The paper improves the classical uniqueness result for the Euler system in the n dimensional case assuming that ∇u E ∈ L 1 (0, T ; BM O(Ω)), only. Moreover the rate of the convergence for the inviscid limit of solutions to the Navier-Stokes equations is obtained, provided the same regularity of the limit Eulerian flow. A key element of the proof is a logarithmic inequality between the Hardy and...

We study here the error of numerical integration on metric measure spaces adapted to a decomposition of the space into disjoint subsets. We consider both the error for a single given function, and the worst case error for all functions in a given class of potentials. The main tools are the classical Marcinkiewicz– Zygmund inequality and ad hoc definitions of function spaces on metric measure sp...

Let H B denote the space of all holomorphic functions on the unit ball B ⊂ C. This paper investigates the following integral-type operator with symbol g ∈ H B , Tgf z ∫1 0 f tz Rg tz dt/t, f ∈ H B , z ∈ B, whereRg z ∑n j 1 zj∂g/∂zj z is the radial derivative of g. We characterize the boundedness and compactness of the integral-type operators Tg from general function spaces F p, q, s to Zygmund-...

In this paper we obtain an explicit formula of Cauchy--Szego kernel for quaternionic Siegel upper half space, and then based on this, prove that the projection Heisenberg group is a Calderon--Zygmund operator via verifying size regularity conditions kernel. Next, also suitable version pointwise lower bound kernel, which further implies characterisations boundedness compactness commutator BMO VM...

We characterize the compactness of commutators in Bloom setting. Namely, for a suitably non-degenerate Calderón–Zygmund operator T, and pair weights σ , ω ∈ A p commutator [ T b ] is compact from L ( ) → if only V M O ν where = / 1 . This extends work first author, Holmes Wick. The weighted VMO spaces are different classical space. In dimension d compactly supported smooth functions dense but t...

It is shown that multilinear Calderón-Zygmund operators are bounded on products of Hardy spaces.

Marcinkiewicz-Zygmund Inequalities involving the zeros {xkn} of Jacobi polynomials for the weight wα,β can take the form

We study the computation complexity of deep ReLU (Rectified Linear Unit) neural networks for approximation functions from Hölder–Zygmund space mixed smoothness defined on d-dimensional unit cube when dimension d may be very large. The error is measured in norm isotropic Sobolev space. For every function f smoothness, we explicitly construct a network having an output that approximates with pres...