本文将Bellman函数方法推广到鞅上一般算子（不限于鞅变换），构造单一Bellman型函数编码Gundy外推定理中几乎所有算子的$L^p$-有界性，并以Haar变换和Walsh系统的Rubio de Francia不等式相关算子为例加以说明。
摘要：
本文证明Bellman函数方法可用于研究鞅上一般算子的$L^p$-范数，即不一定是鞅变换的算子。非正式地说，我们提供了一个单一的Bellman型函数，它“编码”了Gundy外推定理中“几乎所有”算子的$L^p$-有界性。作为此类算子的例子，我们考虑了Haar变换以及其$L^p$-有界性构成Walsh系统的Rubio de Francia不等式基础的算子。
英文摘要：
It is shown that the Bellman function method can be applied to study the $L^p$-norms of general operators on martingales, i.e., of operators that are not necessarily martingale transforms. Informally, we provide a single Bellman-type function that "encodes" the $L^p$-boundedness of "almost all" operators from Gundy's extrapolation theorem. As examples of such operators, we consider the Haar transforms and the operator whose $L^p$-boundedness underlies Rubio de Francia's inequality for the Walsh system.
We consider several weak type estimates for singular operators using the Bellman function approach. We disprove the $A_1$ conjecture of Muckenhoupt, which stayed open after Muckenhoupt--Wheeden's conjecture was disproved by Reguera--Thiele.
We provide some new estimates for Bellman type functions for the dyadic maximal opeator on $R^n$ and of maximal operators on martingales related to weighted spaces. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors we introduce certain conditions on the weight that imply estimate for the maximal operator on the corresponding weighted space. Also using a well known estimate for the maximal operator by a double maximal operators on different m easures related to the weight we give new estimates for the above Bellman type function…
We precisely evaluate Bellman type functions for the dyadic maximal opeator on $R^n$ and of maximal operators on martingales related to local Lorentz type estimates. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors we precisely evaluate the supremum of the Lorentz quasinorm of the maximal operator on a function $ϕ$ when the integral of $ϕ$ is fixed and also the same Lorentz quasinorm of $ϕ$ is fixed. Also we find the corresponding supremum when the integral of $ϕ$ is fixed and several weak type conditions are given.
We prove the boundedness on $L^p$, $1<p<\infty$, of operators on manifolds which arise by taking conditional expectation of transformations of stochastic integrals. These operators include various classical operators such as second order Riesz transforms and operators of Laplace transform-type.
We provide an alternative proof and expression of the Bellman function of the dyadic maximal operator in connection with the Dyadic Carleson Imbedding Theorem, which appears in [10]. We also evaluate the Bellman function of four variables of the dyadic maximal operator
We give a characterization of the extremal sequences for the Bellman function of the dyadic maximal operator.In fact we prove that they behave approximately like eigenfunctions of this operator for a specific eigenvalue.
We prove a necessary condition that has every extremal sequence for the Bellman function of the dyadic maximal operator.This implies the weak-Lp uniqueness for such a sequence.
We precisely compute the Bellman function of two variables of the dyadic maximal operator in relation to Kolmogorov inequality. In this way we give an alternative proof of the results in [5].Additionally, we characterize the sequences of functions that are extremal for this Bellman function.The proof of this is based on that is given in this paper for the Bellman function we are interested in.
We provide a description for the Bellman function related to the Carleson Imbedding theorem, first mentioned in [4], with the use of the Hardy operator.
We prove that the extremal sequences for the Bellman function of the dyadic maximal operator behave approximately as eigenfunctions of this operator for a specific eigenvalue. We use this result to prove the analogous one with respect to the Hardy operator.
This paper proves the $L^p$ boundedness of generalized first order Riesz transforms obtained as conditional expectations of martingale transforms à la Gundy-Varopoulos for quite general diffusions on manifolds and vector bundles. Several specific examples and applications are presented: Lie groups of compact type, the Heisenberg group, SU(2), and Riesz transforms on forms and spinors.
We give a characterization of the extremal sequences for the Bellman function of three variables of the dyadic maximal operator in relation to Kolmogorov's inequality. In fact we prove that they behave approximately like eigenfunctions of this operator for a specific eigenvalue. For this approach we use the one introduced in [4], where the respective Bellman function has been precisely evaluated.
We establish some operator versions of Bellman's inequality. In particular, we prove that if $Φ: \mathbb{B}(\mathscr{H}) \to \mathbb{B}(\mathscr{K})$ is a unital positive linear map, $A,B \in \mathbb{B}(\mathscr{H})$ are contractions, $p>1$ and $0 \leq λ\leq 1$, then {eqnarray*} \big(Φ(I_\mathscr{H}-A\nabla_λB)\big)^{1/p}\geΦ\big((I_\mathscr{H}-A)^{1/p}\nabla_λ(I_\mathscr{H}-B)^{1/p}\big). {eqnarray*}
For p>1 we find the Bellman function of two variables associated with the dyadic maximal operator on Rn.Actually we do that in the more general setting of tree-like maximal operators.We provide a simple and elementary proof,different from those in [4] and [9].
We compute the Bellman function of three integral variables associated to the dyadic maximal operator on a subset of its domain. Additionally, we provide an upper bound for the whole domain of its definition.
We study stability estimates for the almost extremal functions associated with the $L^p$-bound for the real and imaginary parts of the Beurling-Ahlfors operator. The proof exploits probabilistic methods and rests on analogous results for differentially subordinate martingales which are of independent interest. This allows us to obtain stability inequalities for a larger class of Fourier multipliers.
We describe a new operator space structure on $L_p$ when $p$ is an even integer and compare it with the one introduced in our previous work using complex interpolation. For the new structure, the Khintchine inequalities and Burkholder's martingale inequalities have a very natural form:\ the span of the Rademacher functions is completely isomorphic to the operator Hilbert space $OH$, and the square function of a martingale difference sequence $d_n$ is $Σ\ d_n\otimes \bar d_n$. Various inequalities from harmonic analysis are also considered in the same operator valued framework. Moreover, the ne…
We construct quantum stochastic integrals for the integrator being a martingale in a von Neumann algebra, and the integrand -- a suitable process with values in the same algebra, as densely defined operators affiliated with the algebra. In the case of a finite algebra we allow the integrator to be an $L^2$--martingale in which case the integrals are $L^2$--martingales too.
The main aspiration of this note is to construct several different Haar-type systems in euclidean spaces of higher dimensions and prove sharp Lp bounds for the corresponding martingale transforms. In dimension one this was a result of Burkholder. The motivation for working in this direction is the search for Lp estimates of the Ahlfors-Beurling operator.
Let $M$ be a discrete-time normal martingale that has the chaotic representation property. Then, from the space of square integrable functionals of $M$, one can construct generalized functionals of $M$. In this paper, by using a type of weights, we introduce a class of continuous linear operators acting on generalized functionals of $M$, which we call generalized weighted number (GWN) operators. We prove that GWN operators can be represented in terms of generalized annihilation and creation operators (acting on generalized functionals of $M$). We also examine commutation relations between a GW…
The concept of uniform convexity of a Banach space was generalized to linear operators between Banach spaces and studied by Beauzamy [1976]. Under this generalization, a Banach space X is uniformly convex if and only if its identity map I_X is. Pisier showed that uniformly convex Banach spaces have martingale type p for some p>1. We show that this fact is in general not true for linear operators. To remedy the situation, we introduce the new concept of martingale subtype and show, that it is equivalent, also in the operator case, to the existence of an equivalent uniformly convex norm on X. In…
In this paper, we employ the Mond--Pečarić method to establish some reverses of the operator Bellman inequality under certain conditions. In particular, we show \begin{equation*} δI_{\mathscr K}+\sum_{j=1}^nω_jΦ_j\left((I_{\mathscr H}-A_j)^{p}\right)\ge \left(\sum_{j=1}^nω_jΦ_j(I_{\mathscr H}-A_j)\right)^{p} \,, \end{equation*} where $A_j\,\,(1\leq j\leq n)$ are self-adjoint contraction operators with $0\leq mI_{\mathscr H}\le A_j \le MI_{\mathscr H}$, $Φ_j$ are unital positive linear maps on ${\mathbb B}({\mathscr H})$, $ω_j\in\mathbb R_+ \,\,(1\leq j\leq n)$, $0 < p < 1$ and $δ=(1-p)\left(\f…
We establish distributional estimates for noncommutative martingales, in the sense of decreasing rearrangements of the spectra of unbounded operators, which generalises the study of distributions of random variables. Our results include distributional versions of the noncommutative Stein, dual Doob, martingale transform and Burkholder-Gundy inequalities. Our proof relies upon new and powerful extrapolation theorems. As an application, we obtain some new martingale inequalities in symmetric quasi-Banach operator spaces and some interesting endpoint estimates. Our main approach demonstrates a me…
The paper contains the proof of $L^p$-weighted norm inequalities for both, martingales square functions and the classical square functions in harmonic analysis of Littlewood-Paley and Lusin. Furthermore, the bounds are completely explicit and are optimal not only on the dependence of the characteristics of the weight but also on the dependance on $p$, as $p\to\infty$. The proof rests on Bellman function method: the estimates are deduced from the existence of an appropriate and rather complicated function of four variables.
We give necessary and sufficient conditions for two weight norm inequalities for Haar multipliers operators and for square functions. We also give sufficient conditions for two weight norm inequalities for the Hilbert transform.
This paper constructs a class of martingale transforms based on Lévy processes on Lie groups. From these, a natural class of bounded linear operators on the $L^p$-spaces of the group (with respect to Haar measure) for $1<p<\infty$, are derived. On compact groups these operators yield Fourier multipliers (in the Peter-Weyl sense) which include the second order Riesz transforms, imaginary powers of the Laplacian, and new classes of multipliers obtained by taking the Lévy process to have conjugate invariant laws. Multipliers associated to subordination of the Brownian motion on the group are spec…
A positive operator $T:E\to E$ on a Banach lattice $E$ with an order continuous norm is said to be $\mathfrak{B}$-Volterra with respect to a Boolean algebra $\mathfrak{B}$ of order projections of $E$ if the bands canonically corresponding to elements of $\mathfrak{B}$ are left fixed by $T$. A linearly ordered sequence $ξ$ in $\mathfrak{B}$ connecting $\textbf{0}$ to $\textbf{1}$ is called a forward filtration. A forward filtration can be to used to lift the action of the $\mathfrak{B}$-Volterra operator $T$ from the underlying Banach lattice $E$ to an action of a new norm continuous operator $…
A noncommutative Kunita-Watanabe-type representation theorem is established for the martingales of quasifree states of CCR algebras. To this end the basic theory of quasifree stochastic integrals is developed using the abstract Itô integral in symmetric Fock space, whose interaction with the operators of Tomita-Takesaki theory we describe. Our results extend earlier quasifree martingale representation theorems in two ways: the states are no longer assumed to be gauge-invariant, and the multiplicity space may now be infinite-dimensional. The former involves systematic exploitation of Araki's Du…
In this paper we obtain a representation as martingale transform operators for the rearrangement and shift operators introduced by T. Figiel. The martingale transforms and the underlying sigma algebras are obtained explicitly by combinatorial means. The known norm estimates for those operators are a direct consequence of our representation.
