[PDF] - Let d = dim( G ) = 2 d 1 + d 2 and Theorem. suppose ( d 1)

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Lp 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, Lp-estimates for the wave equation on the Heisenberg group, Rev.

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Sub-Laplacian on the Heisenberg group Hd1 is Rd, d = 2d1 + 1 with the group law (x, u) · (x′, u′) = (x + x′, u + u′ + 1 2Jx, x′) where J =

−I

Left invariant vector fields Xi = ∂ ∂xi − xd1+i 2 ∂ ∂u Xd1+i = ∂ ∂xd1+i + xi 2 ∂ ∂u i = 1, . . . , d1. The Sub-Laplacian is the sums

L := −

X2

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IVP for Wave equation: ∂2v (∂t)2 = −Lv v

vt

Formal solution: v(·, t) = cos(t √ L)f + sin(t √ L) √ L g. Makes sense by spectral theorem. Note: While √ L is defined by spectral theory, we do not know a pseudodifferential operator representation. One aims for Lp result of the form v(·, t)p (I + t2L)

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This is accomplished by showing Lp(Hd1) bound- edness of T α

M¨ uller and Stein proved the almost sharp result that T α

is bounded on Lp(Hd1) if α > (d − 1)|1/p − 1/2|.

A more geometric approach is needed.

(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 Lp estimates.

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Heisenberg type groups

dim(g2) = d2, so that [g, g] ⊂ g2 ⊂ z(g).

g1, g2 orthogonal subspaces, and for µ ∈ g∗

define symplectic form ωµ on g1 ωµ(V, W) := µ

transformation Jµ on g1 such that ωµ(V, W) = Jµ(V ), W, and, one a group of Heisenberg type we have J2

Consider g as group G with BCH product, i.e. (x, u) · (x′, u′) = (x + x′, u + u′ + 1 2 Jx, x′) Choose ONB X1, . . . , X2d1 of g1, identify with left invariant vector fields on G and form the Sub-Laplacian L := − 2d1

IVP for the wave equation is then formulated in the same way.

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Theorem. Let d = dim(G) = 2d1 + d2 and suppose α ≥ (d − 1)|1

(I + t2L)−α/2eit

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, t2u) it is sufficient to con- sider the case t = ±1.

for p = 1 (though not invariant under auto- morphic dilations).

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Review of the Euclidean results by Miyachi and Peral, for the wave equation on Rd. For α = d−1

prove that

ei|ξ| (1 + |ξ|2)α/2

defines a bounded operator from the local Hardy space h1 to L1 (in fact bounded on h1). Main step is to choose an atom f supported

where the high frequency part has spectrum in {|ξ| ≥ r−1}. Singular support of kernel: {x : |x| = 1}. Let: Nr =

Hα

complement of Nr.

frequency part.

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Singular support for the convolution kernel Extension of Nachman’s result: The singular support Σ of K = e±i

as the set of all (x, u) ∈ G for which |x| = r(s) :=

s

2s2

This could be computed by looking at the time

at the origin and projecting into space. Unfortunately there are no explicit formulas for the kernel of wave semigroup eit

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Nachman computed the asymptotics of the ker- nel near a generic point in the singular support (no uniformity). M¨ uller and Stein proved estimates for L1 norm

√ L)ei

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¨

semigroup eitL. (Gaveau, Hulanicki, M¨ uller and Ricci, etc.) Motivation for our approach: Bochner’s sub-

e−

1 √π

∞

s−1/2e−se−L/4sds. Complexify?

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Try to express ei

since the Lp operator norms for χ(λ−1√ L)ei

are (supposed to be) much smaller than those for χ(λ−1√ L)eiLf. Here χ smooth and sup- ported in (1/2, 2). We use stationary phase calculations (and stan- dard multiplier results)

χ1(λ−1√ L)ei

= λ3/2

= √ λ

where χ ∈ C∞ supported in (1/4, 4) and EλLp→Lp = O(λ−N).

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Formulas for the Schr¨

M¨ uller and Ricci, ... ). Apply partial Fourier transform fµ(x) =

f(x, u)e−2πiµ·u du. Then (Lf)µ = Lµfµ and eitLµδ = Γµ

where Γµ

e

(det(2 sin(2πitJµ)))1/2

cot(2πitJµ) = iJµ|µ|−1 cot(2πt|µ|) similarly for the sin, and thus

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γµ

sin(2πt|µ|)

d1e−iπ

and also

c (cos(2πt|µ|))d1ei2π

so that γµ

χ(λ−1√ Lµ)ei

and split the integral by localizing where 2πµ/λ is near kπ or near kπ + π

χ(λ−1√ L)ei

Ak,λ +

Bk,λ + Error where on the g2-Fourier transform side Aµ

Bµ

π)ei/4sγµ

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Applying the Fourier transform, then station- ary phase, and changes of variables we see that the convolution kernel Bk,λ(x, u) is the sum of two terms of the form λd1+d2+1

π)td1+d2−1 × eiλs(1−|x|2t cot t±4|u|)dt a(4λs|u|)ds. Here a is symbol of order −d2−1

. Similar ex- pression for Ak,λ(x, u) with a term η0(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|)fµ then for k ≈ 2n the terms Bk,λ and Ak,λ are essentially reproduced by operators η(λ−1√ L)η(λ−12−n|U|).

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Composing with η(λ−1√ L)η(λ−12−n|U|) gives a huge gain if 2n ≥ λ: The convolution kernel of η(λ−1√ L) is of the form λ2d1+2d2Φ(λx, λ2u). The operator η(λ−12−n|U|) corresponds to a standard Euclidean convolution in the central variable; the convolution kernel is 2nλΨ(2nλu) where Ψ has cancellation. This cancellation can be exploited if 2nλ ≫ λ2 and leads to

L)η(λ−12−n|U|)

Now back to the analysis of Bλ,k (and Aλ,k) with focus on k ≤ λ.

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Use polar coordinates r = |x|, v = 4|u| and set νk = kπ + π

λ−(d−1)/2Bλ,k(x, u) ∼ kd1+d2−1λ1+d2/2 (1 + λk|u|)(d2−1)/2 ×

where sφ(t, r, v) = s t(1 t − r2 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 φt(t, r, v) = 0 . These equations are solved by a ‘curve’ in (r, v) = (|x|, |u|) space parametrized by r(t) =

t

t − sin 2t 2t2 for t near 2k + 1 2 π. I.e. Nachman’s description.

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1/k 1/(k+1) 1/k

The values t = kπ (where cot has poles) cor- respond to points in the center. The “cusp points” in the picture correspond to teh values of t = tk where tan t = t; at those points φ, φt, φtt vanish simultaneously (but φttt does not).

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Some lengthy Fourier integral analysis of the

reveals that

k ≤ λ and a more unfavorable bound for k > λ which is however multiplied by (λ/k)N. Similar state- ments for Aλ,k ... Note: Natural uniform estimates are formu- lated not for λ−(d−1)/2Bλ,k but for the non- isotropically rescaled kernels λ−(d−1)/2k2d1+2d2Bλ,k(kx, k2u). Note that the singular set for Bλ,k is essen- tially a (1/k, 1/k2) box at height ≈ k−1, so this scaling is natural.

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Now: Putting dyadic scales in λ together re- quires some Hardy-space theory. There are at least two ways to do this. We define an atomic local Hardy space h1(G) with respect to isotropic dilations.

a standard atom supported in a Euclidean ball

lation if r < 1.

ing Heisenberg left translations. Note that if the distance of P to the center is

1 then these atoms are essentially Euclidean

atoms.

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Start over and decompose ei

(I + L)

=

Wn + Error where more or less W0 =

2−j(d−1)/2η(2−j√ L)η0(2−j|U|)ei

and Wn =

2−j(d−1)/2η(2−j√ L)η(2−j−n|U|)ei

We note that essentially Wn ∼

λ−(d−1)/2

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gument close to the Euclidean (or general FIO) case.

Wn be the operator with nonisotropically rescaled kernel 2−2n(d1+d2)Wn(2−nx, 2−2nu) then a variant of the Euclidean argument ap- plies to the rescaled operator, namely Wnf1 2−n(d1−1

Analytic interpolation with obvious L2 bounds yields nice Lp bounds for p < 2. Note that, in this approach, because of the rescaling we are working not with a single Hardy- space but rather a scale of Hardy spaces.

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we have for the un-rescaled operators Wnf1 (1 + n)2−n(d1−1

(*) So h1(G) is ‘the’ Hardy space for this problem: Corollary: The operator

maps h1(G) to L1. Issue in the proof of (*): Given an isotropic atom centered at the origin, there are O(n) dyadic scales of λ for which cancellation does not help but L2 estimates on the natural ex- ceptional sets are insufficient. But: On those O(n) scales we may simply use the previous uniform L1 bounds.