Let d = dim( G ) = 2 d 1 + d 2 and Theorem. suppose ( d 1) | 1 p - PDF document

L p estimates for the wave equation on Heisenberg type groups Joint work with Detlef M¨ uller. In progress. Two references to previous work: A.I. Nachman, The wave equation on the Heisen- berg group, CPDE 7 (1982), 675–714. uller, E. M. Stein, L p -estimates for the D. M¨ wave equation on the Heisenberg group, Rev. Mat. Iberoamericana 15 (1999), 297–334.

Sub-Laplacian on the Heisenberg group H d 1 is R d , d = 2 d 1 + 1 with the group law ( x, u ) · ( x ′ , u ′ ) = ( x + x ′ , u + u ′ + 1 2 � Jx, x ′ � ) � � 0 I where J = . − I 0 Left invariant vector fields − x d 1 + i ∂ ∂ X i = ∂x i 2 ∂u ∂ + x i ∂ X d 1 + i = ∂x d 1 + i 2 ∂u i = 1 , . . . , d 1 . The Sub-Laplacian is the sums of squares operator 2 d 1 � X 2 L := − j . j =1 2

IVP for Wave equation: ∂ 2 v ( ∂t ) 2 = − Lv � � � � v � t =0 = f, v t � t =0 = g Formal solution: √ √ L ) f + sin( t L ) v ( · , t ) = cos( t √ g. L Makes sense by spectral theorem. √ Note: While L is defined by spectral theory, we do not know a pseudodifferential operator representation. One aims for L p result of the form α α 2 − 1 g � p � v ( · , t ) � p � � ( I + t 2 L ) 2 f � p + � t ( I + t 2 L ) 3

This is accomplished by showing L p ( H d 1 ) bound- edness of √ L . t = ( I + t 2 L ) − α/ 2 e it T α M¨ uller and Stein proved the almost sharp result that T α is bounded on L p ( H d 1 ) if t α > ( d − 1) | 1 /p − 1 / 2 | . • M-S used representation theoretic methods. A more geometric approach is needed. • Can one identify the solution operators as (kind of) Fourier integral operators ? FIO’s and canonical relations etc. are used in Melrose’s approach to the wave equation for subelliptic operators but the parametrices are not adequate for obtaining L p estimates. 4

Heisenberg type groups • Lie algebra g = g 1 ⊕ g 2 , with dim( g 1 ) = 2 d 1 , dim( g 2 ) = d 2 , so that [ g , g ] ⊂ g 2 ⊂ z ( g ) . g 1 , g 2 orthogonal subspaces, and for µ ∈ g ∗ 2 \{ 0 } define symplectic form ω µ on g 1 � � ω µ ( V, W ) := µ [ V, W ] Then there is a unique skew-symmetric linear transformation J µ on g 1 such that ω µ ( V, W ) = � J µ ( V ) , W � , and, one a group of Heisenberg type we have J 2 µ = −| µ | 2 I. Consider g as group G with BCH product, i.e. ( x, u ) · ( x ′ , u ′ ) = ( x + x ′ , u + u ′ + 1 Jx, x ′ � ) 2 � � Choose ONB X 1 , . . . , X 2 d 1 of g 1 , identify with left invariant vector fields on G and form the Sub-Laplacian L := − � 2 d 1 j =1 X 2 j as before. The IVP for the wave equation is then formulated in the same way. 5

Let d = dim( G ) = 2 d 1 + d 2 and Theorem. suppose α ≥ ( d − 1) | 1 p − 1 2 | , 1 < p < ∞ . Then √ � ( I + t 2 L ) − α/ 2 e it L f � p ≤ C p � f � p . This is an analogue of the sharp results of Miy- achi and Peral for the Laplacian. By invariance with respect to the automorphic dilations ( x, u ) → ( tx, t 2 u ) it is sufficient to con- sider the case t = ± 1. • We shall also formulate a Hardy space result for p = 1 (though not invariant under auto- morphic dilations). 6

Review of the Euclidean results by Miyachi and Peral, for the wave equation on R d . For α = d − 1 prove that 2 e i | ξ | � � T α f ( ξ ) = f ( ξ ) (1 + | ξ | 2 ) α/ 2 defines a bounded operator from the local Hardy space h 1 to L 1 (in fact bounded on h 1 ). Main step is to choose an atom f supported on {| x | ≤ r } for r ≪ 1. Split T α f = L α r f + H α r f where the high frequency part has spectrum in {| ξ | ≥ r − 1 } . Singular support of kernel: { x : | x | = 1 } . � � � � � � Let: N r = � | x | − 1 � ≤ Cr x : and estimate r f � L 1 ( N r ) � r 1 / 2 �H α r f � L 2 ( N r ) � r d/ 2 � f � 2 � � f � h 1 . �H α • Use decay estimates for the kernel in the complement of N r . • Use cancellation and L 1 bounds for the low frequency part. 7

Singular support for the convolution kernel Extension of Nachman’s result: √ The singular support Σ of K = e ± i L δ is given as the set of all ( x, u ) ∈ G for which � � sin s � � � � | x | = r ( s ) := � � s for some s ∈ R . � � 2 s − sin 2 s � � � � 4 | u | = v ( s ) := � � 2 s 2 This could be computed by looking at the time one map for the bicharacteristic curves starting at the origin and projecting into space. Unfortunately there are no explicit formulas for √ the kernel of wave semigroup e it L . 8

Nachman computed the asymptotics of the ker- nel near a generic point in the singular support (no uniformity). uller and Stein proved estimates for L 1 norm M¨ √ √ L ) e i L δ . of χ ( Used Gelfand transforms for radial functions, Strichartz projectors, Poisson summation formula, but little geometry (cf. also [MRS I,II]). Logarithmic blowup occurs. However there are explicit formulas for the heat semigroup, and therefore for the Schr¨ odinger semigroup e itL . (Gaveau, Hulanicki, M¨ uller and Ricci, etc.) Motivation for our approach: Bochner’s sub- ordination formula: � ∞ √ 1 L = e − s − 1 / 2 e − s e − L/ 4 s ds. √ π 0 Complexify? 9

√ L by an integral involving Try to express e i e isL . For L p estimates this is counterintuitive √ since the L p operator norms for χ ( λ − 1 √ L ) e i L f are (supposed to be) much smaller than those for χ ( λ − 1 √ L ) e iL f . Here χ smooth and sup- ported in (1 / 2 , 2). We use stationary phase calculations (and stan- dard multiplier results) Lemma. For λ ≫ 1 √ χ 1 ( λ − 1 √ L ) e i L � i = λ 3 / 2 4 s e isL ds + E λ χ ( λs ) e � √ χ ( s ) e i λ 4 s e i s λ L ds + E λ = λ where χ ∈ C ∞ supported in (1 / 4 , 4) and � E λ � L p → L p = O ( λ − N ) . 10

Formulas for the Schr¨ odinger semigroup (from M¨ uller and Ricci, ... ). Apply partial Fourier transform � f ( x, u ) e − 2 πiµ · u du. f µ ( x ) = g 2 Then ( Lf ) µ = L µ f µ and e itL µ δ = Γ µ t ( ω µ ) ∧ ( d 1 ) = γ µ t where π 2 � J µ x, cot(2 πitJ µ ) x ) � e Γ µ t ( x ) = (det(2 sin(2 πitJ µ ))) 1 / 2 • On Heisenberg type groups J 2 µ = −| µ | 2 I , so cot(2 πitJ µ ) = iJ µ | µ | − 1 cot(2 πt | µ | ) similarly for the sin, and thus 11

� � d 1 e − i π | µ | 2 | µ | cot(2 πt | µ | ) | x | 2 γ µ t ( x ) = c sin(2 πt | µ | ) and also (cos(2 πt | µ | )) d 1 e i 2 π c | µ | tan(2 πt | µ | ) | ξ | 2 γ µ � t ( ξ ) = so that γ µ t and then γ t are well defined as dis- tributions. Use subordination � √ χ ( λ − 1 √ L µ δ = λ 3 / 2 χ ( λs ) e i/ 4 s e isL µ ds + Error L µ ) e i and split the integral by localizing where 2 πµ/λ is near kπ or near kπ + π 2 . Thus √ ∞ ∞ χ ( λ − 1 √ � � L ) e i L δ = A k,λ + B k,λ + Error k =0 k =0 where on the g 2 -Fourier transform side � A µ k,λ = λ 3 / 2 χ ( λs ) η (2 πs | µ | − kπ ) e i/ 4 s γ µ s ds � B µ k,λ = λ 3 / 2 χ ( λs ) η (2 πs | µ | − 2 k +1 π ) e i/ 4 s γ µ s ds 2 12

Applying the Fourier transform, then station- ary phase, and changes of variables we see that the convolution kernel B k,λ ( x, u ) is the sum of two terms of the form �� λ d 1 + d 2 + 1 π ) t d 1 + d 2 − 1 × χ ( s ) η 0 ( t − 2 k +1 2 2 e iλs (1 −| x | 2 t cot t ± 4 | u | ) dt a (4 λs | u | ) ds. Here a is symbol of order − d 2 − 1 . Similar ex- 2 pression for A k,λ ( x, u ) with a term η 0 ( s − kπ ). This requires further dyadic decomposition with s − kπ ≈ 2 − l to come to grips with the poles of the cot. Note that there is some hidden cancellation when k ≥ λ . If we use the notation [ m ( | U | ) f ] µ = m ( | µ | ) f µ then for k ≈ 2 n the terms B k,λ and A k,λ are essentially reproduced by operators η ( λ − 1 √ L ) η ( λ − 1 2 − n | U | ) . 13

Composing with η ( λ − 1 √ L ) η ( λ − 1 2 − n | U | ) gives a huge gain if 2 n ≥ λ : The convolution kernel of η ( λ − 1 √ L ) is of the form λ 2 d 1 +2 d 2 Φ( λx, λ 2 u ) . The operator η ( λ − 1 2 − n | U | ) corresponds to a standard Euclidean convolution in the central variable; the convolution kernel is 2 n λ Ψ(2 n λu ) where Ψ has cancellation. This cancellation can be exploited if 2 n λ ≫ λ 2 and leads to � � � η ( λ − 1 √ � � L ) η ( λ − 1 2 − n | U | ) � L p → L p � min { 1 , (2 − n λ ) N } . Now back to the analysis of B λ,k (and A λ,k ) with focus on k ≤ λ . 14

Use polar coordinates r = | x | , v = 4 | u | and set ν k = kπ + π 2 . Assume k ≥ 0. Then k d 1 + d 2 − 1 λ 1+ d 2 / 2 λ − ( d − 1) / 2 B λ,k ( x, u ) ∼ (1 + λk | u | ) ( d 2 − 1) / 2 �� χ ( s ) η ( t − ν k ) e iλsφ ( t,r,v ) dt ds × where sφ ( t, r, v ) = s t (1 t − r 2 cot t + v ) . The singular support is given by the set of all ( x, u ) for which there exists a t such that φ ( t, r, v ) = 0 and φ t ( t, r, v ) = 0 . These equations are solved by a ‘curve’ in ( r, v ) = ( | x | , | u | ) space parametrized by � � sin t � � � � r ( t ) = � � for t near 2 k + 1 t π . v ( t ) = 1 t − sin 2 t 2 2 t 2 I.e. Nachman’s description. 15