In [10], for continuous functions f from the domain of certain discrete operators Ln the inequalities are proved concerning the modulus of continuity of Lnf . Here we present analogues of the results obtained for the Durrmeyer-type modification ̃ Ln of Ln. Moreover, we give the estimates of the rate of convergence of ̃ Lnf in Hölder-type norms

A Hölder continuity result for a class of obstacle problems under non standard growth conditions Michela Eleuteri and Jens Habermann Michela Eleuteri, Dipartimento di Matematica di Trento via Sommarive 14, 38100 Povo (Trento) Italy; e-mail: [email protected] Jens Habermann, Department of mathematics, Friedrich-Alexander University, Bismarckstr. 1 1/2, 91054 Erlangen, Germany; e-mail: ha...

Two new approaches to the infinitesimal characterisation of quantum stochastic cocycles are reviewed. The first concerns mapping cocycles on an operator space and demonstrates the role of Hölder continuity; the second concerns contraction operator cocycles on a Hilbert space and shows how holomorphic assumptions yield cocycles enjoying an infinitesimal characterisation which goes beyond the sco...

We derive sublinear regret bounds for undiscounted reinforcement learning in continuous state space. The proposed algorithm combines state aggregation with the use of upper confidence bounds for implementing optimism in the face of uncertainty. Beside the existence of an optimal policy which satisfies the Poisson equation, the only assumptions made are Hölder continuity of rewards and transitio...

We present the first result for kernel regression where the procedure adapts locally at a point x to both the unknown local dimension of the metric space X and the unknown Hölder-continuity of the regression function at x. The result holds with high probability simultaneously at all points x in a general metric space X of unknown structure.

We study Hölder continuity of solutions to the Dirichlet problem for measures having density in L, p > 1, with respect to Hausdorff-Riesz measures of order 2n− 2 + ǫ for 0 < ǫ ≤ 2, in a bounded strongly hyperconvex Lipschitz domain and the boundary data belongs to C(∂Ω), 0 < α ≤

We show the global existence of weak solutions for a free-boundary problem arising in the non-isothermal crystallization of polymers. In particular, the free interface is shown to be of codimension one for every time t in two space dimensions; Hölder continuity of the temperature u is proven.

We consider a nonlinear (possibly) degenerate elliptic operator Lv = − div a(∇v) + b(x, v) where the functions a and b are, unnecessarly strictly, monotonic. For a given boundary datum φ we prove the existence of the maximum and the minimum of the solutions and formulate a Haar-Rado type result, namely a continuity property for these solutions that may follow from the continuity of φ. In the ho...

In a setting, where only “exit measures” are given, as they are associated with an arbitrary right continuous strong Markov process on a separable metric space, we provide simple criteria for the validity of Harnack inequalities for positive harmonic functions. These inequalities are scaling invariant with respect to a metric on the state space which, having an associated Green function, may be...