Estimates for solutions of the nonstationary stokes problem in anisotropic Sobolev spaces with mixed norm

1997 ◽  
Vol 87 (5) ◽  
pp. 3859-3877 ◽  
Author(s):  
P. Maremonti ◽  
V. A. Solonnikov
Keyword(s):  
Sobolev Spaces ◽  
Stokes Problem ◽  
Mixed Norm ◽  
Anisotropic Sobolev Spaces ◽  
Nonstationary Stokes Problem

scholarly journals A class of non-parametric statistical manifolds modelled on Sobolev space

2019 ◽  
Vol 2 (2) ◽  
pp. 283-312 ◽  
Author(s):  
Nigel J. Newton
Keyword(s):  
Sobolev Spaces ◽  
Nonlinear Filtering ◽  
Linear Growth ◽  
Weighted Sobolev Spaces ◽  
Planck Equation ◽  
Model Space ◽  
Sobolev Embedding ◽  
Mixed Norm ◽  
Nonlinear Superposition ◽  
Non Parametric

AbstractWe construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on $${\mathbb {R}}^d$$Rd, each containing a smoothly embedded submanifold of probability measures. The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert–Sobolev spaces and mixed-norm spaces, and support the Fisher–Rao metric as a weak Riemannian metric. Densities are expressed in terms of a deformed exponential function having linear growth. Unusually for the Sobolev context, and as a consequence of its linear growth, this “lifts” to a nonlinear superposition (Nemytskii) operator that acts continuously on a particular class of mixed-norm model spaces, and on the fixed norm space $$W^{2,1}$$W2,1; i.e. it maps each of these spaces continuously into itself. In contrast with non-parametric exponential manifolds, the density itself belongs to the model space, and the range of the chart is the whole of this space. Some of the results make essential use of a log-Sobolev embedding theorem, which also sharpens existing results concerning the regularity of statistical divergences on the manifolds. Applications to the stochastic partial differential equations of nonlinear filtering (and hence to the Fokker–Planck equation) are outlined.

scholarly journals Some notes on embedding for anisotropic Sobolev spaces

2011 ◽  
Vol 61 (1) ◽  
pp. 97-111 ◽  
Author(s):  
Hongliang Li ◽  
Quinxiu Sun
Keyword(s):  
Sobolev Spaces ◽  
Anisotropic Sobolev Spaces

M-CHANNEL MRA AND APPLICATION TO ANISOTROPIC SOBOLEV SPACES

2005 ◽  
Vol 03 (02) ◽  
pp. 153-166
Author(s):  
ELENA CORDERO
Keyword(s):  
Tensor Product ◽  
Sobolev Spaces ◽  
Unit Interval ◽  
Biorthogonal Wavelet ◽  
Bernstein Type ◽  
Wavelet Bases ◽  
Compactly Supported ◽  
Anisotropic Sobolev Spaces ◽  
Dilation Factor ◽  
Bernstein Type Inequalities

In this paper we construct compactly supported biorthogonal wavelet bases of the interval, with dilation factor M. Next, the natural MRA on the cube arising from the tensor product of a multilevel decomposition of the unit interval is developed. New Jackson and Bernstein type inequalities are proved, providing a characterization for anisotropic Sobolev spaces.

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