Singular integrals on NA groups

Singular integrals on NA groups joint work with Alessio Martini and Alessandro Ottazzi Maria Vallarino Dipartimento di Scienze Matematiche Politecnico di Torino LMS Midlands Regional Meeting and Workshop on Interactions of Harmonic Analysis and Operator Theory, Birmingham, 13-16 September 2016

Singular integrals on R n If T : L 2 (R n ) L 2 (R n ) is a convolution operator with kernel k : R n \ {0} C s.t. sup y =0 x 2 y k(x y) k(x) dx <, then T is of weak type (1, 1) and bounded on L p (R n ), p (1, ). The key ingredient is the classical Calderón Zygmund theory which relies on the doubling property, i.e. V (2B) C V (B) for every ball B.

Singular integrals on R n TWO EXAMPLES: = n j=1 2 j = 0 λ de (λ) Riesz transforms R j = j 1/2 R j f = c n f p.v. x j x n+1 R j is of weak type (1, 1) and bounded on L p (R n ), p (1, ) Spectral multipliers of Mihlin-Hörmander type F : R + C bounded F ( ) = 0 F (λ) de (λ) F ( )f = f F 1 F ( 2 ) If sup t>0 F (t )φ( ) W s 2 < for some s > n 2 and 0 = φ C c (0, ), then F ( ) is of weak type (1, 1) and bounded on L p (R n ), p (1, )

Singular integrals on Lie groups G connected noncompact Lie group µ right Haar measure X 0,..., X q left-invariant vector fields satisfying Hörmander s condition = q j=0 X 2 j (sub)laplacian s.a. on L 2 (G ) = L 2 (G, µ) hypoelliptic and positive QUESTIONS: boundedness of Riesz transforms R j = X j 1/2 boundedness of spectral multipliers of, i.e. find conditions on F : R + C bounded s.t. F ( ) = + 0 F (λ)de (λ) is bounded on L p (G ) for some p = 2

G Lie group of polynomial growth R j is of weak type (1, 1) and bounded on L p (G ), p (1, ) [Christ-Geller (1984), Lohoué-Varopoulos (1985), Alexopoulos (1992)] a Mihlin-Hörmander type theorem holds [Mauceri-Meda (1990), Christ (1991), Alexopoulos (1994),... ] In these cases G is unimodular and G is a doubling measured space the classical CZ theory applies G = R n 1 R H n (R) X 0,..., X n 1 basis of g = Lie(G ) = n 1 j=0 X j 2 complete Laplacian G is a nonunimodular group of exponential growth G is nondoubling Hebisch and Steger (2003) developed a new CZ theory to prove that R j is of weak type (1, 1) and bounded on L p (G ), p (1, 2] (see also Sjögren (1999)) a Mihlin-Hörmander type theorem holds (see also Hebisch (1993), Cowling-Giulini-Hulanicki-Mauceri (1994))

NA groups G = N A N stratified group Lie(N) = n = V 1 V s [V i, V j ] V i+j (V k = 0 if k > s) δ t Aut(n) automorphic dilations s.t. δ t Vj = t j Id Vj t > 0 det δ t = t Q Q = s j=1 j dimv j homogeneous dimension of N A = R acting on N via automorphic dilations (z, u) (z, u ) = (z δ e u z, u + u ) dµ(z, u) = dz du right Haar measure m(z, u) = e Qu modular function E 0 = u basis of a R z, z N, u, u R E 1,..., E q basis of the first layer V 1 of n X 0 = u X j = e u E j j = 1,..., q left invariant v.f. on G = q j=0 X 2 j sublaplacian

G = N A N stratified group of homogeneous dimension Q A = R acting on N via automorphic dilations THEOREM 1. [Martini-Ottazzi-V.] R j = X j 1/2 is of weak type (1, 1) and bounded on L p (G ), p (1, 2]. THEOREM 2. [Martini-Ottazzi-V.] If F : R + C bounded s.t. sup F (t )φ( ) s W 0 < s 0 > 3 2 0<t 1 2 sup F (t )φ( ) W s 2 < s > Q + 1, t 1 2 then F ( ) is of weak type (1, 1) and bounded on L p (G ), p (1, ).

THEOREM 1. [Martini-Ottazzi-V.] R j = X j 1/2 is of weak type (1, 1) and bounded on L p (G ), p (1, 2]. THEOREM 2. [Martini-Ottazzi-V.] If F : R + C bounded s.t. sup F (t )φ( ) s W 0 < s 0 > 3 2 0<t 1 2 sup F (t )φ( ) W s 2 < s > Q + 1, t 1 2 then F ( ) is of weak type (1, 1) and bounded on L p (G ), p (1, ). REMARKS: when N is abelian, G = N A H n (R), is a complete Laplacian related with the hyperbolic Laplacian: THEOREMS 1 and 2 were proved by Hebisch and Steger using spherical harmonic analysis and explicit formulae for the heat kernel if F C c (R) W s 2 (R) with s > 2, then F ( ) is bounded on L1 (G ) [Mustapha (1998), Gnewuch (2002)]

THEOREM 1. [Martini-Ottazzi-V.] R j = X j 1/2 is of weak type (1, 1) and bounded on L p (G ), p (1, 2]. THEOREM 2. [Martini-Ottazzi-V.] If F : R + C bounded s.t. sup F (t )φ( ) s W 0 < s 0 > 3 2 0<t 1 2 sup F (t )φ( ) W s 2 < s > Q + 1, t 1 2 then F ( ) is of weak type (1, 1) and bounded on L p (G ), p (1, ). REMARKS: when N is abelian, G = N A H n (R), is a complete Laplacian related with the hyperbolic Laplacian: THEOREMS 1 and 2 were proved by Hebisch and Steger using harmonic spherical analysis and explicit formulae for the heat kernel if F C c (R) W s 2 (R) with s > 2, then F ( ) is bounded on L1 (G ) [Mustapha (1998), Gnewuch (2002)] boundedness of R j for p > 2?

Ingredients of the proofs X 0,..., X q defines a sub-riemannian structure on G with associated left-invariant Carnot-Carathéodory distance ϱ cosh x ϱ = cosh ϱ ( ) x, e G = cosh u + e u z 2 N /2 x = (z, u) N A, where N is the Carnot-Carathéodory distance on N formula known when N is abelian and stated without proof for stratified N by Hebisch (1999) solutions to Hamilton s equations on N can be re-parametrized and lifted to solutions to Hamilton s equations on G length-minimizing curves need not be solutions to to Hamilton s equations. But points that can be joined to the identity via length-minimizing solution to Hamilton s equations are dense [Agrachev (2009)] { r Q+1 r 1 µ(b ϱ (e G, r)) e Qr (G, ϱ, µ) is nondoubling r > 1 We develop a Calderón Zygmund theory on the nondoubling subriemannian space (G, ϱ, µ)

Ingredients of the proofs Boundedness result for convolution operators on G T right convolution operator bounded on L 2 (G ), i.e., Tf = f k the integral kernel of T is K (x, y) = k(y 1 x)m(y) If k = n Z k n with (i) (ii) G k n (x) (1 + 2 n/2 x ϱ ) dµ(x) 1 k n (y 1 x)m(y) k n (x) dµ(x) 2 n/2 y ϱ y G, G then T is of weak type (1, 1) and bounded on L p (G ), p (1, 2].

Ingredients of the proofs Boundedness result for convolution operators on G T right convolution operator bounded on L 2 (G ), i.e., Tf = f k the integral kernel of T is K (x, y) = k(y 1 x)m(y) If k = n Z k n with (i) (ii) G k n (x) (1 + 2 n/2 x ϱ ) dµ(x) 1 K n (x, y) K n (x, e G ) dµ(x) 2 n/2 y ϱ y G, G then T is of weak type (1, 1) and bounded on L p (G ), p (1, 2].

Ingredients of the proofs Gradient estimates of the heat kernel h t of LEMMA. For every t > 0 G X j h t (x) dµ(x) C t 1/2.

Ingredients of the proofs Gradient estimates of the heat kernel h t of LEMMA. For every t > 0 G X j h t (x) dµ(x) C t 1/2. It follows from the formula [Mustapha 1998, Gnewuch 2006] ( h t (z, u) = Ψ t (ξ) exp cosh u ) h N 0 ξ e u ξ/2 (z)dξ, where ( Ψ t (ξ) = ξ 2 π 2 ) 4π 3 t exp 4t 0 sinh θ sin πθ ( 2t exp θ2 4t cosh θ ) dθ, ξ estimates for the derivative of the heat kernel on N and various integration by parts (delicate cancellation occurs)

Ingredients of the proofs Gradient estimates of the heat kernel h t of LEMMA. For every ε 0 and t > 0 G X j h t (x) exp(εt 1/2 x ϱ ) dµ(x) C ε t 1/2. It follows from the formula [Mustapha 1998, Gnewuch 2006] ( h t (z, u) = Ψ t (ξ) exp cosh u ) h N 0 ξ e u ξ/2 (z)dξ, where ( Ψ t (ξ) = ξ 2 π 2 ) 4π 3 t exp 4t 0 sinh θ sin πθ ( 2t exp θ2 4t cosh θ ) dθ, ξ estimates for the derivative of the heat kernel on N and various integration by parts (delicate cancellation occurs)

Sketch of the proof of THEOREM 1 R j = X j 1/2 1 = t 1/2 X Γ(1/2) j e t dt 0 1 k j = t 1/2 X Γ(1/2) j h t dt = kj n 0 n Z kj n = 1 2 n+1 Γ(1/2) 2 t 1/2 X n j h t dt

Sketch of the proof of THEOREM 1 R j = X j 1/2 1 = t 1/2 X Γ(1/2) j e t dt 0 1 k j = t 1/2 X Γ(1/2) j h t dt = kj n 0 n Z kj n = 1 2 n+1 Γ(1/2) 2 t 1/2 X n j h t dt kj n(x) (1 + 2 n/2 x ϱ ) dµ(x) G 2 n+1 2 n 2 n+1 G t 1/2 X j h t (x) (1 + t 1/2 x ϱ ) dµ(x)dt 2 n t 1/2 t 1/2 dt 1.

Sketch of the proof of THEOREM 1 R j = X j 1/2 1 = t 1/2 X Γ(1/2) j e t dt 0 1 k j = t 1/2 X Γ(1/2) j h t dt = kj n 0 n Z kj n = 1 2 n+1 Γ(1/2) 2 t 1/2 X n j h t dt Using h t = h t/2 h t/2 X j h t = h t/2 X j h t/2 h t/2 (x 1 ) = h t/2 (x)m(x 1 ) kj n(y 1 x)m(y) kj n (x) dµ(x) G 2 n+1 G 2 n t 1/2 X j h t (y 1 x)m(y) X j h t (x) dµ(x)dt q t 1/2 X i h t/2 1 X j h t/2 1 dt 2 n i=0 y ϱ 2 n+1 2 n+1 y ϱ t 1/2 t 1/2 t 1/2 dt 2 n/2 y ϱ. 2 n

Sketch of the proof of THEOREM 1 R j = X j 1/2 1 = t 1/2 X Γ(1/2) j e t dt 0 1 k j = t 1/2 X Γ(1/2) j h t dt = kj n 0 n Z kj n = 1 2 n+1 Γ(1/2) 2 t 1/2 X n j h t dt kj n satisfy the integral estimates of the boundedness theorem for convolution operators R j is of weak type (1, 1) and bounded on L p (G ), p (1, 2]

Related problems we also introduce a Hardy type space H 1 (G ) and prove an endpoint result on H 1 (G ) both for Riesz transforms and MH spectral multipliers of R j : L p (G ) L p (G ) p (2, )? Rj = 1/2 X j : L p (G ) L p (G ) p (1, 2)? if G = R R: R1 : Lp (G ) L p (G ) p (1, 2) and is of weak type (1, 1) [Gaudry-Sjögren (1999)] R0 : Lp (G ) L p (G ) p (1, 2) and is NOT of weak type (1, 1) [Hebisch] if G = R 2 R, then Rj is NOT bounded from H 1 (G ) to L 1 (G ) [Sjögren-V. (2008)] Riesz transforms of the second order X i 1 X j are of weak type (1, 1) and bounded on L p (G, (i, j) = (0, 0) [Martini-Ottazzi-V.] If is a distinguished Laplacian on NA groups arising from the Iwasawa decomposition of a semisimple Lie group of rank one, then X i 1 X j are bounded on L p (G ), p (1, ), and of weak type (1, 1) but X i X j 1 and 1 X i X j are NOT bounded on L p (G ), p [1, ) [Gaudry-Sjögren (1996)]

Related problems we also introduce a Hardy type space H 1 (G ) and prove an endpoint result on H 1 (G ) both for Riesz transforms and MH spectral multipliers of R j : L p (G ) L p (G ) p (2, )? Rj = 1/2 X j : L p (G ) L p (G ) p (1, 2)? Riesz transforms of the second order L p -spectral multipliers for sublaplacian with drift X (holomorphic functional calculus) more general NA groups

Sketch of the proof of THEOREM 2 F ( ) = F ( ) φ(2 n ) = F n (2 n ) n Z n Z φ C c (R + ), n φ(2 n λ) = 0, supp(φ) [1/4, 4] The kernel k n of F n (2 n ) satisfy the conditions of the boundedness theorem. This follows from various facts: and on G = R Q R have the same Plancherel measure, i.e. k F ( ) (z, u) 2 dzdu = F (λ) 2 c Q (λ) 2 dλ F (λ) 2 (λ 2 + λ Q )dλ G 0 0 c Q Harish-Chandra function on H Q+1 (R) weighted L 2 -estimates on G for k F ( ) can be reduced to weighted L 2 -estimates for k F ( ) on G G k F ( ) (z, u) 2 z 2a N dzdu k F ( ) (z, G u) 2 z 2a R dzdu, Q where m 1/2 k F ( ) (z, u) is radial (and one can use spherical analysis on G)

Sketch of the proof of THEOREM 2 F ( ) = F ( ) φ(2 n ) = F n (2 n ) n Z n Z φ C c (R + ), n φ(2 n λ) = 1, supp(φ) [1/4, 4] The kernel k n of F n (2 n ) satisfy the conditions of the boundedness theorem. This follows from various facts: and on G = R Q R have the same Plancherel measure weighted L 2 -estimates on G for k F ( ) can be reduced to weighted L 2 -estimates for k F ( ) on G if the Fourier transform of M is supported in [ R, R], then k M( ) 1 min{r 3/2, R (Q+1)/2 } k M( ) 2 F n (2 n ) can be decomposed as the sum of functions whose Fourier transform is compactly supported in order to apply previous estimate

Sketch of the proof of THEOREM 2 F ( ) = F ( ) φ(2 n ) = F n (2 n ) n Z n Z φ C c (R + ), n φ(2 n λ) = 1, supp(φ) [1/4, 4] The kernel k n of F n (2 n ) satisfy the conditions of the boundedness theorem F ( ) is of weak type (1, 1) and bounded on L p (G ), p (1, )

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