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In this paper we study the regularity properties of fractional maximal operators acting on BVBV-functions. We establish new bounds for the derivative of the fractional maximal function, both in the continuous and in the discrete settings.

Let 0 ≤ α < n , M α be the fractional maximal operator, M ♯ be the sharp maximal operator and b be the locally integrable function. Denote by [ b , M α ] and [ b , M ♯ ] be the commutators of the fractional maximal operator M α and the sharp maximal operator M ♯ . In this paper, we show some necessary and sufficient conditions for the boundedness of the commutators [ b , M α ] and [ b , M ♯ ] on slice spaces when the function b is the Lipschitz function, by which some new characterizations of the non-negative Lipschitz function are obtained.

Let φ:[0,1)×[0,∞)→[0,∞] be a Musielak–Orlicz function and γ,s∈(0,∞). In this article, the authors present some sufficient conditions to ensure that the dyadic maximal operator Uγ,s is bounded on Musielak–Orlicz spaces Lφ[0,1). As applications, the authors establish the characterizations of Musielak–Orlicz Hardy spaces Hφ[0,1) and the boundedness of maximal Fejér operators from Hφ[0,1) to Lφ[0,1). Furthermore, the almost everywhere convergence and the norm convergence of Fejér means of Walsh–Fourier series are also obtained. All these results include, as special cases, the essentially optimal conditions for variable exponent Lebesgue spaces, perturbed variable exponent Lebesgue spaces, and double-phase functionals with variable exponent Lebesgue spaces.

We introduce some new weighted maximal operators of the Fejér means of the Walsh–Fourier series. We prove that for some \"optimal\" weights these new operators are bounded from the martingale Hardy space H p ( G ) to the space weak- L p ( G ) , for 0 < p < 1 / 2 . Moreover, we also prove sharpness of this result. As a consequence we obtain some new and well-known results.

This paper focuses on the operator norm of the truncated Hardy-Littlewood maximal operator Mab and the strong truncated Hardy-Littlewood maximal operator M~ab, respectively. We first present the L1-norm of Mab, and then the L1-norm of M~ab is given. Our study may have some enlightening significance for the research on sharp constant for the classical Hardy-Littlewood maximal inequality.

In this note we study the regularity properties of the discrete maximal operators at endpoint. Precisely, we show that the general discrete centered and non-centered maximal operators are bounded and continuous from ℓ1(Z)\\ell ^1(\\mathbb {Z}) to BV(Z)\\textrm {BV}(\\mathbb {Z}), as well as the non-centered discrete maximal operator maps BV(Z)→BV(Z)\\textrm {BV}(\\mathbb {Z})\\rightarrow \\textrm {BV}(\\mathbb {Z}) boundedly under a more restrictive condition, where BV(Z)\\textrm {BV}(\\mathbb {Z}) denotes the set of functions of bounded variation defined on Z\\mathbb {Z}. As an immediate consequence, we obtain somewhat unexpected endpoint regularities of the discrete fractional maximal functions.

We define the weighted Orlicz-Lorentz-Morrey and weak weighted Orlicz-Lorentz-Morrey spaces to generalize the Orlicz spaces, the weighted Lorentz spaces, the Orlicz-Lorentz spaces, and the Orlicz-Morrey spaces. Furthermore, necessary and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator, generalized fractional integral, and maximal operators on the weighted Orlicz-Lorentz-Morrey and weak Orlicz-Lorentz-Morrey spaces are given, based on the exploration of properties of Young functions, weights, and weights. Specifying the weights and the Young functions, we recover the existing results and we obtain new results in the new and old settings.

We study the fractional integral and Orlicz fractional maximal operator estimates in the second kind of Orlicz-Morrey spaces. This problem originates from the Adams and Olsen inequality in Morrey spaces. We establish the boundedness corresponding to the Adams-type and the Olsen-type inequality of the fractional operator in Orlicz-Morrey spaces under conditions of Young functions. Moreover, we construct and observe the examples of the Young functions’ triplet that satisfies the condition. This paper reinforces the boundedness theory for the fractional integrals and Orlicz fractional maximal operators in Orlicz-Morrey spaces.

We develop novel extensions in the theory of weighted Lorentz spaces. In particular, we generalize classical results by introducing variable-exponent Lorentz spaces, establish sharp constants and quantitative bounds for maximal operators, and extend the framework to encompass fractional maximal operators. Moreover, we analyze endpoint cases through the study of oscillation operators and reveal new connections with weighted Hardy spaces. These results provide a unifying approach that not only refines existing inequalities but also opens new avenues in harmonic analysis and partial differential equations.

In this paper, we find sufficient conditions on functions ω 1 , ω 2 which ensure the boundedness of Riesz potentials and their commutators with BMO functions from one local complementary generalized Orlicz–Morrey spaces to the spaces . As a consequence of the boundedness of the Riesz potential, we give the boundedness the fractional maximal operator in local complementary generalized Orlicz–Morrey spaces.