Joint spectral multipliers for mixed systems of operators

We obtain a general Marcinkiewicz-type multiplier theorem for mixed systems of strongly commuting operators $L=(L_1,...,L_d);$ 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 $A(\mathcal{L}+A)^{-1}$ and $\mathcal{L}(\mathcal{L}+A)^{-1}.$


Introduction
Let (X, ν) be a σ-finite measure space. Consider a system L = (L 1 , . . . , L d ) of strongly commuting non-negative self-adjoint operators on L 2 (X, ν). By strong commutativity we mean that the spectral projections of L j , j = 1, . . . , d, commute pairwise. In this case there exists the joint spectral resolution E(λ) of the system L. Moreover, for a bounded function m : [0, ∞) d → C, the multiplier operator m(L) can be defined on L 2 (X, ν) by By the (multivariate) spectral theorem, m(L) is then bounded on L 2 (X, ν). 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, ν), 1 < p < ∞.
Throughout the paper we assume the L p (X, ν), 1 ≤ p ≤ ∞, contractivity of the heat semigroups corresponding to the operators L j , j = 1, . . . , 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, . . . , d, necessarily has an H ∞ functional calculus on each L p (X, ν), 1 < p < ∞. This means that if m j is a bounded holomorphic function (of one complex variable) in a certain sub-sector S ϕp of the right complex half-plane, then the operator m j (L j ), given initially on L 2 (X, ν) by the spectral theorem, is bounded on L p (X, ν). 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, ∞) → 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, ν), 1 < p < ∞ 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 Section 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] and [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 = (∂ 1 , . . . , ∂ d ), see [26] (the classical Marcinkiewicz multiplier theorem) and [22] (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 [24,Definition 1.1]). In [1] and [2] Albrecht, Franks, and McIntosh studied the existence of an H ∞ 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 [24] 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 [14], [15], [16]. The PhD thesis of Martini, [29] (see also [30] and [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 , µ 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 j A j , t j > 0, j = 1, 2, satisfy certain Gaussian bounds and that the underlying measures µ 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].
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 ∞ 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. [47]) 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, ν). Once we realize that the only assumption we need is that of strong commutativity, the proof follows the scheme developed in [47], [46] and [45]. 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 ∞ 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, . . . , d, is equivalent to having a Marcinkiewicz joint functional calculus by the system A = (A 1 , . . . , 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 ) · · · 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 (3.2) 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 < ∞, 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 ∈ 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 growth of the L p norm of these operators is likely to be of order at least (p − 1) −4 , p → 1 + . In particular, we do not obtain weak type (1, 1) results.
In Section 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: L being the Ornstein-Uhlenbeck operator on L 2 (R d , γ), and A being an operator acting on some other space L 2 (Y, ρ, µ), where (Y, ρ, µ) 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 (L ⊗ I, Here H 1 (Y, µ) 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 L, see [21]) 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.

Preliminaries
Let L = (L 1 , . . . , L d ) be a system of non-negative self-adjoint operators on L 2 (X, ν), for some σ-finite measure space (X, ν). We assume that the operators L j commute strongly, i.e. that their spectral projections E L j , j = 1, . . . , d, commute pairwise. In this case, there exists the joint spectral measure E associated with L and determined uniquely by the condition see [35,Theorem 4.10 and Theorems 5.21,5.23]. Consequently, for a Borel measurable function m on [0, ∞) d , the multivariate spectral theorem allows us to define Here The crucial assumption we make is the L p (X, ν) contractivity of the heat semigroups {e −tL j }, j = 1, . . . , d. More precisely, we impose that, for each 1 ≤ p ≤ ∞, and t > 0, This condition is often phrased as the operator L j generates a symmetric contraction semigroup.
For technical reasons we often also impose Note that under (ATL) the formula (2.1) may be rephrased as A particular instance of strongly commuting operators arises in product spaces, when (X, ν) = (Π d j=1 X j , d j=1 ν j ). In this case, for a self-adjoint or bounded operator T on L 2 (X j , ν j ) we define If T is self-adjoint, then the operators T ⊗ I (j) can be regarded as self-adjoint and strongly commuting operators on L 2 (X, ν), see [35,Theorem 7.23] and [44,Proposition A.2.2]. Once again, let us point out that the general results of Section 3 do not require that the operators act on separate variables. However, in Section 4 we do consider a particular case of operators acting on separate variables. Throughout the paper the following notation is used. The symbols N 0 and N stand for the sets of non-negative and positive integers, respectively, while R d + denotes (0, ∞) d . For a vector of angles ϕ = (ϕ 1 , . . . , ϕ d ) ∈ (0, π/2] d , we denote by S ϕ the symmetric poly-sector (contained in the d-fold product of the right complex half-planes) In the case when all ϕ j are equal to a real number ϕ we abbreviate S ϕ := S (ϕ,...,ϕ) . However, it will be always clear from the context whether ϕ is a vector or a number.
If U is an open subset of C d , the symbol H ∞ (U ) stands for the vector space of bounded functions on U, which are holomorphic in d-variables. The space H ∞ (U ) is equipped with the supremum norm.
For two vectors z, w ∈ C d we set z w = z w 1 1 · · · z w d d , whenever it makes sense. In particular, This notation is also used for operators, i.e. for u ∈ R d and N ∈ N d we set Note that, due to the assumption on the strong commutativity, the order of the operators in the right hand sides of the above equalities is irrelevant.
By z, w , z, w ∈ C d we mean the usual inner product on C d . Additionally, if instead of w ∈ C d we take a vector of self-adjoint operators L = (L 1 , . . . , L d ), then, by z, L we mean d j=1 z j L j . The symbol dλ λ (in some places we write dt t or da a instead) stands for the product Haar measure For a function m ∈ L 1 (R d + , dλ λ ), we define its d-dimensional Mellin transform by It is well known that M satisfies the Plancherel formula and the inversion formula for m such that both m ∈ L 1 (R d + , dλ λ ) and M(m) ∈ L 1 (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 b means that a ≤ Cb, 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 → B 2 is bounded, if it has a (unique) bounded extension to B 1 .

General multiplier theorems
Throughout this section, for the sake of brevity, we write L p instead of L p (X, ν) and · p instead of · L p (X,ν) . The symbol · p→p denotes the operator norm on L p .
The first n operators in the system L 1 , . . . , L n , 0 ≤ n ≤ d are assumed to have an H ∞ functional calculus. We say that a single operator L has an H ∞ functional calculus on L p , 1 < p < ∞, whenever we have the following: there is a sector S ϕp = {z ∈ C : | Arg(z)| < ϕ p }, ϕ p < π/2, such that, if m is a bounded holomorphic function on S ϕp , then m(L) L p (X,ν)→L p (X,ν) ≤ C p m H ∞ (Sϕ p ) . The phrase 'L has an H ∞ functional calculus' means that L has an H ∞ functional calculus on L p for every 1 < p < ∞. An analogous terminology is used when considering a system of operators L = (L 1 , . . . , L d ) instead of a single operator. We say that L has an H ∞ joint functional calculus, whenever the following holds: for each 1 < p < ∞ there is a poly-sector S ϕp , The last l operators in the system L, i.e. L n+1 , . . . , 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, . . . , l. In order to define the Marcinkiewicz functional calculus and formulate the main theorem of the paper we need the following definition.
If m satisfies the Marcinkiewicz condition of order ρ, then we set m M ar,ρ := sup γ≤ρ m (γ) .
We say that a single operator A has a Marcinkiewicz functional calculus 3 of order ρ > 0, whenever the following holds: if the multiplier function m satisfies the one-dimensional (i.e. with d = 1) Marcinkiewicz condition (3.2) of order ρ, then the multiplier operator m(A) is bounded on all L p (X, ν), 1 < p < ∞, and m(A) L p (X,ν)→L p (X,ν) ≤ C p m M ar,ρ . Similarly, to say that a system A = (A 1 , . . . , A l ) has a Marcinkiewicz joint functional calculus of order ρ = (ρ 1 , . . . , ρ l ) ∈ R l + we require the following condition to be true: if the multiplier function m satisfies the ddimensional Marcinkiewicz condition (3.2) of order ρ = (ρ 1 , . . . , ρ d ), then the multiplier operator m(L) is bounded on L p (X, ν), 1 < p < ∞, and m(L) L p (X,ν)→L p (X,ν) ≤ C p m M ar,ρ . What concerns the operators L 1 , . . . , L n , we assume that there exist θ = (θ 1 , . . . , θ n ) ∈ [0, ∞) n and φ p = (φ 1 p , . . . , φ n p ) ∈ (0, π/2) n , such that It can be deduced that the above condition is (essentially) equivalent to each L j , j = 1, . . . , n, having an H ∞ functional calculus on L p in the sector see [9,Section 5]. Moreover, by a recent result of Carbonaro and Dragičević [5] (see also [8]), every operator for which (CTR) holds satisfies (3.3) with the optimal angle φ j p = φ * p := arcsin |2/p − 1| and θ j = θ = 3. Put in other words every operator generating a symmetric contraction semigroup has an H ∞ functional calculus on L p in every sector larger than S φ * p . The angle φ * p is optimal among general operators satisfying (CTR), however in many concrete cases it can be significantly sharpened.
When it comes to the operators A 1 , . . . , A l , we impose that there is a vector of positive real numbers σ = (σ 1 , . . . , σ l ), such that for every 1 < p < ∞ and j = 1, . . . , l Condition Theorem 3.1. Fix 1 < p < ∞ and let m : S φp × R l → C be a bounded function with the following property: for each fixed a ∈ R l + , m(·, a) ∈ H ∞ (S φp ), and all the functions In the latter case we do not require m to be holomorphic. We only assume that it satisfies (3.2) of some order ρ > |1/p − 1/2|σ + 1.

Remark 2. Examples of multiplier functions satisfying the assumptions of the corollary include
. . , d, are intimately connected with the Riesz transforms, see [43].
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 , . . . , A d ) has a Marcinkiewicz joint functional calculus of a finite order if and only if each A j , j = 1, . . . , d, has a Marcinkiewicz functional calculus of a finite order. Corollary 3.3. We have the following: (i) If, for each j = 1, . . . , d, the operator A j has a Marcinkiewicz functional calculus of order ρ j , then the system A = (A 1 , . . . , A d ) has a Marcinkiewicz joint functional calculus of every order greater than ρ + 1. (ii) If the system A = (A 1 , . . . , A d ) has a Marcinkiewicz joint functional calculus of order ρ, then, for each j = 1, . . . , l, the operator A j has a Marcinkiewicz functional calculus of order ρ j .
Proof. To prove item (i), note that having a Marcinkiewicz functional calculus of order ρ j implies satisfying (3.4) with every σ j > 2ρ j . This observation follows from the bounds A iv j j p→p ≤ C p (1 + |v j |) ρ j , 1 < p < ∞, and A iv j j 2→2 ≤ 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, . . . , d, which depend only on the variable λ j .
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 , ν j ). Moreover, there are many results in the literature, see e.g. [3,4,11,12,20,27,41], 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(λ) = 1 − (λ 1 + · · · + λ d ) δ χ λ 1 +···λ d ≤1 , for δ > 0 large enough, thus obtaining the boundedness of the Bochner-Riesz means for the operator A 1 + · · · + A d . 4 However, because of the assumed generality, these results are by no means optimal.
To prove Theorem 3.1 we need two auxiliary results which seem interesting on their own. First we need the L p boundedness of the square function (3.5) g This will be proved as a consequence of a 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.
In the terminology of [2] this means that h j ∈ Ψ(S µ ), for every µ < π/2. Observe also that our square function is of the form Fix j = 1, . . . , d, and denote T = L j . By referring to the d-dimensional version of [2, Theorem 5.3] we are left with verifying that: T is of a type ω < π/2 (see [2, p. 293] for a definition), T is one-one, and both Dom T and Ran T are dense in the Banach space B := L p (X, ν). The reader is kindly referred to consult the proof of [43,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 [44,Corollary 4 Recall that the Mellin transform M is given by (2.3), while L iu = L iu 1 · · · L iu d , with L n+j = A j and u n+j = v j , for j = 1, . . . , l. Theorem 3.1 will be deduced from the following.
then the multiplier operator m(L) is bounded on L p (X, ν) and Proof. The proof follows the scheme developed in the proof of [32, Theorem 1] and continued in the proof of [47, Theorem 2.2], however, for the convenience of the reader we provide details.
All the needed quantities are defined on L 2 ∩ L p by the multivariate spectral theorem. From the inversion formula for the Mellin transform and the multivariate spectral theorem we see that Note that, for each fixed t ∈ R d + , both the integrals in (3.6) and (3.7) can be considered as Bochner integrals of (continuous) functions taking values in L 2 .
Then, at least formally, from Theorem 3.4 followed by (3.7), we obtain Hence, using Minkowski's integral inequality, it follows that m(L)f p is bounded by and using once again Theorem 3.4 (this time with N = 1), we arrive at Thus, the proof of Theorem 3.5 is finished, provided we justify the formal steps above. This however can be done almost exactly as in [32, p. 642]. We omit the details here and kindly refer the interested reader to [44, p. 24].
Remark. The proof of Theorem 3.5 we present here is modeled over the original proof of [ bounded function on 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.
The proof of (3. Defining R n ε = {x ∈ R d : ε j x j ≥ 0, j = 1, . . . , d}, with ε ∈ {−1, 1} n , we note that it suffices to obtain (3.8) separately on each R n ε × R l . Thus, till the end of the proof we fix ε ∈ {−1, 1} n and take u ∈ R n −ε . By our assumptions, for each fixed a ∈ R l + , N ∈ N d , t ∈ R n + and u ∈ R n , the function is bounded and holomorphic on Thus, for each ε ∈ {−1, 1} n , we can use (multivariate) Cauchy's integral formula to change the path of integration in the first n variables of the integral defining M(m N,t )(u, v) to the poly-ray {(e iε 1 φ 1 p λ 1 , . . . , e iεnφ n p λ n ) : λ ∈ R n + }. Then, denotingm := m φp ε and εφ p = (ε 1 φ 1 p , . . . , ε n φ n p ), we obtain (3.9) In te second to the last equality above it is understood that u ∈ R d and λ ∈ R d + with λ n+j = a j , u n+j = v j , for j = 1, . . . , l; while dλ λ denotes the Haar measure on (R d + , ·). We claim that, for u ∈ R d , Once the claim is proved, coming back to (3.9) we obtain (3.8) for u ∈ R n −ε and v ∈ R l , hence, finishing the proof of Theorem 3.1.
Thus, till the end of the proof we focus on justifying (3.10). Let N ∈ N d , N > ρ, and ψ be a nonnegative, Changing variables t j λ j → λ j and integrating by parts ρ j times in the j-th variable, j = 1, . . . , d, we see that where w ∈ C n × R l + is the vector w = (e iε 1 φ 1 p , · · · , e iεnφ n p , 1, . . . , 1). For further reference note that Re(w j ) > 0, for each j = 1, . . . , d.
Leibniz's rule allows us to express the derivative ∂ ρ as a weighted sum of derivatives of the form where γ = (γ 1 , . . . , γ d ) and δ = (δ 1 , . . . , δ d ) are multi-indices such that γ + δ ≤ ρ. Proceeding further as in the proof of [32, Theorem 4], we denote Set p k = p k 1 · · · p k d with p k j , j = 1, . . . , d, given by Observe that it is enough to verify the bound (3.11) |I k,N,γ,δ (t, u)| ≤ C N,γ,δ m M ar,ρ p k , k ∈ Z d , uniformly in t ∈ R d + and u ∈ R d . Indeed, assuming (3.11) we obtain and (3.10) follows. Thus, it remains to show (3.11). From the change of variable 2 k j λ j → λ j we have Thus, applying Schwarz's inequality we obtain Moreover, since Re(w j ) > 0, for j = 1, . . . , d, it is not hard to see that Now, coming back to (3.12), we use the assumption thatm satisfies the Marcinkiewicz condition of order ρ together with (3.13) (recall that γ + δ ≤ ρ < N ) to obtain (3.11). The proof of Theorem 3.1 is thus finished.

Weak type results for the system (L, A)
Here we consider the pair of operators (L ⊗ I, I ⊗ A), where 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, ζ, µ). The main theorem of this section is Theorem 4.1. It states that Laplace transform type multipliers of (L ⊗ I, I ⊗ A) are bounded from the H 1 (Y, µ)-valued L 1 (R d , γ) to L 1,∞ (γ ⊗ µ). Here H 1 (Y, µ) is the atomic Hardy space in the sense of Coifman and Weiss [7], while γ is the Gaussian measure on R d given by dγ(x) = π −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 A. 1.
In what follows we denote by L the d-dimensional Ornstein-Uhlenbeck operator It is easily verifiable that L is symmetric on C ∞ c (R d ) with respect to the inner product on L 2 (R d , γ). The operator L is also essentially self-adjoint on C ∞ c (R d ), and we continue writing L for its unique self-adjoint extension.
It is well known that L can be expressed in terms of Hermite polynomials by Here |k| = k 1 + · · · + k d is the length of a multi-index k ∈ N d 0 ,H k denotes the L 2 (R d , γ) normalized d-dimensional Hermite polynomial of order k, while is the projection onto the eigenspace of L with eigenvalue j.
For a bounded function m : N 0 → C, the spectral multipliers m(L) are defined by (2.1) with d = 1. In the case of the Ornstein-Uhlenbeck operator they are given by Let m be a function, which is bounded on [0, ∞) and continuous on R + . We say that m is an Observe that by the spectral theorem the above bound clearly holds for p = 2. Using [21, Theorem 3.5 (i)] it follows that, if m is an L p (R d , γ)-uniform multiplier of L for some 1 < p < ∞, p = 2, then m necessarily extends to a holomorphic function in the sector S φ * p (recall that φ * p = arcsin |2/p−1|). Assume now that m(tL) is of weak type (1, 1) with respect to γ, with a weak type constant which is uniform in t > 0. Then, since the sector S φ * p approaches the right half-plane S π/2 when p → 1 + , using the Marcinkiewicz interpolation theorem we see that the function m is holomorphic (but not necessarily bounded) in S π/2 . An example of such an m is a function of Laplace transform type in the sense of Stein [39, pp. 58, 121], i.e. m(z) = z ∞ 0 e −zt κ(t) dt, with κ ∈ L ∞ (R + , dt). 5 Let now A be a non-negative, self-adjoint operator defined on a space L 2 (Y, µ), where Y is equipped with a metric ζ such that (Y, ζ, µ) is a space of homogeneous type, i.e. µ is a doubling measure. For simplicity we assume that µ(Y ) = ∞, and that for all x 2 ∈ Y, the function (0, ∞) ∋ R → µ(B ζ (x 2 , R)) is continuous and lim R→0 µ(B ζ (x 2 , R)) = 0. We further impose on A the assumptions (CTR) and (ATL) of Section 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 ∈ Y, which is continuous on R + × Y × Y, and satisfies the following Gaussian bounds.
We also impose that for some δ > 0, if 2ζ(y 2 , y ′ 2 ) ≤ ζ(x 2 , y 2 ), then while in general, From [36, Theorem 2.1] (or rather its version for a single operator), it follows that, under (4.1), the operator A has a finite order Marcinkiewicz functional calculus on L p (Y, µ), 1 < p < ∞. Denote by H 1 = H 1 (Y, ζ, µ) the atomic Hardy space in the sense of Coifman-Weiss [7]. More precisely, we say that a measurable function b is an

Examples of operators
The space H 1 is defined as the set of all g ∈ L 1 (Y, µ), which can be written as g = ∞ j=1 c j b j , where b j are atoms and ∞ j=1 |c j | < ∞, c j ∈ C. We equip H 1 with the norm f H 1 = inf ∞ j=1 |c j |, where the infimum runs over all absolutely summable {c j } j∈N , for which g = ∞ j=1 c j b j , with b j being H 1 -atoms. Note that from the very definition of H 1 we have g L 1 (Y,µ) ≤ g H 1 .
It can be shown that under (4.1), (4.2), and (4.3), the space coincides with the atomic H 1 , i.e., there is a constant C µ such that The proof of (4.4) is similar to the proof of [13, Proposition 4.1 and Lemma 4.3]. The main trick is to replace the metric ζ with the measure distance (see [7]) ζ(x 2 , y 2 ) = inf{µ(B) : B is a ball in Y, x 2 , y 2 ∈ B}, change the time t via µ(B(y, √ t)) = s, y ∈ Y, t, s > 0, and apply Uchiyama's Theorem, see [42, Corollary 1']. We omit the details. Note that by taking r = e −t , the equation (4.4) can be restated as For fixed 0 < ε < 1/2, define M A,ε (g)(x) = Y sup ε<r<1−ε |r A (x 2 , y 2 )||g(y 2 )| dµ(y 2 ). Then, a short reasoning using the Gaussian bound (4.1) and the doubling property of µ gives Denote by L 1 γ (H 1 ) the Banach space of those Borel measurable functions f on R d × Y such that the norm is finite. In other words L 1 γ (H 1 ) is the L 1 (γ) space of H 1 -valued functions. Moreover, it is the closure of in the norm given by (4.7). From now on in place of L and A we consider the tensor products L ⊗ I and I ⊗ A. Slightly abusing the notation we keep writing L and A for these operators. For the sake of brevity we write L p , · p and · p→p , instead of L p (R d ⊗ Y, γ ⊗ µ), · L p , and · L p →L p , respectively. We shall also use the space L 1,∞ := L 1,∞ (R d × Y, γ ⊗ µ), equipped with the quasinorm Let S be an operator which is of weak type (1, 1) with respect to γ ⊗ µ. Then, S L 1 →L 1,∞ = sup f 1 =1 Sf L 1,∞ is the best constant in its weak type (1, 1) inequality. Let m be a bounded function defined on [0, ∞) × σ(A), and let m(L, A) be a joint spectral multiplier of (L, A), as in (2.1). Assume that for each t > 0, the operator m(tL, A) is of weak type (1, 1) with respect to γ ⊗ µ, 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 ∈ σ(A) the function m(·, a) has a holomorphic extension to the right half-plane. We limit ourselves to m being of the following Laplace transform type: Moreover, m κ (0, a) = 0 for a > 0, and, consequently, the function m κ (λ, a)χ {a>0} is bounded on [0, ∞) × R + . Now, using the multivariate spectral theorem we see that m κ (L, A) is bounded on The operator m κ (L, A) is also bounded on all L p spaces, 1 < p < ∞. This follows from Corollary 3.2. Moreover, we have m p→p ≤ C p , with universal constants C p , 1 < p < ∞.
However, the following question is left open: is m κ (L, A) also of weak type (1, 1)? The main theorem of this section is a positive result in this direction.
Theorem 4.1. Let L be the Ornstein-Uhlenbeck operator on L 2 (R d , γ) and let A be a non-negative self-adjoint operator on L 2 (Y, ζ, µ), satisfying all the assumptions of Section 2 and such that its heat kernel satisfies (4.1), (4.2) and (4.3), as described in this section. Let κ be a bounded function on R + and let m κ be given by (4.9). Then the multiplier operator m κ (L, A) is bounded from Remark 1. Observe that L 2 ∩ L 1 γ (H 1 ) is dense in L 1 γ (H 1 ). Thus, it is enough to prove (4.10) for f ∈ L 2 ∩ L 1 γ (H 1 ).
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 κ (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 γ ⊗ µ) in the ordinary sense. For both the local and global parts we use ideas and some estimates from García-Cuerva, Mauceri, Sjögren, and Torrea [18] and [19].
In the proof of Theorem 4.1 the variables with subscript 1, e.g. x 1 , y 1 , are elements of R d , while the variables with subscript 2, e.g. x 2 , y 2 , are taken from Y.
We start with introducing some notation and terminology. Define , we see that for each 1 ≤ p < ∞, L ∞ c is a dense subspace of both L p and L p (Λ ⊗ µ). In particular, any operator which is bounded on L 2 or L 2 (Λ ⊗ µ) is well defined on L ∞ c . We also need the weak space L 1,∞ (Λ ⊗ µ) := L 1,∞ (R d × Y, Λ ⊗ µ) equipped with the quasinorm given by (4.8) with γ replaced by Λ. An operator S is of weak type (1, 1) precisely when Let η be the product metric on R d × Y, Then it is not hard to see that the triple (R d × Y, η, Λ ⊗ µ) is a space of homogeneous type.
Definition 4.12. We say that a function S(x, y) defined on the product Remark 1. We do not restrict to x ∈ 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 κ is supported away from 0 and ∞.
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 dγ(y 1 ). The reason for this convention is the form of Mehler's formula we use, see (4.14).
The change of variable r = e −t leads to the formal equality Suggested by the above we define the kernel with r A (x 2 , y 2 ) = e (log r)A (x 2 , y 2 ). Then we have. Proof (sketch). It is enough to show that for f, h ∈ L ∞ c we have From the multivariate spectral theorem together with Fubini's theorem we see that Now, by the multivariate spectral theorem Lr L−1 (r A f ) = (∂ r r L )(r A f ), where on right hand side we have the Fréchet derivative in L 2 . Thus, Lr L−1 r A f, h L 2 is the limit (as δ → 0) of Since f, g ∈ L ∞ c , using (4.6), (4.16), and the dominated convergence theorem we justify taking the limit inside the integral in (4.19) and obtain Plugging the above formula into (4.18), and using Fubini's theorem (which is allowed by (4.6), (4.16) and the fact that supp κ ⊆ [ε, 1 − ε]), we arrive at (4.17), as desired.
Let N s , s > 0, be given by We call N s the local region with respect to the Gaussian measure γ on R d . This set (or its close variant) is very useful when studying maximal operators or multipliers for L. After being applied by Sjögren in [37], it was used in [17], [18], [19], and [28], among others. The local and global parts of the operator T are defined, for f ∈ L ∞ c , by and , respectively. The estimates from Proposition 4.3 demonstrate that the integral (4.20) defining T glob is absolutely convergent for a.e. x, whenever f ∈ L 1 .
Note that the cut-off considered in (4.20) is the rough one from [18, p. 385] (though only with respect to x 1 , y 1 ) rather than the smooth one from [19, p. 288]. In our case, using a smooth cut-off with respect to 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 .
We begin with proving the desired weak type (1, 1) property for T glob . Since (4.21) Moreover, the following proposition holds.
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 γ ⊗ µ.
Proposition 4.4. The operator T loc is of weak type (1, 1) with respect to γ⊗µ, and T loc 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 kernelK 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 γ ⊗ µ is comparable with Λ ⊗ µ.  , such that: i) the family {B j : j ∈ N} covers R d ; ii) the balls { 1 4 B j : j ∈ N} are pairwise disjoint; iii) for any β > 0, the family {βB j : j ∈ N} has bounded overlap, i.e.; sup j χ βB j ( The next lemma we need is a two variable version of [19,Lemma 3.3] (see also the following remark). The proof is based on Lemma 4.5 and proceeds as in [19]. 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 ν we denote one of the measures γ or Λ. Lemma 4.6. Let S be a linear operator defined on L ∞ c and set where B j is the family of balls from Lemma 4.5. We have the following: i) If S is of weak type (1, 1) with respect to the measure ν ⊗ µ, then S 1 is of weak type (1, 1) with respect to both γ ⊗ µ and Λ ⊗ µ; moreover, , for some 1 < p < ∞, then S 1 is bounded on both L p and L p (Λ ⊗ µ); moreover, We proceed with the proof of Proposition 4.4. Decompose T = D +T , where, with ∆ being the self-adjoint extension of the Laplacian on L 2 (R d , Λ). Observe that, by the multivariate spectral theorem applied to the system (−∆, A), the operatorT is bounded on L 2 (Λ⊗ µ). Consequently,T and thus also D = T −T , are both well defined on L ∞ c . We start with considering the operatorT . First we demonstrate that T = 1 0 κ(r) ∂ r e 1 4 (1−r 2 )∆ r A dr is a Calderón-Zygmund operator on the space of homogeneous type (R d × Y, η, Λ ⊗ µ); recall that η is defined by (4.11). In what followsK is given bỹ In the proof of Lemma 4.7 we often use the following simple bound and its kernel satisfies standard Calderón-Zygmund estimates, i.e. the growth estimate , η(x, y))) , x = y, and, for some δ > 0, the smoothness estimate , η(x, y))) , 2η(y, y ′ ) ≤ η(x, y).

(4.26)
ConsequentlyT is of weak type (1, 1) with respect to Λ ⊗ µ, and Proof. As we have already remarked, by spectral theoryT is bounded on L 2 (Λ ⊗ µ), and we easily see that (4.24) holds. Additionally, an argument similar to the one used in the proof of Lemma 4.2 shows thatT is associated with the kernelK even in the sense of Definition 4.12.
We now pass to the proofs of the growth and smoothness estimates and start with demonstrating (4. 25). An easy calculation shows that Hence, we have for For further use we remark that the above bound implies From (4.28) we see that Thus, coming back to the variable t = − log r and then using (4.1), we arrive at A standard argument using the doubling property of µ (cf. (4.32)) shows that we can further estimate The last integral is bounded by a constant times η d (x, y), which equals C d Λ(B |·| (x 1 , η(x, y))). Thus, (4.25) follows once we note that 1 Λ(B |·| (x 1 , η(x, y))µ(B ζ (x 2 , η(x, y))) = 1 (Λ ⊗ µ)(B(x, η(x, y))) .
We now focus on the smoothness estimate (4.26), which is enough to obtain the desired weak type (1, 1) property ofT . We decompose the difference in (4.26) as Till the end of the proof of (4.26) we assume η(x, y) ≥ 2η(y, y ′ ), so that η(x, y) ≈ η(x, y ′ ).
Hence, proceeding similarly as in the previous case (this time we use (4.3) instead of (4.2)), we obtain The latter quantity has already appeared in (4.31) and has been estimated by the right hand side of (4.33). Now we pass to I 1 . A short computation based on (4.27) gives From the above inequality it is easy to see that and consequently, after the change of variable e −t = r, Hence, from the mean value theorem it follows that for |x 1 − y 1 | ≥ 2|y 1 − y ′ 1 |, while for arbitrary x 1 , y 1 , Moreover, at the cost of a constant in the exponent, the expression |y 1 − y ′ 1 |/ √ t from the right hand sides of (4.34) and (4.35) can be replaced by (|y 1 − y ′ 1 |t −1/2 ) δ , for arbitrary 0 < δ ≤ 1. If this is a consequence of (4.34) and (4.35), while if |y 1 − y ′ 1 | ≥ √ t it can be deduced from (4.34) and (4.30). Similarly as it was done for I 2 , to estimate I 1 we consider two cases.
Lemma 4.8. If (x 1 , y 1 ) ∈ N 2 , then we have Proof. We proceed similarly to the proof of [18,Lemma 3.9]. Since for (x 1 , y 1 ) from the local region N 2 we have Thus, using (4.15) we obtain for (x 1 , y 1 ) ∈ N 2 , Note that the above inequality implies Using (4.40) and (4.28) we easily see that which is even better then the estimate we want to prove. Now we consider the integral over (1/2, 1). Denoting r(x 1 ) = max(1/2, 1 − |x 1 | 2 ) and using once again (4.40) and (4.28) we obtain The above quantity is exactly the one estimated by the right hand side of (4.37) in the second paragraph of the proof of [18,Lemma 3.9]. It remains to estimate the integral taken over (r(x 1 ), 1). Using the formulae (4.15) and (4.27) together with (4.39) we write The quantity J 2 has been already estimated in the proof of [18, Lemma 3.9, p.12], thus we focus on J 1 . For fixed r, x 1 , y 1 denote by using (4.38) and (4.39) with r replaced by s, we obtain Thus, by the mean value theorem Recalling that J 2 was estimated before, we conclude the proof.
As a corollary of Lemma 4.8 we now prove the following.
Lemma 4.9. The operator D loc is bounded on all the spaces L p (Λ ⊗ µ). Moreover, Proof. Observe that D loc may be expressed as at least for f ∈ L ∞ c . Moreover, the estimates below imply that the integral defining D loc is actually absolutely convergent, whenever f ∈ L p (Λ ⊗ µ), for some 1 ≤ p ≤ ∞.
The last lemma of this section shows that the local parts of T andT inherit their boundedness properties. Moreover, it says that the operators T loc ,T loc , and D loc are bounded on appropriate spaces with regards to both the measures Λ ⊗ µ and γ ⊗ µ.  Moreover, both S =T loc and S = D loc are of weak type (1, 1) with respect to ν ⊗ µ, with ν = γ or ν = Λ, and Proof. In what follows S(x, y) denotes the kernel K(x, y) of T, or the kernelK(x, y) ofT , 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.
Summarizing, since T loc =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 γ ⊗ µ and Λ ⊗ µ. Moreover, the weak type (1, 1) constant is less than or equal to C d,µ κ ∞ . Hence, after combining Propositions 4.3 and 4.4, the proof of Theorem 4.1 is completed.
Theorem A.1. Let S be an operator which is bounded from L 1 γ (H 1 ) to L 1,∞ (γ ⊗ µ), and from L 2 to L 2 . Then S is bounded on all L p spaces, 1 < p < 2.
The main ingredient of the proof is a Calderón-Zygmund decomposition of a function f (x 1 , x 2 ), with respect to the variable x 2 , when x 1 is fixed, see Lemma A.2. For the decomposition we present it does not matter that we consider R d with the measure γ. The important assumption is that (Y, ζ, µ) is a space of homogeneous type. Therefore till the end of the proof of Lemma A.2 we consider a more general space L 1 := L 1 (X × Y, ν ⊗ µ). Here ν is an arbitrary σ-finite Borel measure on X. Recall that, by convention, elements of X are denoted by x 1 , y 1 , while elements of Y are denoted by x 2 , y 2 .
It is known that in every space of homogeneous type in the sense of Coifman-Weiss there exists a family of disjoint 'dyadic' cubes, see [23,Theorem 2.2]. Here we use [23, Theorem 2.2] to (Y, ζ, µ). Let Q l be the set of all dyadic cubes of generation l in the space (Y, ζ, µ). Note that l → ∞ corresponds to 'small' cubes, while l → −∞ to 'big' cubes. We define the l-th generation dyadic average and the dyadic maximal function with respect to the second variable, by f (x 1 , y 2 ) dµ(y 2 ) χ Q (x 2 ), It remains to prove the property (iii). The inequality j µ(S j (x 1 )) ≤ s −1 Y f (x) dµ(x 2 ) follows from (A.4). If S j (x 1 ) = ∅ then obviously, b j (x 1 , ·) = 0. If S j (x 1 ) is not empty, then S j (x 1 ) = Q j (x 1 ), for some j(x 1 ), so that supp b j (x 1 , ·) ⊂ Q j (x 1 ). In either case Y b j (x) dµ(x 2 ) = S j (x 1 ) b j (x) dµ(x 2 ) = 0.
Using Lemma A.2 we now prove Theorem A.1. The proof follows the scheme from [7, Theorem D, pp. 596, 635-637] by Coifman and Weiss.
Proof of Theorem A.1. Fix 1 < q < p and set D q (f ) = (D(|f | q )) 1/q , with D given by (A.1). Then, since D is bounded on L p and 1 < q < p, the same is true for D q . From item (v) of Lemma A.2 it follows that where the sets S j satisfy properties (i)-(iv) from Lemma A.2 with s q in place of s and f q in place of f. In particular Decompose f = g s + b s = g s + j b j,s with f (x 1 , y 2 ) dµ(y 2 ) χ S j .
Since the spaces H 1,q (Y, µ) and H 1 (Y, µ) = H 1,∞ (Y, µ) coincide, cf. [7, Theorem A], using Fubini's theorem and the disjointness of S j we obtain By the layer-cake formula we have and, consequently, Passing to E 2 , the layer-cake formula together with the L 2 boundedness of S and Chebyshev's inequality produce From (A.5), (A.6) and the definition of g s we see that |g s | ≤ Cs, and consequently, The above quantity has already been estimated, see (A.10). Now we focus on E 2,2 . Since g s = f outside of Θ s and f ≤ D q (f ), using Fubini's theorem we have thus obtaining the desired estimate for E 2 and hence, finishing the proof of Theorem A.1.