Let G be a finite group of order n and let A be a (real or complex) Banach algebra. Rudin and Schneider [3] ask whether a mapping f-.G^A satisfying \\f(x)\\ = 1 and/(x) = (\/n)S.yeJ(xy-')f(y) is necessarily a homomorphism (Question 1, p. 602). They give an affirmative answer if A is either commutative and semisimple or strictly convex. Here, we prove this result for general Banach algebras, and...

We present uniform and $L_p$ mixed Caputo-Bochner abstract generalized fractional Landau inequalities over $\mathbb{R}$ of orders $2 < \alpha \leq 3 $. These estimate the size first second derivatives a composition with Banach space valued function $\mathbb{R}$. give applications when $? = 2.5$.

The starting point of this work is the Bochner theorem on harmonics 1-forms stated at 1946. We show that many results on minimal foliations of codimension one and two on compact pseudo-Riemannian manifolds are at the origin of this theorem. We also prove the non existence of minimal Riemannian foliations of codimension one defined by a 1-form with finite global norm on complete non compact Riem...

A quantum version of Bochner’s theorem characterising Fourier transforms of probability measures on locally compact Abelian groups gives a characterisation of the Fourier transforms of Wigner quasi-joint distributions of position and momentum. An analogous quantum Bochner theorem characterises quasi-joint distributions of components of spin. In both cases quantum states in which a true distribu...

and Applied Analysis 3 Proposition 1.2. Function σ t is μ-measurable. Proof. Pick a dense set {ri}i 1 in 0,∞ and set Bk { t ∈ T : M ( t, 1 2 rk ) 1 2 M t, rk } , qk t rkχBk t k ∈ N . 1.7 It is easy to see that for all k ∈ N, σ t ≥ qk t μ-a.e on T . Hence, supk≥1qk t ≤ σ t . For μ-a.e t ∈ T , arbitrarily choose ε ∈ 0, σ t . Then, there exists rk ∈ σ t − ε, σ t such that M t, 1/2 rk 1/2 M t, rk ,...

For an arbitrary innnite-dimensional Banach space X, we construct examples of strongly-measurable X-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly diierentiable; thus, for these functions the Lebesgue Diierentiation Theorem fails rather spectacularly. We also relate the degree of nondiierentiability of the indeenite Pettis integral to the cotype of X, fr...

A formal treatment of Killing 1-form and 2-Killing on Riemannian Poisson manifold, warped product space are presented. In this way, we obtain Bochner type results compact for 1-form. Finally, give the characterization a (R2, 1,?).