By Edwin Hewitt, Kenneth A. Ross

ISBN-10: 3540583181

ISBN-13: 9783540583189

This ebook is a continuation of vol. I (Grundlehren vol. one hundred fifteen, additionally on hand in softcover), and incorporates a distinct remedy of a few very important elements of harmonic research on compact and in the neighborhood compact abelian teams. From the experiences: "This paintings goals at giving a monographic presentation of summary harmonic research, way more entire and complete than any booklet already present at the subject...in reference to each challenge handled the ebook bargains a many-sided outlook and leads as much as most recent advancements. Carefull awareness is additionally given to the historical past of the topic, and there's an in depth bibliography...the reviewer believes that for a few years to come back this may stay the classical presentation of summary harmonic analysis." Publicationes Mathematicae

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A formula and an estimate for e−uLτ , u > 0 Let τ ∈ R\{0}. Then a formula for e−uLτ , u > 0, is given by the following theorem. 1. Let f ∈ L2 (C). Then for u > 0, e−uLτ f = (2π)1/2 ∞ e−(2k+1)|τ |u Vτ (Wfτˆ eτk , eτk ), k=0 where the convergence of the series is understood to be in L2 (C). Proof. Let f ∈ L2 (C). 1) k=0 j=0 where the series is convergent in L2 (C). 5), (f, eτj,k ) = C = C for j, k = 0, 1, 2, . . 2) The Heat Equation for the Sub-Laplacian 39 Similarly, for j, k = 0, 1, 2, . . , and g in L2 (C), we get (eτj,k , g) = (g, eτj,k ) = (2π)1/2 (Wgˆτ eτk , eτj ) = (2π)1/2 (eτj , Wgˆτ eτk ).

14) Proof. We have cosh t ≤ et while for every ε ∈]0, 1[ we can ﬁnd με > 0, νε > 0 such that νε + (1/2 − ε)et ≤ cosh t ≤ (1/2 + ε)et + με , t > 0. 17) t 0 +∞ K0 (t) ≥ e−t(1/2+ε)e r −tμε +∞ dr = e−με t t 0 The asymptotic analysis of the integral above for t 0 and t → +∞ yield the desired estimates in i) and ii). 18) t and we use estimates for the Euler gamma function. This gives iii) (and smoothing eﬀects of K1 ). 2 2 2 . Let k = (k1 , k2 ) ∈ Z+ . We have iv) Put g(τ, η) = (4π)−1 K0 τ +η 4 ∞ (4π)g (k) (τ, η) = e− τ 2 +η2 4 (−1/2)k1 +k2 (cosh tk1 +k2 (−1/2)k1 +k2 τ k1 η k2 0 + cosh tk1 +k2 −1 τ k1 η k2 −1 + cosh tk1 +k2 −1 τ k1 −1 η k2 + cosh tk1 +k2 −2 τ k1 −1 η k2 −1 )dt.

4] B. Gaveau, Principe de moindre action, propagation de la chaleur et estim´ees sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), 95–153. [5] E. S. Phillips, Functional Analysis and Semi-groups, Third Printing of Revised Version of 1957, American Mathematical Society, 1974. 42 A. W. Wong [6] A. Hulanicki, The distribution of energy in the Brownian motion in the Gaussian ﬁeld and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group, Studia Math.

### Abstract Harmonic Analysis: Structure and Analysis, Vol.2 by Edwin Hewitt, Kenneth A. Ross

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