Let g be a holomorphic map of B, where B is the unit ball of C. Let 0 < p < +∞,−n − 1 < q < +∞, q > −1 and α > 0. This paper gives some necessary and sufficient conditions for the Extended Cesáro Operators induced by g to be bounded or compact between generalized Besov space B(p, q) and αBloch space B.

We want to recover a continuous function f : (0, 1)d → C using only its function values. Let us assume, that f is from the unit ball of some function space (for example a fractional Sobolev space or a Besov space) and the precision of the reconstruction is measured in the norm of another function space of this type. We describe the rate of convergence of the optimal sampling method (linear as w...

We prove that the multiplier algebra of the Drury-Arveson Hardy space H n on the unit ball in C n has no corona in its maximal ideal space, thus generalizing the famous Corona Theorem of L. Carleson in the unit disk. This result is obtained as a corollary of the Toeplitz Corona Theorem and a new Banach space result: the Besov-Sobolev space B p has the ”baby corona property” for all 0 ≤ σ < n p ...

This paper studies the global well-posedness of solutions to the Cauchy problem of incompressible magneto-hydrodynamics system for large initial data in homogeneous Besov space Ḃ 2 p −1 p,r (R) for 2 < p < ∞ and 1 ≤ r < ∞. In the case of spatial dimension n ≥ 3 we establish the global well-posedness of solution for small data and the local one for large data in Besov space Ḃ n p −1 p,r (R), 1 ≤...

This paper is concerned with problems in the context of the theoretical foundation of adaptive (wavelet) algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Never...

The representational change of input along the intermediate layers is an important aspect of understanding deep learning architectures. To this end, we propose an approach that relies on the foundation of Function Space theory. In particular, we argue that a weak-type Besov smoothness index can quantify the geometry of the clustering in the feature space of each layer. Therefore, our approach m...