We give the complete solutions to the diophantine equation X+Y+Z=W in coprime positive integers X, Y, Z, W such that each of the numbers X, Y, Z, W has prime factors 2 and 3 only. The solution with the largest value of W is 2333 + 29 + 1= 36. The method works for any pair of primes (p, q) in place of (2, 3).
We prove an improved version of Korotkov's characterization of Carleman operators and we determine those bounded linear operators U: Lϱ(μ)→L2(ν) (Lϱ(μ) being a Banach function space) with the property that for every bounded linear operator B on L2(ν), the restriction of BU to a given order ideal E in Lϱ(μ) is integral or regular as an operator from E to L0(ν).
The author characterizes two Carleson-type measures relative to Hardy spaces of functions, holomorphic in the unit ball of CN.
A bounded subset X of a Banach space over a non-archimedean field ϰ is a compactoid if and only if each basic sequence in X tends to zero (Theorem 2). As a consequence the notions “weakly precompact’ and ‘precompact’ are identical for members of a wide class of ϰ-Banach spaces (Theorem 3).
Suppose U is a domain in ℂn, not necessarily pseudoconvex, and D is a derivation on the algebra %plane1D;4AA;(U) of holomorphic functions on U, i.e. D : %plane1D;4AA;(U)→%plane1D;4AA;(U) is additive and satisfies Dfg=fDg+gDf for all ƒ,g ε %plane1D;4AA;(U). It is shown that there are h1,hnε%plane1D;4AA;(U) such that Df = Σi=1nhi ∂ƒ/∂Zi for all ƒ ε %plane1D;4AA;(U). The same techniques are then applied to show that, for a Stein manifold Ω, the natural map from the space of global holomorphic sections of the holomorphic tangent bundle of Ω to the space of derivations on %plane1D;4AA;(Ω) is a bijection.
The set of multipliers from one vector space to another vector space may be seen as a generalized dual space in the sense of Köthe. We give some properties of this kind of duality and prove precise estimates concerning generalized duality of XP-spaces, Lebesgue, Lorentz, Marcinkiewicz and Orlicz spaces. We complement and unify several previous results of this kind.
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