# Joint Spectral Multipliers for Mixed Systems of Operators

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## Abstract

We obtain a general Marcinkiewicz-type multiplier theorem for mixed systems of strongly commuting operators \(L=(L_1,\ldots ,L_d);\) where some of the operators in *L* have only a holomorphic functional calculus, while others have additionally a Marcinkiewicz-type functional calculus. Moreover, we prove that specific Laplace transform type multipliers of the pair \((\mathcal {L},A)\) are of certain weak type (1, 1). Here \(\mathcal {L}\) is the Ornstein-Uhlenbeck operator while *A* is a non-negative operator having Gaussian bounds for its heat kernel. Our results include the Riesz transforms \(A(\mathcal {L}+A)^{-1},\) \(\mathcal {L}(\mathcal {L}+A)^{-1}\).

## Keywords

Joint functional calculus Multiplier operator Ornstein-Uhlenbeck operator## Mathematics Subject Classification

47A60 42B15 60G15## 1 Introduction

*L*. Moreover, for a bounded function \(m:[0,\infty )^d\rightarrow \mathbb {C},\) the multiplier operator

*m*(

*L*) can be defined on \(L^2(X,\nu )\) by

*m*(

*L*) is then bounded on \(L^2(X,\nu ).\) In this article we investigate under which assumptions on the multiplier function

*m*is it possible to extend

*m*(

*L*) to a bounded operator on \(L^p(X,\nu ),\) \(1<p<\infty .\)

Throughout the paper we assume the \(L^p(X,\nu ),\) \(1\le p\le \infty ,\) contractivity of the heat semigroups corresponding to the operators \(L_j,\) \(j=1,\ldots ,d.\) If this condition holds then we say that \(L_j\) generates a symmetric contraction semigroup.

Then, by Cowling’s [8, Theorem 3], each of the operators \(L_j,\) \(j=1,\ldots ,d,\) necessarily has an \(H^{\infty }\) functional calculus on each \(L^p(X,\nu ),\) \(1<p<\infty .\) This means that if \(m_j\) is a bounded holomorphic function (of one complex variable) in a certain sub-sector \(S_{\varphi _p}\) of the right complex half-plane, then the operator \(m_j(L_j),\) given initially on \(L^2(X,\nu )\) by the spectral theorem, is bounded on \(L^p(X,\nu ).\) However, it may happen that some of our operators also have the stronger Marcinkiewicz functional calculus. We say that \(L_j\) has a Marcinkiewicz functional calculus, if every bounded function \(m_j:[0,\infty )\rightarrow \mathbb {C},\) which satisfies a certain Marcinkiewicz-type condition, see Definition 3.1 (with \(d=1\)) gives rise to a bounded operator \(m_j(L_j)\) on all \(L^p(X,\nu ),\) \(1<p<\infty \) spaces. Throughout the paper we use letter *A* to denote operators which have a Marcinkiewicz functional calculus. The formal definitions of the two kinds of functional calculi are given in Sect. 3.

Perhaps the most eminent difference between these functional calculi is the fact that the Marcinkiewicz functional calculus does not require the multiplier function to be holomorphic. In fact, every function which is sufficiently smooth, and compactly supported away from 0 does satisfy the Marcinkiewicz condition.

For the single operator case various kinds of multiplier theorems have been proved in a great variety of contexts. The literature on the subject is vast; let us only name here [9, 32] as the papers which have directly influenced our research.

As for the joint spectral multipliers for a system of commuting self-adjoint operators there are relatively fewer results. The first studied case was the one of partial derivatives \(L=(\partial _1,\ldots ,\partial _d),\) see [26] (the classical Marcinkiewicz multiplier theorem) and [23] (the classical Hörmander multiplier theorem). The two theorems differ in the type of conditions imposed on the multiplier function *m*. The Marcinkiewicz multiplier theorem requires a product decay at infinity of the partial derivatives of *m*, while the Hörmander multiplier theorem assumes a radial decay. However, neither of the theorems is stronger than the other. Our paper pursues Marcinkiewicz-type multiplier theorems in more general contexts.

One of the first general cases of commuting operators, investigated in the context of a joint functional calculus, was that of sectorial operators (see [25, Definition 1.1]). In [1, 2] Albrecht, Franks, and McIntosh studied the existence of an \(H^{\infty }\) joint functional calculus for a pair \(L=(L_1,L_2)\) of commuting sectorial operators defined on a Banach space *B*. For some other results concerning holomorphic functional calculus for a pair of sectorial operators see [25] by Lancien, Lancien, and Le Merdy.

Marcinkiewicz-type (multivariate) multiplier theorems for specific commuting operators (i.e sublaplacians and central derivatives) on the Heisenberg (and related) groups were investigated by Müller, Ricci, and Stein in [33, 34], and by Fraser in [15, 16, 17]. The PhD thesis of Martini, [29] (see also [30, 31]), is a treatise of the subject of joint spectral multipliers for general Lie groups of polynomial growth. He proves various Marcinkiewicz-type and Hörmander-type multiplier theorems, mostly with sharp smoothness thresholds.

In [36] Sikora proved a Hörmander-type multiplier theorem for a pair of non-negative self-adjoint operators \(A_j\) acting on \(L^2(X_j,\mu _j),\) \(j=1,2,\) i.e. on separate variables.^{1} In this article the author assumes that the kernels of the heat semigroup operators \(e^{-t_jA_j},\) \(t_j>0,\) \(j=1,2,\) satisfy certain Gaussian bounds and that the underlying measures \(\mu _j\) are doubling. Corollary 3.3 of our paper is, in some sense, a fairly complete answer to a question posed in [36, Remark 4].

Quite recently, Chen, Duong, Li, Ward, and Yan in [6] and Duong, Li, and Yan in [12], proved Marcinkiewicz-type multiplier theorems in settings close to the one considered in [36].

The main purpose of the the present article is to prove (multivariate) multiplier theorems in the case when some of the considered operators have a Marcinkiewicz functional calculus, while others have only an \(H^{\infty }\) functional calculus. Let us underline that, for the general results of Section 3, we only require strong commutativity and do not need that the operators in question arise from orthogonal expansions (cf. [46]) nor that they act on separate variables (cf. [36]). In Theorem 3.1 we show that under a certain Marcinkiewicz-type assumption on a bounded multiplier function *m*, the multiplier *m*(*L*) extends to a bounded operator on \(L^p(X,\nu ).\) Once we realize that the only assumption we need is that of strong commutativity, the proof follows the scheme developed in [44, 45, 46]. The argument we use relies on Mellin transform techniques, together with \(L^p\) bounds for the imaginary power operators, and square function estimates. For the convenience of the reader, we give a fairly detailed proof of Theorem 3.1.

From Theorem 3.1 we derive two seemingly interesting corollaries. The first of these, Corollary 3.2, gives a close to optimal \(H^{\infty }\) joint functional calculus for a general system of strongly commuting operators that generate symmetric contraction semigroups. The second, Corollary 3.3, states that having a Marcinkiewicz functional calculus by each of the operators \(A_j,\) \(j=1,\ldots ,d,\) is equivalent to having a Marcinkiewicz joint functional calculus by the system \(A=(A_1,\ldots ,A_d).\) Thus, in a sense, Corollary 3.3 provides a most general possible Marcinkiewicz-type multiplier theorem for commuting operators.

The prototypical multipliers which fall under our theory have a product form \(m_1(L_1)\cdots m_d(L_d).\) However the reader should keep in mind that Theorem 3.1 applies to a much broader class of multiplier functions. Our condition (3.1) does not require *m* to have a product form, but rather assumes it has a product decay. In particular Theorem 3.1 implies \(L^p,\) \(1<p<\infty ,\) boundedness of the imaginary power operators and Riesz transforms. In the case of a pair (*L*, *A*) by imaginary powers we mean the operators \((L+A)^{iu},\) \(u\in \mathbb {R},\) while by Riesz transforms we mean the operators \(L(L+A)^{-1},\) \(A(L+A)^{-1}\). Note however that due to the methods we use the upper bound we obtain for the \(L^p\) norm of these operators is likely to be of order \((p-1)^{-4},\) \(p\rightarrow 1^+.\) In particular, we do not obtain weak type (1, 1) results.

In Sect. 4 we pursue a particular instance of our general setting in which some weak type (1, 1) results can be proved. Namely, we restrict to the case of two operators: \(\mathcal {L}\) being the Ornstein–Uhlenbeck operator on \(L^2(\mathbb {R}^d,\gamma )\), and *A* being an operator acting on some other space \(L^2(Y,\rho ,\mu ),\) where \((Y,\rho ,\mu )\) is a space of homogeneous type. We also assume that the heat semigroup \(e^{-tA}\) has a kernel satisfying Gaussian bounds and some Lipschitz estimates, see (4.1), (4.2), (4.3). Here the operators do act on separate variables. The main result of this section is Theorem 4.1, which states that certain ’Laplace transform type’ multipliers of the system \((\mathcal {L}\otimes I,I\otimes A)\) are not only bounded on \(L^p(\mathbb {R}^d\times Y, \gamma \otimes \mu ),\) \(1<p<\infty ,\) but also from \(L^1_{\gamma }(H^1(Y,\mu ))\) to \(L^{1,\infty }_{\gamma \otimes \mu }.\) Here \(H^1(Y,\mu )\) denotes the atomic Hardy space \(H^1\) in the sense of Coifman–Weiss. Section 4 gives weak type (1, 1) results for joint multipliers in the case when one of the operators (the Ornstein–Uhlenbeck operator \(\mathcal {L}\), see [22]) does not have a Marcinkiewicz functional calculus. It seems that so far such results were proved only for systems of operators all having a Marcinkiewicz functional calculus.

The paper is organized as follows. First, in Sect. 2 we present the setting of the article and introduce the needed notation. Then, in Sect. 3 a general result for joint spectral multipliers of mixed systems of operators on \(L^p(X, \nu )\) spaces, \(1<p<\infty ,\) is proved, see Theorem 3.1. As a corollary we obtain analogous results in the case when all the operators considered have only a holomorphic functional calculus, see Corollary 3.2, or in the case when all of these operators have a stronger Marcinkiewicz functional calculus, see Corollary 3.3. Next, in Sect. 4 we treat the system \((\mathcal {L}\otimes I,I\otimes A),\) described in the previous paragraph, and prove the boundedness of ’Laplace transform type’ multipliers from \(L^1_{\gamma }(H^1(Y,\mu ))\) to \(L^{1,\infty }_{\gamma \otimes \mu },\) see Theorem 4.1. Finally, in the Appendix we prove that \(L^1_{\gamma }(H^1(Y,\mu ))\) interpolates with \(L^2,\) see Theorem 5.1.

## 2 Preliminaries

*E*associated with

*L*and determined uniquely by the condition

*m*on \([0,\infty )^d,\) the multivariate spectral theorem allows us to define

*the operator*\(L_j\)

*generates a symmetric contraction semigroup*. For technical reasons we often also impose

*T*on \(L^2(X_j,\nu _j)\) we define

*T*is self-adjoint, then the operators \(T\otimes I_{(j)}\) can be regarded as self-adjoint and strongly commuting operators on \(L^2(X,\nu ),\) see [35, Theorem 7.23] and [43, Proposition A.2.2]. Once again, let us point out that the general results of Sect. 3 do not require that the operators act on separate variables. However, in Sect. 4 we do consider a particular case of operators acting on separate variables.

Throughout the paper the following notation is used. The symbols \(\mathbb {N}_0\) and \(\mathbb {N}\) stand for the sets of non-negative and positive integers, respectively, while \(\mathbb {R}_+^d\) denotes \((0,\infty )^d\).

*d*-fold product of the right complex half-planes)

If *U* is an open subset of \(\mathbb {C}^d,\) the symbol \(H^{\infty }(U)\) stands for the vector space of bounded functions on *U*, which are holomorphic in *d*-variables. The space \(H^{\infty }(U)\) is equipped with the supremum norm.

If \(\gamma \) and \(\rho \) are real vectors (e.g. multi-indices), by \(\gamma <\rho \) (\(\gamma \le \rho \)) we mean that \(\gamma _j<\rho _j\) (\(\gamma _j\le \rho _j\)), for \(j=1,\ldots ,d.\) For any real number *x* the symbol \(\mathbf{x}\) denotes the vector \((x,\ldots ,x)\in \mathbb {R}^d.\)

By \(\langle z, w \rangle ,\) \(z,w\in \mathbb {C}^d\) we mean the usual inner product on \(\mathbb {C}^d.\) Additionally, if instead of \(w\in \mathbb {C}^d\) we take a vector of self-adjoint operators \(L=(L_1,\ldots ,L_d),\) then, by \(\langle z, L \rangle \) we mean \(\sum _{j=1}^d z_j L_j.\)

*d*-dimensional Mellin transform by

*m*such that both \(m\in L^1(\mathbb {R}_+^d,\frac{d\lambda }{\lambda })\) and \(\mathcal {M}(m)\in L^1(\mathbb {R}^d,du).\)

Throughout the paper we use the variable constant convention, i.e. the constants (such as *C*, \(C_p\) or *C*(*p*), etc.) may vary from one occurrence to another. In most cases we shall however keep track of the parameters on which the constant depends, (e.g. *C* denotes a universal constant, while \(C_p\) and *C*(*p*) denote constants which may also depend on *p*). The symbol \(a\lesssim b\) means that \(a\le C b,\) with a constant *C* independent of significant quantities.

Let \(B_1,B_2\) be Banach spaces and let *F* be a dense subspace of \(B_1.\) We say that a linear operator \(T:F\rightarrow B_2\) is bounded, if it has a (unique) bounded extension to \(B_1.\)

## 3 General Multiplier Theorems

Throughout this section, for the sake of brevity, we write \(L^p\) instead of \(L^p(X,\nu )\) and \(\Vert \cdot \Vert _p\) instead of \(\Vert \cdot \Vert _{L^p(X,\nu )}.\) The symbol \(\Vert \cdot \Vert _{p\rightarrow p}\) denotes the operator norm on \(L^p.\)

The first *n* operators in the system \(L_1,\ldots ,L_n,\) \(0\le n\le d\) are assumed to have an \(H^{\infty }\) functional calculus. We say that a single operator *L* has an \(H^{\infty }\) *functional calculus on* \(L^p,\) \(1<p<\infty \), whenever we have the following: there is a sector \(S_{\varphi _p}=\{z\in \mathbb {C}:|{{\mathrm{Arg}}}(z)|<\varphi _p\},\) \(\varphi _p<\pi /2,\) such that, if *m* is a bounded holomorphic function on \(S_{\varphi _p},\) then \(\Vert m(L)\Vert _{L^p(X,\nu )\rightarrow L^p(X,\nu )}\le C_{p}\Vert m\Vert _{H^{\infty }(S_{\varphi _p})}.\) The phrase ’*L* has an \(H^{\infty }\) functional calculus’ means that *L* has an \(H^{\infty }\) functional calculus on \(L^p\) for every \(1<p<\infty .\) An analogous terminology is used when considering a system of operators \(L=(L_1,\ldots ,L_d)\) instead of a single operator. We say that *L* has an \(H^{\infty }\) *joint functional calculus*, whenever the following holds: for each \(1<p<\infty \) there is a poly-sector \(\mathbf{S}_{\varphi _p},\) \(\varphi _p=(\varphi _p^1,\ldots ,\varphi _p^d)\in [0,\pi /2)^d,\) such that if *m* is a bounded holomorphic function in several variables on \(\mathbf{S}_{\varphi _p},\) then \(\Vert m(L)\Vert _{L^p(X,\nu )\rightarrow L^p(X,\nu )}\le C_{p}\Vert m\Vert _{H^{\infty }(\mathbf{S}_{\varphi _p})}.\)

The last *l* operators in the system *L*, i.e. \(L_{n+1},\ldots ,L_d,\) with \(n+l=d,\) are assumed to have additionally a Marcinkiewicz functional calculus. Therefore, according with our convention, we use letter *A* to denote these operators, i.e. \(A_j=L_{n+j},\) \(j=1,\ldots ,l.\) In order to define the Marcinkiewicz functional calculus and formulate the main theorem of the paper we need the following definition.

**Definition 3.1**

*m*is a bounded function having partial derivatives up to order \(\rho \),

^{2}and for all multi-indices \(\gamma =(\gamma _1,\ldots ,\gamma _d)\le \rho \)

*m*satisfies the Marcinkiewicz condition of order \(\rho ,\) then we set

*A*has a

*Marcinkiewicz functional calculus*

^{3}of order \(\rho \in \mathbb {N}_0\), whenever the following holds: if the multiplier function

*m*satisfies the one-dimensional (i.e. with \(d=1\)) Marcinkiewicz condition (3.1) of order \(\rho ,\) then the multiplier operator

*m*(

*A*) is bounded on all \(L^p(X,\nu ),\) \(1<p<\infty ,\) and \(\Vert m(A)\Vert _{L^p(X,\nu )\rightarrow L^p(X,\nu )}\le C_{p}\Vert m\Vert _{Mar,\rho }.\) Similarly, to say that a system \(A=(A_1,\ldots ,A_l)\) has a

*Marcinkiewicz joint functional calculus*of order \(\rho =(\rho _1,\ldots ,\rho _l)\in \mathbb {N}_0^l\) we require the following condition to be true: if the multiplier function

*m*satisfies the

*d*-dimensional Marcinkiewicz condition (3.1) of order \(\rho =(\rho _1,\ldots ,\rho _d),\) then the multiplier operator

*m*(

*L*) is bounded on \(L^p(X,\nu ),\) \(1<p<\infty ,\) and

\(\Vert m(L)\Vert _{L^p(X,\nu )\rightarrow L^p(X,\nu )}\le C_{p}\Vert m\Vert _{Mar,\rho }.\)

Throughout this section we impose the assumptions of Sects. 2 and 3; in particular both (ATL) and (CTR) as well as (3.2) and (3.3). The following is our main theorem.

**Theorem 3.1**

*d*-dimensional Marcinkiewicz condition (3.1) of some order \(\rho >|1/p-1/2|(\theta ,\sigma )+\mathbf{1},\) where \(\rho =(\rho _1,\ldots ,\rho _n,\rho _{n+1},\ldots ,\rho _d).\) Then the multiplier operator

*m*(

*L*,

*A*) is bounded on \(L^p\) and

*Remark 1*

If \(l=0\) (\(n=d\)) then we consider only operators \(L_1,\ldots ,L_d\) with an \(H^{\infty }\) functional calculus, while if \(n=0\) (\(l=d\)) then we consider only operators \(A_1,\ldots ,A_d,\) with a Marcinkiewicz functional calculus. In the latter case we do not require *m* to be holomorphic. We only assume that it satisfies (3.1) of some order \(\rho >|1/p-1/2|\sigma +{ \mathbf 1}.\)

*Remark 2*

From the theorem it follows that if \(m(e^{i\varepsilon \phi _p}\lambda ,a),\) \(\varepsilon \in \{-1,1\}^{n},\) satisfy the Marcinkiewicz condition of some order \(\rho >\frac{1}{2}(\theta ,\sigma )+\mathbf{1},\) then *m*(*L*, *A*) is in fact bounded on all \(L^p\) spaces, \(1<p<\infty .\)

Before proving Theorem 3.1 let us first state and prove two corollaries.

The first of these corollaries provides an \(H^{\infty }\) joint functional calculus for a general system of strongly commuting operators \(L_j,\) \(j=1,\ldots ,d,\) satisfying (CTR) and (ATL). Corollary 3.2 generalizes [5, Theorem 1] to systems of commuting operators; although it is slightly weaker than [5, Theorem 1] in the case \(d=1.\) Recall that \(\phi _p^{*}=\arcsin |2/p-1|.\)

**Corollary 3.2**

*m*be a bounded holomorphic functions of

*d*-variables in \(\mathbf{S}_{\phi _p^{*}}.\) If for some \(\rho >(5/2,\ldots ,5/2)\) we have

*m*(

*L*) is bounded on \(L^p\) and

*Proof*

Using [5, Theorem 1] to the imaginary powers \(L_j^{iu_j},\) \(j=1,\ldots ,d,\) and interpolating with the bound \(\Vert L_j^{iu_j}\Vert _{2\rightarrow 2}\le 1,\) we obtain (3.2) with arbitrary \(\theta _j/2>3/2\) and \(\phi _p^j=\phi _p^{*}.\) Now, an application of Theorem 3.1 (with \(n=d\)) gives the desired boundedness.\(\square \)

*Remark 1*

Note that, as we do not require *m* to be holomorphic in a bigger sector, our theorem is stronger than a combination of [2, Theorem 5.4] and [5, Theorem 1] given in [42, Proposition 3.2].

*Remark 2*

Examples of multiplier functions satisfying the assumptions of the corollary include \(m_{j}^{\sigma }(\lambda )=\lambda _j^{\sigma }/(\lambda _1+\cdots + \lambda _d)^{-\sigma },\) where \(\sigma >0.\) The operators \(m_{j}^{\sigma }(L),\) \(j=1,\ldots ,d,\) are intimately connected with the Riesz transforms, see [42].

The second corollary treats the case when all the considered operators have a Marcinkiewicz functional calculus, i.e. \(n=0\) and \(l=d.\) It implies that a system \(A=(A_1,\ldots ,A_d)\) has a Marcinkiewicz joint functional calculus of a finite order if and only if each \(A_j,\) \(j=1,\ldots ,d,\) has a Marcinkiewicz functional calculus of a finite order.

**Corollary 3.3**

- (i)
If, for each \(j=1,\ldots ,d,\) the operator \(A_j\) has a Marcinkiewicz functional calculus of order \(\rho _j,\) then the system \(A=(A_1,\ldots ,A_d)\) has a Marcinkiewicz joint functional calculus of every order greater than \(\rho +\mathbf{1}.\)

- (ii)
If the system \(A=(A_1,\ldots ,A_d)\) has a Marcinkiewicz joint functional calculus of order \(\rho ,\) then, for each \(j=1,\ldots ,d,\) the operator \(A_j\) has a Marcinkiewicz functional calculus of order \(\rho _j.\)

*Proof*

To prove item (i), note that having a Marcinkiewicz functional calculus of order \(\rho _j\) implies satisfying (3.3) with every \(\sigma _j>2\rho _j\). This observation follows from the bounds \(\Vert A_j^{iv_j}\Vert _{p\rightarrow p}\le C_p (1+|v_j|)^{\rho _j},\) \(1<p<\infty ,\) and \(\Vert A_j^{iv_j}\Vert _{2\rightarrow 2}\le 1,\) together with an interpolation argument. Now, Theorem 3.1 (with \(n=0\) and \(l=d\)) implies the desired conclusion.

The proof of item (ii) is even more straightforward, we just need to consider functions \(m_j,\) \(j=1,\ldots ,d,\) which depend only on the variable \(\lambda _j.\) \(\square \)

*Remark*

The most typical instance of strongly commuting operators arises on product spaces, when each \(A_j\) initially acts on some \(L^2(X_j,\nu _j).\) Moreover, there are many results in the literature, see e.g. [3, 4, 11, 13, 21, 27, 40], which imply that a single operator has a Marcinkiewicz functional calculus. Consequently, using the corollary we obtain a joint Marcinkiewicz functional calculus for a vast class of systems of operators acting on separate variables. In particular, we may take \(m(\lambda )=1-(\lambda _1+\cdots + \lambda _d)^{\delta }\chi _{\lambda _1+\cdots \lambda _d\le 1},\) for \(\delta >0\) large enough, thus obtaining the boundedness of the Bochner–Riesz means for the operator \(A_1+\cdots + A_d\).^{4} However, because of the assumed generality, these results are by no means optimal.

*d*-dimensional variant of [2, Theorem5.3] due to Albrecht, Franks and McIntosh.

**Theorem 3.4**

*Proof (sketch)*

Even though [2, Theorem 5.3] is given only for \(d=2\) it readily generalizes to systems of *d* operators, with the same assumptions. Hence, we just need to check that these assumptions are satisfied.

*d*-dimensional version of [2, Theorem 5.3] we are left with verifying that:

*T*is of a type \(\omega <\pi /2\) (see [2, p. 293] for a definition),

*T*is one-one, and both \({{\mathrm{Dom}}}T\) and \({{\mathrm{Ran}}}T\) are dense in the Banach space \(B:=L^p(X,\nu ).\) The reader is kindly referred to consult the proof of [42, Proposition 3.2], where a justification of these statements is contained

A more detailed and slightly different proof of the proposition can be given along the lines of the proof of [43, Corollary 4.1.2].\(\square \)

**Theorem 3.5**

*m*(

*L*) is bounded on \(L^p(X,\nu )\) and

*Proof*

The proof follows the scheme developed in the proof of [32, Theorem 1] and continued in the proof of [46, Theorem 2.2], however, for the convenience of the reader we provide details.

*Remark*

The proof of Theorem 3.5 we present here is modeled over the original proof of [32, Theorem 1] for the one-operator case. In [10, Theorem 2.1] the authors gave a simpler proof of [32, Theorem 1]. However, a closer look at their method reveals that it does not carry over to our multivariate setting. The reason is that we initially do not know whether multivariate multipliers of Laplace transform type \(\mathbb {R}_+^d\ni \lambda \mapsto \lambda _1\cdots \lambda _d \int _{\mathbb {R}_+^d}\exp (-t_1\lambda _1+\cdots +t_d \lambda _d)\, \kappa (t)\,dt,\) with \(\kappa \) being a bounded function on \(\mathbb {R}_+^d\) that may not have a product form, produce bounded multiplier operators on \(L^p.\)

Having proved Theorem 3.5 we proceed to the proof of our main result.

*Proof of Theorem 3.1*

Throughout the proof we will often denote by \(\xi \) the vector \((\lambda _1,\ldots ,\lambda _n,a_1,\ldots ,a_l)\in \mathbb {R}_+^d,\) and by \(\frac{d\xi }{\xi }\) the measure \(\frac{d\lambda _1}{\lambda _1}\cdots \frac{d\lambda _n}{\lambda _n}\frac{da_1}{a_1}\cdots \frac{da_l}{a_l}.\)

The proof of (3.7) is an appropriately adjusted combination of the proofs of [44, Theorem 4.2] and [46, Theorem 4.1], based on the usage of Theorem 3.5. The main idea is to change the path of integration in the first *n* variables under the integral in (3.7). This approach originates in [18, Theorem 2.2]. The proof we present here is a multivariate generalization of both the proofs of [18, Theorem 2.2] and [32, Theorem 4]. For the sake of completeness we give details.

*n*variables of the integral defining \(\mathcal {M}(m_{N,t})(u,v)\) to the poly-ray \(\{(e^{i\varepsilon _1\phi _p^1}\lambda _1,\ldots ,e^{i\varepsilon _n\phi _p^n}\lambda _n):\lambda \in \mathbb {R}_+^n\}.\) Then, denoting

*u*,

*v*), hence finishing the proof of Theorem 3.1.

*j*-th variable, \(j=1,\ldots ,d,\) we see that

For further reference note that \({{\mathrm{Re}}}(w_j)>0,\) for each \(j=1,\ldots ,d.\)

## 4 Weak Type Results for the System \((\mathcal {L},A)\)

Here we consider the pair of operators \((\mathcal {L}\otimes I,I\otimes A),\) where \(\mathcal {L}\) is the *d*-dimensional Ornstein–Uhlenbeck (OU) operator, while *A* is an operator having certain Gaussian bounds on its heat kernel (which implies that *A* has a Marcinkiewicz functional calculus). We also assume that *A* acts on a space of homogeneous type \((Y,\zeta ,\mu ).\) The main theorem of this section is Theorem 4.1. It states that Laplace transform type multipliers of \((\mathcal {L}\otimes I,I\otimes A)\) are bounded from the \(H^1(Y,\mu )\)-valued \(L^1(\mathbb {R}^d,\gamma )\) to \(L^{1,\infty }(\gamma \otimes \mu ).\) Here \(H^1(Y,\mu )\) is the atomic Hardy space in the sense of Coifman and Weiss [7], while \(\gamma \) is the Gaussian measure on \(\mathbb {R}^d\) given by \(d\gamma (x)=\pi ^{-d/2}e^{-|x|^2}dx.\) Additionally, in the Appendix we show that the considered weak type (1, 1) property interpolates well with the boundedness on \(L^2,\) see Theorem 5.1.

*d*-dimensional Ornstein-Uhlenbeck operator

*d*-dimensional Hermite polynomial of order

*k*, while

*j*.

*m*be a function, which is bounded on \([0,\infty )\) and continuous on \(\mathbb {R}_+.\) We say that

*m*is an \(L^p(\mathbb {R}^d,\gamma )\)-uniform multiplier of \(\mathcal {L},\) whenever

*m*is an \(L^p(\mathbb {R}^d,\gamma )\)-uniform multiplier of \(\mathcal {L}\) for some \(1<p<\infty ,\) \(p\ne 2,\) then

*m*necessarily extends to a holomorphic function in the sector \(S_{\phi _p^{*}}\) (recall that \(\phi _p^{*}=\arcsin |2/p-1|\)). Assume now that \(m(t\mathcal {L})\) is of weak type (1, 1) with respect to \(\gamma ,\) with a weak type constant which is uniform in \(t>0.\) Then, since the sector \(S_{\phi _p^{*}}\) approaches the right half-plane \(S_{\pi /2}\) when \(p\rightarrow 1^+,\) using the Marcinkiewicz interpolation theorem we see that the function

*m*is holomorphic (but not necessarily bounded) in \(S_{\pi /2}\). An example of such an

*m*is a function of Laplace transform type in the sense of Stein [38, p. 58, 121], i.e. \(m(z)=z\int _0^{\infty }e^{-zt}\kappa (t)\,dt,\) with \(\kappa \in L^{\infty }(\mathbb {R}_+,dt).\)

^{5}

*A*be a non-negative, self-adjoint operator defined on a space \(L^2(Y,\mu ),\) where

*Y*is equipped with a metric \(\zeta \) such that \((Y,\zeta ,\mu )\) is a space of homogeneous type, i.e. \(\mu \) is a doubling measure. For simplicity we assume that \(\mu (Y)=\infty ,\) and that for all \(x_2\in Y,\) the function \((0,\infty )\ni R\mapsto \mu (B_{\zeta }(x_2,R))\) is continuous and \(\lim _{R\rightarrow 0}\mu (B_{\zeta }(x_2,R))=0.\) We further impose on

*A*the assumptions (CTR) and (ATL) of Sect. 2. Throughout this section we also assume that the heat semigroup \(e^{-tA}\) has a kernel \(e^{-tA}(x_2,y_2),\) \(x_2,y_2\in Y,\) which is continuous on \(\mathbb {R}^+\times Y\times Y,\) and satisfies the following Gaussian bounds.

*A*has a finite order Marcinkiewicz functional calculus on \(L^p(Y,\mu ),\) \(1<p<\infty \). Examples of operators

*A*satisfying (4.1), (4.2), and (4.3) include, among others, the Laplacian \(-\Delta \) and the harmonic oscillator \(-\Delta +|x|^2\) on \(L^2(\mathbb {R}^d,dx),\) or the Bessel operator \(-\Delta -\sum _{j=1}^d \frac{2\alpha _j}{x_j}\partial _j\) (see [14, Lemma 4.2]).

*b*is an \(H^1\)-atom, if there exists a ball \(B=B_{\zeta }\subseteq Y\), such that \({{\mathrm{supp}}}\, b \subset B,\) \(\Vert b\Vert _{L^{\infty }(Y,\mu )}\le 1/ \mu (B),\) and \(\int _{Y}b(x_2)d\mu (x_2) =0.\) The space \(H^1\) is defined as the set of all \(g\in L^1(Y,\mu ),\) which can be written as \(g= \sum _{j=1}^{\infty } c_j b_j,\) where \(b_j\) are atoms and \(\sum _{j=1}^{\infty } |c_j|<\infty ,\) \(c_j\in \mathbb {C}.\) We equip \(H^1\) with the norm

*t*via

*f*on \(\mathbb {R}^d\times Y\) such that the norm

*A*we consider the tensor products \(\mathcal {L}\otimes I\) and \(I \otimes A.\) Slightly abusing the notation we keep writing \(\mathcal {L}\) and

*A*for these operators. For the sake of brevity we write \(L^p,\) \(\Vert \cdot \Vert _{p}\) and \(\Vert \cdot \Vert _{p \rightarrow p},\) instead of \(L^p(\mathbb {R}^d \otimes Y, \gamma \otimes \mu ),\) \(\Vert \cdot \Vert _{L^p},\) and \(\Vert \cdot \Vert _{L^p \rightarrow L^p},\) respectively. We shall also use the space \(L^{1,\infty }:=L^{1,\infty }(\mathbb {R}^d\times Y,\gamma \otimes \mu ),\) equipped with the quasinorm

*S*be an operator which is of weak type (1, 1) with respect to \(\gamma \otimes \mu .\) Then, \(\Vert S\Vert _{L^1\rightarrow L^{1,\infty }}=\sup _{\Vert f\Vert _{1}=1}\Vert Sf\Vert _{L^{1,\infty }}\) is the best constant in its weak type (1, 1) inequality.

*m*be a bounded function defined on \([0,\infty )\times \sigma (A),\) and let \(m(\mathcal {L},A)\) be a joint spectral multiplier of \((\mathcal {L},A),\) as in (2.1). Assume that for each \(t>0,\) the operator \(m(t\mathcal {L},A)\) is of weak type (1, 1) with respect to \(\gamma \otimes \mu ,\) with a weak type (1, 1) constant uniformly bounded with respect to

*t*. Then, from what was said before, we may conclude

^{6}that for each fixed \(a\in \sigma (A)\) the function \(m(\cdot ,a)\) has a holomorphic extension to the right half-plane. We limit ourselves to

*m*being of the following Laplace transform type:

*A*, the function \(m_{\kappa }\) gives a well defined bounded operator \(m_{\kappa }(\mathcal {L},A)\) on \(L^2.\) Indeed, since \(\chi _{\{a=0\}}(\mathcal {L},A)=0,\) we have

The operator \(m_{\kappa }(\mathcal {L},A)\) is also bounded on all \(L^p\) spaces, \(1<p<\infty .\) This follows from Corollary 3.2. Moreover, we have \(\Vert m\Vert _{p\rightarrow p}\le C_p,\) with universal constants \(C_p,\) \(1<p<\infty .\)

However, the following question is left open: is \(m_{\kappa }(\mathcal {L},A)\) also of weak type (1, 1)? The main theorem of this section is a positive result in this direction.

**Theorem 4.1**

*A*be a non-negative self-adjoint operator on \(L^2(Y,\zeta ,\mu ),\) satisfying all the assumptions of Sect. 2 and such that its heat kernel satisfies (4.1), (4.2) and (4.3), as described in this section. Let \(\kappa \) be a bounded function on \(\mathbb {R}_+\) and let \(m_{\kappa }\) be given by (4.9). Then the multiplier operator \(m_{\kappa }(\mathcal {L},A)\) is bounded from \(L^1_{\gamma }(H^1)\) to \(L^{1,\infty }(\gamma \otimes \mu ),\) i.e.

*Remark 1*

Observe that \(L^2 \cap L^1_{\gamma }(H^1)\) is dense in \(L^1_{\gamma }(H^1).\) Thus, it is enough to prove (4.10) for \(f\in L^2 \cap L^1_{\gamma }(H^1).\)

*Remark 2*

Examples of multiplier operators of the form \(m_{\kappa }(\mathcal {L},A)\) include the Riesz transforms \(\mathcal {L}(\mathcal {L}+A)^{-1}\) (here \(\kappa \equiv 1\)) or the partial imaginary powers \(\mathcal {L}(\mathcal {L}+A)^{-iu-1},\) \(u\in \mathbb {R}\) (here \(\kappa (t)=t^{iu}/\Gamma (iu+1)\)). Note that since \(I=\mathcal {L}(\mathcal {L}+A)^{-1}+A(\mathcal {L}+A)^{-1},\) the boundedness of \(\mathcal {L}(\mathcal {L}+A)^{-1}\) implies also the boundedness of \(A(\mathcal {L}+A)^{-1}\) from \(L^1_{\gamma }(H^1)\) to \(L^{1,\infty }(\gamma \otimes \mu ).\)

Altogether, the proof of Theorem 4.1 is rather long and technical, thus for the sake of the clarity of the presentation we do not provide all details. We use a decomposition of the kernel of the operator \(T:=m_{\kappa }(\mathcal {L},A)\) into the global and local parts with respect to the Gaussian measure in the first variable. The local part will turn out to be of weak type (1, 1) (with respect to \(\gamma \otimes \mu \)) in the ordinary sense. For both the local and global parts we use ideas and some estimates from García-Cuerva et al. [19, 20].

Set \(\kappa ^{\varepsilon }=\kappa \chi _{[\varepsilon ,1/\varepsilon ]},\) \(0<\varepsilon <1.\) Then, using the multivariate spectral theorem together with the fact that *A* satisfies (ATL), we see that \(\lim _{\varepsilon \rightarrow 0^+}m_{\kappa ^{\varepsilon }}((\mathcal {L},A))=m_{\kappa }((\mathcal {L},A)),\) strongly in \(L^2.\) Consequently, we also have convergence in the measure \(\gamma \otimes \mu \). Since, clearly \(\Vert \kappa ^{\varepsilon }\Vert _{L^{\infty }(\mathbb {R}^+)}\le \Vert \kappa \Vert _{\infty },\) it suffices to prove (4.10) for \(\kappa \) such that \({{\mathrm{supp}}}\kappa \subseteq [\varepsilon ,1/\varepsilon ].\) ^{7} Thus, throughout the proof of Theorem 4.1 we assume (often without further mention) that \(\kappa \) is supported away from 0 and \(\infty .\) Additionally, the symbol \(\lesssim \) denotes that the estimate is independent of \(\kappa .\)

In the proof of Theorem 4.1 the variables with subscript 1, e.g. \(x_1,y_1,\) are elements of \(\mathbb {R}^d,\) while the variables with subscript 2, e.g. \(x_2,y_2,\) are taken from *Y*.

*S*is of weak type (1, 1) precisely when

**Definition 4.12**

*S*(

*x*,

*y*) defined on the product \((\mathbb {R}^{d}\times Y)\times (\mathbb {R}^d\times Y)\) is a kernel of a linear operator

*S*defined on \(L^{\infty }_c\) if, for every \(f\in L^{\infty }_c\) and a.e. \(x\in \mathbb {R}^d\times Y,\)

*Remark 1*

We do not restrict to \(x\not \in {{\mathrm{supp}}}f;\) the operators we consider later on are well defined in terms of their kernels for all *x*. This is true because of the assumption that \(\kappa \) is supported away from 0 and \(\infty .\)

*Remark 2*

The reader should keep in mind that the inner integral defining *Sf*(*x*) is taken with respect to the Lebesgue measure \(dy_1\) rather than the Gaussian measure \(d\gamma (y_1).\) The reason for this convention is the form of Mehler’s formula we use, see (4.13).

**Lemma 4.2**

The function *K* is a kernel of *T* in the sense of Definition 4.12.

*Proof (sketch)*

*T*are defined, for \(f\in L_c^{\infty },\) by

*x*, whenever \(f\in L^1.\)

Note that the cut-off considered in (4.19) is the rough one from [19, p. 385] (though only with respect to \(x_1,y_1\)) rather than the smooth one from [20, p. 288]. In our case, using a smooth cut-off with respect to \(\mathbb {R}^d\) does not simplify the proofs. That is because, even a smooth cut-off with respect to \(x_1,y_1\) may not preserve a Calderón-Zygmund kernel in the full variables (*x*, *y*). Moreover, the rough cut-off has the advantage that \((T^{loc})^{loc}=T^{loc}.\)

**Proposition 4.3**

*Proof*

By (4.20) it clearly suffices to focus on \(T_*^{glob}.\)

Now we turn to the local part \(T^{loc}.\) As we already mentioned, \(T^{loc}\) turns out to be of (classical) weak type (1, 1) with respect to \(\gamma \otimes \mu .\)

**Proposition 4.4**

From now on we focus on the proof of Proposition 4.4. The key ingredient is a comparison (in the local region) of the kernel *K* with a certain convolution kernel \(\tilde{K}\) in the variables \((x_1,y_1),\) i.e. depending on \((x_1-y_1,x_2,y_2).\) We also heavily exploit the fact that in the local region \(N_2\) the measure \(\gamma \otimes \mu \) is comparable with \(\Lambda \otimes \mu .\)

For further reference we restate [20, Lemma 3.1]. The first five items of Lemma 4.5 are exactly items i)-v) from [20, Lemma 3.1], item vi) is [20, eq. (3.2), p. 289], while item vii) is [20, eq. (3.3), p. 289].

**Lemma 4.5**

- (i)
the family \(\{B_j:j\in \mathbb {N}\}\) covers \(\mathbb {R}^d\);

- (ii)
the balls \(\{\frac{1}{4} B_j :j\in \mathbb {N}\}\) are pairwise disjoint;

- (iii)
for any \(\beta > 0\), the family \(\{\beta B_j :j\in \mathbb {N}\}\) has bounded overlap, i.e.; \(\sup \sum _j \chi _{\beta B_j}(x_1)\le C\);

- (iv)
\(B_j \times 4B_j \subseteq N_1\) for all \(j\in \mathbb {N}\);

- (v)
if \(x_1 \in B_j,\) then \(B(x_1,\frac{1}{20(1+|x_1|)}) \subseteq 4B_j\);

- (vi)
for any measurable \(V\subseteq 4B_j,\) we have \(\gamma (V)\approx e^{-|x_1^j|^2}\Lambda (V);\)

- (vii)
\(N_{1/7}\subseteq \bigcup _j B_j \times 4B_j \subseteq N_2.\)

The next lemma we need is a two variable version of [20, Lemma 3.3] (see also the following remark). The proof is based on Lemma 4.5 and proceeds as in [20]. We omit the details, as the only ingredient that needs to be added is an appropriate use of Fubini’s theorem. In Lemma 4.6 by \(\nu \) we denote one of the measures \(\gamma \) or \(\Lambda .\)

**Lemma 4.6**

Let *S* be a linear operator defined on \(L_c^{\infty }\) and

- (i)If
*S*is of weak type (1, 1) with respect to the measure \(\nu \otimes \mu ,\) then \(S_1\) is of weak type (1, 1) with respect to both \(\gamma \otimes \mu \) and \(\Lambda \otimes \mu ;\) moreover,$$\begin{aligned} \Vert S_1\Vert _{L^1\rightarrow L^{1,\infty }}+\Vert S_1\Vert _{L^1(\Lambda \otimes \mu )\rightarrow L^{1,\infty }(\Lambda \otimes \mu )}\lesssim \Vert S\Vert _{L^1(\nu \otimes \mu )\rightarrow L^{1,\infty }(\nu \otimes \mu )}. \end{aligned}$$ - (ii)If
*S*is bounded on \(L^p(\mathbb {R}^d\times Y,\nu \otimes \mu ),\) for some \(1<p<\infty ,\) then \(S_1\) is bounded on both \(L^p\) and \(L^p(\Lambda \otimes \mu );\) moreover,$$\begin{aligned} \Vert S_1\Vert _{p\rightarrow p}+\Vert S_1\Vert _{L^p(\Lambda \otimes \mu )\rightarrow L^p(\Lambda \otimes \mu )}\lesssim \Vert S\Vert _{L^p(\nu \otimes \mu )\rightarrow L^p(\nu \otimes \mu )}. \end{aligned}$$

**Lemma 4.7**

*Proof*

As we have already remarked, by spectral theory \(\tilde{T}\) is bounded on \(L^2(\Lambda \otimes \mu ),\) and we easily see that (4.23) holds. Additionally, an argument similar to the one used in the proof of Lemma 4.2 shows that \(\tilde{T}\) is associated with the kernel \(\tilde{K}\) even in the sense of Definition 4.12.

If \(|y_1-y'_1|\le \sqrt{t},\) this is a consequence of (4.33) and (4.34), while if \(|y_1-y'_1|\ge \sqrt{t}\) it can be deduced from (4.33) and (4.29). Similarly as it was done for \(I_2,\) to estimate \(I_1\) we consider two cases.

Finally, (4.25) follows after collecting the bounds (4.32) and (4.35), thus finishing the proof of Lemma 4.7. \(\square \)

*T*and \(\tilde{T}\) are associated with the kernels

*K*and \(\tilde{K},\) respectively,

*D*is associated with

We shall need an auxiliary lemma. Recall that \(\mathcal {M}_r\) and \(\mathcal {W}_r\) are given by (4.12) and (4.22), respectively.

**Lemma 4.8**

*Proof*

We proceed similarly to the proof of [19, Lemma 3.9].

*r*replaced by

*s*, we obtain

As a corollary of Lemma 4.8 we now prove the following.

**Lemma 4.9**

*Proof*

*C*is independent of \(y_1.\) Thus, applying Fubini’s theorem we obtain \(\Vert D^{loc}\Vert _{L^{1}(\Lambda \otimes \mu )\rightarrow L^{1}(\Lambda \otimes \mu )}\le C\Vert \kappa \Vert _{\infty }\). Since in the local region \( |x_1|\le 2+ |y_1|\le 4+|x_1|\) and \(\chi _{N_2}(x_1,y_1)=\chi _{N_2}(y_1,x_1),\) the singularity of \(\chi _{N_2}D_I(x_1,y_1)\) is also integrable in \(y_1.\) Hence, using Fubini’s theorem and the \(L^{\infty }(Y,\mu )\) contractivity of \(r^A,\) we have \(\Vert D^{loc}\Vert _{L^{\infty }(\Lambda \otimes \mu )\rightarrow L^{\infty }(\Lambda \otimes \mu )}\le C\Vert \kappa \Vert _{\infty }.\) Interpolating between the \(L^1(\Lambda \otimes \mu )\) and \(L^{\infty }(\Lambda \otimes \mu )\) bounds for \(D^{loc}\) we finish the proof of (4.41). \(\square \)

The last lemma of this section shows that the local parts of *T* and \(\tilde{T}\) inherit their boundedness properties. Moreover, it says that the operators \(T^{loc},\) \(\tilde{T}^{loc},\) and \(D^{loc}\) are bounded on appropriate spaces with regards to both the measures \(\Lambda \otimes \mu \) and \(\gamma \otimes \mu .\)

**Lemma 4.10**

*S*denote one of the operators

*T*, \(\tilde{T},\) or \(D^{loc}\) and let \(\nu \) be any of the measures \(\gamma \) or \(\Lambda .\) Then \(S^{loc}\) is bounded on \(L^2(\nu \otimes \mu ):=L^2(\mathbb {R}^d\times Y,\nu \otimes \mu ),\) and

*Proof*

In what follows *S*(*x*, *y*) denotes the kernel *K*(*x*, *y*) of *T*, or the kernel \(\tilde{K}(x,y)\) of \(\tilde{T},\) or the kernel \(D^{loc}(x,y)\) of \(D^{loc}.\) Recall that in all the cases the integral defining \(S^{glob}f(x)\) is absolutely convergent.

*j*, we arrive at the inequality

*T*is bounded on \(L^2,\) while \(\tilde{T}\) and \(D^{loc}\) are bounded on \(L^2(\Lambda \otimes \mu ).\) Hence, taking

*S*equal to \(T,\tilde{T},\) or \(D^{loc},\) and using Lemma 4.6 we see that in all the considered cases \(S_1\) is bounded on \(L^2(\nu \otimes \mu ),\) and \(\Vert S_1\Vert _{L^2(\nu \otimes \mu )\rightarrow L^2(\nu \otimes \mu )}\lesssim \Vert \kappa \Vert _{\infty }.\) Moreover, from Lemmata 4.7 and 4.9, we know that both \(\tilde{T}\) and \(D^{loc}\) are of weak type (1, 1) with respect to \(\Lambda \otimes \mu ,\) and

It remains to consider \(S_2,\) for which we show boundedness on both \(L^1(\nu \otimes \mu )\) and \(L^{\infty }(\nu \otimes \mu ),\) hence, by interpolation on all \(L^p(\nu \otimes \mu )\) spaces, \(1\le p\le \infty .\)

\( \Vert S_2(f)\Vert _{L^1(\nu \otimes \mu )}\lesssim \int _{\mathbb {R}^d}g(y_1)\,d\nu (y_1)= \Vert f\Vert _{L^1(\nu \otimes \mu )}. \)

Summarizing, since \(T^{loc}=\tilde{T}^{loc}+D^{loc},\) from Lemma 4.10 it follows that the local part \(T^{loc}\) is of weak type (1, 1) with respect to both \(\gamma \otimes \mu \) and \(\Lambda \otimes \mu .\) Moreover, the weak type (1, 1) constant is less than or equal to \(C_{d,\mu }\Vert \kappa \Vert _{\infty }.\) Hence, after combining Propositions 4.3 and 4.4, the proof of Theorem 4.1 is completed.

## Footnotes

- 1.
Then, the tensor products \(A_1\otimes I\) and \(I\otimes A_2\) commute strongly on \(L^2(X_1\times X_2,\mu _1\otimes \mu _2)\).

- 2.
i.e. \(\partial ^{\gamma }(m)\) exist for \(\gamma =(\gamma _1,\ldots ,\gamma _d)\le \rho \).

- 3.
In the single operator case it might seem better to use the term ’Hörmander functional calculus’, cf. [32, Theorem 2]. We use the name of Marcinkiewicz to accord with the naming of the multi-dimensional condition.

- 4.
More formally, we mean here \(A_1\otimes I_{(1)}+\cdots + A_d\otimes I_{(d)},\) with the summands given by (2.2).

- 5.
Taking \(\kappa (t)=e^{-it},\) so that \(m(z)=z/(z+i),\) we see that these multipliers may be unbounded on \(S_{\pi /2}.\)

- 6.
At least in the case when

*A*has a discrete spectrum. - 7.
This reduction was suggested to us by Prof. Fulvio Ricci.

## Notes

### Acknowledgments

Most of the material of this paper is a part of the PhD thesis of the author [43, Chapter 6]. The thesis was written under a cotutelle agreement between Scuola Normale Superiore, Pisa, and Uniwersytet Wrocławski, and was jointly supervised by Prof. Fulvio Ricci and Prof. Krzysztof Stempak. I am most grateful to both the advisors for all their help and encouragement. Specifically, I thank Prof. Ricci, for suggesting the topic of Sect. 4. I would like to thank the referees for their careful reading of the article and useful comments. The manuscript was prepared while the author was a PostDoc at Università di Milano-Bicocca, working under the mentorship of Prof. Stefano Meda. The research was partially supported by Polish funds for sciences, National Science Centre (NCN), Poland, Research Projects 2011/01/N/ST1/01785, and 2014/15/D/ST1/00405, by Foundation for Polish Science - START scholarship, and by the Italian PRIN 2011 project *Real and complex manifolds: geometry, topology and harmonic analysis*.

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