Download An Introduction to Harmonic Analysis on Semisimple Lie by V. S. Varadarajan PDF

By V. S. Varadarajan

Now in paperback, this graduate-level textbook is a superb creation to the illustration concept of semi-simple Lie teams. Professor Varadarajan emphasizes the advance of significant issues within the context of specific examples. He starts with an account of compact teams and discusses the Harish-Chandra modules of SL(2,R) and SL(2,C). next chapters introduce the Plancherel formulation and Schwartz areas, and convey how those result in the Harish-Chandra idea of Eisenstein integrals. the ultimate sections give some thought to the irreducible characters of semi-simple Lie teams, and contain specific calculations of SL(2,R). The booklet concludes with appendices sketching a few simple issues and with a accomplished consultant to additional examining. This impressive quantity is extremely appropriate for college kids in algebra and research, and for mathematicians requiring a readable account of the subject.

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Extra resources for An Introduction to Harmonic Analysis on Semisimple Lie Groups

Example text

Using other fixed point theorems, solvability results have been obtained in [10] where f may contain some superlinear terms in the nonresonance case. An existing problem is: Can one relax the order of growth for the superlinear term in f in [10]? Also one is interested in extending the result to the resonance case. CHAPTER III The Initial Value Problem for a Nonlinear Schrodinger Equation [ 11 §1. Background materials THREE-LINES THEOREM IN ONE COMPLEX VARIABLE. Let f(z) (z = x+iy) be bounded, analytic in Q, and continuous in 11, where SZ is a strip { z Ixl < X < x2}.

A solution for this nonlinear problem can be regarded as a fixed point w + W = V for the following two problems: I iwt + Ow = 0, wlt=o=(P and iW+OW=F(V), 1 W I t=0 = 0. We shall first derive estimates for w and W and then decide which fixed point theorem is more appropriate. III. A NONLINEAR SCHRODINGER EQUATION 50 Using the Riesz-Thorin theorem we have (47Lt)n/q-n/2II9,IIL4 IIwlIL9 = II R(t)(OII L° :5 < KStnlq-nl2(II SOIIL + II0IL2) This estimate is not effective for small t. II , q>2. To compensate it we use Dkw I I L2 = I I Dk rp 11 L2 and the embedding theorem to derive IIlIL9 +K,IIDxwIIL2)(q-2)lgIIwIIL9 (K(IIwIIL2 L2)(g-2)IgII (IILq _ (K6II0IL2 + K,II Dk(9 II Combining these two estimates we have IIIWIIIq t)n/2-n/gIIwIILe] = sup[(1 + t>0 < Kg(IISOIIL + IIcoIIL2 + IIDk(pIIL2).

Lluotll = Ilutll :5 010 Let KI = minA j>C (' i - C) and K2 = min, by u I , by integration we have KIIIu1112 <- J2 x [o , 2n] 1:

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