On noncommutative distributional Khintchine type inequalities

The purpose of this paper is to provide distributional estimates for the series of the form ∑k=1∞xk⊗rk$\sum _{k=1}^\infty x_k\otimes r_k$ with {xk}k⩾1$\lbrace x_k\rbrace _{k\geqslant 1}$ being elements from noncommutative Lorentz spaces Λlog1/2(M)$\Lambda _{\log ^{1/2}}(\mathcal {M})$ and {rk}k⩾1$\lbrace r_k\rbrace _{k\geqslant 1}$ being Rademacher functions. To this end, we introduce a novel class of operators {Pα}α>0$\lbrace P_{\alpha }\rbrace _{\alpha >0}$ that are closely related to the dual Cesáro operator C*$C^\ast$ and construct a new extrapolation theorem that is of independent interest.


INTRODUCTION
Khintchine inequalities have been long of interest as an essential tool for bounding the growth of sums of independent random variables.Recall that the classical Khintchine inequalities assert that for every 1 <  < ∞ there exists constant   depending only on  such that where {  } ⩾1 is a sequence of scalars and {  } ⩾1 is a sequence of Rademacher functions defined on [0,1] (see, for example, the papers [12,27] for details).
In [30], Lust-Piquard and Pisier successfully formulated Khintchine inequalities in the noncommutative context.Let  be a semifinite von Neumann algebra equipped with a normalized semifinite faithful trace .By replacing scalars {  } ⩾1 in (1) with operators {  } ⩾1 from the noncommutative Lebesgue space   () for 1 <  < ∞, Lust-Piquard and Pisier [30] established that where   is a constant depending only on .Since then, noncommutative Khintchine inequalities have garnered significant attention and have been extended to numerous directions, notably to various symmetric quasi-norms (  , Orlicz, Lorentz quasi-norms, generalized symmetric quasinorms, etc.), see, for example, [1,4,7,16,19].In particular, Junge [20] and Pisier [34] determined the optimal order of   in (2), for all sequences {  } ⩾1 in   ().Here, the order √  is the best possible.The purpose of this paper is to establish a distributional Khintchine type estimate involving singular value functions in the noncommutative setting.Our study deeply roots in the work of Bennett and Sharpley [2,Theorem 3.4.7],where distributional inequalities related to singular value functions of Hilbert transforms were provided.In [37], the last two authors and Tulenov extended Bennett and Sharpley's result to the context of operator-valued functions.More recently, distributional inequalities of noncommutative martingales have been established (see [15,18]).The advantage of the distributional estimates is that they enable us to obtain some interesting endpoint estimates and various new inequalities in symmetric quasi-Banach function (or operator) spaces.We refer to [5, 23-25, 33, 38] for more information.
Our main result of this paper is stated below.It provides the distributional estimate of the series ∑ ⩾1   ⊗   .For any unexplained notation, we refer to the next section.  .
The proof of Theorem 1.1 is accomplished via a new and powerful extrapolation theorem that is of independent interests.We also should mention that the optimal order achieved in (3) is of special importance for our proof.To construct the extrapolation theorem, we introduce a novel family of operators {  } >0 that are closely related to the dual Cesáro operator and discuss some interesting properties of these operators.Our result and the method are new even in the commutative case.
The paper is organized as follows.In Section 2, we recall some background on the subject.The operators {  } >0 and their useful properties are collected in Section 3. A noncommutative extrapolation theorem is stated and proved in Section 4. In the same section, we exploit the extrapolation theorem to prove Theorem 1.1.

PRELIMINARIES
Throughout, we use  abs to denote some absolute constant that may change from line to line.We write  ≲  if there is some absolute constant  abs such that  ⩽  abs .We say that  is equivalent to  (written as  ≈ ) if there exists some absolute constant  abs such that  −1 abs  ⩽  ⩽  abs .The notation  ≲  (or,  ≈  ) means that the inequality (or, equivalence) holds true for some constant depending on the parameter .

Generalized singular value functions
In what follows, ℍ is a separable Hilbert space and  ⊂ (ℍ) denotes a semifinite von Neumann algebra equipped with a faithful normal semifinite trace .The pair (, ) is called a noncommutative measure space.A closed and densely defined operator  on ℍ is said to be affiliated with  if  *  =  for each unitary operator  in the commutant  ′ of .The operator  is called -measurable if  is affiliated with  and for every  > 0, there exists a projection  ∈  such that (ℍ) ⊂ dom() and (1 − ) < .The set of all -measurable operators will be denoted by (, ).Given a self-adjoint operator  ∈ (, ) and a Borel set  ⊂ ℝ, we denote by   () its spectral projection.For a projection  ∈ , if () < ∞, we say that  is -finite.
The function  ↦ (, ) is decreasing and right-continuous.Given ,  ∈ (, ), we say that  is submajorized by  in the sense of Hardy-Littlewood-Pólya (written as  ≺≺ ) if In the case that  is the abelian von Neumann algebra  ∞ (0, ) (0 <  ⩽ ∞) with the trace given by integration with respect to the Lebesgue measure, (, ) is the space of all measurable functions with nontrivial distribution, and () is the decreasing rearrangement of a measurable function ; see [26,27].In the abelian case, we write (0, ) instead of ( ∞ (0, )) (0 <  ⩽ ∞).For more discussion on generalized singular value functions, we refer the reader to [11,28].
As Köthe dual will be used in the following paragraph, we give the definition of the Köthe dual of symmetric Banach operator spaces.Assume that  is a symmetric Banach function space.The Köthe dual of (, ) is defined as

Lorentz spaces and Marcinkiewicz spaces
A function  on the semiaxis [0, ∞) is said to be quasi-concave if (1) () = 0 if and only if  = 0; (2)  is positive and increasing; ( is decreasing.For a quasi-concave function , the Lorentz space Λ  is defined by setting The Lorentz space Λ  (ℤ + ) is defined as Its norm is given by the formula The Marcinkiewicz space   (0, ∞) is defined by setting The Marcinkiewicz space   (ℤ + ) is defined as Its norm is given by the formula Note that the Köthe dual of the Lorentz space Λ  is the Marcinkiewicz space   .The Köthe dual of Marcinkiewicz space   is the Lorentz space Λ  .We refer to [26] for more information on Lorentz and Marcinkiewicz spaces.

Cesáro and dual Cesáro operators
To begin with, we recall the definition of the weak space  1,∞ :

Now we introduce Cesáro and dual Cesáro operators. The Cesáro operator 𝐶
We define the operator  * defined by the setting It acts boundedly from Λ log (0, ∞) to ( 1 +  ∞ )(0, ∞).These operators also enjoy the following properties: where  ′ is the conjugate index of .We point out here that the case  = ∞ in (5) (respectively,  = 1 in ( 6)) is trivial and the remaining estimates, see, for example, [14, Theorems 327 and 328].

NEW OPERATORS 𝑷 𝜶 AND RELATED PROPERTIES
To formulate the extrapolation theorem and provide the proof of our main result, we introduce a new collection of operators {  } >0 and verify some elementary but crucial properties of these operators.Given  > 0, the operator   is defined by setting where Γ is the Gamma function.As we will see, the operators {  } >0 are closely related with the dual Cesáro operator  * and share many similar properties with  * .In fact, in some sense the operators {  } >0 could be viewed as powers of  * .First, we show the semigroup property of the operators   .
Proof.We have It is immediate from changing variables that where  is the beta function.Thus, As the desired assertion follows.□ The lemma below shows that   acts boundedly from   to   for 1 ⩽  < ∞.
Proof.First, we recall that (see, e.g., [11,Lemma 4.1]) for any fixed  > 0, Let  ∈ Λ log  (0,∞) .We write = sup Noting that we have Making the substitution  = , we obtain This completes the proof.□ The next lemma plays a crucial role in the proof of our extrapolation theorem.
Proof.We concentrate on the first part of the assertion.The second part follows immediately from the first one and from Proposition 3.3.
As  is positive and decreasing, it follows from (7) that    is positive and decreasing.Thus, we have Moreover, we may further write For the former term, by changing variables, we obtain Similarly, for the latter term, we have Thus, it follows from ( 9), (10), and ( 11) that Now set Observe that 1 ⩽ 1 + On the other hand, by the L'Hopital rule, This implies that )  > 0.
Therefore, we conclude that Combining ( 12), (13), and ( 14), we obtain This completes the proof.□ Remark 3.5.Lemma 3.1 shows that the operators (  ) >0 satisfy a semigroup property.Noting that  1 is exactly the dual Cesáro operator  * , we conclude from the semigroup property that . Lemma 3.4 describes the maximal domain of   (which means this domain is the maximal symmetric quasi-Banach function space on (0, ∞) such that   is bounded from that space to ( 1 +  ∞ )(0, ∞)).

MAIN RESULTS
This section is devoted to the proof of the distributional Khintchine inequalities (i.e., Theorem 1.1).
To begin with, we state and verify a new type of extrapolation theorem, which underpins the solution to Theorem 1.1.
The desired assertion follows.□ Now, we are ready to prove the extrapolation theorem.
As  > 0 is arbitrary, the assertion follows.□ By using the extrapolation theorem established above, we now present the proof of Theorem 1.1.))

Proof of Theorem
.