Dyadic Martingale Research Articles

Dyadic BMO on the bidisk

We give several new characterizations of the dual of the dyadic Hardy space H^{1,d}(\mathbb{T}^2) , the so-called dyadic BMO space in two variables and denoted \mathrm{BMO}^{\mathit d}_{\mathrm{prod}} . These include characterizations in terms of Haar multipliers, in terms of the “ymmetrised paraproduct” \Lambda_b , in terms of the rectangular BMO norms of the iterated “weeps”, and in terms of nested commutators with dyadic martingale transforms. We further explore the connection between \mathrm{BMO}^{\mathit d}_{\mathrm{prod}} and John–Nirenberg type inequalities, and study a scale of rectangular BMO spaces.

Oscillation in ergodic theory: Higher dimensional results

In this paper we continue our investigations of square function inequalities. The results in [9] are primarily one dimensional, and here we extend all the results to multi-dimensional averages. Our basic tool is still a comparison of the ergodic averages with various dyadic (reversed) martingales, but the Fourier transform arguments are replaced by more geometric almost orthogonality arguments.

A version of Burkholder’s theorem for operator-weighted spaces

Let W W be an operator weight, i.e. a weight function taking values in the bounded linear operators on a Hilbert space H \mathcal {H} . We prove that if the dyadic martingale transforms are uniformly bounded on L R 2 ( W ) L^2_{\mathbb {R}}(W) for each dyadic grid in R \mathbb {R} , then the Hilbert transform is bounded on L R 2 ( W ) L^2_{\mathbb {R}}(W) as well, thus providing an analogue of Burkholder’s theorem for operator-weighted L 2 L^2 -spaces. We also give a short new proof of Burkholder’s theorem itself. Our proof is based on the decomposition of the Hilbert transform into “dyadic shifts”.

LOGARITHMIC GROWTH FOR MATRIX MARTINGALE TRANSFORMS

An example is given of an operator weight W that satisfies the dyadic operator Hunt–Muckenhoupt–Wheeden condition [Aopf ]d2 for which there exists a dyadic martingale transform on L2 (W) that is unbounded. The construction relates weighted boundedness to the boundedness of dyadic vector Hankel operators.

Convergence of Double Walsh-Fourier Series and Hardy Spaces

It is proved that the maximal operator of the Marczinkiewicz-Fejer meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space Hpto Lp (2/3<p<∞) and is of weak type (1,1). As a consequence we obtain that the Marczinkiewicz-Fejer means of a function f∈L1converge a.e. to the function in question. Moreover, we prove that these means are uniformly bounded on Hpwhenever 2/3<p<∞. Thus, in case f∈Hp, the Marczinkiewicz-Fejer means conv f in Hpnorm. The same results are proved for the conjugate means, too.

( C ,α) Summability of Walsh—Fourier Series

The one-dimensional dyadic martingale Hardy spaces Hp are introduced and it is proved that the maximal operator of the (C,α) means of a Walsh—Fourier series is bounded from Hp to Lp (1/(α + 1) < p < ∞) and is of weak type (L1,L1). As a consequence, we obtain the summability result due to Fine; more exactly, the (C,α) means of the Walsh—Fourier series of a function f ∈ L1 converge a.e. to the function in question. Moreover, we prove that the (C,α) means are uniformly bounded on Hp whenever 1/(α + 1) < p < ∞. We define the two-dimensional dyadic hybrid Hardy space H1♯ and verify that the maximal operator of the (C,α,β) means of a two-dimensional function is of weak type H1♯,L1). Consequence, the Walsh—Fourier series of every function f ∈ H1♯ is (C,α,β) summable to the function f.

The maximal (C, α, β) operator of two-parameter walsh-fourier series

The two-parameter dyadic martingale Hardy spacesH p are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from Hp to Lp (1/(α+1), 1/(β+1)<p<∞) and is of weak type (H 1 # , L1), where the Hardy space H 1 # is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f∈H 1 # converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on Hp whenever 1/(α+1), 1/(β+1)<p<∞. Thus in case f∈Hp, the (C, α, β) means converge to f in Hp norm. The same results are proved for the conjugate (C, α, β) means, too.

Oscillation in ergodic theory

In this paper we establish a variety of square function inequalities and study other operators which measure the oscillation of a sequence of ergodic averages. These results imply the pointwise ergodic theorem and give additional information such as control of the number of upcrossings of the ergodic averages. Related results for differentiation and for the connection between differentiation operators and the dyadic martingale are also established.

On the characterization of certain similarly ordered super-additive functionals

Functionals which behave (sub-, super-) additively on similarly ordered functions occur quite naturally in many contexts. In the present paper we characterize (super-) additive functionals which are defined on a family of functions with the Stone-property in terms of their naturally adjoined dyadic martingales. As corollaries we obtain essential generalizations of integral representations as derived by Schmeidler (1986) and discussed in a recent monograph of Denneberg (1994).

Cesàro Summability of Two-Parameter Walsh–Fourier Series

A new atomic decomposition of the two-parameter dyadic martingale Hardy spacesHpdefined by the quadratic variation is given. We introduceHp-quasi-local operators and prove that if a sublinear operatorVisHp-quasi-local and bounded fromL2toL2then it is also bounded fromHptoLp(0<p⩽1). By an interpolation theorem we get thatVis of weak type (H#1, L1) where the Hardy spaceH#1is defined by the hybrid maximal function. As an application it is shown that the maximal operator of the Cesàro means of a two-parameter martingale is bounded fromHptoLp(4/5<p⩽∞) and is of weak type (H#1, L1). So we obtain that the Cesàro means of a functionf∈H#1converge a.e. to the function in question. Finally, it is verified that if the supremum is taken over all two-powers, only, then the maximal operator of the Cesàro means is bounded fromHptoLpfor every 2/3<p⩽∞.

Some maximal lnequalities with respect to two-parameter dyadic derivative and cesàro summability

We consider two-dimensional convolution operators with general kernel functions and give a sufficient condition for the two-parameter maximal operator to be bounded from the dyadic martingale Hardy space Hp to Lp. Especially, the boundedness of the maximal operator of the twoparameter dyadic derivative of the dyadic integral function and the maximal operator of the two-parameter Cesaro means are verified. As a consequence we obtain that every function f ∊ L log+ L[O, 1)2 is Cesho summable and the dyadic integral of it is dyadically differentiable.

A Martingale Inequality Related to Exponential Square Integrability

We present an inequality for dyadic martingales (together with its continuous analog for functions on ${\mathbb {R}^n}$) which is shown to be equivalent to a result of Chang-Wilson-Wolff on exponential square integrability. The analog of this weighted inequality for double dyadic martingales is also proven. Finally, we discuss a possible connection between these inequalities and a theorem of Garnett.

A martingale inequality related to exponential square integrability

We present an inequality for dyadic martingales (together with its continuous analog for functions on R n {\mathbb {R}^n} ) which is shown to be equivalent to a result of Chang-Wilson-Wolff on exponential square integrability. The analog of this weighted inequality for double dyadic martingales is also proven. Finally, we discuss a possible connection between these inequalities and a theorem of Garnett.

Dyadic martingale Hardy and VMO spaces on the plane

Dyadic martingale Hardy and VMO spaces on the plane

A Sharp Estimate for Dyadic Martingales with Multiple Indices

A Sharp Estimate for Dyadic Martingales with Multiple Indices

A sharp estimate for dyadic martingales with multiple indices

A variant of Doob’s maximal inequality is obtained for dyadic martingales with multiple indices. The inequality furnishes a precise estimate of the L p {L^p} norm of the maximal function in terms of the L p {L^p} norms of the jumps, p ≥ 2 p \geq 2 .

Bounded double square functions

We extend some recent work of S. Y. Chang, J. M. Wilson and T. Wolff to the bidisc. For f∈L loc 1 (R 2 ), we determine the sharp order of local integrability obtained when the square function of f is in L ∞ . The Calderón-Torchinsky decomposition reduces the problem to the case of double dyadic martingales. Here we prove a vector-valued form of an inequality for dyadic martingales that yields the sharp dependence on p of C p in ∥f∥ p ≤C p ∥Sf∥ p .