An $L^p$-spectral multiplier theorem with sharp $p$-specific regularity bound on Heisenberg type groups

We prove an $L^p$-spectral multiplier theorem for sub-Laplacians on Heisenberg type groups under the sharp regularity condition $s>d\left|1/p-1/2\right|$, where $d$ is the topological dimension of the underlying group. Our approach relies on restriction type estimates where the multiplier is additionally truncated along the spectrum of the Laplacian on the center of the group.

1. Introduction 1.1.Statement of the main result.Let G be a two-step stratified Lie group, that is, a connected, simply connected, two-step nilpotent Lie group whose Lie algebra g admits a decomposition g = g 1 ⊕ g 2 with [g 1 , g 1 ] = g 2 and g 2 ⊆ g being contained in the center of g.Let X 1 , . . ., X d1 be a basis of g 1 .This basis can be identified with a system of left-invariant vector fields on G via the Lie derivative.Let L be the sub-Laplacian associated with the vector fields X 1 , . . ., X d1 , that is, the second order, left-invariant differential operator given by ).This operator is positive and self-adjoint on L 2 (G), where G is endowed with a left-invariant Haar measure.Via functional calculus, one can define for every Borel measurable function F : R → C the operator F (L) on L 2 (G), which is a bounded operator on L 2 (G) whenever the spectral multiplier F is bounded.Regarding multipliers F for which F (L) extends to a bounded operator on L p (G), sufficient conditions can be given in terms of differentiability properties of the multiplier F , usually expressed by the scale-invariant localized Sobolev norms • L 2 s,sloc , s ≥ 0, given by F L 2 s,sloc := sup t>0 Here, η : R → C is a smooth non-zero function with compact support in (0, ∞) and L 2 s (R) ⊆ L 2 (R) denotes the Sobolev space of (fractional) order s ≥ 0. Due to a celebrated theorem of Christ [Chr91] and of Mauceri and Meda [MM90], F (L) extends to a bounded operator on all L p -spaces for 1 < p < ∞ whenever F L 2 s,sloc < ∞ for some s > Q/2, where Q = dim g 1 + 2 dim g 2 is the homogeneous dimension of the underlying Lie group.Moreover, F (L) is of weak type (1, 1), i.e., bounded from L 1 (G) to the Lorentz space L 1,∞ (G).(Actually, the theorem holds true for stratified Lie groups of arbitrary step, but our focus lies on stratified Lie groups of step two.) In the case of Heisenberg (-type) groups, Müller and Stein [MS94], and independently Hebisch [Heb93] showed that the threshold s > Q/2 can even be pushed down to s > d/2, where d is the topological dimension of the underlying group.This result has been extended to other specific classes of two-step stratified Lie groups [Mar12,MM14b,Mar15] (and also to other settings, cf.[CS01, MM14a, CCMS17, ACMM20]), but up to now, it is still open whether the threshold s > d/2 is sufficient for any two-step stratified Lie group.However, Martini and Müller [MM16] were able to show that for all two-step stratified Lie groups and left-invariant sub-Laplacians, the sharp threshold is strictly less than Q/2, but not less than d/2.
On the other hand, instead of asking for boundedness on all L p -spaces for 1 < p < ∞ simultaneously, one can ask for the minimal threshold s p ∈ [0, d/2] such that F (L) is bounded on L p whenever F L 2 s,sloc < ∞ for some s > s p .In [MMNG23], Martini, Müller, and Nicolussi Golo showed for a large class of smooth secondorder real differential operators associated with a sub-Riemannian structure on smooth d-dimensional manifolds that at least regularity of order s ≥ d |1/p − 1/2| is necessary for having L p -spectral multiplier estimates.On the opposite, one expects this threshold also to be essentially sufficient.Sufficiency results featuring the regularity condition s > Q |1/p − 1/2|, where Q ≥ d is the homogeneous dimension, are available in various settings [COSY16,LRS14,See86], but beyond the Euclidean setting (where d = Q), to the best of my knowledge, sufficiency of the threshold s ≥ d |1/p − 1/2| has so far only been proven in exceptional cases, see [CO16,Nie22].
The purpose of this paper is to extend the results for Grushin operators of [Nie22] to sub-Laplacians on Heisenberg type groups, which is the following subclass of twostep stratified Lie groups: Suppose that g is endowed with an inner product •, • for which the stratification g = g 1 ⊕ g 2 is orthogonal.Let g * 2 denote the dual of g 2 .For any µ ∈ g * 2 , we have a skew-symmetric bilinear form ω µ : g 1 × g 1 → R given by ω µ (x, x ′ ) := µ([x, x ′ ]), x, x ′ ∈ g 1 . (1.1) Then, for any µ ∈ g * 2 , there is a skew-symmetric endomorphism J µ on g 1 such that ω µ (x, x ′ ) = J µ x, x ′ for all x, x ′ ∈ g 1 .Let | • | denote the norm on g * 2 induced by the inner product •, • .Then the group G is called a Heisenberg type group if J µ is orthogonal for |µ| = 1, that is, Note that this implies in particular that g 2 = [g 1 , g 1 ] is the center of g.Given any orthonormal basis X 1 , . . ., X d1 of g 1 with respect to the inner product •, • , we consider the associated sub-Laplacian Our main result is the following spectral multiplier estimate, together with a corresponding result for Bochner-Riesz multipliers.By the result of [MMNG23], the threshold s > d (1/p − 1/2) is optimal up to the endpoint and cannot be decreased.Note that the required order of regularity in the second part of Theorem 1.1 is the same as in the Bochner-Riesz conjecture, see for example [Tao99].However, note also that Theorem 1.1 only makes a statement in the range 1 ≤ p ≤ 2(d 2 + 1)/(d 2 + 3), where d 2 is the dimension of the center of the group.
(1) If p > 1 and if F : R → C is a bounded Borel function such that If s > d/2 in the first part of Theorem 1.1, then the operator F (L) is of weak type (1, 1) by [Heb93].The condition 1 ≤ p ≤ 2(d 2 + 1)/(d 2 + 3) in Theorem 1.1 derives from the Stein-Tomas restriction estimate on g 2 , which is used for the proof of the spectral multiplier estimates.Due to this assumption, the first part of the theorem only gives results when d 2 ≥ 2, which means that G must be a Heisenberg type group which is not a Heisenberg group.This reflects the phenomenon that there are no good analogues of Fourier restriction estimates available in the case of Heisenberg groups, except for the trivial L 1 -L ∞ estimates, which is due to the fact that the Heisenberg group admits only a one-dimensional center [Mül90].It remains an open question whether such p-specific spectral multiplier estimates as above also hold in the setting of Heisenberg groups.
The spectral multiplier results of Theorem 1.1 are extended in a follow-up paper [Nie23] to the class of Métivier groups.However, the approach of [Nie23] requires additional methods due to the fact that the matrices J µ from above are no longer orthogonal and the spectral decomposition into eigenspaces, which depend on µ, is more complicated.1.2.Structure of the proof.Building on methods of [CC13,CO16,Nie22], the proof of the spectral multiplier estimates of Theorem 1.1 relies on restriction type estimates, a fundamental connection that was first discovered by C. Fefferman [Fef73] and has since then been exploited by many other authors, see [SS89,GHS13,COSY16].The key idea can be illustrated as follows.Suppose that we want to derive L p -boundedness of the Bochner-Riesz means (1 − L) δ + .If F : R → C is the multiplier given by where |τ | ∼ 2 ι whenever τ ∈ supp F (ι) for ι ≥ 1.It suffices to show Let K (ι) be the convolution kernel associated with where * denotes the group convolution.By the Fourier inversion formula, we have where χ : R → C is some smooth function which is compactly supported in (0, ∞).
Thanks the frequency localization of the dyadic pieces F (ι) , since solutions of the wave equation associated with L possess finite propagation speed, the convolution kernel centered at the origin with radius R ∼ 2 ι .Now having a restriction type estimate which is essentially of the form then, given a function f with supp f ⊆ B R × B R 2 (which one can assume without loss of generality), Hölder's inequality provides where Q = d 1 + 2d 2 and 1/q = 1/p − 1/2.On the other hand, one can show that whence the last term of (1.4) can be estimated via However, this approach does in general only provide thresholds featuring the homogeneous dimension Q in place of the topological one.Employing the approach of [Nie22], we additionally decompose the multipliers F (ι) dyadically along the spectrum of the Laplacian , where U 1 , . . ., U d2 is a basis of the second layer g 2 of the Lie algebra g.More precisely, we decompose F (ι) such that where ) can be explicitly written down in terms of the Fourier transform and rescaled Laguerre functions ϕ |µ| k (for their definition, see (2.9) below), namely, ℓ , whence the truncation afforded by the functions χ ℓ is also referred to as a truncation along the spectrum of U .Pointwise estimates for Laguerre functions (cf.Eq. (1.1.44) 1 , (1.3.41), and Lemma 1.5.3 of [Tha93] or alternatively the table on page 699 of [AW65]) suggest that ϕ k (x) := ϕ 1 k (x) has exponential decay for R ℓ , up to some exponentially decaying term.This means that the kernel at the origin.Now, instead of (1.3), suppose we had a restriction type estimate of the form We distinguish the cases 0 ≤ ℓ ≤ ι and ℓ > ι.In the former case, we decompose the ball B R × B R 2 into a grid of balls B (ℓ) 1 There is a small typo in Eq. (1.1.44):The factor e −x has to be replaced by e −x/2 .
to the Euclidean distance on the layers g 1 and g 2 , respectively.Correspondingly, we decompose the function f supported in B R × B R 2 into a sum of functions f m supported in balls of dimension R ℓ × R 2 .Then Hölder's inequality, applied with 1/p = 1/q + 1/2, combined with the restriction type estimate (1.7) would provide Since G is a Heisenberg type group, we have . Hence we may estimate the last term of (1.8) by The case ℓ > ι can be treated similarly by using (1.4) in conjunction with (1.7) and summing the geometric series over all ℓ > ι.As a consequence, only regularity δ > d/q−1/2 of the multiplier Thus, at least by our heuristics, the truncation of (1.5) ultimately provides thresholds with the topological dimension instead of the homogeneous one.
It should be emphasized that the present approach benefits from the fact that the dimension d 2 of the second layer g 2 of the Lie algebra g is smaller than the dimension of the first layer.Doing the decomposition in the first layer reflects a phenomenon that has already been prominent in the setting of Grushin operators −∆ x − |x| 2 ∆ u , (x, u) ∈ R d1 × R d2 , which are closely related to sub-Laplacians on Heisenberg type groups.In [MS12], Martini and Sikora proved a Mikhlin-Hörmander type result with threshold s > D/2 for Grushin operators, where D := max{d 1 +d 2 , 2d 2 }, which was later improved by Martini and Müller [MM14a] to hold for the topological dimension d in place of D. Both approaches rely on weighted Plancherel estimates for the associated integral kernels, but differ by the employed weights.While [MS12] uses the weight |x| γ in the first layer in conjunction with a sub-elliptic estimate, [MM14a] employs the weight |u| γ in the second layer.A similar phenomenon occurred later in the articles [CO16] and [Nie22], where spectral multiplier theorems with p-specific regularity bounds were proved.The approach of Chen and Ouhabaz [CO16] relies on weighted restriction type estimates using the weight |x| γ in the first layer, while [Nie22] employs the weighted Plancherel estimate of [MM14a] with weight in the center to express support conditions of integral kernels.In accordance with the phenomenon of [MS12] and [MM14a], the result of [CO16] needs s > D (1/p − 1/2) as regularity condition, while [Nie22] only needs regularity of order s > d (1/p − 1/2).The present article relies on the same key idea as in [Nie22], but uses instead a weighted Plancherel estimate featuring the weight |x| γ .However, this approach still provides the optimal threshold s > d (1/p − 1/2) in Theorem 1.1 and is in line with the phenomena described above, since d 2 < d 1 and thus D = d, due to the fact that G is a Heisenberg type group.1.3.Structure of the paper.Section 2 and 5 are preliminary sections dealing with the spectral theory and sub-Riemannian geometry of sub-Laplacians.In Section 3, we prove the previously mentioned truncated restriction type estimates.Section 4 is devoted to proving a weighted Plancherel estimate with weight |x| γ .In Section 6 we reduce the proof of Theorem 1.1 to spectral estimates for multipliers whose Fourier transform is supported on dyadic scales.The proof of this reduced version of Theorem 1.1 is given in Section 7. Section 8 is an additional section, where we show that a sub-elliptic estimate for the sub-Laplacian with the help of which one could directly transfer the approach of [CO16] to the setting of Heisenberg type groups is in general false.
1.4.Notation.We let N = {0, 1, 2, . . .}.The space of (equivalence classes of) integrable simple functions on a two-step stratified Lie group G will be denoted by D(G), while S(G) shall denote the space of Schwartz functions on G ∼ = R d .The indicator function of a subset A of some measurable space will be denoted by 1 A .Given two suitable functions f 1 , f 2 on G, let f * g denote the group convolution given by while the inverse Fourier transform f is given by Moreover, we fix the following dyadic decomposition throughout this article: Let χ : R → [0, 1] be an even and smooth function such that 1/2 ≤ |λ| ≤ 2 for all λ ∈ supp χ and j∈Z χ j (λ) = 1 for λ = 0, where χ j is given by χ j (λ) = χ(λ/2 j ) for j ∈ Z.
(1.9) 1.5.Acknowledgments.I am deeply grateful to my advisor Detlef Müller for constant support and numberless helpful suggestions.I am also very grateful to Alessio Martini for bringing an error in Lemma 2.1 of an earlier version of this article to my attention.I would also like to thank the anonymous referee for carefully reading the paper and making a number of helpful suggestions.

Spectral theory of sub-Laplacians on Heisenberg type groups
Let G be a two-step stratified Lie group.Via exponential coordinates, we may identify G with its Lie algebra g, which is the tangent space at the identity of G. Since G is stratified of step 2, g can be decomposed as g = g 1 ⊕ g 2 with [g 1 , g 1 ] = g 2 and g 2 ⊆ g being contained in the center of g.Let ġ * 2 = g * 2 \ {0}.For any µ ∈ g * 2 , let ω µ be the skew-symmetric bilinear form given by Then G is called a Heisenberg type group if there is an inner product •, • on g with respect to which the decomposition g = g 1 ⊕ g 2 is orthogonal, and the skewsymmetric endomorphisms J µ given by ω µ (x, x ′ ) = J µ x, x ′ for all x, x ′ ∈ g 1 satisfy where | • | is the norm on g * 2 induced by the inner product •, • .For the rest of this section, we assume that G is a Heisenberg type group.In particular, this implies that Let d 1 = dim g 1 and d 2 = dim g 2 .We fix an orthonormal basis X 1 , . . ., X d1 of g 1 and an orthonormal basis U 1 , . . ., U d2 of g 2 .In the following, to simplify our notation, we identify G and g = g 1 ⊕ g 2 with R d1 × R d2 via the chosen basis.Note that the group multiplication is then given by As usual, the tangent space g is in turn identified with the Lie algebra of (smooth) left-invariant vector fields on G via the Lie-derivative.Given a smooth function f on G, we have The sub-Laplacian L associated with the vector fields X 1 , . . ., X d1 is the second order differential operator given by 2 , let f µ denote the µ-section of the partial Fourier transform along the second layer g 2 given by (2.1) Up to some constant, this defines an isometry 2 ) in the following.For fixed µ ∈ g * 2 , we have (X j f ) µ = X µ j f µ , where X µ j is the differential operator on g 1 given by ).Let L µ be the µ-twisted Laplacian on g 1 given by Let n := d 1 /2 and S := {µ ∈ g * 2 : |µ| = 1}.Note that J µ is orthogonal with eigenvalues ±i for µ ∈ S. Hence, for any µ ∈ ġ * 2 , there is an orthogonal matrix By compactness of S, we may assume that the map ġ * 2 ∋ µ → T μ is measurable, see for instance [Azo74, Thm.1].Let v µ j be the j-th column of T −1 μ and ∇ denote 2 A prominent choice in the literature is also the symplectic form (z, w) → Im(z w) = −ω(z, w), which would correspond to the transpose of the matrix in (2.3).The reason for choosing ω as above instead of (z, w) → Im(z w) is that we want to have the same sign for the third summand in (2.5) as in (1.3.14) of [Tha93].
the usual gradient on R 2n .Then, given a smooth function g on R 2n , we obtain ⊤ is the identity matrix on R d1 thanks to orthogonality, the µ-twisted Laplacian L µ of (2.2) transforms into where L λ 0 is the λ-twisted Laplacian on R 2n given by where we write z ∈ R 2n as z = (a 1 , . . ., a n , b 1 , . . ., b n ).The λ-twisted Laplacian L λ 0 admits a complete orthonormal system of eigenfunctions, which are given by the matrix coefficients of the Schrödinger representation, see [Tha93, Section 1.3].More precisely, let π λ : is the group of unitary operators on L 2 (R n ), denote the Schrödinger representation of the Heisenberg group where h ℓ shall denote the ℓ-th Hermite function on R given by Then, by Theorems 1.3.2 and 1.3.3 of [Tha93], the matrix coefficients Φ λ ν,ν ′ , ν, ν ′ ∈ N n given by where (•, •) is the inner product on L 2 (R 2n ), and ) decomposes into eigenspaces of L λ 0 , where the orthogonal projection Λ λ k onto the eigenspace of the eigenvalue (2k Via the transformation T μ, the spectral decomposition of the µ-twisted Laplacian L µ of (2.2) can be expressed in terms of the spectral decomposition of L λ 0 , which follows directly from (2.4) and (2.7).In the following, we put 2 , the operator L µ on L 2 (g 1 ) admits an orthonormal basis of eigenfunctions associated with the eigenvalues (2.8) Remark.Lemma 2.1 is essentially Proposition 4.5 of [CC13].Note, however, that the definitions of Λ λ k and Π µ k in [CC13] differ from ours by the factors λ n and |µ| n .Moreover, [CC13] even states that the above result holds for left-invariant sub-Laplacians on the larger class of Métivier groups.Unfortunately, this is not the case in general, since the eigenvalues of the twisted Laplacian L µ are not necessarily of the form [k]|µ| if G is only assumed to be a Métivier group.An example may be found for instance in [MS94, Eq. (2.3), (2.4)], where the corresponding eigenvalues are of the form See also [MM14b, Sec.2] for a further discussion.However, the results of [CC13] remain true under the additional hypothesis that G is a Heisenberg type group, and the restriction theorem of [CC13] can be expected to hold in greater generality.
A restriction type estimate that holds beyond Heisenberg type groups can be found in the follow-up paper [Nie23], but unfortunately the estimate there does not seem to be sufficient to recover the result claimed in [CC13].
The projection Λ λ k can be written in a more explicit form as a twisted convolution with a Laguerre function.For λ > 0, let f × λ g be the λ-twisted convolution given by where ω is again the standard symplectic form induced by the matrix J in (2.3).Moreover, let ϕ λ k be the Laguerre function given by where L n−1 k denotes the k-th Laguerre polynomial of type n − 1.Then, since Hence, by (2.1.5) of [Tha93, p. 30], Λ λ k may be rewritten as (2.12) Remark.Our definition of ϕ λ k differs from that of [CC13] by the factor |λ| n/2 .The operators L, −iU 1 , . . ., −iU d2 (where U 1 , . . ., U d2 is the chosen basis of the second layer g 2 ) form a system of formally self-adjoint, left-invariant and pairwise commuting differential operators, whence they admit a joint functional calculus [Mar11].It is well-known [MRS96, Section 1] that for suitable functions F : R × R d2 → C, the operator F (L, U) with U := (−iU 1 , . . ., −iU d2 ) possesses a convolution kernel that can be expressed in terms of the Fourier transform and Laguerre functions ϕ λ k .We provide a direct argument here, although alternatively, the convolution kernel can also be computed by using the Fourier inversion formula of the group Fourier transform on G and the fact that the unitary group representations and the joint functional calculus of L, −iU 1 , . . ., −iU d2 are compatible, see Proposition 1.1 and Lemma 2.2 of [Mül90].
Proposition 2.2.Let F : R × R d2 → C be a bounded Borel function.Then which is given by for almost all (x, u) ∈ G.
Proof.The identity (2.13) can be proved by the same approach as in the proof of Proposition 5 of [MS12] by writing down the corresponding functional calculi in terms of the Fourier transform and the orthogonal projections provided by the eigenfunctions of the µ-twisted Laplacian L µ .To prove (2.14), we observe that (2.13) and Lemma 2.1 yield (2.15) By Lemma 2.1, (2.12), and the fact that where ω denotes again the standard symplectic form on R 2n associated with the matrix J from (2.3).Plugging (2.16) into (2.15),unboxing the Fourier transform f µ , and rearranging the order of integration yields By definition, we have This yields (2.14).

Truncated restriction type estimates
In this section, we prove the truncated restriction type estimates for the sub- , given that G is a Heisenberg type group and X 1 , . . ., X d1 is an orthonormal basis on the first layer g 1 of the stratification g = g 1 ⊕ g 2 .As in [CC13] (and similarly in [CO16,Nie22]), the idea of the proof is to first apply a restriction type estimate in the variable x ∈ g 1 for the µ-twisted Laplacian L µ given by (2.2) and then the Stein-Tomas restriction estimate in the central variable u ∈ g 2 .The restriction type estimate for the orthogonal projection Π µ k onto the k-th eigenspace of the µ-twisted Laplacian L µ is given by the following lemma, which is Lemma 4.7 of [CC13].Let again In view of (2.10), rescaling with λ 1/2 yields Hence, together with (2.8) and a substitution, we obtain (3.1).
Choosing a basis U 1 , . . ., U d2 of the second layer g 2 , we define the operator )) 1/2 .Now we state the truncated restriction type estimates for the sub-Laplacian L.

A weighted Plancherel estimate
In this section, we prove a weighted Plancherel estimate for convolution kernels associated with the sub-Laplacian L on the Heisenberg type group G. Usually, those estimates are the crux of the matter when proving Mikhlin-Hörmander results featuring the threshold s > d/2, where d is the topological dimension of the underlying space, see for example [Mar12, Thm.4.6] or [Mar15, Prop.3].However, in the present setting, the weighted Plancherel estimate (4.1) will serve a different purpose, namely turning support conditions in conjunction with convolution kernels into some sort of rapid decay.
Proposition 4.1.Let F and F ℓ be defined as in Theorem 3.2, and K ℓ be the convolution kernel of the operator F ℓ (L, U ).Then, for all α ≥ 0, for all ℓ ∈ N. (4.1) Proof.Let α ≥ 0. Using (2.14) in combination with Plancherel's theorem, we obtain Given µ ∈ ġ * 2 , we consider the rescaled Hermite operator On the other hand, by Equation (1.3.25) of [Tha93], the functions Φ |µ| ν,ν ′ defined by (2.6) are also eigenfunctions of H µ , with By the definition (2.11) of ϕ |µ| k , this implies in particular (Alternatively, one could use (2.7) by exploiting that ϕ |µ| k is radial-symmetric by (2.9) and that the operators H µ and L |µ| 0 coincide on such functions.)Hence, together with (4.3), the right-hand side of (4.2) can be dominated by a constant times | and orthogonality of the functions ϕ |µ| k , (4.4) can be estimated by a constant times Using polar coordinates and a substitution in the integral over ġ * 2 shows that (4.6) in turn is comparable to This proves (4.1).

The sub-Riemannian geometry of the sub-Laplacian
In this section we summarize the main properties of the sub-Riemannian geometry associated with left-invariant sub-Laplacians on two-step stratified groups.Let G be a two-step stratified Lie group and g = g 1 ⊕ g 2 be a stratification of its Lie algebra.Let X 1 , . . ., X d1 be a basis of g 1 and L = −(X 2 1 + • • • + X 2 d1 ) be the associated sub-Laplacian.We again identify G ∼ = g via the exponential map and g ∼ = R d by means of the basis X 1 , . . ., X d1 of g 1 and a basis U 1 , . . ., U d2 of g 2 .
Let d CC denote the Carnot-Carathéodory distance associated with the vector fields X 1 , . . ., X d1 .By definition, this means that for g, h ∈ G, the distance d CC (g, h) is given by the infimum over all lengths of horizontal curves γ : [0, 1] → G joining g with h, see for instance [VSCC92,Section III.4].Since the vector fields X 1 , . . ., X d1 are left-invariant and [g 1 , g 1 ] = g 2 , they satisfy Hörmander's condition [Hör67], that is, the vector fields X 1 , . . ., X d1 along with their iterated commutators span the tangent space g = T e G of G at the identity e ∈ G, and hence at every point g ∈ G. (In our two-step setting, the vector fields X 1 , . . ., X d1 together with their commutators [X i , X j ] already span the tangent space.)Hence, due to the Chow-Rashevskii theorem [VSCC92, Proposition III.4.1], d CC is indeed a metric on M , which induces the (Euclidean) topology of G = R d .Since X 1 , . . ., X d1 are left-invariant vector fields, d CC is left-invariant, that is, d CC (ag, ah) = d CC (g, h) for all a, g, h ∈ G. (5.1) On the other hand, (x, u) := (|x| 4 + |u| 2 ) 1/4 , (x, u) ∈ G defines a homogeneous norm in the sense of Folland and Stein [FS82] with respect to the dilations δ R given by Since any two homogeneous norms on a homogeneous Lie group are equivalent [FS82, Lemma 1.4], we have (5.3) Let B dCC R (g) denote the ball of radius R ≥ 0 centered at g ∈ G with respect to d CC .Then (5.1) and (5.2) yield where we identify 0 with the identity element e ∈ G via G ∼ = g, and Q = d 1 + 2d 2 is the homogeneous dimension.Note that (5.4) yields in particular that the metric space (G, d CC ) equipped with the the Lebesgue measure (which is a biinvariant Haar measure on G) is a space of homogeneous type with homogeneous dimension Q.Furthermore, the sub-Laplacian L possesses the finite propagation speed property with respect to the Carnot-Carathéodory distance d CC , which will be of fundamental importance in the proof of Theorem 1.1.

Reduction of Theorem 1.1 to dyadic spectral multipliers
To prove Theorem 1.1, we use the following general spectral multiplier result of [COSY16], which allows us to reduce the spectral multiplier estimates of Theorem 1.1 to estimates for spectral multipliers whose Fourier transforms are supported on dyadic scales.Very similar arguments are used in [CO16, Section 5] and [Nie22, Section 4], but we give a detailed discussion for the convenience of the reader.Given a suitable multiplier F : R → C, we use the notation where • and • ∨ denote the Fourier transform and its inverse on R, respectively, and (χ ι ) ι∈Z is the dyadic decomposition from (1.9).
Proposition 6.1.[COSY16, Proposition I.22] Let (X, ρ, µ) be a metric measure space of homogeneous type and Q ≥ 0 such that where B s (x) denotes the ball of radius s > 0 centered at x ∈ X.Let 1 ≤ p 0 < p < 2. Suppose that L is a positive self-adjoint operator on L 2 (X) such that the following statements are satisfied: (i) L possesses the finite propagation speed property, that is, (ii) L satisfies the Stein-Tomas restriction type condition (ST ∞ p0,2 ) of [COSY16], that is, for any R > 0 and all bounded Borel functions for all x ∈ X, all r ≥ 1/R, and all f ∈ L p0 (X).(iii) There is some Suppose that F : R → C is an even bounded Borel function and that there is a bounded sequence (α(ι)) ι∈Z with ι≥0 (ι + 1)α(ι) < ∞ such that Remark.To be precise, in [COSY16], Proposition I.22 requires the condition (E p0,2 ) in place of the Stein-Tomas type restriction condition (ST ∞ p0,2 ).However, both conditions are equivalent by Proposition I.3 of the same paper.Moreover, the notation of our decomposition indexed by i and j differs slightly from that of [COSY16] since (F η i ) (j) = (F χ −i ) (j) , where (η i ) i∈Z is the dyadic decomposition from [COSY16, Eq. (I.3.3)].The somewhat artificial requirement that F shall be an even function is linked to the finite propagation speed property.This will become apparent at the beginning of the proof of Proposition 7.1.
We apply Proposition 6.1 in the setting where X = G is a two-step stratified Lie group and is the sub-Laplacian associated with a basis X 1 , . . ., X d1 of the first layer of the stratification g = g 1 ⊕ g 2 of G.The measure µ in Proposition 6.1 will be the Lebesgue measure on G.
For our purposes of proving Theorem 1.1, we only need Corollary 6.2 for the case where L is a sub-Laplacian on a Heisenberg type group, but the more general version for arbitrary two-step stratified Lie groups is readily available.Note, however, that Proposition 6.1 requires the restriction type condition (ST ∞ p0,2 ) in the setting of arbitrary two-step stratified Lie groups.However, instead of using the restriction type estimates of Theorem 3.2, we use a Plancherel estimate for the associated convolution kernel, which in turn implies a restriction type estimate from L 1 to L 2 .Corollary 6.2.Let G be a two-step stratified Lie group and L be a sub-Laplacian as in (6.4).Let p * ∈ [1, 2] and s > 1/2.Suppose that for all 1 ≤ p ≤ p * there exists some ε > 0 such that and all even bounded Borel functions

Then the statements (1) and (2) of Theorem 1.1 hold for all
However, in Corollary 6.2, we only require 1 ≤ p ≤ p * for some p * ∈ [1, 2], which is why we additionally assume s > 1/2 to make sure that F | (0,∞) ∞ F L 2 s,sloc .Proof.The result for Bochner-Riesz multipliers in the second part of Theorem 1.1 is a direct consequence of (6.5) without Proposition 6.1 involved, which can be seen in the same way as in the proof of Theorem 1.2 in [Nie22, Section 4].
For the first part of Theorem 1.1, we observe that Thus, we may pass from F (L) to the operator F ( √ L).We use Proposition 6.1 for p 0 = 1.Proposition 6.1 only shows that the operator F ( √ L) is of weak type (p, p), but the boundedness on L p can be easily recovered as follows: First, note that the case p = 1 is excluded in the first part of Theorem 1.1.Thus, suppose that the interval (1, p * ] is non-empty and let 1 < p ≤ p * .Suppose that F : R → C is a bounded Borel function satisfying ) is of weak type (p, p).On the other hand, the assumption s > 1/2 in Corollary 6.2 ensures that Hence, via interpolation with the L 2 -L 2 bound provided by the spectral theorem, we may conclude that F ( √ L) is bounded on L p .The claimed estimate in the first part of Theorem 1.1 follows by the closed graph theorem applied to the map F → F ( √ L).Alternatively, this estimate can also be derived by inspecting the arguments of [COSY16] (see also [Nie21,Section 7]).Now we verify the assumptions of Proposition 6.1.Let p 0 = 1.The finite propagation speed property in (i) holds due to Lemma 5.1, and the estimate in (iii) is automatically fulfilled by Theorem I.5 of [COSY16].Since p 0 = 1, the restriction type condition (ST ∞ p0,2 ) is a consequence of a Plancherel estimate for the associated convolution kernel, see also [COSY16, Section III.5].More precisely, given a bounded Borel measurable function there is some constant C > 0 such that we have the Plancherel estimate Thus, we obtain By (5.4), we have Hence, for p 0 = 1, the factor on the right-hand side of (6.2) is given by which verifies the restriction type condition (ST ∞ p0,2 ).Thus the assumptions of Proposition 6.1 are satisfied.Now suppose that the dyadic estimate (6.5) of Corollary 6.2 holds.Let F : R → C be a bounded Borel function such that ) is of weak type (p, p), we verify the required estimate (6.3) of Proposition 6.1.Note that we may assume without loss of generality that F is an even function since L is a positive operator.For i ∈ Z, we let F i := F χ i , where (χ i ) i∈Z is the dyadic decomposition from (1.9).Given i, j ∈ Z, let ι := i + j and where χ is the bump function from (1.9).Then G is an even function, and Let again δ R be the dilation given by δ This implies Hence, for ι ≥ 0, (6.5) yields L 2 s 2 −ει F L 2 s,sloc .The case ι < 0 can be treated by the Mikhlin-Hörmander type result of [Chr91] and [MM90].Suppose ι < 0. Let ψ := i≤2 χ i .Then ψ is supported in [−8, 8].We decompose G (ι) as Let Q denote again the homogeneous dimension of G. Choosing N := 0 in (6.6) and using 2 ι(α+1) ≤ 1, we obtain On the other hand, choosing N := α + 1 in (6.6) yields in particular Since all derivatives of 1 − ψ are Schwartz functions, Leibniz rule yields Hence, applying the Mikhlin-Hörmander type result of [Chr91] and [MM90] yields . This establishes the required condition (6.3) of Proposition 6.1.Thus we can apply Proposition 6.1 and we get that the operator F ( √ L) is of weak type (p, p).

7.
Proof of the reduction of Theorem 1.1 ) is a sub-Laplacian on a Heisenberg type group G, where X 1 , . . ., X d1 is an orthonormal basis of the first layer g 1 of the stratification g = g 1 ⊕ g 2 .Let again d CC denote the Carnot-Carathéodory distance associated with the vector fields X 1 , . . ., X d1 , let d = d 1 + d 2 be the topological dimension, and Q = d 1 + 2d 2 be the homogeneous dimension of G.By Corollary 6.2, the proof of Theorem 1.1 can be reduced to proving the statement.Given a suitable multiplier F : R → C, we write again where (χ ι ) ι∈Z is the dyadic decomposition of (1.9).Proposition 7.1.Suppose that 1 ≤ p ≤ 2(d 2 + 1)/(d 2 + 3).If s > d (1/p − 1/2), then there exists some ε > 0 such that L 2 s for all ι ∈ N and any even bounded Borel function Proof.Let ι ∈ N and R := 2 ι .We proceed in several steps.
(1) Reduction to compactly supported functions.Let f ∈ D(G) be an integrable simple function on G.We first show that we may restrict to the case where f is supported in B dCC R (0).Since the metric space (G, d CC ) endowed with the Lebesgue measure is a space of homogeneous type and separable, we may choose a decomposition into disjoint sets B j ⊆ B dCC R (x (j) , u (j) ), j ∈ N, (x (j) , u (j) ) ∈ G such that for every λ ≥ 1, the number of overlapping dilated balls B dCC λR (x (j) , u (j) ) is bounded by a constant C(λ) ∼ λ Q , which is independent of ι.We decompose f as Since F is even, so is F .As χ ι is even as well, the Fourier inversion formula provides Since L satisfies the finite propagation speed property, j) , u (j) ) by the formula above.Together with the bounded overlap of these balls, we obtain Altogether, since L is left-invariant, it suffices to show (2) Localizing the multiplier.Next we show that we may replace the multiplier F (ι) by F (ι) ψ, where ψ is a smooth cut-off function which is compactly supported away from the origin.Using the dyadic decomposition (χ ι ) ι∈Z from (1.9), we put Then 1/8 ≤ |λ| ≤ 8 whenever λ ∈ supp ψ, and |λ| / ∈ (1/4, 4) if λ ∈ supp(1 − ψ).We decompose F (ι) as The second part of this decomposition can be treated by the Mikhlin-Hörmander type result of [Chr91] and [MM90].Note that F (ι) = F * χι , and χ ∈ S(R).Thus, given α ∈ N and N ∈ N, we have This implies Hence, the Mikhlin-Hörmander type result of [Chr91] and [MM90] yields Thus, instead of (7.1), we are left proving ℓ (λ, ρ) = 0 else.We decompose the function on the left-hand side of (7.3) as The sum over ℓ > ι can be treated directly by the restriction type estimate of Theorem 3.2.Recall that |B dCC R (0)| ∼ R Q by (5.4).Hence, Hölder's inequality with 1/q = 1/p − 1/2 and the restriction type estimate (3.2) imply s due to the localization in frequency.)Thus, we are done once we have also treated the sum over ℓ ∈ {−1, . . ., ι}, that is, it remains to show ℓ be the convolution kernel of the operator F (ι) ℓ (L, U ).By (5.3), there is a constant C > 0 such that B dCC R (0) ⊆ B CR (0) × B CR 2 (0).Hence the function f is supported in a Euclidean ball of dimension R × R 2 .In view of the finite propagation speed property which we exploited in part (1) of the proof, we may think of K (ι) ℓ being supported in a ball of dimension R × R 2 as well (which is of course not quite true since we replaced the multiplier F (ι) by F (ι) ψ).In the following, we show that the convolution kernel K (ι) ℓ of the truncated multiplier essentially supported in an even smaller ball of dimension R ℓ R γ × R 2 , where R ℓ := 2 ℓ and γ > will be a number chosen sufficiently small, depending only on the parameters s, p, d 1 , d 2 .For convenience, we introduce the following notation: We will write A ι B whenever A ≤ R C(p,d1,d2)γ B for some constant C(p, d 1 , d 2 ) > 0 depending only on the parameters p, d 1 , d 2 .Given ℓ ∈ {0, . . ., ι}, we split the Euclidean ball B CR (0) × B CR 2 (0) into a grid with respect to the first layer, which gives a decomposition of supp f where Moreover, given γ > 0, the number of overlapping balls In the following, we show that the function is essentially supported in the ball B(ℓ) m .We decompose the function g ≤ι of (7.4) as ≤ι + g (2) ≤ι . (7.7) To show that the second summand is negligible (in the sense of (7.13)), we interpolate between L 1 and L 2 via the Riesz-Thorin interpolation theorem.For the ℓ be the convolution kernel associated with ℓ (x, u), and we obtain where Given N ∈ N, the Cauchy-Schwarz inequality yields . (7.9) In the last line we used again that |B dCC 3R (0)| ∼ R Q by (5.4).By Proposition 4.1, the second factor of (7.9) can be estimated by Hence Altogether, with (7.8), we have Together with (7.6) and the Sobolev embedding 2 f 2 for α ′ > α + d 1 /2.(7.12) Applying the Riesz-Thorin interpolation theorem with (7.10) and (7.12) and choosing N = N (γ) ∈ N sufficiently large in (7.10) yields g ).
On a formal level, this means that we may indeed assume that the convolution kernel K Plugging this estimate into the right-hand side of (7.15) and using the fact that the (R d1−d2 ℓ R 2d2 ) p/q F (ι) p 2 f p p .

Remarks on weighted restriction type estimates for sub-Laplacians
In [CO16], Chen and Ouhabaz proved a spectral multiplier theorem for the Grushin operator G = −∆ x − |x| 2 ∆ u acting on R d1 × R d2 by using a weighted restriction type estimate of the form where α > 0 and F : R → C is a bounded Borel function supported in [1/4, 1].Let L denote again a sub-Laplacian on a Heisenberg type group G with Lie algebra g = g 1 ⊕ g 2 with layers of dimension d 1 and d 2 , respectively.Then Lf = Gf for any g 1 -radial function on the Heisenberg type group G, i.e., a function on G which only depends on |x| (with x ∈ g 1 ) and u ∈ g 2 (where we identify again G with its Lie algebra g, which is in turn identified with R d1 × R d2 ).In view of this close relationship, one might hope that the approach of Chen and Ouhabaz can also be applied in the setting of Heisenberg type groups.However, a crucial ingredient of their approach is the sub-elliptic estimate where H µ = −∆ x + 1 4 |x| 2 |µ| 2 denotes again the rescaled Hermite operator on R 2n .Unfortunately, the analogous estimate of (8.1) in our setting, where H µ is replaced by the µ-twisted Laplacian L µ of (2.2), fails.We will prove in the following that the estimate (8.1)where H µ is replaced by L µ is false for α = 1.(The approach of [CO16] requires to choose 0 < α < d 2 (1/p − 1/2) as large as possible, so large values of α are the crucial ones.)Via (2.4) and a linear substitution, the estimate is equivalent to where L (8.2) 0). (3) Truncation along the spectrum of U .Next we decompose the operator (F (ι) ψ)( √ L) by a dyadic decomposition of U .For ℓ ∈ N, let the function F (ι) ℓ : R × R → C be given by

ℓ
is supported in a ball of dimension R ℓ R γ × R 2 .(5)The main contribution.Hölder's inequality and the bounded overlapping property of the balls B(ℓ)