Theory Of Orlicz Space Applied In The Inequalities For Differential Forms

Differential forms, as a class of the tensor fields with anti-symmetry, are an extensionof multi-function. Such tensor fields have been widely applied in physics, mechanics,engineering science and mathematics. For example, the gradient, divergence, curl, andGreen, Gauss and Stokes’theorem in the classic analysis can be uniformly represented bymeans of differential forms. Harmonic equations in function, as a special kind of partialdifferential equations, describes the nature of variety of physical fields as the form ofthe potential function. There is also a special class of harmonic equation in differentialforms, that is, the （?）-harmonic equation, which is a generation of many classic harmonicequations, such as the p-harmonic equations and the Laplace equation. Since differentialforms and their （?）-harmonic equation have such a wide range of applications, they havecaused a great deal of interest for scholars. Over the last decade, the research on theLp-norm inequalities for the operators applied in differential forms has achieved fruitfulresults. However, the study on the inequalities for differential forms with the basic theoryof the Orlicz space, as which is the promotation of the Lp-space space, is still yet to bedeveloped.In this dissertation, we mainly study the integral and norm inequalities with a fewspecial class of Young function of the composite operators T οdοH, GοT and GοP, whichconsists of the differential operator d, the homotopy operator T, the projection operatorH, the Green operator G and the potential operator P. Then, we obtain the local weightedwersions and global forms in L （μ）-average domains of theses inequalities. Finally, weprove the inequalities of the fractional integral operator F_αapplied in differential formand defined in this paper, which meets the certain Orlicz condition（ie. Lα,-H o¨rmandercondition）. Specifically, the study in this paper contains the following aspects:First of all, based on the work of the potential operator P defined by H. Bi, we fur-ther study the Lp-norm inequality and its local weighted version of the composite operatorGοP, which consists of the potential operator P and Green’s operator G, and then obtaina variety of Lipschtz and BMO norm inequalities of the operator GοP and their weightedversions. Noting that the Lp（log L）α-norm for differential forms is a more complicatedkind of specific Orlicz norm compared to the Lp-norm for that, we study the Lp（log L）α-norm inequalities of the composite operator G ο T by two different kind of methods and further obtain the local weighted version and the global form in the L^?（μ）-average do-mains.Secondly, we study some integral and norm inequalities with two types of abstractiveYoung functions （ie. the G（p,q,C）-class of Young functions and the p-class of Youngfunctions）of a few kinds of composite operators. In the study of the weighted integralinequality with the G（p,q,C）-class of Young function of the operator T ο d ο H, we takefull advantage of the properties of the G（p,q,C）-class of Young functions. However,In the study of the integral inequality with the G（p,q,C）-class of Young function of theoperator GοP, we still use the segment discussion and placed under the transformation ofideas apart from this. In the study of the Orlicz norm inequality in the Lebesgue averagemeasure sense with the（?）_p-class of Young functions of the operator G ο T, we use a lotof important norm inequalities in the theory of Orlicz space. Then, we obtain the integralinequality in the Lebesgue average measure sense with the（?）_p-class of Young functions ofthe operator G ο T, according to the comparison between the Orlicz norm and the integralof function in the Orlicz space and acted on by the double Young function. Then, wemake some numerical estimates with the particular solution of the-harmonic equationand obtain the global form in the L^?（μ）-average domains of the integral inequality ofG ο T.Finally, we define a fractional integral operator F_αfor differential forms, whosenuclear function satisfies Sα-conditions. If the operator F_αmeets the special Orlicz con-ditions (ie. L^α-（?）-Hrmander conditions), we obtian the Coifman type inequality of theoperator with the basic inequality, the Poincare′inequality, the global form in the Lp（m）-average domains, Lipschitz and BMO norm inequalities. Then, we give two specificapplications of the Coifman type inequality.