Some Harmonic Problems On Sch(o|¨)dinger Operators With Constant Magnetic Fields

As the development of the Schrodinger operator theory, the harmonic analysis has become an important tool in the research for it. In2006, when he reviewed the progress in the Schrodinger operator theory in the twentieth century (Schrodinger operators in the twentieth century) B. Simon showed that the mathematics most closely related to the modern theory of Schrodinger operators is functional anal-ysis, harmonic analysis and complex analysis. The classical harmonic analysis in a certain sense can be regarded as the mathematical theory closely related to the Laplacean operator. The research for harmonic analysis problems associated to Schrodinger operators is further development of the classical harmonic anal-ysis theory and become more popular for a direction in the field of the modern mathematics.In this dissertation, we fully combine the spectral theory with classical har-monic analysis methods. On the basis of well known conclusions we study some harmonic problems on Schrodinger operators with constant magnetic fields in depth. This dissertation can be divided into following parts.In the introduction, we present the background and the recent developmen-t of harmonic analysis associated to the Schrodinger operators, as well as the Schrodinger operators with magnetic fields.In Chapter2, we research the LP boundedness of Marcinkiewicz-type multipli-er for the Schrodinger operator with constant magnetic fields, where the p is near2. The main tool used for the research is Littlewood-Paley g-functions of the spectral representation of Schrodinger operators with constant magnetic fields. Based on the spectrum of Schrodinger operators with constant magnetic fields, we use the Riesz means of its spectral expansion to construct the two g-functions. By using the restriction theorem of the spectral project operator, we give the relation be-tween the index p for which the g-functions are bounded on Lp and the summation one of Riesz means. According to the relation, we give the Marcinkiewicz criteri-on for the spectral multipliers related to the Schrodinger operators with constant magnetic fields.In Chapter3, we consider the almost everywhere convergence of Riesz means of the spectral expansion related to the Schrodinger operator with constant mag-netic fields. Persons used only to research the problems related Laplacean oper-ator, Hermite operator, and so on. Nobody has investigated the one relevant to the Schrodinger operator with constant magnetic fields. By establishing the L2estimate for some g-functions, the boundedness of maximal operators is reduced, which implies that if the summation index β>0then the Reisz means are con-vergent almost in L2. By the weighted L2norm estimate for compactly supported multipliers, we obtain the results for Riesz means with index0<β<n-1/2and Lp for some2≤p<2n/n-1-2β similar to those for the classical Fourier integrals.In Chapter4,we research Hardy spaces related to the Schrodinger operator with constant magnetic fields. People have fully researched Hardy spaces related to convolution and twisted convolution and gotten the good results. However, nobody has thought over the Hardy spaces associated to the heat kernel in which the convolution and the twisted convolution are mixed. Hence, this inquiry is a new attempt. We study atomic decomposition of the Hardy space. We find that the atoms are of the vanishing condition with its twisted factor. By introducing the relevant Heisenberg type group, we describe the Hardy space on the group with the maximal function related to its heat kernel. And we also study the relationship between the two kinds of Hardy spaces. Furthermore, we investigate the characterization of Hardy spaces in terms of Riesz transforms associated to the Schrodinger operator.In chapter5, we study the mapping properties of singular integrals on prod-uct domains with kernels in L(log+L)ε(Sm-1×Sn-1)(ε=1or2) supported by hyper-surfaces. The Lp bounds for such singular integral operators as well as the related Marcinkiewicz integral operators are established, provided that the lower dimensional maximal function is bounded on Lq(R3) for all q>1.The condition on the integral kernels is known to be optimal.In the end, we summarize the main achievements in this paper. And, we introduce innovative conclusions and methods. Furthermore, we prospect and imagine on the later research on the Schrodinger operators with constant magnetic fields.