ON NON-INCREASING REARRANGEMENTS OF THE GENERALIZED FRACTIONAL MAXIMAL FUNCTION - Abek, А.; Khairkulova, А.; Turgumbayev, M. (Bulletin of Abai KazNPU. Series of Physical and Mathematical sciences, 2023)
The purpose of this article is to consider the symmetric rearrangement and non-increasing rearrangement of generalized fractional maximal functions. Concepts of rearrangement-invariant spaces and concepts of ideal spaces are considered. A generalized Lorentz-Morrey type space, in which the norm is determined by a symmetric rearrangement of functions, is considered. The equivalent norm for the function from the generalized Lorentz-Morrey space obtained. It is proved that in the definition of the norm in the generalized Lorentz-Morrey space, the internal norm from a symmetric rearrangement of a function over a ball centered at the point can be replaced by the norm from a symmetric rearrangement of a function over a ball centered at an arbitrary point . A generalized fractional-maximal function a special case of which is a classical fractional-maximal function is considered. Estimates obtained for the non-increasing rearrangement of the generalized fractional maximal function. A pointwise estimate of the generalized fractional-maximal function by the generalized Riesz potential is obtained.