Introduction to Maximal Functions

What is/are Maximal Functions?

Maximal Functions - Assuming that the powered Hardy--Littlewood maximal operator satisfies some Fefferman--Stein vector-valued maximal inequality on $X$ as well as it is bounded on both the weak ball quasi-Banach function space $WX$ and the associated space, the authors then establish several real-variable characterizations of $WH_X({\mathbb R}^n)$, respectively, in terms of various maximal functions, atoms and molecules. ^[1] Our results imply new $L^2$-estimates for Kakeya-type maximal functions with tubes pointing along polynomial directions. ^[2] In this paper, for general plane curves γ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus L p ( R 2 ) -boundedness of the Hilbert transforms H U , γ ∞ along the variable plane curves ( t , U ( x 1 , x 2 ) γ ( t ) ) and the L p ( R 2 ) -boundedness of the corresponding maximal functions M U , γ ∞ , where p > 2 and U is a measurable function. ^[3] Explicit upper and lower bounds for the two-dimensional discrete Hardy–Littlewood maximal functions from $$\ell ^{1}({\mathbb {Z}}^{2}) $$ to the space of functions of bounded variation $$\mathrm {BV}({\mathbb {Z}}^{2}) $$ are obtained. ^[4] The main ingredients are sharp estimates of noncommutative Kakeya maximal functions and geometric estimates in the plane. ^[5] Depending on the class, estimates of the Fourier transform of charges are obtained in terms of Randol maximal functions. ^[6] We establish the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. ^[7] In this paper, we study Sobolev type inequalities for fractional maximal functions in the half space. ^[8] The Hardy-Morrey spaces related to Laplace-Bessel differential equations are introduced in terms of maximal functions. ^[9] This is a survey article about recent developments in dimension-free estimates for maximal functions corresponding to the Hardy--Littlewood averaging operators associated with convex symmetric bodies in $\mathbb R^d$ and $\mathbb Z^d$. ^[10] Our approach involves a duality argument and maximal functions of $$L^p({\mathbb {S}}_n)$$ functions on the unit ball of $${\mathbb {C}}^n$$. ^[11] In this article, we discuss the connections of function of bounded mean oscillations with weight functions, sharp maximal functions and Carleson measure. ^[12] In this paper, we establish $$L^{p}$$Lp estimates for certain class of maximal functions on product domains with rough kernels in $${L^{q}}(\mathbf S ^{n-1}\times \mathbf S ^{m-1})\,(n,m\ge 2). ^[13] A characterization of -maximal functions is also shown. ^[14] In this paper, we study the Hardy type space H ℒ p ⁢ ( ℝ n ) {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} by means of local maximal functions associated with the heat semigroup e - t ⁢ ℒ {e^{-t\mathcal{L}}} generated by - ℒ {-\mathcal{L}} , where ℒ = - Δ + μ {\mathcal{L}=-\Delta+\mu} is the generalized Schrödinger operator in ℝ n {{\mathbb{R}^{n}}} ( n ≥ 3 {n\geq 3} ) and μ ≢ 0 {\mu\not\equiv 0} is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. ^[15] In this work, the boundedness of the spherical maximal function, the mapping properties of the fractional spherical maximal functions, the variation and oscillation inequalities of Riesz transforms on Herz spaces have been established. ^[16] Moreover, it is well-known that for the particular case $\Phi(t)=t(1+\log^+t)^m$ with $m\in\mathbb{N}$ these maximal functions control, in some sense, certain operatos in Harmonic Analysis. ^[17] Moreover, the L2$L^{2}$-bounds of the maximal functions related to the above integrals are also established. ^[18] Given $1\leq qrough singular integrals are obtained. ^[19] We build a connection to cut-generating functions in the Gomory–Johnson and related models, complete the characterization of maximal functions, and prove analogues of the Gomory–Johnson 2-slope theorem and the Basu–Hildebrand–Molinaro approximation theorem. ^[20] And the main ingredients are the action of the cut-off fractional maximal functions and some local interior and boundary comparison estimates developed in previous works \cite{55QH4, MPT2018, MPT2019} and references therein. ^[21] Furthermore, we calculate the rate of convergence by means of the modulus of continuity, Lipschitz type maximal functions, Peetre’s K-functional and the second order modulus of continuity. ^[22] We establish Korovkin type result in weighted spaces and also study approximation properties with the help of modulus of continuity of order one, Lipschitz type maximal functions, and Peetre’s K-functional. ^[23] In this paper, the weighted grand Lebesgue spaces with mixed-norms are introduced and boundedness criteria in these spaces of strong maximal functions and Riesz transforms are presented. ^[24] We discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. ^[25] Our positive results are improvements over bounds for the corresponding full maximal functions which were initially studied by Magyar. ^[26] By using different maximal functions and Littlewood-Paley g function on distinct variables, in this paper, some characterizations for functions in the product Hardy space $$H_{{L_1},{L_2}}^1$$(ℝn × ℝm) associated to operators L1 and L2 are obtained. ^[27] Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. ^[28] In this work, we study the \(L^p\) estimates for a certain class of rough maximal functions with mixed homogeneity associated with the surfaces of revolution. ^[29] Maximal functions are among the most important operators in harmonic analysis and are some of those that are most studied. ^[30] Weighted grand Lebesgue spaces with mixed norms are introduced, and criteria for the boundedness of strong maximal functions and Riesz transforms in these spaces are given. ^[31] In this note we establish certain weighted estimates for a class of maximal functions with rough kernels along “polynomial curves” on. ^[32]

Type Maximal Functions

Our results imply new $L^2$-estimates for Kakeya-type maximal functions with tubes pointing along polynomial directions. ^[1] Furthermore, we calculate the rate of convergence by means of the modulus of continuity, Lipschitz type maximal functions, Peetre’s K-functional and the second order modulus of continuity. ^[2] We establish Korovkin type result in weighted spaces and also study approximation properties with the help of modulus of continuity of order one, Lipschitz type maximal functions, and Peetre’s K-functional. ^[3]

Spherical Maximal Functions

In this work, the boundedness of the spherical maximal function, the mapping properties of the fractional spherical maximal functions, the variation and oscillation inequalities of Riesz transforms on Herz spaces have been established. ^[1] We discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. ^[2]

Fractional Maximal Functions

In this paper, we study Sobolev type inequalities for fractional maximal functions in the half space. ^[1] And the main ingredients are the action of the cut-off fractional maximal functions and some local interior and boundary comparison estimates developed in previous works \cite{55QH4, MPT2018, MPT2019} and references therein. ^[2]

Strong Maximal Functions

In this paper, the weighted grand Lebesgue spaces with mixed-norms are introduced and boundedness criteria in these spaces of strong maximal functions and Riesz transforms are presented. ^[1] Weighted grand Lebesgue spaces with mixed norms are introduced, and criteria for the boundedness of strong maximal functions and Riesz transforms in these spaces are given. ^[2]