By Sundaram Thangavelu
Motivating this fascinating monograph is the advance of a few analogs of Hardy's theorem in settings coming up from noncommutative harmonic research. this is often the primary subject matter of this work.
Specifically, it truly is dedicated to connections between quite a few theories bobbing up from summary harmonic research, concrete challenging research, Lie thought, distinctive services, and the very attention-grabbing interaction among the noncompact teams that underlie the geometric items in query and the compact rotation teams that act as symmetries of those objects.
A educational creation is given to the mandatory heritage fabric. the second one bankruptcy establishes a number of types of Hardy's theorem for the Fourier remodel at the Heisenberg team and characterizes the warmth kernel for the sublaplacian. In bankruptcy 3, the Helgason Fourier rework on rank one symmetric areas is handled. many of the effects provided listed below are legitimate within the normal context of solvable extensions of H-type groups.
The ideas used to end up the most effects run the gamut of contemporary harmonic research comparable to illustration idea, round capabilities, Hecke-Bochner formulation and specified functions.
Graduate scholars and researchers in harmonic research will drastically make the most of this book.
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Extra resources for An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups
2. 2 is still unknown. 2 are the best possible since faster growth assumptions on P and Q will make the theorem incorrect. 1 can be proved on jRn if we replace the pointwise estimates on f and by integral conditions. 2, sn - I whe re the fun ctions P and Q are such that . log P (r ) . log Q ().. ) hm sup 2 = hm sup 2 = O. ->- 00 i, and when ab = i, ).. 40 1. Euclidean Spaces where p and q are such that their L 2 -norms on and Q(r) respectively. Ixl = r do not grow faster than per) This theorem is proved in  where the authors have also proved analogues of Hardy's theorem for the Dunk!
X) where A and B are real, symmetric matrices and P is a polynomial. (l Iyl)-N d xdy < 00 ; jR" jR" (iii) f f If(x)llf(y)lel(x ,Y)I(l + Ixl)- ~(l + Iyl)-~dxdy < 00 . jR" jR" Proof. If f is as in the proposition with A symmetric , then by diagonalising A we see that A has to be positive definite since f is square integrable. 2) where Q is another polynomial with deg P = deg Q = m. It is easy to show that (i) implies (iii). 3) 00. jR" jR" Whenlxl2: 3lylorlyl2: 3Ixltheintegralisfinite,sincethenlx-Ye 2: c(lxI 2+IYI2) for some constant c > O.
Rr 2a 1 }. < - , a> - 2 Suppose F(I;) is holomorphic in Q and continuous up to the boundary and satisfies the estimates W(I;)I :s M on oQ, W(I;)I :s K ell;Jtl , fJ < a on Q. Then W(I;)I < M on the whole of Q . Proof. Let E > 0 and choose y such that fJ < y < a. Consider the function F€ (I;) = e-€I;Y F(I;) so that IF€(I;)I = e-€I I;IY cos(ye)IF(I;)I . On the boundary lines 0 = ±~ we have cos(yO) > 0 since y < a . Therefore, on these lines IF€(I;)I :s W(I;)I :s M. 4. Hardy's theorem on jRn 21 which goes to 0 as R ~ 00 since f3 < y.
An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups by Sundaram Thangavelu