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Mixed norms and Sobolev type inequalities

V. I. Kolyada (2006)

Banach Center Publications

We study mixed norm spaces that arise in connection with embeddings of Sobolev and Besov spaces. We prove Sobolev type inequalities in terms of these mixed norms. Applying these results, we obtain optimal constants in embedding theorems for anisotropic Besov spaces. This gives an extension of the estimate proved by Bourgain, Brezis and Mironescu for isotropic Besov spaces.

Mixed-norm spaces and interpolation

Joaquín Ortega, Joan Fàbrega (1994)

Studia Mathematica

Let D be a bounded strictly pseudoconvex domain of n with smooth boundary. We consider the weighted mixed-norm spaces A δ , k p , q ( D ) of holomorphic functions with norm f p , q , δ , k = ( | α | k ʃ 0 r 0 ( ʃ D r | D α f | p d σ r ) q / p r δ q / p - 1 d r ) 1 / q . We prove that these spaces can be obtained by real interpolation between Bergman-Sobolev spaces A δ , k p ( D ) and we give results about real and complex interpolation between them. We apply these results to prove that A δ , k p , q ( D ) is the intersection of a Besov space B s p , q ( D ) with the space of holomorphic functions on D. Further, we obtain several properties of the mixed-norm...

Model selection for regression on a random design

Yannick Baraud (2002)

ESAIM: Probability and Statistics

We consider the problem of estimating an unknown regression function when the design is random with values in k . Our estimation procedure is based on model selection and does not rely on any prior information on the target function. We start with a collection of linear functional spaces and build, on a data selected space among this collection, the least-squares estimator. We study the performance of an estimator which is obtained by modifying this least-squares estimator on a set of small probability....

Model selection for regression on a random design

Yannick Baraud (2010)

ESAIM: Probability and Statistics

We consider the problem of estimating an unknown regression function when the design is random with values in k . Our estimation procedure is based on model selection and does not rely on any prior information on the target function. We start with a collection of linear functional spaces and build, on a data selected space among this collection, the least-squares estimator. We study the performance of an estimator which is obtained by modifying this least-squares estimator on a set of small...

Modifications of the double arrow space and related Banach spaces C(K)

Witold Marciszewski (2008)

Studia Mathematica

We consider the class of compact spaces K A which are modifications of the well known double arrow space. The space K A is obtained from a closed subset K of the unit interval [0,1] by “splitting” points from a subset A ⊂ K. The class of all such spaces coincides with the class of separable linearly ordered compact spaces. We prove some results on the topological classification of K A spaces and on the isomorphic classification of the Banach spaces C ( K A ) .

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