| Harmonic analysis originated from the study of heat conduction equation by mathematicians such as Euler and Fourier.After more than 200 years of research and development,harmonic analysis has become an important branch of modern analytical mathematics.Due to the continuous in-depth research of harmonic analysis,some problems encountered in engineering science have been solved,making signal processing,quantum mechanics and other scientific fields develop rapidly.The problem of boundedness of operators is a research hotspot of harmonic analysis.Many scholars have made significant achievements in the study of boundedness of operators in function space,which provides theoretical support for some problems in natural science and engineering technology.In recent years,many scholars have been looking for new function spaces and studied many new theoretical results,which makes the theory of boundedness of operators more and more perfect.In recent years,the grand variable function space has become a hot research topic.The study of boundedness of operators in the grand variable function space is a very meaningful research topic.Based on the above research background,this dissertation studies the boundedness of several kinds of operators and commutators in grand variable Herz space.The main results are as follows:Firstly,the research background,domestic and international research status,and research significance of variable index Herz spaces are introduced.Secondly,by using the theory of variable exponent function space and function decomposition,it is proved that the intrinsic square function and its commutator are bounded in the grand variable Herz space when the variable exponent satisfies certain conditions.Once again,by using the theory of variable exponent function space and the generalized BMO norm property,it is proved that the commutator of Marcinkiewicz integral operators is bounded in grand variable Herz space when the variable exponent satisfies certain conditions.Finally,by using the Minkowski inequality,Holder’s inequality,and vector-valued inequality,it is proved that vector-valued sublinear operators are bounded in grand variable Herz space. |
