In this paper we deal with the interpolation from Lebesgue spaces and , into an Orlicz space , where and for some concave function , with special attention to the interpolation constant . For a bounded linear operator in and , we prove modular inequalities, which allow us to get the estimate for both the Orlicz norm and the Luxemburg norm,
where the interpolation constant depends only on and . We give estimates for , which imply . Moreover, if either or , then . If , then , and, in particular, for the case this gives the classical Orlicz interpolation theorem with the constant .
Let be an ideal of over a -finite measure space and let be the Köthe dual of with . Let be a real Banach space, and the topological dual of . Let be a subspace of the space of equivalence classes of strongly measurable functions and consisting of all those for which the scalar function belongs to . For a subset of for which the set is -bounded the following statement is equivalent to conditional -compactness: the set is conditionally -compact and is a conditionally weakly compact subset of for each , with . Applications to Orlicz-Bochner spaces are given.
This article presents sufficient conditions for the positive definiteness of radial functions , , in terms of the derivatives of . The criterion extends and unifies the previous analogues of Pólya's theorem and applies to arbitrarily smooth functions. In particular, it provides upper bounds on the Kuttner-Golubov function which gives the minimal value of such that the truncated power function , , is positive definite. Analogous problems and criteria of Pólya type for -dependent functions, 0$">, are also considered.
Let be a convex curve in the plane and let be the arc-length measure of Let us rotate by an angle and let be the corresponding measure. Let . Then This is optimal for an arbitrary . Depending on the curvature of , this estimate can be improved by introducing mixed-norm estimates of the form where and are conjugate exponents. 相似文献
We define a quotient of bounded operators and on a Hilbert space with a kernel condition as the mapping , . A quotient is said to be positive symmetric if . In this paper, we give a simple construction of positive selfadjoint extensions of a given positive symmetric quotient .
Let , where and is a Banach space. Let be an extension of to all of (i.e., ) such that has minimal (operator) norm. In this paper we show in particular that, in the case and the field is R, there exists a rank- such that for all if and only if the unit ball of is either not smooth or not strictly convex. In this case we show, furthermore, that, for some , there exists a choice of basis such that ; i.e., each is a Hahn-Banach extension of .
Let and be compact Hausdorff spaces and let . A linear mapping is called -disjointness preserving if implies that . If is a continuous or surjective -disjointness preserving linear mapping, we prove that there exists a disjointness preserving linear mapping satisfying . We also prove that every unbounded -disjointness preserving linear functional on is disjointness preserving.
Let be a compact manifold, and let be a transitive homologically full Anosov flow on . Let be a -cover for , and let be the lift of to . Babillot and Ledrappier exhibited a family of measures on , which are invariant and ergodic with respect to the strong stable foliation of . We provide a new short proof of ergodicity.
A variety is a class of Banach algebras , for which there exists a family of laws such that is precisely the class of all Banach algebras which satisfies all of the laws (i.e. for all , . We say that is an -variety if all of the laws are homogeneous. A semivariety is a class of Banach algebras , for which there exists a family of homogeneous laws such that is precisely the class of all Banach algebras , for which there exists 0$"> such that for all homogeneous polynomials , , where . However, there is no variety between the variety of all -algebras and the variety of all -algebras, which can be defined by homogeneous laws alone. So the theory of semivarieties and the theory of varieties differ significantly. In this paper we shall construct uncountable chains and antichains of semivarieties which are not varieties.
Let be a covariant system and let be a covariant representation of on a Hilbert space . In this note, we investigate the representation of the covariance algebra and the -weakly closed subalgebra generated by and in the case of or when there exists a pure, full, -invariant subspace of .
We show that a family of functions meromorphic in some domain is normal, if for all the derivative omits the value and if the values that can take at the zeros of satisfy certain restrictions. As an application we obtain a new proof of a theorem of Langley which classifies the functions meromorphic in the plane such that and have no zeros.