Strong Maximal Operator Research Articles

On the regularity and continuity of the multilinear fractional strong maximal operators

AbstractLet , , and denote the family of all rectangles in with sides parallel to the axes. The main purpose of this paper is to study the regularity and continuity properties of the multilinear fractional strong maximal operators associated with rectangles which covers the multilinear strong maximal operator , corresponding to the critical index . Boundedness and continuity for the map are proved, provided that , and . As an application of the above boundedness result, the q‐quasicontinuity for is established. In addition, we also prove that a vector function with each implies almost everywhere approximately differentiability of when , which is stronger than the best result known for .

Weighted Lorentz spaces: Sharp mixed Ap − A∞ estimate for maximal functions

We prove the sharp mixed Ap−A∞ weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely‖M‖Lp,q(w)≲p,q,n[w]Ap1p[σ]A∞1min⁡(p,q), where σ=w11−p. Our method is rearrangement free and can also be used to bound similar operators, even in the two-weight setting. We use this to also obtain new quantitative bounds for the strong maximal operator and for M in a dual setting.

Regularity of local multilinear fractional maximal and strong maximal operators

Regularity of local multilinear fractional maximal and strong maximal operators

Sobolev inequality with non-uniformly degenerating gradient

In this paper we prove the following weighted Sobolev inequality in a bounded domain Ω ⊂ R n , n ≥ 1 , of a homogeneous space ( R n , ρ , w d x ) , under suitable compatibility conditions on the positive weight functions ( v , w , ω 1 , ω 2 , … , ω n ) and on the quasi-metric ρ , ( ∫ Ω | f | q v w d z ) 1 q ≤ C ∑ i = 1 N ( ∫ Ω | f z i | p ω i M S w d z ) 1 p , f ∈ L i p 0 ( Ω ¯ ) , where q ≥ p > 1 and M S denotes the strong maximal operator. Some corollaries on non-uniformly degenerating gradient inequalities are derived.

The limiting weak-type behaviors of the strong maximal operators

The limiting weak-type behaviors of the strong maximal operators

Extrapolation to product Herz spaces and some applications

Abstract This paper extends the extrapolation theory to product Herz spaces. To prove the main result, we first investigate the dual space of the product Herz space, and then show the boundedness of the strong maximal operator on product Herz spaces. By using this extrapolation theory, we establish the John–Nirenberg inequality, the characterization of little bmo, the Fefferman–Stein vector-valued inequality, the boundedness of the bi-parameter singular integral operator, the strong fractional maximal operator, and the bi-parameter fractional integral operator on product Herz spaces. We also give a new characterization of little bmo via the boundedness of the commutators of some bi-parameter operators on product Herz spaces. Even in the one-parameter setting, some of our results are new.

Weighted Sobolev inequality in grand mixed norm Lebesgue spaces

We derive necessary and sufficient conditions on weights guaranteeing the one-weight Sobolev-type inequality for strong fractional maximal and multiple fractional integral operators in grand mixed norm Lebesgue spaces. As a consequence, appropriate statements for fractional integral operators defined on regular curves and domains in $${{\mathbb {R}}}^n$$ are obtained.

Multilinear strong maximal operators on weighted mixed norm spaces

Multilinear strong maximal operators on weighted mixed norm spaces

Regularity and continuity of the multilinear strong maximal operators

Let m≥1, in this paper, our object of investigation is the regularity and continuity properties of the following multilinear strong maximal operatorMR(f→)(x)=supR∋xR∈R⁡∏i=1m1|R|∫R|fi(y)|dy, where x∈Rd and R denotes the family of all rectangles in Rd with sides parallel to the axes. When m=1, denote MR by MR. Then, MR coincides with the classical strong maximal function initially studied by Jessen, Marcinkiewicz and Zygmund. We showed that MR is bounded and continuous from the product Sobolev spaces W1,p1(Rd)×⋯×W1,pm(Rd) to W1,p(Rd), from the product Besov spaces Bsp1,q(Rd)×⋯×Bspm,q(Rd) to Bsp,q(Rd), from the product Triebel-Lizorkin spaces Fsp1,q(Rd)×⋯×Fspm,q(Rd) to Fsp,q(Rd). As a consequence, we further showed that MR is bounded and continuous from the product fractional Sobolev spaces to fractional Sobolev space. As an application, we obtain a weak type inequality for the Sobolev capacity, which can be used to prove the p-quasicontinuity of MR. In addition, we proved that MR(f→) is approximately differentiable a.e. when f→=(f1,⋯,fm) with each fj∈L1(Rd) being approximately differentiable a.e.

On the boundedness of multilinear fractional strong maximal operators with multiple weights

On the boundedness of multilinear fractional strong maximal operators with multiple weights

The Fefferman–Stein Type Inequalities for Strong and Directional Maximal Operators in the Plane

AbstractThe Fefferman–Stein type inequalities for strongmaximal operators and directional maximal operators are verified with an additional composition of the Hardy–Littlewood maximal operator in the plane.

Corrigendum to “On multilinear fractional strong maximal operator associated with rectangles and multiple weights” [Rev. Mat. Iberoam. 33 (2017), no. 2, 555–572

Corrigendum to “On multilinear fractional strong maximal operator associated with rectangles and multiple weights” [Rev. Mat. Iberoam. 33 (2017), no. 2, 555–572

On multilinear fractional strong maximal operator associated with rectangles and multiple weights

In this paper, we introduce the multilinear fractional strong maximal operator \mathcal{M}_{\mathcal{R},\alpha} and the corresponding multiple weights A_{({\vec{\hskip.5pt p}},q),\mathcal{R}} associated with rectangles. Under the dyadic reverse doubling condition, a necessary and sufficient condition for two-weight inequalities is given. As consequences, we first obtain a necessary and sufficient condition for one-weight inequalities. Then, we present a new proof for the weighted estimates of multilinear fractional integral operator and fractional maximal operator associated with cubes, which is quite different from and simpler than the proof that has been presented previously.

Strong maximal operator and singular integral operators in weighted Morrey spaces on product domains

AbstractWe introduce the Morrey spaces on product domains and extend the boundedness of strong maximal operator and singular integral operators on product domains to Morrey spaces.

Strong maximal operator on mixed-norm spaces

We establish the boundedness of the strong maximal operator on mixed-norm spaces by using extrapolation. In particular, we obtain a non-trivial boundedness result for the strong maximal operator in variable exponent analysis.

Boundedness of multi-parameter Fourier multiplier operators on Triebel–Lizorkin and Besov–Lipschitz spaces

The main purpose of this paper is three-fold. First, we prove that under the limited smoothness conditions, multi-parameter Fourier multiplier operators are bounded on multi-parameter Triebel–Lizorkin and Besov–Lipschitz spaces by the Littlewood–Paley decomposition and the strong maximal operator. Second, we offer a different and more direct method to deal with the boundedness instead of transforming Fourier multiplier operators into multi-parameter Calderón–Zygmund operators. Third, we also prove the boundedness of multi-parameter Fourier multiplier operators on weighted multi-parameter Triebel–Lizorkin and Besov–Lipschitz spaces when the Fourier multiplier is only assumed with limited smoothness.

Solyanik estimates in ergodic theory

Let $U_1, \ldots, U_n$ be a collection of commuting measure preserving transformations on a probability space $(\Omega, \Sigma, \mu)$. Associated with these measure preserving transformations is the ergodic strong maximal operator $\mathsf M ^\ast _{\mathsf S}$ given by \[ \mathsf M ^\ast _{\mathsf S} f(\omega) := \sup_{0 \in R \subset \mathbb{R}^n}\frac{1}{\#(R \cap \mathbb{Z}^n)}\sum_{(j_1, \ldots, j_n) \in R\cap \mathbb{Z}^n}\big|f(U_1^{j_1}\cdots U_n^{j_n}\omega)\big|, \] where the supremum is taken over all open rectangles in $\mathbb{R}^n$ containing the origin whose sides are parallel to the coordinate axes. For $0 0}}\frac{1}{\mu(E)}\mu(\{\omega \in \Omega : \mathsf M ^\ast _{\mathsf S} \chi_E (\omega) > \alpha\}). \] Motivated by previous work of A. A. Solyanik and the authors regarding Solyanik estimates for the geometric strong maximal operator in harmonic analysis, we show that the Solyanik estimate \[ \lim_{\alpha \rightarrow 1}\mathsf C ^\ast _{\mathsf S}(\alpha) = 1 \] holds, and that in particular we have \[\mathsf C ^\ast _{\mathsf S}(\alpha) - 1 \lesssim_n (1 - \frac{1}{\alpha})^{1/n}\] provided that $\alpha$ is sufficiently close to $1$. Solyanik estimates for centered and uncentered ergodic Hardy-Littlewood maximal operators associated with $U_1, \ldots, U_n$ are shown to hold as well. Further directions for research in the field of ergodic Solyanik estimates are also discussed.

Two-Weight Norm Estimates for Multilinear Fractional Integrals in Classical Lebesgue Spaces

Abstract We derive criteria governing two-weight estimates for multilinear fractional integrals and appropriate maximal functions. The two and one weight problems for multi(sub)linear strong fractional maximal operators are also studied; in particular, we derive necessary and sufficient conditions guaranteeing the trace type inequality for this operator. We also establish the Fefferman-Stein type inequality, and obtain one-weight criteria when a weight function is of product type. As a consequence, appropriate results for multilinear Riesz potential operator with product kernels follow.

Solyanik estimates and local Hölder continuity of halo functions of geometric maximal operators

Let B be a homothecy invariant basis consisting of convex sets in Rn, and define the associated geometric maximal operator MB byMBf(x):=supx∈R∈B⁡1|R|∫R|f| and the halo function ϕB(α) on (1,∞) byϕB(α):=supE⊂Rn:0<|E|<∞⁡1|E||{x∈Rn:MBχE(x)>1/α}|. It is shown that if ϕB(α) satisfies the Solyanik estimate ϕB(α)−1≤C(1−1α)p for α∈(1,∞) sufficiently close to 1 then ϕB lies in the Hölder class Cp(1,∞). As a consequence we obtain that the halo functions associated with the Hardy–Littlewood maximal operator and the strong maximal operator on Rn lie in the Hölder class C1/n(1,∞).

Weighted Solyanik estimates for the strong maximal function

Let $\mathsf M_{\mathsf S}$ denote the strong maximal operator on $\mathbb R^n$ and let $w$ be a non-negative, locally integrable function. For $\alpha\in(0,1)$ we define the weighted sharp Tauberian constant $\mathsf C_{\mathsf S}$ associated with $\mathsf M_{\mathsf S}$ by $$ \mathsf C_{\mathsf S} (\alpha):= \sup_{\substack {E\subset \mathbb R^n \\ 0<w(E)<+\infty}}\frac{1}{w(E)}w(\{x\in\mathbb R^n:\, \mathsf M_{\mathsf S}(\mathbf{1}_E)(x)>\alpha\}). $$ We show that $\lim_{\alpha\to 1^-} \mathsf C_{\mathsf S} (\alpha)=1$ if and only if $w\in A_\infty ^*$, that is if and only if $w$ is a strong Muckenhoupt weight. This is quantified by the estimate $\mathsf C_{\mathsf S}(\alpha)-1\lesssim_{n} (1-\alpha)^{(cn [w]_{A_\infty ^*})^{-1}}$ as $\alpha\to 1^-$, where $c>0$ is a numerical constant; this estimate is sharp in the sense that the exponent $1/(cn[w]_{A_\infty ^*})$ can not be improved in terms of $[w]_{A_\infty ^*}$. As corollaries, we obtain a sharp reverse H\"older inequality for strong Muckenhoupt weights in $\mathbb R^n$ as well as a quantitative imbedding of $A_\infty^*$ into $A_{p}^*$. We also consider the strong maximal operator on $\mathbb R^n$ associated with the weight $w$ and denoted by $\mathsf M_{\mathsf S} ^w$. In this case the corresponding sharp Tauberian constant $\mathsf C_{\mathsf S} ^w$ is defined by $$ \mathsf C_{\mathsf S} ^w \alpha) := \sup_{\substack {E\subset \mathbb R^n \\ 0<w(E)<+\infty}}\frac{1}{w(E)}w(\{x\in\mathbb R^n:\, \mathsf M_{\mathsf S} ^w (\mathbf{1}_E)(x)>\alpha\}).$$ We show that there exists some constant $c_{w,n}>0$ depending only on $w$ and the dimension $n$ such that $\mathsf C_{\mathsf S} ^w (\alpha)-1 \lesssim_{w,n} (1-\alpha)^{c_{w,n}}$ as $\alpha\to 1^-$ whenever $w\in A_\infty ^*$ is a strong Muckenhoupt weight.