On the consequences of a Mihlin-H\"ormander functional calculus: maximal and square function estimates

We prove that the existence of a Mihlin-H\"ormander functional calculus for an operator $L$ implies the boundedness on $L^p$ of both the maximal operators and the continuous square functions build on spectral multipliers of $L.$ The considered multiplier functions are finitely smooth and satisfy an integral condition at infinity. In particular multipliers of compact support are admitted.


Introduction
Consider the Laplacian −∆ on R d and let m : [0, ∞) → C be a bounded compactly supported function having [d/2] + 1 continuous derivatives. The Mihlin-Hörmander multiplier theorem [20], [28], implies the boundedness of the Fourier multiplier operator m(−∆) on all L p , 1 < p < ∞, spaces. Moreover it is well known that the maximal function M m (f ) := sup t>0 |m(−t∆)(f )| has the same boundedness properties, as well as the square function S m (f ) 2 = ∞ 0 |m(−t∆)f | 2 dt t (under the additional assumption m(0) = 0). The boundedness of M m on L p spaces, 1 < p < ∞, follows from majorizing this operator by the Hardy-Littlewood maximal function. As for the square function S m , the boundedness on all L p spaces, 1 < p < ∞, can be proved by applying the vector-valued Caldeón-Zygmund theory. We remark that all the proofs referred to in this paragraph are using dilation properties of R d and the Fourier transform in a decisive manner.
On the other hand the topic of L p estimates for the general maximal operators M m (f ) := sup t>0 |m(tL)(f )| and square functions S m (f ) 2 = ∞ 0 |m(tL)f | 2 dt t has attracted considerably less attention. The main aim of this paper is to prove that, to a large degree, these estimates can be deduced from the Mihlin-Hörmander functional calculus itself.
Let ϕ be a compactly supported function with continuous α + 2 derivatives and such that 0 ∈ supp ϕ. One of the well known consequences of a Mihlin-Hörmander functional calculus on L p , 1 < p < ∞, is the L p boundedness of the discrete square function The aforementioned boundedness follows easily from Khintchine's inequality. As a consequence, for such a ϕ, the discrete maximal function M disc ϕ (f ) = sup k∈Z |ϕ(2 −k L)f | is also bounded on all L p spaces, 1 < p < ∞. For the Fourier case we know that the assumption 0 ∈ supp ϕ is superfluous. We prove that the same is true for general maximal operators M ϕ (f ) = sup t>0 |ϕ(tL)f |, where it might well be that ϕ(0) = 0. This is done in Theorem 3.1 and Corollary 3.2.
What concerns the square function S ψ (f ) 2 = ∞ 0 |ψ(tL)f | 2 dt t , the paper [11] by Cowling, Doust, McIntosh, and Yagi treats exhaustively the case when ψ is holomorphic. Our article relaxes this assumption to some finite order of smoothness, see Theorem 4.1 and Corollary 4.2. For instance we may take ψ to be a compactly supported function having continuous α + 2 derivatives and satisfying ψ(0) = 0.
The methods we use to treat maximal functions are based on Mellin transform techniques and have their roots in Cowling's [10]. These techniques were employed by Alexopoulos and Lohué in [4] and by Gunawan and Sikora in [17]. The paper [4] focuses on the specific maximal functions associated with Bochner-Riesz means on general Lie groups of polynomial volume growth or on Riemannian manifolds of non-negative curvature. The report [17] studies maximal functions associated with more general multipliers for elliptic operators on R d . Our contribution is the observation that similar methods can be used in a far bigger generality. The techniques we employ to examine square functions are an adaptation of those from [11] and [27].
It seems that the boundedness of maximal and square functions (M ϕ and S ψ ) for general operators L having a Mihlin-Hörmander functional calculus has not yet been fully considered. In the present paper we prove some results in this direction. Theorems 3.1 and Corollary 3.2 as well as 4.2 and Corollary 4.2 apply to all the operators considered in [1], [2], [3], [8], [9], [13], [14], [18], [21], [23], [24], [26], [29]. In particular L can be: a Laplacian on a general Lie group of polynomial volume growth (the Mihlin-Hörmander functional calculus follows from [2]), a Laplacian on a discrete groups with polynomial volume growth (the Mihlin-Hörmander functional calculus can be deduced from [1]), or a non-negative operator having Davies-Gaffney estimates for its heat semigroup (the Mihlin-Hörmander functional calculus is a consequence of [29]).
The arguments used in Theorems 3.1 and 4.1 could be easily refined to yield sharper results. We decided to stick with integral conditions N (ϕ) < ∞ andÑ (ϕ) < ∞ (see (2.6)) as they can often be directly verified. Finally, let us underline that an important motivation for our research was to obtain maximal and square functions estimates for operators based on compactly supported C α+2 multipliers. This has been accomplished in Corollaries 3.2 and 4.2.

Preliminaries
The following terminology is used throughout the paper. The Melin transform of a function m is given by The inversion formula for the Melin transform reads, for m ∈ L 1 ((0, ∞), ds s ) and M(m) ∈ L 1 (R, du), as The Plancherel formula for the Melin transform is Definition 2.3. We say that m : [0, ∞) → C satisfies the Mihlin-Hörmander condition of order α ∈ N if m is a bounded function having partial derivatives up to order α, and for all non-negative integers j ≤ α If m satisfies the Mihlin-Hörmander condition of order α, then we set Let L be a non-negative self-adjoint operator on L 2 (X, ν), for some σ-finite measure space (X, ν), with ν being a Borel measure. Then, for m : [0, ∞) → C, the spectral theorem allows us to define the multiplier operator m(L) Here E is the spectral measure of L, while E f,f is the complex measure given by E f,f (·) = E(·)f, f L 2 (X,ν) . Mainly for notational convenience throughout the paper we also assume that L has trivial kernel (or in other words that the spectral projection satisfies E({0}) = 0). In this case m(L) can be rewritten as m(L) = (0,∞) m(λ)dE(λ).
We say that L has the Mihlin-Hörmander (MH) functional calculus of order α > 0 if the following holds: every bounded multiplier function m that satisfies the Mihlin-Hörmander condition (MH) of order α, gives rise to an operator m(L) (defined initially on L 2 (X, ν) by the spectral theorem), which satisfies Clearly, in this case m(L) extends to a bounded operator on all L p (X, ν) spaces, 1 < p < ∞. Throughout the paper we assume that L is a non-negative self-adjoint operator that has the Mihlin-Hörmander funcional calculus of order α on every L p (X, ν), 1 < p < ∞. Among the well-known consequences of a Mihlin-Hörmander functional calculus the one especially important to us is the bound for the imaginary powers We say that L generates a symmetric contraction semigroup whenever Note that by Meda's [26,Theorem 4], for operators generating symmetric contraction semigroups, condition (2.4) is, up to the order of differentiability, equivalent with L having a Mihlin-Hörmander functional calculus on all L p spaces, 1 < p < ∞.
We will often consider Banach spaces L p (X, ν) for 1 < p < ∞. For brevity we write L p and · p instead of L p (X, ν) and · L p (X,ν) , 1 ≤ p ≤ ∞. If T is a sublinear operator then the symbol T p→p denotes the norm of T acting on L p . Slightly abusing the terminology we say that a sublinear operator is bounded on L p if it has a unique bounded extension to L p . In particular it is enough to prove the boundedness on note that it may happen that h C β (Y ) = ∞. In this paper Y equals R, [0, ∞) or [0, 1]; in the last two cases only one-sided derivatives on the boundary are considered.
For non-negative numbers A and B by A ω B we mean that A ≤ C ω B, where C ω is a constant that may depend only on ω. In our case C ω is always independent of the function f ∈ L 2 ∩ L p , though it may depend on both p ∈ (1, ∞) and the multiplier ϕ or ψ.
The maximal functions investigated here are of the form In particular, for ϕ(λ) = exp(−λ) we have the maximal operator associated with the heat semigroup M exp(−·) . The following is the main result of Section 3. Note that formally the supremum in (3.1) should be restricted to a countable set, however, from the proof of Theorem 3.1 it will follow that the supremum may be taken over all of (0, ∞). We recall that N (ϕ) is defined by (2.6).

Remark 1.
A natural question is whether it is enough to assume that ϕ satisfies the Mihlin-Hörmander condition (MH) instead of imposing N (ϕ) < ∞. In [7] the authors proved that this is not the case even for radial Fourier multipliers (which correspond to L being the Laplacian). Therefore some other assumptions are indeed necessary.
Remark 2. The required smoothness threshold can be improved if we use a Sobolev (continuous) variant of the Mihlin-Hörmander norm (MH) and interpolate (2.4) with the obvious L 2 (X, ν) bound L iu L 2 (X,ν)→L 2 (X,ν) ≤ 1. However, in general the obtained result is still far from being optimal. Thus, for the clarity of the presentation we decided to phrase Theorem 3.1 in terms of the integer Mihlin-Hörmander norm. L . This will be one of the problems addressed in a forthcoming paper by Jacek Dziubański and the author.
Before proceeding to the proof of Theorem 3.1 let us note the following useful corollary.
Now we pass to the proof of the main result of this section.
Proof of Theorem 3.1. We will use Melin transform techniques here, to estimate the maximal operator by a certain integral of imaginary powers L iu . This idea is due to Cowling [10,Section 3]. Remark that essentially all that is needed to write the proof below rigorously is the strong L p continuity (for 1 < p < ∞) of the group of imaginary powers. From [10,Theorem 7] it follows that M exp(−·) is bounded on all L p . Thus, it is enough to show that M ϕ(·)−ϕ(0) exp(−·) is also bounded on L p . For each fixed λ, t > 0, we have and, consequently, by spectral theory, Hence, using (2.1) we formally write which together with Minkowski's integral inequality imply We remark that the validity of (3.4) was exactly the reason for subtracting the term ϕ(0) exp(−s). Now, in view of (2.4) it is enough to prove the bound Assume first that ϕ is compactly supported. Then, clearly, |A ϕ (u)| N (ϕ). Moreover, repeated integration by parts produces
In summary, the proof is completed, provided we justify that the formal expression (3.4) converges as an L p -valued integral. This follows from the well-known L p continuity of u → L iu f and the finiteness of R |A ϕ (u)| L iu f p du. Observe also that the above argument together with the Lebesgue's dominated convergence theorem shows that the mapping t → R A ϕ (u)t iu L iu f (x) du is continuous, for a.e. x ∈ X. Hence, in view of (3.4) the supremum in the definition of the maximal function may be taken over all (0, ∞).

Continuous square functions
Now we pass to square functions. These are given by The spectral theorem implies that S ψ is bounded on L 2 (in which case it is an isometry) if and only if The next theorem establishes the boundedness of S ψ on other L p spaces. Recall thatÑ (ψ) is defined by (2.6).
We proceed with the proof of the main result of this section.
Proof of Theorem 4.1. SinceÑ (ψ) < ∞ implies the boundedness of ψ, we see that (4.2) holds. Therefore the square function S ψ is bounded on L 2 . We use Melin transform techniques here together with ideas from [11]. By a polarization argument it is enough to prove the right hand side inequality in (4.4). Note that, since ψ(0) = 0, the function M(ψ)(u) is well defined. Moreover, for fixed λ > 0 it holds M(ψ(·λ))(u) = M(ψ)(u)λ iu . Thus using the spectral theorem and Plancherel's formula (2.2) the square function given by (4.1) can be reexpressed as (4.5) S We start by following the scheme from [27]. Take a smooth function h on R, supported in [−π, π], and such that For each j ∈ Z we define the functions b j,k on [0, ∞) by It is not hard to see that the functions b j,k are bounded on [0, ∞), thus it makes sense to consider b j,k (L). From Parseval's formula for the Fourier series it follows that, for each k ∈ Z, the set {(2π) −1/2 e −ij(u−πk) } j∈Z forms an orthonormal basis in the space L 2 (Q k , du), with Q k = [kπ − π, kπ + π]. Therefore, since supp(h k ) ⊂ Q k and |e iπ j,k | = 1, for each fixed λ > 0 we have (4.7) Now, using (4.5), (4.6) and Minkowski's integral inequality, followed by the spectral theorem and (4.7), we obtain x − a.e., and consequently, Then from Khintchine's inequality it follows that We shall now focus on estimating the L p operator norm of each of the operators m a k (L), defined in (4.8). We claim that each of the functions If the claim is true, then, using the assumption that the operator L has a Mihlin-Hörmander functional calculus of order α, and coming back to (4.8) we finish the proof of Theorem 4.1.
Thus we focus on proving (4.9). Let n(y) = m(e y ), y ∈ R. Then, using e.g. Faá di Bruno's formula, it is not hard to see that Consequently, denoting it is enough to show that (4.10) n C α (R) = j a j C j,k C α (R) 1 1 + k 2 .
In order to prove (4.10) we will need the bound It suffices to show (4.11) for compactly supported ψ as the general case follows from an approximation argument similar to the one performed in the proof of Theorem 3.1. For small |u| < 1 the finiteness ofÑ (ψ) implies For large |u| > 1 integration by parts (which is legitimate since ψ is compactly supported) produces ds α+2 ψ(s) ds, and, consequently, ds α+2 ψ(s) ds.
Coming back to the proof of (4.10) we observe that for each fixed β ≤ α it holds (4.13) d β dy β C j,k (y) = i β R h k (u)M(ψ)(u)u β e i(y−j)u du.