By Antoine Derighetti (auth.)
This quantity is dedicated to a scientific learn of the Banach algebra of the convolution operators of a in the community compact crew. encouraged through classical Fourier research we give some thought to operators on Lp areas, arriving at an outline of those operators and Lp types of the theorems of Wiener and Kaplansky-Helson.
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Additional resources for Convolution Operators on Groups
G/. p . a/Pfor every a 2 G. G/. b S / '. 6 The Fourier Transform of a Convolution Operator 21 Proof. This Theorem is a consequence of Theorem 2 and Corollary 3 of Sect. 3 and Theorem 4 of Sect. 5. b may be considered as a tempered distribution. For Remark. T b interpreted as the inverse Fourier transform of the distribution T . Corollary 2. Z/. Z/ if p 2. m/ D 2 nD 1 2 n/Â dÂ; m 2 Z: 0 The following proposition gives a characterization of the Fourier transform of a p-convolution operator. Proposition 3.
G/, we have jp . G . // D G . /. T /' D T '. 18 1 Elementary Results 0 Proof. G/. Consider '1 2 '. x/j for every x 2 X and n 2 N. x/ mG 0 almost everywhere. G/. rn '1 / D 0. T /'. The following corollary improves Theorem 1. Corollary 6. G/. G/ and kT 'kp0 Ä jjjT jjjp k'kp0 . Corollary 7. G/. '/ Ä 1 : > ˇ ˇ ˇ ; G Remark. e. , Corollary 1, p. 12/. See also the notes to Chap. 1. G / Theorem 1 (Riesz–Thorin). X I /. Let 0 < ˛ Ä Ä 1 and let the map T W E ! G/ 19 kT 'k1=˛ Ä M1 k'k1=˛ and kT 'k1= Ä M2 k'k1= : Then T W E !
1 vn kAp D 0: Remarks. G/ has been first considered by A. Fig`aTalamanca in 1965  for G abelian, for G compact but non necessarily commutative and also for G unimodular non-commutative, non-compact and p D 2. The above definition is due to Eymard . G/ and kukAp D kukAp . G/ and k kA0p Ä k k. Lemma 5. Let E be a C-vector space, p a seminorm on E, F a C-subspace of E, x in the closure of F in E and " > 0. x/ C ". nD1 Proof. Let 0 < "1 < minf1; "g. G/ 37 Let y1 D v1 and yn D vn vn 1 p x for n > 2.
Convolution Operators on Groups by Antoine Derighetti (auth.)