Dipartimento di Matematica

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1 Dipartimento di Matematica V. CASARINO TWO-PARAMETER ESTIMATES FOR JOINT SPECTRAL PROJECTIONS ON S n Rapporto interno N. 3, febbraio 7 Poitecnico di Torino Corso Duca degi Abruzzi, 4-9 Torino-Itaia

2 TWO-PARAMETER ESTIMATES FOR JOINT SPECTRAL PROJECTIONS ON S n VALENTINA CASARINO Abstract. We prove sharp two-parameter estimates for the L p -L norm, p, of the joint spectra projectors associated to the Lapace-Betrami operator and to the Kohn Lapacian on the unit sphere S n in C n. Then, by using the notion of contraction of Lie groups, we deduce anaogous estimates for joint spectra projections on the reduced Heisenberg group h.. Introduction In this paper we prove two-parameter bounds for the L p L -norm, p, of joint spectra projection operators on the compex unit sphere S n in C n. Then we show how to deduce from our resuts on S 3 anaogous estimates for joint spectra projections on the reduced Heisenberg group h, by using the notion of contraction of Lie groups. In the ast two decades, L p L -norm of spectra custers has been deepy investigated and the first bounds proved by C. Sogge for the harmonic projectors associated to the Lapace-Betrami operator Σ n on the unit sphere Σ n in R n [So86] have been generaized in various directions, unti to cover the case of a generic eiptic operator on a compact riemannian manifod with or without boundary [So93], [So]. Anyway, it is worth noticing that the case of Σ n, Σ n pays a specia roe; indeed, in spite of the symmetry properties of Σ n and of the invariance of Σ n under the action of the orthogona group On, this case, due to the high concentration of some spherica harmonics, represents in some sense the worst one. Recenty, in a non-compact framework H. Koch and D. Tataru [KoT5] proved L p eigenfunction bounds for the Hermite operator in R n, whie Koch and F. Ricci obtained 99 Mathematics Subject Cassification. 43A85 33C55, 4B5. Key words and phrases. Joint spectra projectors, compex spheres, L p eigenfunction bounds, Heisenberg group, contraction.

3 VALENTINA CASARINO sharp estimates for the twisted Lapacian in R n [KoR]. L p eigenfunction bounds find appication in the context of Riesz summabiity [So87], [So] and in the strong unique continuation probem for second order eiptic operators [JKe], [So9], [KoT] In a aforementioned cases, the authors start from an operator P, acting on a manifod M, decompose L M as a direct sum of eigenspaces of P and then estimate the L p L - norm of the spectra projection operators. A different, interesting situation occurs when two essentiay sef-adjoint strongy commuting operators P and Q act on M and it is possibe to work out a joint spectra decomposition of L M. In this case a reasonabe mode space is given by the compex unit sphere S n in C n. The agebra of Un-invariant differentia operators on S n is commutative and generated by two eements; a possibe choice for a basis is given by the Lapace-Betrami operator S n and the Kohn Lapacian L on S n. One can define a joint spectrum and the decomposition of L S n as. L S n = +, = H, where H, is the space of compex spherica harmonics of bidegree, see [KV, Ch.], may be viewed as a joint spectra decomposition; indeed, H, is an eigenspace both for S n w.r. to the eigenvaue µ, := n, and for L w.r. to the eigenvaue λ, = n + [K]. By the symbo π we denote the joint spectra projector from L S n onto H. In [Ca] we proved estimates for π, p, when, beongs to a proper anguar sector in N N and when, ies on one of the axes. More specificay, we showed that. π, f Cn, p µ, γ p,n f p, for a, such that ε M, for some < ε M, with γ given by {.3 γ n p, n := + n p n 3 p and that n if p n+ if n n+ p,.4 π, f Cn, p n p f p for a p. Here we compete the picture, by proving sharp bounds for π, p, when ε, for some ε >. As a consequence, we show that.5 π, p, C + α p,n + Q β p,n, where Q := max{, }, := min{, } and α and β are the affine functions represented in Figure. Observe that and Q are reated to the eigenvaues λ, and µ,, since they grow, respectivey, as λ, and µ +,. An appication of these estimates to the probem of Bochner-Riesz summabiity on compex spheres wi be the subject of a subsequent paper, written in coaboration with Marco M. Peoso [CaP].

4 JOINT SPECTRAL PROJECTIONS ON S n 3 Let us consider now the reduced Heisenberg group h n := C n [, π], with product defined by z, tw, s := z + w, t + s + Im z w, z, w C n, t, s [, π]. The set { m n + k, m : m Z \ {}, k N} gives the joint spectrum of the eftinvariant subapacian L and of i t. Recenty, as a consequence of estimates of spectra projections for the twisted Lapacian in R n, Koch and Ricci [KoR] proved the foowing bounds for the joint orthogona projectors P m,k : P m,k L p h n,l h n C k + n β p,n m α p,n where α and β are the same exponents as in.5. Here we deduce the estimates for P m,k p,, in the case n =, from the bounds for the joint spectra projections π, on S 3, by using the notion of contraction of Lie groups. Reca that a Lie group G is caed a contraction or a continuous deformation of a Lie group G if there exists a famiy {π ε } ε> of oca diffeomorphisms π ε : G G satisfying π ε x e and π ε π ε xπ ε y xy when ε that is, π ε are approximate homomorphisms. Geometrica aspects of the contraction from SU to the three-dimensiona Heisenberg group H have been discussed in particuar by Ricci [R]. The idea of contraction has been argey used in the ast years to prove de Leeuw type theorems for Fourier mutipiers [R], [RRu], [D], [DG]. Here we show how this notion may be used to transfer L p eigenfunction bounds as we. It is a peasure to thank Professor M. Cowing for providing me with the Master s Thesis Anaysis of a subeiptic operator on the sphere in compex n-space by O. Kima [K] and Professor F. Ricci for many hepfu conversations on the subject of this paper. n β α n O n n+ n p Figure. The exponents α and β as functions of p

5 4 VALENTINA CASARINO. Notation and definitions For n, et S n denote the unit sphere in C n, that is S n := {z = z,..., z n C n : < z, z >= }. The symbo wi denote the north poe of S n, that is :=,...,,. We wi denote by the symbo H, the space of the restrictions to S n of harmonic poynomias pz, z = pz,..., z n, z,..., z n of homogeneity degree in z,..., z n and of homogeneity degree in z,..., z n. The space L S n, endowed with the inner product f, g := fξgξdσξ, S n where dσ is the measure invariant under the action of Un, turns out to be the direct sum of the pairwise orthogona spaces H,,,. Each subspace H, is Un-invariant and the representation of Un on H, is irreducibe. A specia roe in H, is payed by the function, caed zona, which is constant on the orbits of the stabiizer of which is isomorphic to Un. If f is zona, we may associate to f a map b f defined on the unit disk fξ = b f< ξ, >, ξ S n, by using the notation in [5, Section..5] we have < ξ, >= ξ n = e iϕ cos θ, where ϕ [, π] and θ [, π ]. Then the convoution between a zona function f and an arbitrary function g on S n may be defined as f g ξ := S n b f< ξ, η >gηdση. To simpify notation, we sha sometimes use fθ, ϕ to denote b fe iϕ cos θ. An expicit formua for the zona function Z, in H, is given by. b Z, θ, ϕ := d,!n! ϕ ω n + n! ei cos θ P n, cos θ,,, ϕ [, π], θ [, π ]. where = min,, ω n denotes the surface area of S n, P n, poynomia and d, := dimh, = n + + n + n + n is the Jacobi for a,. Reca moreover that H, consists of hoomorphic poynomias, and H, consists of poynomias whose compex conjugates are hoomorphic. In this case, the dimension of the space is given by + n dim H, = and the zona function is Z, θ, ϕ := ω n + n e iϕ cos θ, ϕ [, π], θ [, π ].

6 JOINT SPECTRAL PROJECTIONS ON S n 5 Finay, we denote by π if f L S n, then the orthogona projector onto H. It is a cassica fact that, π f = Z, f. 3. The main estimate To prove our main resut, we sha need the foowing emma see [Sz, pgs.7-73]. Lemma 3.. Let α, β, µ be rea numbers, each greater than. Then, as j, x µ P α,β j j α µ if µ < α 3 x dx = j n j if µ = α 3 j if µ > α 3. Proposition 3.. Let n and et, be positive integers such that ε for some ε,. Then { C n q + p n Q n p if p < p 3. π, p, C q 4 p Q n p if p p, where p = n n+, := min{, } and Q := max{, }. Proof. For p = and p = the inequaity 3. becomes 3. π, f C n Q n f and 3.3 π, f f respectivey. Now 3. foows from Young s inequaity and the fact that 3.4 Z, = O dimh, n n = O Q, whie 3.3 is a consequence of the fact that π, is an orthogona projection operator. When is bounded by some positive constant M, then 3. reduces to π, p, C + n p for a p, which may be proved by interpoating between 3. and 3.3. When tends to +, by the Riesz-Thorin convexity theorem in order to prove 3. it suffices to consider the specia case p = p, that is 3.5 π, f C n Q n By appying Young s inequaity, we obtain n f n n+ 3.6 π, f = Z, f C Z, n f n n. n+ Since < n n < for a n, standard interpoation impies that Z, n n C Z, n 3 n Z, n,.

7 6 VALENTINA CASARINO so that 3.5 wi foow from 3.4 and from the estimate 3.7 Z, C when +. We sha now prove 3.7 by using essentiay Lemma 3.. We have Z, = π ω n = π =: ω n + π d, n d, n ω n π x d, n cos θ cos θ sin θ n 3 cos θdθ P n, + x P,n I + I, P n, n x x dx n + x x dx since P q α,β x = P q β,α x. In order to estimate I, observe that Lemma 3. and the Mean Vaue Theorem yied I = = C P n, x x n dx+ n + x P n, x x dx q + + x x n P n, x q + x x dx for some < x <. Now the condition ε for some ε, and the fact that < +x < impy that π ω n d, n I C + n I C Since simiar estimates hod for I, 3.7 is proved. In order to prove that our bounds are sharp, we need a finer anaysis than in [Ca] of the norm both of the zona functions and of the functions z z S in H n,, since we need to distinguish, in particuar, the contribute given in the norm by the minimum and the maximum between and. Proposition 3.3. Let n. Set Q, z := z z S n. Then for a r. 3.8 Z, r C + n n r Q n n r Z, 3.9 Q, r Q, C + r 4 Q n n r.

8 Proof. Observe that Z, r r = d, r ω n r q n r Cd, r JOINT SPECTRAL PROJECTIONS ON S n 7 π +q + d, r C n, + + cos θ r P n, cos θ r sin θ n 3 cos θdθ cos θ r+ sin θ n 3 dθ where we used the fact that cos θ r m, for some positive constant m >, and that P n, cos θ q n if θ [, ]. Thus by using 3.4 we obtain 3.8. In order to prove 3.9, we reca that in [Ca, Lemma 3.4] we proved that Q, r Γ r r = C + Γ r n +. + r Γ + r + Now a standard appication of the Stiring s formua shows that where so that yieding 3.9. Γ r + Γ r + r + f,, r, Γ + r + f,, r := + + r + + r Q, r + n r r f,, r,, r, By coecting Proposition 3.,. and.4, we finay obtain the foowing resut. Theorem 3.4. Let n and et, be non-negative integers. Then { C + q n + p n + Q n p if p < p 3. π, p, C + 4 p + Q n p if p p, where p = n n+, := min{, } and Q := max{, }. 4. Projections on the Heisenberg group We consider the group SU, formed by the compex matrices β α α, β :=, α β with α + β =. In an obvious way, we may identify SU with the unit sphere S 3 in C. Let P n denote the space of poynomias in one compex variabe of degree at most n, with norm P n := n + P z + z n dz, P P n. π C

9 8 VALENTINA CASARINO An irreducibe unitary representation π n of SU acts on P n in the foowing way 4. π n α, βp z := iαz + β n βz iα P iαz + β for P P n. An orthonorma basis for P n is given by {e n j } j n, where e n j z := nj z j. By writing in an expicit way the matrix coefficients t n jk of 4. with respect to {en j } j n 4. t n jkα, β :=< π n α, βe n k, e n j >, it is easy to see that the poynomia t n jk is homogeneous of degree k w.r. to α, β and homogeneous of degree n k w.r. to α, β and that t n j,k is an eigenvector for the Lapace- Betrami operator on the sphere, with eigenvaue nn +. Moreover, if we denote by S the Kohn subapacian on the unit sphere S 3, considered as SU, {t n j,k } j,k n, n, yieds a compete set of eigenfunctions of S on the sphere, w.r. to the eigenvaue kn k n. The three-dimensiona Heisenberg group H is a nipotent Lie group whose underying manifod is C R, with product defined by z, tw, s := z + w, t + s + Im z w, z, w C, t, s R. Let F λ be the Fock space consisting of entire functions such that F F := λ F z e λ z dz < +. λ π C For every λ > two non-equivaent, irreducibe, unitary representations of H act on F λ in the foowing way: 4.3 σ λ ζ, tf z := e λit+ζz+ ζ F z + ζ and σ λ ζ, tf z := e λ it ζz+ ζ F z ζ. An orthonorma basis of C -vectors for F λ is given by the functions ηj λ λ z := j j! zj. The matrix coefficients of 4.3 with respect to the basis {η λ j } are defined by 4.4 τ ±λ jk ζ, t :=< σ±λ ζ, tη λ k, η λ j >. It is a we-known fact that the function τ ±λ jk is an eigenvector both for the operator T := i w.r. to the eigenvaue λ and for the subapacian L on t H w.r. to λ j +. The reduced Heisenberg group centra group of H given by is defined as the quotient group H /Γ, where Γ is the Γ := {o, kπ : k Z}.

10 JOINT SPECTRAL PROJECTIONS ON S n 9 In [R] Ricci studied the asymptotic reations between the matrix entries 4. and 4.4 and proved, in particuar, that each matrix eement τ ±λ jk may be obtained as imit of a certain sequence of t n jk. As a consequence of this fact, he obtained a cassica Meher-Heine formua, reating Jacobi and Laguerre poynomias. Proposition 4.. Let n and et x be a rea number. Fix k N. Then x P,n k x k cos = L k x e x. n k n k im n + cosn k Define now a map ψ : H SU in the foowing way. Given the eement ζ, t = ρe iϕ, t H, we set ψ ρe iϕ, t := i sin ρ e iϕ, cos ρ e it SU. In this way we define coordinates ρ, ϕ, t on SU, when ρ, ϕ, t are restricted, respectivey, to [, π ], [, π] and [ π, π]. With respect to these coordinates the normaized Haar measure on SU turns out to be 4π sin ρ dρ dϕ dt. Let now f be a function on h, with compact support. Let f be the function f extended by periodicity on R w.r. to the variabe t. Then define the function f on SU f ρ, ϕ, t := f ρ, ϕ, t, N. Lemma 4.. Let f be an integrabe function on h with compact support. If p +, then p f L p SU < f L p h and im p f L + p SU = f L p h. Proof. The proof is very simiar to that of Lemma in [RRu] and it is omitted. We wi use the foowing estimate, due to Darboux and Szegö [Sz, pgs. 69,98]. Lemma 4.3. Let α, β >. Fix < c < π. Then O α if θ c, P α,β cos θ = kθ cos N θ + γ + sin θ O if c θ π c O β if π c θ π, where kθ := π sin θ α cos θ β, N := + α+β+, γ := α + π. As it is we-known, the pairs m k +, m, with m Z \ {} and k N, give the discrete joint spectrum of the eft-invariant subapacian L and i t on h. Let P m,k be the orthogona projector onto the joint eigenspace. We wi now show that P m,k may be obtained as imit in the L -norm of a sequence of joint spectra projectors on S 3. Proposition 4.4. Let f be a continuous, compacty supported function on h. m N \ {} and k N. For every N et n N such that 4.5 im + n = m. Take

11 VALENTINA CASARINO Then 4.6 P m,k f L h = im π k,n k f L + SU, and 4.7 P m,k f L h = im π n k,k f L + SU. Proof. By orthogonaity we obtain P m,k f L h =< P m,kf, f > L h = f τk,kz, m t fz, t dz dt h = τk,k m z w, t t Imwz fw, t dw dt fz, t dz dt h h = m e i mt t Imwz L k m z w e m z w fw, t dw dt fz, t dz dt, h h where we used the expicit expression for τk,k m given, for instance, in [F]. Moreover π k,n k f L SU =< π k,n kf, f > L SU sin ρ = π k,n k f ρ, ϕ, t f ρ, ϕ, t dρ dϕ dt SU 4π = b sin ρ Z k,n k f ρ, ϕ, t f ρ, ϕ, t dρ dϕ dt SU 4π = π π π b Z k,n k < ρ, ϕ, t, ρ, ϕ, t > f ρ, ϕ, t = A π SU sin ρ 4π dρ dϕ dt A b Z k,n k fρ, ϕ, t sin ρ f ρ, ϕ, t sin ρ ρ ρ π < ρ, ϕ, t ρ,, ϕ, t ρ ρ π dρ dϕ dt f ρ, ϕ, t dρ dϕ dt > sin ρ ρ ρ dρ dϕ dt, π where the integration set A is defined by A := {ρ, ϕ, t : ρ π, ϕ π, π t π}. Now an expicit computation yieds < ρ, ϕ, t ρ,, ϕ, t > = sin ρ sin ρ e iϕ ϕ + cos ρ cos ρ e it t = r e iψ,

12 JOINT SPECTRAL PROJECTIONS ON S n where r = ρ + ρ ρρ cosϕ ϕ + o and ψ = arctan ρρ sinϕ ϕ + t t + o, +. By setting z = ρe iϕ and w = ρ e iϕ, we have r = cos z w + o and ψ = t t Imwz + o, so that by using formua. b Z k,n k < ρ, ϕ, t ρ,, ϕ, t > = n + e in k t t Imw z+o n k cos z w P, n k cos z w + o, +. As a consequence of 4.5 and the Mean Vaue Theorem, we have π k,n kf L SU = n + e i mt t Imw z A A n k cos P, n k fρ, ϕ, t z w ρ sin ρ ρ k π dρ dϕ dt k f ρ, ϕ, t cos z w sin ρ ρ ρ π dρ dϕ dt + R, +, with im + R =. The main term above may be written as + π π + π π n + e i mt t Imw z π π 4.8 cos z w fρ, ϕ, t P, n k k f ρ, ϕ, t sin ρ ρ ρ dρ dϕ dt, π n k cos z w sin ρ ρ ρ π dρ dϕ dt since f has compact support and the integrand is periodic w.r. to t. In order to appy the Dominated Convergence Theorem to 4.8 we have to study the pointwise convergence of the integrand for + and to ook for a uniform majorant for it. Since the expressions sin ρ ρ and sin ρ ρ tend to when + and are bounded on [, +, we may ignore them. We easiy find a uniform majorant by using the pointwise estimates for the Jacobi poynomias contained in Lemma 4.3.

13 VALENTINA CASARINO In order to study the pointwise convergence of the integrand, we use the Meher-Heine formua, as stated in Lemma 4., and we see that n k n + im cos z w P, n k + k cos z w = ml k m z w e m z w, yieding 4.6. The proof for 4.7 is competey anaogous. Our aim is to prove the foowing bound for the norm of P m,k : { C k + + p m p if p < 6 5 P m,k p, C k + 4 p m p if 6 p. 5 By interpoation, it suffices to estimate P m,k L,L and P m,k L 6 5,L Coroary 4.5. Take m Z \ {} and k N. Then 4.9 P m,k f L h m f L h and 4. P m,k f L h k 6 m 3 f L 6 5 h. Proof. One has P m,k f L h = im + π k,n k f L SU im + = m im + f L SU = m f L SU, im + n f L SU where we used first Proposition 4.4 and then Theorem 3.4 and Lemma 4.. In the same way, we see that P m,k f L h = π k,n k f L SU = im n 3 k 6 6 f L h = im + 3 im + n 3 k 6 f L 6 5 h = m 3 k 6 f L 6 5 h. n 3 k 6 f L 6 5 SU References [Ca] V. Casarino, Norms of compex harmonic projection operators, Canad. J. Math., 55 3, [CaP] V. Casarino and M. M. Peoso, in preparation. [D] A. W. Dooey, Contractions of Lie groups and appications to anaysis, In Topics in Modern Harmonic Anaysis, Roma, Ist. di Ata Matematica 983, [DG] A. H. Dooey and G. I. Gaudry, An extension of deleeuw s theorem to the n-dimensiona rotation group, Ann. Inst. Fourier, , -35. [F] G. B. Foand, Harmonic Anaysis in Phase Space, Annas of Math. Stud., Princeton University Press, 989.

14 JOINT SPECTRAL PROJECTIONS ON S n 3 [JKe] D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvaues for Schrödinger operators, with an appendix by E. M. Stein, Ann. of Math. 985, [K] O. Kima, Anaysis of a subeiptic operator on the sphere in compex n-space, Thesis, Schoo of Mathematics, University of New South Waes, 3. [KV] A. U. Kymik and N. Ja. Vienkin, Representation of Lie groups and specia functions, Kuwer Academic Pubishers, 993. [KoR] H. Koch and F. Ricci, Spectra projections for the twisted Lapacian, to appear on Studia Math.. [KoT] H. Koch and D. Tataru, Careman estimates and unique continuation for second-order eiptic equations with non-smooth coefficients Comm. Pure App. Math. 54, [KoT5] H. Koch and D. Tataru, L p eigenfunction bounds for the Hermite operator, Duke Math. J.8 5, [R] F. Ricci, A Contraction of SU to the Heisenberg group, Mh Math., 986, -5. [RRu] F. Ricci and R. L. Rubin, Transferring Fourier mutipiers from SU to the Heisenberg group, Amer. J. Math., 8 986, [So86] C. Sogge, Osciatory integras and spherica harmonics, Duke Math. J , [So87] C. Sogge, On the convergence of Riesz means on compact manifods, Ann. of Math , [So9] C. Sogge, Strong uniqueness theorems for second order eiptic differentia equations, Amer. J. Math. 99, [So93] C. Sogge, Fourier integras in cassica anaysis, Cambridge Tracts in Mathematics. 5, Cambridge University Press, Cambridge, 993. [So] C. Sogge, Eigenfunction and Bochner Riesz estimates on manifods with boundary, Math. Res. Lett. 9, 5-6. [Sz] G. Szegö, Orthogona Poynomias, Amer. Math. Soc. Cooq. Pub., vo.3, Amer. Math. Soc., 4th ed. Providence, R.I.974. Dipartimento di Matematica, Poitecnico di Torino, Corso Duca degi Abruzzi 4, 9 Torino E-mai address: casarino@cavino.poito.it