• Title/Summary/Keyword: Hardy Littlewood maximal function

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ON THE CONTINUITY OF THE HARDY-LITTLEWOOD MAXIMAL FUNCTION

  • Park, Young Ja
    • Journal of the Chungcheong Mathematical Society
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    • v.31 no.1
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    • pp.43-46
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    • 2018
  • It is concerned with the continuity of the Hardy-Little wood maximal function between the classical Lebesgue spaces or the Orlicz spaces. A new approach to the continuity of the Hardy-Littlewood maximal function is presented through the observation that the continuity is closely related to the existence of solutions for a certain type of first order ordinary differential equations. It is applied to verify the continuity of the Hardy-Littlewood maximal function from $L^p({\mathbb{R}}^n)$ to $L^q({\mathbb{R}}^n)$ for 1 ${\leq}$ q < p < ${\infty}$.

WEAK HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENTS AND APPLICATIONS

  • Souad Ben Seghier
    • Journal of the Korean Mathematical Society
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    • v.60 no.1
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    • pp.33-69
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    • 2023
  • Let α ∈ (0, ∞), p ∈ (0, ∞) and q(·) : ℝn → [1, ∞) satisfy the globally log-Hölder continuity condition. We introduce the weak Herz-type Hardy spaces with variable exponents via the radial grand maximal operator and to give its maximal characterizations, we establish a version of the boundedness of the Hardy-Littlewood maximal operator M and the Fefferman-Stein vector-valued inequality on the weak Herz spaces with variable exponents. We also obtain the atomic and the molecular decompositions of the weak Herz-type Hardy spaces with variable exponents. As an application of the atomic decomposition we provide various equivalent characterizations of our spaces by means of the Lusin area function, the Littlewood-Paley g-function and the Littlewood-Paley $g^*_{\lambda}$-function.

WAVELET CHARACTERIZATIONS OF VARIABLE HARDY-LORENTZ SPACES

  • Yao He
    • Bulletin of the Korean Mathematical Society
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    • v.61 no.2
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    • pp.489-509
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    • 2024
  • In this paper, let q ∈ (0, 1]. We establish the boundedness of intrinsic g-functions from the Hardy-Lorentz spaces with variable exponent Hp(·),q(ℝn) into Lorentz spaces with variable exponent Lp(·),q(ℝn). Then, for any q ∈ (0, 1], via some estimates on a discrete Littlewood-Paley g-function and a Peetre-type maximal function, we obtain several equivalent characterizations of Hp(·),q(ℝn) in terms of wavelets.

FOURIER TRANSFORM OF ANISOTROPIC MIXED-NORM HARDY SPACES WITH APPLICATIONS TO HARDY-LITTLEWOOD INEQUALITIES

  • Liu, Jun;Lu, Yaqian;Zhang, Mingdong
    • Journal of the Korean Mathematical Society
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    • v.59 no.5
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    • pp.927-944
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    • 2022
  • Let $\vec{p}{\in}(0,\;1]^n$ be an n-dimensional vector and A a dilation. Let $H^{\vec{p}}_A(\mathbb{R}^n)$ denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of $H^{\vec{p}}_A(\mathbb{R}^n)$ and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of $f{\in}H^{\vec{p}}_A(\mathbb{R}^n)$ coincides with a continuous function F on ℝn in the sense of tempered distributions. Moreover, the function F can be controlled pointwisely by the product of the Hardy space norm of f and a step function with respect to the transpose matrix of A. As applications, the authors obtain a higher order of convergence for the function F at the origin, and an analogue of Hardy-Littlewood inequalities in the present setting of $H^{\vec{p}}_A(\mathbb{R}^n)$.

COMMUTATORS OF THE MAXIMAL FUNCTIONS ON BANACH FUNCTION SPACES

  • Mujdat Agcayazi;Pu Zhang
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.5
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    • pp.1391-1408
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    • 2023
  • Let M and M# be Hardy-Littlewood maximal operator and sharp maximal operator, respectively. In this article, we present necessary and sufficient conditions for the boundedness properties for commutator operators [M, b] and [M#, b] in a general context of Banach function spaces when b belongs to BMO(?n) spaces. Some applications of the results on weighted Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces and Musielak-Orlicz spaces are also given.

Lq-ESTIMATES OF MAXIMAL OPERATORS ON THE p-ADIC VECTOR SPACE

  • Kim, Yong-Cheol
    • Communications of the Korean Mathematical Society
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    • v.24 no.3
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    • pp.367-379
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    • 2009
  • For a prime number p, let $\mathbb{Q}_p$ denote the p-adic field and let $\mathbb{Q}_p^d$ denote a vector space over $\mathbb{Q}_p$ which consists of all d-tuples of $\mathbb{Q}_p$. For a function f ${\in}L_{loc}^1(\mathbb{Q}_p^d)$, we define the Hardy-Littlewood maximal function of f on $\mathbb{Q}_p^d$ by $$M_pf(x)=sup\frac{1}{\gamma{\in}\mathbb{Z}|B_{\gamma}(x)|H}{\int}_{B\gamma(x)}|f(y)|dy$$, where |E|$_H$ denotes the Haar measure of a measurable subset E of $\mathbb{Q}_p^d$ and $B_\gamma(x)$ denotes the p-adic ball with center x ${\in}\;\mathbb{Q}_p^d$ and radius $p^\gamma$. If 1 < q $\leq\;\infty$, then we prove that $M_p$ is a bounded operator of $L^q(\mathbb{Q}_p^d)$ into $L^q(\mathbb{Q}_p^d)$; moreover, $M_p$ is of weak type (1, 1) on $L^1(\mathbb{Q}_p^d)$, that is to say, |{$x{\in}\mathbb{Q}_p^d:|M_pf(x)|$>$\lambda$}|$_H{\leq}\frac{p^d}{\lambda}||f||_{L^1(\mathbb{Q}_p^d)},\;\lambda$ > 0 for any f ${\in}L^1(\mathbb{Q}_p^d)$.

HARDY-LITTLEWOOD MAXIMAL FUNCTIONS IN ORLICZ SPACES

  • Yoo, Yoon-Jae
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.2
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    • pp.225-231
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    • 1999
  • Let Mf(x) be the Hardy-Littlewood maximal function on $\mathbb{R}^n$. Let $\Phi$ and $\Psi$ be functions satisfying $\Phi$(t) = ${\int^t}_0$a(s)ds and $\Psi(t)$ = ${\int^t}_0$b(s)ds, where a(s) and b(s) are positive continuous such that ${\int^\infty}_0\frac{a(s)}{s}ds$ = $\infty$ and b(s) is quasi-increasing. We show that if there exists a constant $c_1$ so that ${\int^s}_0\frac{a(t)}{t}dt\;c_1b(c_1s)$ for all $s\geq0$, then there exists a constant $c_1$ such that(0.1) $\int_{\mathbb{R^{n}}$ $\Phi(Mf(x))dx\;\leq\;c_2$ $\int_\mathbb{R^{n}}$$\Psi(c_2\midf(x)\mid)dx$ for all $f\epsilonL^1(R^n_$. Conversely, if there exists a constant $c_2$ satisfying the condition (0.1), then there exists a constant $c_1$ so that ${\int^s}_\delta\frac{a(t)}{t}dt=;\leq\;c_1b(c_1s$ for all $\delta$ > 0 and $s\geq\delta$.

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SHARP FUNCTION AND WEIGHTED $L^p$ ESTIMATE FOR PSEUDO DIFFERENTIAL OPERATORS WITH REDUCED SYMBOLS

  • Kim, H.S.;Shin, S.S.
    • East Asian mathematical journal
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    • v.6 no.2
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    • pp.133-144
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    • 1990
  • In 1982, N. Miller [5] showed a weighted $L^p$ boundedness theorem for pseudo differential operators with symbols $S^0_{1.0}$. In this paper, we shall prove the pointwise estimates, in terms of the Fefferman, Stein sharp function and Hardy Littlewood maximal function, for pseudo differential operators with reduced symbols and show a weighted $L^p$-boundedness for pseudo differential operators with symbol in $S^m_{\rho,\delta}$, 0{$\leq}{\delta}{\leq}{\rho}{\leq}1$, ${\delta}{\neq}1$, ${\rho}{\neq}0$ and $m=(n+1)(\rho-1)$.

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[ $L^p$ ] NORM INEQUALITIES FOR AREA FUNCTIONS WITH APPROACH REGIONS

  • Suh, Choon-Serk
    • East Asian mathematical journal
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    • v.21 no.1
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    • pp.41-48
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    • 2005
  • In this paper we first introduce a space of homogeneous type X, and then consider a kind of generalized upper half-space $X{\times}(0,\;\infty)$. We are mainly considered with inequalities for the $L^p$ norms of area functions associated with approach regions in $X{\times}(0,\;\infty)$.

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SINGULAR AND MARCINKIEWICZ INTEGRAL OPERATORS ON PRODUCT DOMAINS

  • Badriya Al-Azri;Ahmad Al-Salman
    • Communications of the Korean Mathematical Society
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    • v.38 no.2
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    • pp.401-430
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    • 2023
  • In this paper, we prove Lp estimates of a class of singular integral operators on product domains along surfaces defined by mappings that are more general than polynomials and convex functions. We assume that the kernels are in L(log L)2 (𝕊n-1 × 𝕊m-1). Furthermore, we prove Lp estimates of the related class of Marcinkiewicz integral operators. Our results extend as well as improve previously known results.