# Maximal theorems and square functions for analytic operators on -spaces

## Abstract.

Let be a contraction, with , and assume that is analytic, that is, . Under the assumption that is positive (or contractively regular), we establish the boundedness of various Littlewood-Paley square functions associated with . In particular we show that satisfies an estimate for any integer . As a consequence we show maximal inequalities of the form , for any integer . We prove similar results in the context of noncommutative -spaces. We also give analogs of these maximal inequalities for bounded analytic semigroups, as well as applications to -boundedness properties.

2000 Mathematics Subject Classification : 47B38, 46L52, 46A60.

## 1. Introduction.

Let be a measure space, let and let be a positive contraction. Then Akcoglu’s Theorem [1] asserts that satisfies a maximal ergodic inequality,

(1.1) |

A well-known question is to determine which operators satisfy a stronger maximal inequality,

(1.2) |

In this paper we show that this holds true provided that is analytic, that is, there exists a constant such that

for any (see Section 2 for some background). More generally, we show that for any integer , analytic positive contractions satisfy a maximal inequality

(1.3) |

Note that for any , the sequence of operators appearing here is the -th order discrete derivative of the original sequence . The proofs of these inequalities rely on the boundedness of certain discrete Littlewood-Paley square functions of independent interest that we establish in Section 3. In particular we will show that for as above, we have an estimate

(1.4) |

These maximal theorems and square function estimates extend Stein’s famous results [35, 36] which show that (1.2), (1.3) and (1.4) hold true in the case when acts as a contraction for any and its -realization is a positive selfadjoint operator.

Let be a von Neumann algebra equipped with a normal semifinite faithful trace and for any , let be the associated noncommutative -space. Let be a positive contraction whose restriction to extends to a contraction . Recall that in this case, actually extends to a contraction for any . It is shown in [17] that satisfies a noncommutative analog of (1.1). In the latter paper, a large part of Stein’s work mentioned above is also transfered to the noncommutative setting. Indeed it is shown that if the -realization is a positive selfadjoint operator, then for any , satisfies noncommutative analogs of (1.2) and (1.3). This is generalized in [3] under an appropriate condition on the numerical range of . We extend these results by showing that for any , the noncommutative analogs of (1.2) and (1.3) hold true provided that is merely analytic (which is a much weaker assumption).

Besides investigating the behaviour of operators and their powers (discrete semigroups), we consider continuous semigroups , both in the commutative and in the noncommutative settings. The continuous analog of the maximal inequality (1.2) reads as follows:

(1.5) |

We prove that such an estimate holds true whenever is a bounded analytic semigroup on (with ) such that is a positive contraction for any . Likewise we show that the noncommutative analog of (1.5) holds true whenever is a semigroup of positive contractions on for any and is a bounded analytic semigroup on (with ). These results both extend Stein’s classical maximal theorem [35, 36] for semigroups and its recent noncommutative counterpart from [17]. Finally we extend some results from [18, Chapter 5] concerning -boundedness in the noncommutative setting.

In the above presentation and later on in the paper, stands for an inequality up to a constant which may depend on and , but not on .

## 2. Preliminaries.

An operator is called regular if there is a constant such that

for any finite sequence in . Then we let denote the smallest for which this holds. The set of all regular operators on is a vector space on which is a norm. We say that is contractively regular if . Clearly any positive operator is regular and in this case. Thus all statements given for contractively regular operators apply to positive contractions. It is well-known that conversely, is regular with if and only if there is a positive operator with , such that for any (see [27, Chap. 1]). Furthermore, is contractively regular if acts as a contraction for any .

We recall some definitions and simple facts about sectorial operators and analyticity. Throughout we let denote an arbitrary (complex) Banach space and we let denote the algebra of all bounded operators on . Next for any angle , we introduce

the open sector of angle around .

Let be a (possibly unbounded) closed linear operator, with dense domain . We let denote the spectrum of and for any , we let denote the corresponding resolvent operator. We say that is sectorial if there exists an angle such that is contained in the closed sector and

Then we let be the infimum of all such that holds, and this real number is called the type of . It is well-known that if holds true for some , then there exists such that holds true as well. Thus,

(2.1) |

Let be a bounded strongly continuous semigroup on . We call it a bounded analytic semigroup if there exists a positive angle and a bounded analytic family extending . Let be the infinitesimal generator of . Analyticity has two classical characterizations in terms of that operator. First, is a bounded analytic semigroup if and only if for any and there exists a constant such that for any . Note here that since , we have

(2.2) |

Second, is a bounded analytic semigroup if and only if is sectorial and . According to (2.1), this is also equivalent to saying that satisfies . We refer e.g. to [15, 30] for proofs and complements on semigroups.

We will make a crucial use of -calculus and square functions for sectorial operators. Here are the basic notions and results which will be needed. For more information, we refer e.g. to [11, 19, 21, 23].

For any , we define

This is a Banach algebra with the norm

Then let be the subalgebra of all for which there exist two constants such that

For any sectorial operator , for any and for any , we define

where and is the boundary oriented counterclockwise. This integral is well-defined, its definition does not depend on and the resulting mapping is an algebra homomorphism from into . We say that has a bounded functional calculus if the latter homomorphism is bounded, that is, there exists a constant such that

Consider now the specific case when , with . On such a space, Cowling, Doust, McIntosh and Yagi have proved a remarkable equivalence result between the boundedness of functional calculus and certain square function estimates. In particular they established the following key result.

###### Proposition 2.1.

[11] Let be a sectorial operator on and assume that there exists such that admits a bounded functional calculus for any . Then for any and any , there exists a constant such that

(2.3) |

Let us now turn to discrete semigroups. Let . We say that is power bounded if the set

(2.4) |

is bounded. Then we say that is analytic if moreover the set

(2.5) |

is bounded. This notion of discrete analyticity goes back to [10]. Since is the ‘discrete derivative’ of the sequence , we can regard as a discrete analog of . In view of (2.2), the boundedness of (2.5) is therefore a natutal discrete analog of the boundedness of .

The most important result concerning discrete analyticity is perhaps the following characterization: an operator is power bounded and analytic if and only if

(2.6) |

This property is called the ‘Ritt condition’. The key argument for this characterization is due to O. Nevanlinna [29], however we refer to [25, 28] for a complete proof and complements. Let us gather a few observations which will be used later on in the paper. First we note that (2.6) implies that

(2.7) |

Indeed, for any . Second, (2.6) implies the existence of a constant such that whenever This means that

satisfies . According to (2.1), this implies that is a sectorial operator of type . Hence

(2.8) |

In this case, the bounded analytic semigroup generated by is given by

(2.9) |

We now recall the definition of -boundedness (see [4, 7]). Let be a sequence of independent Rademacher variables on some probability space . Let be the closure of in the Bochner space . Thus for any finite family in , we have

By definition, a set is -bounded if there is a constant such that for any finite families in , and any in , we have

Obviously any -bounded set is bounded and if is isomorphic to a Hilbert space, then all bounded subsets of are automatically -bounded. However if is not isomorphic to a Hilbert space, then contains bounded subsets which are not -bounded [2, Prop. 1.13].

Let be a measure space and let . Then . Hence a set is -bounded if and only if we have an estimate

for finite families in and in .

We shall now consider these general definitions for specific sets of operators. Let be a bounded analytic semigroup on . We say that this is an -bounded analytic semigroup if there exists a positive angle such that is -bounded. It was observed in [37] that this holds true if and only if the two sets

are -bounded.

Accordingly we will say that an operator is an -analytic power bounded operator if the two sets and from (2.4) and (2.5) are -bounded.

The above notions of -analyticity were introduced by Weis [37] for the continuous case and Blunck [5] for the discrete one. In both cases they played a crucial role in the solution of maximal regularity problems on UMD Banach spaces, see the above papers for more information. -boundedness for sectorial operators is also a key tool for various questions regarding functional calculus, see in particular [19, 21, 18].

The next result is well-known to specialists.

###### Proposition 2.2.

Let be a bounded analytic semigroup on , with , and assume that for any . Let be the generator of . Then there exists such that admits a bounded functional calculus.

###### Proof.

We end this section with a few notation. For any complex number and any , we will let denote the open disc of center and radius . We let be the usual unit disc. Also we let denote the algebra of complex polynomials in one variable.

## 3. Square functions on .

Throughout the next two sections we let be a measure space and we fix some . We will establish general square function estimates for analytic contractively regular operators on (see Theorem 3.3 below).

We will need the following elementary fact.

###### Lemma 3.1.

Let be an open set and let be a compact -curve. Let be an analytic function. Then there exists a contant such that

for any .

###### Proof.

Let . Write as the juxtaposition of -curves of length . Then for each , choose and set

Let

(3.1) |

be the Taylor expansion of about . Then by Cauchy’s inequalities. Any satisfies (3.1) hence we have

However for any , we have

and for any . Thus

Consequently,

Since

we obtain the result with . ∎

For any , let

Alternatively, is the convex hull of and the disc .

Following usual terminology, these sets will be called ‘Stolz domains’ in the sequel. We will use the fact that for any , there exists a constant such that

(3.2) |

Let be an integer and let be an matrix of polynomials, that is, belongs to for any . Then for any , we set

###### Proposition 3.2.

Let be any analytic contractively regular operator. Then there exists an angle and a constant satisfying the following property. For any , for any matrix of polynomials and for any in , we have

(3.3) |

###### Proof.

We let be the conjugate number of . Let and let be the semigroup defined by (2.9), whose generator is . We noticed in Section 2 that this is a bounded analytic semigroup. Furthermore for any , we have

Hence by Proposition 2.2, admits a bounded functional calculus for some . By (2.7) and (2.8), there exists such that . Equivalently,

We now fix . Then we let be the boundary of oriented counterclockwise.

We claim that we have estimates

(3.4) |

and

(3.5) |

Recall that we let denote the boundary of oriented counterclockwise. Thus the contour is the juxtaposition of a part of and the curve going from to counterclockwise along the circle of center and radius . Obviously we have

Since , Lemma 3.1 ensures that we can control the last integral by a constant times . Hence to prove (3.4), it suffices to prove an estimate

(3.6) |

Likewise, to prove (3.5), it suffices to prove an estimate

(3.7) |

Consider , and define two functions by letting

For any , we have

Likewise,

Hence

Applying Proposition 2.1 to and , we deduce the estimate (3.4). Now note that also admits a bounded functional calculus (see e.g. [11] for this duality principle). Hence arguing as above with the two functions

we get (3.5).

The estimates (3.4) and (3.5) can be formally strengthened as follows. There is a constant such that for any integer , we have

(3.8) |

for any in and similarly,

(3.9) |

for any in . Indeed (3.8) (resp. (3.9)) can be deduced from (3.4) (resp. (3.5)) by applying Khintchine’s inequality and Fubini’s Theorem. The argument is similar to the one in the proof of [22, Lemma 5.4] so we omit it.

In the sequel, we let be the space of polynomials vanishing at . The function is well-defined and bounded on , and the same is true for whenever . It therefore follows from the Dunford functional calculus that

for any . Likewise,

for any . Hence

that is,

(3.10) |

Let be an integer, let be an matrix of polynomials, and let be in . For any , we set , so that . Also we assume that , so that . For any in , we have

by (3.10). Applying Cauchy-Schwarz and Hölder’s inequalities, we deduce that

Furthermore,

is less than or equal to