By Antoine Derighetti (auth.)
This quantity is dedicated to a scientific learn of the Banach algebra of the convolution operators of a in the neighborhood compact crew. encouraged by way of classical Fourier research we think about operators on Lp areas, arriving at an outline of those operators and Lp models of the theorems of Wiener and Kaplansky-Helson.
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Extra info for Convolution Operators on Groups
With this assumption, we have p f D fL. Lemma 3. G/. f /. Proof. s . z/jd zA dy < 1: G This implies s G . s / rQ . Theorem 4. Let G be a locally compact unimodular group. G/. Proof. f /S . G/ and " > Z0. G/ and " > 0. AccordingZto Proposition 2 of Sect. G / Using the commutation theorem of Sect. G/ is the limit of 2G . / for G a locally compact unimodular group. H/ the involutive Banach algebra of all continuous operators of H. H/, kT k is the norm of the operator T . E 0 /0 . H/ with T x ˇx 2 H; T 2 B dense in H .
3, p. 789/. , Th´eor`eme 1, 32 2 The Commutation’s Theorem p. 280, , Chap. I, Sect. 5, p. 71, Th´eor`eme 1 and Exercice 5 p. 80/. , p. / Theorem 5. G/. f˛ / 2 Ä jjjT jjj2 for every ˛. Proof. According to Theorem 1 But by Theorem 5 of Sect. G//. G/: Remark. We will extend this result to p 6D 2 for certain classes of locally compact groups. f˛ /. Chapter 3 The Figa–Talamanca Herz Algebra Let G be a locally compact group. G/, is a Banach algebra for the b pointwise product on G. G/. G / 0 Let G be a locally compact group and 1 < p < 1.
6002, Th´eor`eme 1, , p. 244, , p. 72, Corollary/. G/ is a Banach algebra . 1, p. , p. 54/. See the notes to Chap. 3 for Herz’ approach and various generalizations. Corollary 6. Let G be a locally compact group and 1 < p < 1. G/ . G/ of all limits of convolution operators associated to bounded measures. G/. G/. G /: The Notion of Pseudomeasure Proposition 1. Let G be an abelian locally compact group. u/j Ä kbk1 kukA2 : Proof. We have Z ! u/. x/. u/. x/ d : G GO ! u/k1 D kbk1 kukA2 : ˇˇˇ ˇˇˇ p According to Theorem 2 of Sect.
Convolution Operators on Groups by Antoine Derighetti (auth.)