We study the boundedness of Hardy–Littlewood maximal function on the spaces defined in terms of Choquet integrals associated with weighted Bessel and Riesz capacities. As a consequence, we obtain a class of weighted Sobolev inequalities.
We study the boundedness of Hardy–Littlewood maximal function on the spaces defined in terms of Choquet integrals associated with weighted Bessel and Riesz capacities. As a consequence, we obtain a class of weighted Sobolev inequalities.
In this paper, we investigate some reflexivity-type properties of separable measurable Banach bundles over a (sigma )-finite measure space. Our two main results are the following:
The fibers of a bundle are uniformly convex (with a common modulus of convexity) if and only if the space of its (L^p)-sections is uniformly convex for every (pin (1,infty )).
The fibers of a bundle are reflexive if and only if the space of its (L^p)-sections is reflexive for every (pin (1,infty )).
They generalise well-known results for Lebesgue–Bochner spaces.
We study linear combinations of two composition operators induced by linear symbols on the Hilbert space of Dirichlet series. Based on partial reproducing kernels, we obtain an equivalent inscription of the compactness of a single composition operator and describe the compact linear combinations of composition operators.
Our focus is the operators of multivariable stochastic calculus, i.e., systems of transfer operators, covariance operators, conditional expectations, stochastic integrals, and the counterpart infinite-dimensional stochastic derivatives. In this paper, we present a new operator algebraic framework which serves to unify the analysis and the interrelations for the operators in question. Our approach uses Rokhlin decompositions, and it applies to both general classes of Gaussian processes, and white noise probability space, in commutative probability, as well as to the analogous operators in the framework of quantum (non-commutative) probability.
It is known that the invariant subspace problem for Hilbert spaces is equivalent to the statement that all minimal non-trivial invariant subspaces for a universal operator are one dimensional. In this paper, we first give a characterization of the boundedness of composition operators on weighted Dirichlet spaces ({mathcal {D}}_{alpha }(Pi ^{+})) over the upper half-plane (Pi ^{+}) using generalized Nevanlinna counting functions, where (alpha >-1.) As an application, we discuss the boundedness of composition operators on ({mathcal {D}}_{alpha }(Pi ^{+})) induced by linear fractional self-maps of (Pi ^{+}.) Second, we characterize composition operators and their adjoints induced by affine self-maps of (Pi ^{+}) that have universal translates on ({mathcal {D}}_{alpha }(Pi ^{+}).) Moreover, we investigate which composition operators and their adjoints induced by hyperbolic non-automorphism self-maps of the open unit disk ({mathbb {D}}) have universal translates on weighted Dirichlet spaces ({mathcal {D}}_{alpha }({mathbb {D}})) for (alpha >-1.) Finally, we consider the minimal invariant subspaces of the composition operators that have universal translates.
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