Article
Mathematics, Applied
Long Huang, Ferenc Weisz, Dachun Yang, Wen Yuan
Summary: In this paper, the authors establish the boundedness of the mixed centered Hardy-Littlewood maximal operator and prove the almost everywhere convergence of the theta-mean of f to f over the diagonal in the mixed-norm Lebesgue space under certain assumptions. They also extend the convergence to the unrestricted case and show that it holds at (p) over bar-Lebesgue points as well. Furthermore, the authors demonstrate that the Herz spaces to which the Fourier transform of theta belongs are the best choice in these results.
ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Samia Bashir, Babar Sultan, Amjad Hussain, Aziz Khan, Thabet Abdeljawad
Summary: This paper proves that the fractional Hardy-type operators of variable order are bounded mappings from the grand Herz spaces Ka(& BULL;),u),& theta; p(& BULL;) (Rn) with variable exponent into the weighted space Ka(& BULL;),u),& theta; & rho;,q(& BULL;) (Rn), where the weight function & rho; = (1 +|z1|)-& lambda; and 1 q(z)= when p(z) is not necessarily constant at infinity.
Article
Mathematics
Yichun Zhao, Jiang Zhou
Summary: In this article, we introduce anisotropic mixed-norm Herz spaces and investigate their properties. We also establish the Rubio de Francia extrapolation theory to resolve the boundedness problems of certain operators on these spaces. Additionally, we gain insights into the Littlewood-Paley characterizations of anisotropic mixed-norm Herz spaces. Moreover, the introduction of anisotropic mixed-norm Herz-Hardy spaces allows for atomic and molecular decompositions, as well as the boundedness of classical Calderon-Zygmund operators.
Article
Mathematics
Dalimil Pesa
Summary: This paper introduces and studies the concept of Wiener-Luxemburg amalgam spaces, a modification of classical Wiener amalgam spaces, aiming to address their shortcomings in rearrangement-invariant Banach function spaces. The study includes properties, normability, embeddings, and associate spaces, as well as a generalization of the concept for quasi-Banach function spaces. The application of this theory resolves the question of the Hardy-Littlewood-Polya principle for all r.i. quasi-Banach function spaces.
JOURNAL OF FUNCTIONAL ANALYSIS
(2022)
Article
Mathematics, Applied
Babar Sultan, Mehvish Sultan, Qian-Qian Zhang, Nabil Mlaiki
Summary: In this paper, the concept of grand variable weighted Herz spaces K & alpha;(& BULL;),),& theta; q(& BULL;) (& tau;) is introduced, where & alpha; is also a variable. The main purpose of this paper is to prove the boundedness of Hardy operators on grand variable weighted Herz spaces.
Article
Mathematics
Ferenc Weisz
Summary: We prove that the triangular Cesàro means of two-dimensional functions f in L1(T2) converge to f at each strong (1, α)-Lebesgue point. Moreover, if f is in Lp(T2) with 1 < p < α, then the Cesa`ro means converge to f at each (p, α)-Lebesgue point. This generalizes the well-known classical Lebesgue's theorem.
MATHEMATICAL INEQUALITIES & APPLICATIONS
(2022)
Article
Mathematics, Applied
Babar Sultan, Mehvish Sultan, Ilyas Khan
Summary: The higher order commutators of the variable order fractional integral operator are shown to be bounded from the Grand variable Herz spaces to the weighted space.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Claudia Capone
Summary: The boundedness of Calderon singular operator and Hardy-Littlewood maximal operator in generalized weighted Grand Lebesgue spaces is proved when the subset of R is a bounded interval.
RICERCHE DI MATEMATICA
(2022)
Article
Mathematics
Divyang G. Bhimani, Saikatul Haque
Summary: In this study, norm inflation (NI) is established for the Benjamin-Bona-Mahony (BBM) equation with general initial data in Fourier and Wiener amalgam spaces with negative regularity. This result strengthens known NI results at zero initial data and aligns with the local well-posedness result in modulation spaces.
Article
Mathematics, Applied
Yueping Zhu, Yan Tang, Lixin Jiang
Summary: This paper introduces weighted Morrey-Herz spaces with variable exponents and proves the boundedness of multilinear Calderon-Zygmund singular operators on these spaces.
Article
Mathematics, Applied
Humberto Rafeiro, Stefan Samko
Summary: This study focuses on the embeddings of Morrey type spaces and global Morrey type spaces into weighted Lebesgue spaces, aiming to better understand the behavior of functions locally and at infinity. It is proved under certain conditions that the local Morrey type space can be embedded into the weighted Lebesgue space, with some limitations when q > p. Other embeddings and inverse embeddings into Stummel spaces are also explored, as well as the relationships between Herz and Morrey type spaces for obtaining similar embeddings for Herz spaces.
BANACH JOURNAL OF MATHEMATICAL ANALYSIS
(2021)
Article
Mathematics
Kwok-Pun Ho
Summary: This paper establishes the mapping properties of sublinear operators on Herz-Hardy spaces with variable exponents. These mapping properties are obtained by extending the extrapolation theory to Herz-Hardy spaces with variable exponents. In particular, the mapping properties of multiplier operators, Littlewood-Paley functions, and parametric Marcinkiewicz integrals on Herz-Hardy spaces with variable exponents are studied.
MATHEMATISCHE NACHRICHTEN
(2022)
Article
Mathematics, Applied
Hua Zhu
Summary: In this article, the denseness of the Schwartz class in the modulation spaces with variable smoothness and integrability is proven. The dual spaces of such modulation spaces are also studied.
JOURNAL OF FUNCTION SPACES
(2022)
Article
Mathematics, Applied
Jing Zhou, Fayou Zhao
Summary: The strong type boundedness of the fractional Hardy-Littlewood maximal operator from weighted Morrey spaces to certain L-q spaces is proven, and the weak type estimate is obtained under certain conditions.
ANALYSIS AND MATHEMATICAL PHYSICS
(2022)
Article
Mathematics
Rovshan A. Bandaliyev, Kamala H. Safarova
Summary: This paper proves the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces L-p)(0, 1) (0 < p <= 1), and similar results for the Hardy operator in weighted Lebesgue spaces. It also demonstrates that the grand Lebesgue space L-p)(0, 1) is a quasi-Banach function space, and establishes necessary and sufficient conditions for the boundedness of some integral operator in weighted quasi-Banach Lebesgue spaces.
LINEAR & MULTILINEAR ALGEBRA
(2022)
Article
Mathematics
Kristof Szarvas, Ferenc Weisz
Summary: In this paper, martingale Hardy spaces defined with mixed Lp ->-norm are considered. Five mixed martingale Hardy spaces are investigated, and several results are proved, including atomic decompositions, Doob's inequality, boundedness, martingale inequalities, and the generalization of the Burkholder-Davis-Gundy inequality.
JOURNAL OF GEOMETRIC ANALYSIS
(2021)
Article
Mathematics, Applied
Ferenc Weisz
Summary: The paper discusses the properties of a measurable function p(.) defined on a probability space and its applications in inequalities, proving the validity of certain inequalities under specific conditions.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Yong Jiao, Ferenc Weisz, Guangheng Xie, Dachun Yang
Summary: In this article, the authors introduce five martingale Musielak-Orlicz-Lorentz Hardy spaces and prove that these new spaces have important features such as atomic characterizations, the boundedness of sigma-sublinear operators, and martingale inequalities. The authors also demonstrate the fundamental properties of the Musielak-Orlicz-Lorentz space, including completeness, convergence, real interpolations, and the Fefferman-Stein vector-valued inequality for the Doob maximal operator.
JOURNAL OF GEOMETRIC ANALYSIS
(2021)
Article
Mathematics, Applied
Ferenc Weisz
Summary: A new type of dyadic maximal operator is introduced, and it is proved to be bounded under certain conditions. The space generated by this operator’s norms is equivalent to the Hardy space, including some existing maximal operators.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2021)
Article
Mathematics, Applied
Long Long, Ferenc Weisz, Guangheng Xie
Summary: The authors in this article prove that the real interpolation spaces between martingale Orlicz Hardy spaces and martingale BMO spaces are martingale Orlicz-Lorentz Hardy spaces. They also establish the characterizations of martingale Orlicz Hardy spaces using sharp maximal functions.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
L. -E. Persson, F. Schipp, G. Tephnadze, F. Weisz
Summary: In this paper, we discuss and prove an analogy of the Carleson-Hunt theorem with respect to Vilenkin systems. We use the theory of martingales to provide a new and shorter proof of the almost everywhere convergence of Vilenkin-Fourier series for p > 1, in case the Vilenkin system is bounded. Moreover, we also prove the sharpness by stating an analogy of the Kolmogorov theorem for p = 1 and construct a function in L-1(G(m)) such that the partial sums with respect to Vilenkin systems diverge everywhere.
JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Ferenc Weisz, Guangheng Xie, Dachun Yang
Summary: This article establishes the atomic characterization of martingale Musielak-Orlicz Lorentz Hardy spaces under the assumption that the Doob maximal operator is bounded on Musielak-Orlicz spaces. Using atomic characterizations, the authors clarify the relations among five martingale Musielak-Orlicz Lorentz Hardy spaces and construct the generalized martingale BMO type spaces. As applications, the authors further investigate John-Nirenb erg inequalities using the dual method.
BULLETIN DES SCIENCES MATHEMATIQUES
(2022)
Article
Mathematics
Ferenc Weisz
Summary: This article investigates variable Hardy and HardyLorentz spaces satisfying the log-Holder condition and containing Vilenkin martingales. It proves the convergence of the partial sums of the Vilenkin-Fourier series in L-p((.))- and L-p(.),L-q-norm, as well as the boundedness of the maximal operator of the Fejer means in H(p(.)) to L-p(.) and H-p(.),H-q to L-p(i)q. The condition for boundedness in the last result is surprising and not required for Fourier series or Fourier transforms.
MATHEMATISCHE NACHRICHTEN
(2022)
Article
Mathematics
Mikhail Dyachenko, Erlan Nursultanov, Sergey Tikhonov, Ferenc Weisz
Summary: This article obtains Fourier inequalities in the weighted L-p spaces for any 1 < p < infinity involving the Hardy-Cesaro and Hardy-Bellman operators. These results are extended to product Hardy spaces for p <= 1. Additionally, the boundedness of the Hardy-Cesaro and Hardy-Bellman operators in various spaces (Lebesgue, Hardy, BMO) is discussed. A key tool used is an appropriate version of the Hardy-Littlewood-Paley inequality.
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Mathematics, Applied
Ferenc Weisz
Summary: In this study, the usual Doob maximal operator and the fractional maximal operator are generalized, and a new fractional maximal operator M-gamma, M-s, M-alpha is introduced. It is proven mathematically that these maximal operators are bounded under certain conditions.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2022)
Article
Mathematics, Applied
Jianzhong Lu, Ferenc Weisz, Dejian Zhou
Summary: In this paper, we mainly investigate the real interpolation spaces for variable Lebesgue spaces and martingale Hardy spaces using the decreasing rearrangement function. Our three main results involve the relationship between different variables and the measurable function p(.).
BANACH JOURNAL OF MATHEMATICAL ANALYSIS
(2023)
Article
Mathematics
Ferenc Weisz
Summary: In this paper, the authors generalize the Hardy-Littlewood maximal operator by considering rectangles with side lengths in a cone-like set defined by a given function psi, and using Lq-means instead of integral means. They then prove that the maximal operator is bounded on Lp when p=q*r and satisfies the log-Hulder condition.
MATHEMATISCHE NACHRICHTEN
(2023)
Article
Mathematics, Applied
Ferenc Weisz
Summary: This paper proves that under the log-Holder continuity condition of the variable exponent p(.), a new type of maximal operators, U-?, U-s, are bounded from the variable martingale Hardy-Lorentz space H-p(.), H-q to L-p(.), L-q, whenever 0 < p- = p+ < 8, 0 < q = 8, 0 < ?, s < 8, and 1/p- - 1/p+ < ? + s. Moreover, the operator U-?, U-s generates equivalent quasi-norms on the Hardy-Lorentz spaces H-p(.), H-q.
REVISTA MATEMATICA COMPLUTENSE
(2023)
Article
Computer Science, Theory & Methods
Ferenc Weisz
Summary: This paper extends the classical Lebesgue's theorem to multi-dimensional functions and Fourier series, introducing new concepts of Lebesgue points and corresponding Hardy-Littlewood type maximal functions, and demonstrating that almost every point is a Lebesgue point. For four different types of summability and convergences investigated in the literature, it is shown that the Cesaro means of the Fourier series of a multi-dimensional function converge to the function at each Lebesgue point as n approaches infinity.
MATHEMATICAL FOUNDATIONS OF COMPUTING
(2022)
Article
Mathematics
Ferenc Weisz
Summary: The concept of Lebesgue points is introduced in the context of the convergence of Fourier series of multidimensional functions, as a generalization of classical Lebesgue's theorem. It is proven that the Cesaro means of the Fourier series converge to the function at each omega-Lebesgue point as n approaches infinity.
ACTA SCIENTIARUM MATHEMATICARUM
(2021)
