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In this article, motivated by the regularity theory of the solutions of doubly nonlinear parabolic partial differential equations, the authors introduce the off-diagonal two-weight version of the parabolic Muckenhoupt class with time lag. Then the authors introduce the uncentered parabolic fractional maximal operator with time lag and characterize its two-weighted boundedness (including the endpoint case) in terms of these weights under an additional mild assumption (which is not necessary for one-weight case). The most novelty of this article exists in that the authors further introduce a new parabolic shaped domain and its corresponding parabolic fractional integral with time lag and, moreover, applying the aforementioned (two-)weighted boundedness of the parabolic fractional maximal operator with time lag, the authors characterize the (two-)weighted boundedness (including the endpoint case) of these parabolic fractional integrals in terms of the off-diagonal (two-weight) parabolic Muckenhoupt class with time lag; as applications, the authors further establish a parabolic weighted Sobolev embedding and a priori estimate for the solution of the heat equation. The key tools to achieve these include the parabolic Calderón–Zygmund-type decomposition, the chaining argument, and the parabolic Welland inequality, which is obtained by making the utmost of the geometrical relation between the parabolic shaped domain and the parabolic rectangle.

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 mapping properties of the Hardy–Littlewood fractional maximal operator between Lorentz spaces of the homogeneous tree and discuss the optimality of all the results.

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.

In this study, firstly we define the weighted modified Morrey spaces L ~ p , λ ( ℝ n , μ ) for the weight function 𝜇 in the class A p ( ℝ n ) and we prove the boundedness of some classical operators as the generalized fractional maximal operator 𝑀𝜌 and the generalized fractional integral operator 𝐼𝜌 from the weighted modified Morrey spaces L ~ p , λ ( ℝ n , μ p ) to L ~ q , λ ( ℝ n , μ q ) with μ q ∈ A 1 + q p ' ( ℝ n ) and from L ~ 1 , λ ( ℝ n , μ ) to weighted weak modified Morrey spaces W L ~ q , λ ( ℝ n , μ q ) , with μ ∈ A 1 , q ( ℝ n ) by proving the appropriate weighted norm inequalities.