Dyadic Maximal Function Research Articles

Boundedness of the dyadic maximal function on graded Lie groups

Abstract Let $1 \lt p\leq \infty$ and let $n\geq 2.$ It was proved independently by Calderón, Coifman and Weiss that the dyadic maximal function $$ \mathcal{M}^{d\sigma}_Df(x)=\sup_{j\in\mathbb{Z}}\left|\smallint\limits_{\mathbb{S}^{n-1}}f(x-2^jy)d\sigma(y)\right| \\[4pt] $$ is a bounded operator on $L^p(\mathbb{R}^n)$, where $d\sigma(y)$ is the surface measure on $\mathbb{S}^{n-1}.$ In this paper we prove an analogue of this result on arbitrary graded Lie groups. More precisely, to any finite Borel measure $d\sigma$ with compact support on a graded Lie group $G,$ we associate the corresponding dyadic maximal function $\mathcal{M}_D^{d\sigma}$ using the homogeneous structure of the group. Then, we prove a criterion in terms of the order (at zero and at infinity) of the group Fourier transform $\widehat{d\sigma}$ of $d\sigma$ with respect to a fixed Rockland operator $\mathcal{R}$ on G that assures the boundedness of $\mathcal{M}_D^{d\sigma}$ on $L^p(G)$ for all $1 \lt p\leq \infty.$

Zygmund type and flag type maximal functions, and sparse operators

We prove that the maximal functions associated with a Zygmund dilation dyadic structure in three-dimensional Euclidean space, and with the flag dyadic structure in two-dimensional Euclidean space, cannot be bounded by multiparameter sparse operators associated with the corresponding dyadic grid. We also obtain supplementary results about the absence of sparse domination for the strong dyadic maximal function.

Variation of the Dyadic Maximal Function

Abstract We prove that for the dyadic maximal operator $\textrm {M}$ and every locally integrable function $f\in L^1_{{\textrm {loc}}}(\mathbb R^d)$ with bounded variation, also $\textrm {M} f$ is locally integrable and $\mathop {\textrm {var}}\textrm {M} f\leq C_d\mathop {\textrm {var}} f$ for any dimension $d\geq 1$. It means that if $f\in L^1_{{\textrm {loc}}}(\mathbb R^d)$ is a function whose gradient is a finite measure then so is $\nabla \textrm {M} f$ and $\|\nabla \textrm {M} f\|_{L^1(\mathbb R^d)}\leq C_d\|\nabla f\|_{L^1(\mathbb R^d)}$. We also prove this for the local dyadic maximal operator.

Extensions of the John–Nirenberg theorem and applications

The John–Nirenberg theorem states that functions of bounded mean oscillation are exponentially integrable. In this article we give two extensions of this theorem. The first one relates the dyadic maximal function to the sharp maximal function of Fefferman–Stein, while the second one concerns local weighted mean oscillations, generalizing a result of Muckenhoupt and Wheeden. Applications to the context of generalized Poincaré type inequalities and to the context of the C p C_p class of weights are given. Extensions to the case of polynomial BMO \operatorname {BMO} type spaces are also given.

Sharp integral inequalities for the dyadic maximal operator and applications

We prove a sharp integral inequality for the dyadic maximal function of $$\phi \in L^p$$ . This inequality connects certain quantities related to integrals of $$\phi $$ and the dyadic maximal function of $$\phi $$ , under the hypothesis that the variables $$\int _X\phi \, \mathrm {d}\mu =f,$$ $$\int _X\phi ^q\, \mathrm {d}\mu =A,$$ $$1<q<p,$$ are given, where $$0<f^q \le A.$$ Additionally, it contains a parameter $$\beta >0$$ which when it attains a certain value depending only on f, A, q, the inequality becomes sharp. Using this inequality we give an alternative proof of the evaluation of the Bellman function related to the dyadic maximal operator of two integral variables.

Weighted square function inequalities

For an integrable function $f$ on $[0,1)^d$, let $S(f)$ and $Mf$ denote the corresponding dyadic square function and the dyadic maximal function of $f$, respectively. The paper contains the proofs of the following statements. (i) If $w$ is a dyadic $A_1$ weight on $[0,1)^d$, then $$ ||S(f)||_{L^1(w)}\leq \sqrt{5}[w]_{A_1}^{1/2}||Mf||_{L^1(w)}. $$ The exponent $1/2$ is shown to be the best possible. (ii) For any $p>1$, there are no constants $c_p$, $\alpha_p$ depending only on $p$ such that for all dyadic $A_p$ weights $w$ on $[0,1)^d$, $$ ||S(f)||_{L^1(w)}\leq c_p[w]_{A_p}^{\alpha_p}||Mf||_{L^1(w)}. $$

Weak-type inequalities for maximal operators acting on Lorentz spaces

We prove sharp a priori estimates for the distribution function of the dyadic maximal function Mφ, when φ belongs to the Lorentz space L, 1 < p < ∞, 1 ≤ q < ∞. The approach rests on a precise evaluation of the Bellman function corresponding to the problem. As an application, we establish refined weak-type estimates for the dyadic maximal operator: for p, q as above and r ∈ [1, p], we determine the best constant Cp,q,r such that for any φ ∈ L p,q , ||Mφ||r,∞ ≤ Cp,q,r||φ||p,q .

On Muckenhoupt–Wheeden Conjecture

Let M denote the dyadic Maximal Function. We show that there is a weight w, and Haar multiplier T for which the following weak-type inequality fails sup t > 0 t w ( { x ∈ R : | T f ( x ) | > t } ) ⩽ C ∫ R | f | M w ( x ) d x . (With T replaced by M, this is a well-known fact.) This shows that a dyadic version of the so-called Muckenhoupt–Wheeden Conjecture is false. This accomplished by using current techniques in weighted inequalities to show that a particular L 2 consequence of the inequality above does not hold.

Extremal problems related to maximal dyadic-like operators

We obtain sharp estimates for the localized distribution function of the dyadic maximal function M d ϕ , when ϕ belongs to L p , ∞ . Using this we obtain sharp estimates for the quasi-norm of M d ϕ in L p , ∞ given the localized L 1 -norm and certain weak L p -conditions.

Radial maximal function characterizations of Hardy spaces on RD-spaces and their applications

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a dimension n .F or α ∈ (0, ∞ )d enote byH p(X), H p d(X), and H ∗ ,p (X) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calderon reproducing formula, it is shown that all these Hardy spaces coincide with L p (X )w henp ∈ (1, ∞) and with each other when p ∈ (n/(n +1 ) ,1). An atomic characterization for H ∗ ,p (X )w ithp ∈ (n/(n +1 ),1) is also established; moreover, in the range p ∈ (n/(n +1 ), 1), it is proved that the spaceH ∗ ,p (X), the Hardy space H p (X) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman and Weiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X )t o some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩(1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.

Maximal function characterizations of Hardy spaces on RD-spaces and their applications

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a “dimension” n. For α ∈ (0, ∞) denote by H (X), H d (X), and H *,p (X) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calderon reproducing formula, it is shown that all these Hardy spaces coincide with L p (X) when p ∈ (1,∞] and with each other when p ∈ (n/(n + 1), 1]. An atomic characterization for H ∗,p (X) with p ∈ (n/(n + 1), 1] is also established; moreover, in the range p ∈ (n/(n + 1),1], it is proved that the space H *,p (X), the Hardy space H p (X) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman andWeiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X) to some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩[1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.

On weak type inequalities for dyadic maximal functions

We obtain sharp estimates for the localized distribution function of the dyadic maximal function M ϕ d , given the local L 1 norms of ϕ and of G ○ ϕ where G is a convex increasing function such that G ( x ) / x → + ∞ as x → + ∞ . Using this we obtain sharp refined weak type estimates for the dyadic maximal operator.

Comparison of Hardy–Littlewood and dyadic maximal functions on spaces of homogeneous type

We obtain a comparison of the level sets for two maximal functions on a space of homogeneous type: the Hardy–Littlewood maximal function of mean values over balls and the dyadic maximal function of mean values over the dyadic sets introduced by M. Christ in [M. Christ, A T ( b ) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990) 601–628]. As applications to the theory of A p weights on this setting, we compare the standard and the dyadic Muckenhoupt classes and we give an alternative proof of reverse Hölder type inequalities.