on the Heisenberg group (see [6, (7.36, p. 7], or [5, (5.] p (x, y, t (4π n+ (4π n+ λ n cos tλ ( λ(x (sinh λ exp + y coth λ n 4 λ n e itλ ( λ(x (sinh λ exp + y coth λ n 4 + (4π n+ (P (x, y, t + P (x, y, t, (4π n+ λ n e itλ ( λ(x (sinh λ exp + y coth λ n 4 say. We note that, if λ > and m N, then ( m + (e λ e λ m e (+mλ, where ( (4 ( if and otherwise. Now n P (x, y, t ( n + ( λb coth λ λ n e itλ e (+nλ exp ( n + λ n exp ( λ(a n ( λb(coth λ exp ( n + λ n exp ( λ(a n (λb(coth λ j j j! j ( n + λ n exp ( λ(a j n (λb j j!(e λ e λ j j ( ( n + j + l λ n+j exp ( λ(a n ( + j + l B j l j! j l ( ( ( n + j + l n + j B j n! l j (n + ( + j + l A n+j+. j l Note that, for any compact subset K of H n (C \ S, the sum in the last line of (4 is absolutely uniformly convergent on K and that for all (x, y, t H n (R all expressions in the formula (4 are absolutely convergent. Indeed, there exists a constant C K such that (n + ( + j + l A n j C K ( + j + l + n j (x, y, t K. Next, ( n + C n n and ( j + l (j + + l + j /j!. l

4 4 ADAM SIKORA AND JACEK ZIENKIEWICZ Hence ( ( ( n + j + l n + j B j l j (n + ( + j + l A n+j+ j l n j n B j C n C n+ C K j!( + j + l + n+ C nc K C nc K C nc K C K,n. j l j j j n j n B j j!( + j + n+ B j j n j!( + j + B j j n (j +! Now we note that, for any (x, y, t H n (R, (n + ( + j + l A (n + ( + j + l B, and sup λ R+ λ(coth λ, so ( ( ( n + j + l n + j B j n! l j (n + ( + j + l A n+j+ j l ( ( ( n + j + l n + j ( B j n! l j (n + ( + j + l B/ n+j+ j l ( n + λ n exp ( λ( + n B/ ( λb(coth λ exp n λ n ( B(λ (sinh λ exp. n The Lebesgue monotone convergence theorem proves the absolute convergence for all (x, y, t H n (R. We obtain ( by virtue of the identity ( ( ( n + m + l n + m + h l +lh l and (4. To conclude the proof of (3 we note that L is a homogeneous operator so (3 holds when s > (see [6, Proposition 7.3, p. 7] and both sides of (3 are analytic as a function of s when Re s >. Corollary. When s R, the convolution ernel of the Schrödinger operator p is is smooth on H n (R \ S s, where S s {(,, ±s H n (R : n, n +, n + 4,... }. At every point of S s, the ernel p is has a singularity of Calderón Zygmund type. Proof. Corollary is a straightforward consequence of (3, and (4 or (. h

5 A NOTE ON THE HEAT KERNEL ON THE HEISENBERG GROUP 5 Remar. There is an alternative proof of Theorem. One can use the formula [5, (.8] to prove that P (x, y, t is equal to ( ( (n +! n (B + ( A + n + + A n.! ( A + n + n++ n + B + ( A + n + An argument similar to that in the proof of Theorem shows that the above sum is uniformly absolutely convergent on any compact subset of H n (C \ S. Remar. In [6, (7.53, p. 73], Taylor noticed that, when s > and Re α < n, (5 K e slα (x, y, t p s (x, y, t isα, where K e slα is the convolution ernel of the operator e slα and L α L + iαt. By virtue of Theorem, (5 holds also when Re α n. Note that, when Re α n, the ernel K e slα, s > is no longer smooth and it has a singularity of Calderón Zygmund type at the point (,, s Im α. We would lie to end with another observation, concerning a full Laplace operator on the Heisenberg group. We define this operator by the following formula L n r X r + Y r T L T. Theorem 3. For any s R \ {}, the convolution ernel of the operator e is L extends to an analytic function on H n (C. Proof. Write γ for e iπ/4. We denote the convolution ernels of the operators e is L and e s(γl T by K e is and K L e s(γl T respectively. Note that (see [6, (7.36] and (3 λ n cos(tλ/(γs ( λ(x + y coth λ K e s(γl+t (x, y, t exp e sλ. (4πγ n+ (sinh λ n 4γs Hence K e s(γl+t is an analytic function on H n (C. Now to finish the proof it is enough to note that K L(x, e is y, t K e s(γ γl+γ T (x, y, t γ K n+ e s(γl+t ( x y,, t γ γ γ for all s > and (x, y, t H n (R (see [4, (8]. Remar 3. When Re s >, one can use the theory of analytic hypoellipticity to investigate the smoothness and analyticity of the ernel K e s see, e.g., [, ]. However, L analytic hypoellipticity cannot be used directly to investigate the convolution ernel of K e s when Re s. L References [] Michael Christ, Analytic hypoellipticity, representations of nilpotent groups, and a nonlinear eigenvalue problem, Due Math. J., 7 (993, [] Michael Christ, Nonexistence of invariant analytic hypoelliptic differential operators on nilpotent groups of step greater than two, in Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 99, Princeton Univ. Press, Princeton, NJ (995, [3] G. B. Folland and E. M. Stein, Estimates for the b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 7 (974, [4] Adam Siora, On the L L norms of spectral multipliers of quasi-homogeneous operators on homogeneous groups, Trans. Amer. Math. Soc., 35 (999, [5] Robert S. Strichartz, L p harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Anal., 96 (99,

6 6 ADAM SIKORA AND JACEK ZIENKIEWICZ [6] Michael E. Taylor, Noncommutative harmonic analysis (American Mathematical Society, Providence, R.I., 986. Adam Siora, Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT, Australia address: Jace Zieniewicz, Instytut Matematyczny, Uniwersytet Wroc lawsi, Wroc law, pl. Grunwaldzi /4, Poland address: