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We introduce p-quasilocal operators and prove that if a sublinear operator T is p-quasilocal and bounded from to then it is also bounded from the classical Hardy space to (0 < p ≤ 1). As an application it is shown that the maximal operator of the one-parameter Cesàro means of a distribution is bounded from to (3/4 < p ≤ ∞) and is of weak type . We define the two-dimensional dyadic hybrid Hardy space and verify that the maximal operator of the Cesàro means of a two-dimensional...
We consider the maximal function where and 0 < a < 1. We prove the global estimate
, s > a/4,
with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.
We characterize the UMD-property of a Banach space X by sequences of ideal norms associated with trigonometric orthonormal systems. The asymptotic behavior of those numerical parameters can be used to decide whether X is a UMD-space. Moreover, if this is not the case, we obtain a measure that shows how far X is from being a UMD-space. The main result is that all described sequences are not only simultaneously bounded but are also asymptotically equivalent.
We present an estimation of the and means as approximation versions of the Totik type generalization (see [5], [6]) of the result of G. H. Hardy, J. E. Littlewood. Some corollaries on the norm approximation are also given.
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