# On the consequences of a Mihlin-Hörmander functional calculus: maximal and square function estimates

## Abstract

We prove that the existence of a Mihlin-Hörmander functional calculus for an operator *L* implies the boundedness on \(L^p\) of both the maximal multiplier 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.

## Keywords

Maximal multiplier Square function Spectral multiplier## Mathematics Subject Classification

47A60 42B25 42B15## 1 Introduction

Consider the Laplacian \(-\Delta \) on \(\mathbb {R}^d\) and let \(m:[0,\infty )\rightarrow \mathbb {C}\) 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 \(m(-\Delta )\) on all \(L^p,\) \(1<p<\infty ,\) spaces. Moreover it is well known that the maximal multiplier \(M_m(f):=\sup _{t>0}|m(-t\Delta )(f)|\) has the same boundedness properties, as well as the square function \(S_m(f)^2=\int _0^{\infty }|m(-t\Delta )f|^2\,\frac{dt}{t}\) (under the additional assumption \(m(0)=0\)).

The boundedness of \(M_m\) on \(L^p\) spaces, \(1<p<\infty ,\) 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, \(1<p<\infty ,\) 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 \(\mathbb {R}^d\) and the Fourier transform in a decisive manner.

Replacing the Laplacian \(-\Delta \) by some other non-negative self-adjoint operator *L* on \(L^2(X)\) leads to the spectral multipliers *m*(*L*). If \(m:[0,\infty )\rightarrow \mathbb {C}\) is Borel measurable and bounded then *m*(*L*) is initially well defined (and bounded) on \(L^2(X)\) by the spectral theorem. Throughout the paper we assume that *L* has a Mihlin-Hörmander functional calculus on \(L^p:=L^p(X),\) i.e. the Mihlin-Hörmander multiplier theorem holds for *L*, regarded as an operator on \(L^p\) spaces, \(1<p<\infty \). More precisely, we impose that every function satisfying the Mihlin-Hörmander condition \(\sup _{\lambda >0}\left| \lambda ^\beta \frac{d^\beta }{d\lambda ^\beta }m(\lambda )\right| \le C_\beta ,\) \(\beta =0,\ldots ,\alpha ,\) of order \(\alpha \) (here \(\alpha =\alpha (L)\) is a fixed parameter) gives rise to a bounded operator *m*(*L*) on all \(L^p\) spaces, \(1<p<\infty .\) There is a vast literature giving (or implying) the existence of a Mihlin-Hörmander functional calculus for more general operators, see e.g. [1, 2, 3, 8, 9, 13, 14, 18, 21, 23, 24, 25, 26, 29], and references therein.

On the other hand the topic of \(L^p\) estimates for the general maximal multipliers \(M_m(f):=\sup _{t>0}|m(tL)(f)|\) and square functions \(S_m(f)^2=\int _0^{\infty }|m(tL)f|^2\,\frac{dt}{t}\) 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 us underline that we do not require any structure (e.g. dilations or translations) on the underlying space *X*.

What concerns the square function \(S_{\psi }(f)^2=\int _0^{\infty }|\psi (tL)f|^2\,\frac{dt}{t}\), the paper [11] by Cowling, Doust, McIntosh, and Yagi treats exhaustively the case when \(\psi \) 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 \(\psi \) to be a compactly supported function having continuous \(\alpha +2\) derivatives and satisfying \(\psi (0)=0.\)

The methods we use to treat maximal multipliers 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 \(\mathbb {R}^d.\) 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].

The main results of our paper, Theorem 3.1 and Corollary 3.2 as well as Theorem 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 \(N(\varphi )<\infty \) and \(\tilde{N}(\varphi )<\infty \) (see (2.6)) as they can often be directly verified. Finally, let us highlight that an important motivation for our research was to obtain maximal and square functions estimates for operators based on compactly supported \(C^{\alpha +2}\) multipliers. This has been accomplished in Corollaries 3.2 and 4.2.

## 2 Preliminaries

*m*is given by

### Definition 2.3

*m*is a bounded function having partial derivatives up to order \(\alpha ,\) and for all non-negative integers \(j\le \alpha \)

*m*satisfies the Mihlin-Hörmander condition of order \(\alpha ,\) then we set

*L*be a non-negative self-adjoint operator on \(L^2(X,\nu ),\) for some \(\sigma \)-finite measure space \((X,\nu ),\) with \(\nu \) being a Borel measure. Then, for \(m:[0,\infty )\rightarrow \mathbb {C},\) the spectral theorem allows us to define the multiplier operator \(m(L)=\int _{[0,\infty )}m(\lambda )dE(\lambda )\) on the domain

*E*is the spectral measure of

*L*, while \(E_{f,f}\) is the complex measure given by \(E_{f,f}(\cdot )=\langle E(\cdot )f,f\rangle _{L^2(X,\nu )}.\) 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 \(m(L)=\int _{(0,\infty )}m(\lambda )dE(\lambda ).\)

*L*has the Mihlin-Hörmander (MH) functional calculus of order \(\alpha >0\) if the following holds: every bounded multiplier function

*m*that satisfies the Mihlin-Hörmander condition (MH) of order \(\alpha ,\) gives rise to an operator

*m*(

*L*) (defined initially on \(L^2(X,\nu )\) by the spectral theorem), which satisfies

*m*(

*L*) extends to a bounded operator on all \(L^p(X,\nu )\) spaces, \(1<p<\infty .\) Throughout the paper we assume that

*L*is a non-negative self-adjoint operator that has the Mihlin-Hörmander funcional calculus of order \(\alpha \) on every \(L^p(X,\nu ),\) \(1<p<\infty .\)

*L*generates a symmetric contraction semigroup whenever

*L*having a Mihlin-Hörmander functional calculus on all \(L^p\) spaces, \(1<p<\infty \).

We will often consider Banach spaces \(L^p(X,\nu )\) for \(1<p<\infty \). For brevity we write \(L^p\) and \(\Vert \cdot \Vert _p\) instead of \(L^p(X,\nu )\) and \(\Vert \cdot \Vert _{L^p(X,\nu )},\) \(1\le p\le \infty .\) If *T* is a sublinear operator then the symbol \(\Vert T\Vert _{p\rightarrow p}\) 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 \(L^2\cap L^p.\)

*Y*with continuous \(\beta \) derivatives. For \(h\in C^{\beta }(Y)\) we define

*Y*equals \(\mathbb {R}\), \([0,\infty )\) 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 \(A\lesssim _\omega B\) we mean that \(A\le C_{\omega } B,\) where \(C_{\omega }\) is a constant that may depend only on \(\omega \). In our case \(C_{\omega }\) is always independent of the function \(f\in L^2\cap L^p,\) though it may depend on both \(p\in (1,\infty )\) and the multiplier \(\varphi \) or \(\psi \).

## 3 Maximal multiplier operators

In this section we additionally impose that *L* generates a symmetric contraction semigroup, i.e. (2.5) holds. This assumption holds for all the operators studied in [1, 2, 3, 8, 9, 13, 14, 18, 21, 23, 24, 25, 26, 29].

### Theorem 3.1

### Remark 1

A natural question is whether it is enough to assume that \(\varphi \) satisfies the Mihlin-Hörmander condition (MH) instead of imposing \(N(\varphi )<\infty \). 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 \(L^2(X,\nu )\) bound \(\Vert L^{iu}\Vert _{L^2(X,\nu )\rightarrow L^2(X,\nu )}\le 1.\) 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.

### Remark 3

The proof presented here can be adjusted to operators that have a Mihlin-Hörmander functional calculus on other Banach spaces *B* than \(L^p.\) For instance the results of [15] imply the existence of such a calculus, on the Hardy space \(B=H^1_L\) (introduced in [19]), for operators *L* with Davies-Gafney estimates on their heat semigroups. We remark that a proper adjustment requires the strong continuity on \(H^1_L\) of the imaginary powers \(\{L^{iu}\}_{u\in \mathbb {R}}\); this is proved in [16, Corollary 3.3].

Before proceeding to the proof of Theorem 3.1 let us note the following useful corollary.

### Corollary 3.2

*K*, then, for each \(1<p<\infty ,\) we have

### Proof

Use Theorem 3.1 and the estimate \(N(\varphi )\lesssim _K \Vert \varphi \Vert _{C^{\alpha +2}([0,\infty ))}.\) \(\square \)

### Remark

Exemplary functions satisfying the assumptions of the corollary are \(\varphi ^{\delta }=(1-\lambda )^{\delta }\chi _{\{\lambda <1\}},\) with \(\delta \ge \alpha +2.\) In this case \(M_{\varphi ^{\delta }}\) is the maximal function connected with the Bochner-Riesz means.

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 multiplier 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 \(1<p<\infty \)) of the group of imaginary powers.

In summary, the proof is completed, provided we justify that the formal expression (3.2) converges as an \(L^p\)-valued integral. This follows from the well-known \(L^p\) continuity of \(u\mapsto L^{iu}f\) and the finiteness of \(\int _{\mathbb {R}}|A_{\varphi }(u)|\Vert L^{iu}f\Vert _p\,du.\) Observe also that the above argument together with the Lebesgue’s dominated convergence theorem shows that the mapping \(t\mapsto \int _{\mathbb {R}}A_{\varphi }(u)t^{iu}L^{iu}f(x)\,du\) is continuous, for a.e. \(x\in X.\) Hence, in view of (3.2) the supremum in the definition of the maximal multiplier may be taken over all \((0,\infty ).\) \(\square \)

## 4 Continuous square functions

### Theorem 4.1

Before proving the theorem we state a seemingly interesting corollary.

### Corollary 4.2

*K*. Then, for each \(1<p<\infty ,\) we have

### Proof

Simply observe that \(\tilde{N}(\psi )<\infty .\) \(\square \)

### Remark

Examples of functions admitted by the corollary are \(\psi ^{\delta }(\lambda )=\lambda (1-\lambda )^{\delta }\chi _{\{\lambda <1\}},\) with \(\delta \ge \alpha +2.\) The usefulness of \(S_{\psi ^{\delta }}\) lies in the fact that sharp \(L^p,\) \(p>2,\) bounds for this square function imply sharp \(L^p\) bounds for maximal multipliers \(M_{\varphi ^{\delta }}\) (defined on p. 5) associated with Bochner-Riesz means, see [5, 6, 22 p. 274], and [12]. This is true not only in the Fourier case but also in general as the reasoning from [6, p. 54] can be repeated mutatis mutandis.

We proceed with the proof of the main result of this section.

### Proof of Theorem 4.1

Since \(\tilde{N}(\psi )<\infty \) implies the boundedness of \(\psi ,\) we see that (4.2) holds. Therefore \(S_{\psi }\) is bounded on \(L^2.\)

*h*on \(\mathbb {R},\) supported in \([-\pi ,\pi ],\) and such that

*a*, and with a quadratic decay in

*k*, i.e.

*L*has a Mihlin-Hörmander functional calculus of order \(\alpha ,\) and coming back to (4.7) we finish the proof of Theorem 4.1.

## Notes

### Acknowledgements

Part of the research presented in this paper was carried over while the author was *Assegnista di ricerca* at the Università di Milano-Bicocca, working under the mentorship of Prof. Stefano Meda. The research was supported by Italian PRIN 2010 “Real and complex manifolds: geometry, topology and harmonic analysis”; Polish funds for sciences, National Science Centre (NCN), Poland, Research Project 2014/15/D/ST1/00405; and by the Foundation for Polish Science START Scholarship.

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