Commutator of fractional maximal function on Lorentz spaces

In the paper we study the fractional maximal commutators Mb,α and the commutators of the fractional maximal operator [b, Mα] in the Lorentz spaces Lp,r (Rn). The study of maximal operators is one of the most important topics in harmonic analysis. These significant non-linear operators, whose behavior are very informative in particular in differentiation theory, provided the understanding and the inspiration for the development of the general class of singular and potential operators. The commutator estimates play an important role in studying the regularity of solutions of elliptic, parabolic and ultraparabolic partial differential equations of second order, and their boundedness can be used to characterize certain function spaces. Our main aim is to characterize the commutator functions b, involved in the boundedness on Lorentz spaces of the fractional maximal commutator Mb,α and the commutator of the fractional maximal operator [b, Mα]. We give necessary and sufficient conditions for the boundedness of the operators Mb,α and [b, Mα] on Lorentz spaces Lp,r (Rn) when b belongs to BMO (Rn) spaces, whereby some new characterizations for certain subclasses of BMO (Rn) spaces are obtained. We can apply this boundedness of fractional-maximal commutators in Lorentz spaces to study the regularity in Lorentz spaces of of the Navier-Stokes equations. Solutions to the Navier–Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering. Keywords: Lorentz space; fractional maximal operator; commutator; space.