scholarly journals H-Distributions: An Extension ofH-Measures to anLp-LqSetting

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Nenad Antonić ◽  
Darko Mitrović
Keyword(s):  
Fourier Multiplier ◽  
Fourier Multiplier Operators ◽  
Multiplier Operators

We use the continuity of Fourier multiplier operators onLpto introduce theH-distributions—an extension ofH-measures in theLpframework. We apply theH-distributions to obtain anLpversion of the localisation principle and reprove the MuratLp-Lp′variant of div-curl lemma.

scholarly journals Some Weighted Estimates for Multilinear Fourier Multiplier Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Zengyan Si
Keyword(s):  
Sobolev Spaces ◽  
Fourier Multiplier ◽  
Multilinear Operators ◽  
Multiplier Theorem ◽  
Weighted Estimates ◽  
Fourier Multiplier Operators ◽  
Multiplier Operators

We first provide a weighted Fourier multiplier theorem for multilinear operators which extends Theorem 1.2 in Fujita and Tomita (2012) by usingLr-based Sobolev spaces (1<r≤2). Then, by using a different method, we obtain a result parallel to Theorem 6.2 which is an improvement of Theorem 1.2 under assumption (i) in Fujita and Tomita (2012).

scholarly journals Smoothness of functions versus smoothness of approximation processes

2020 ◽  
Vol 10 (03) ◽  
pp. 2030002
Author(s):  
Yu. S. Kolomoitsev ◽  
S. Yu. Tikhonov
Keyword(s):  
Banach Spaces ◽  
Fourier Multiplier ◽  
Moduli Of Smoothness ◽  
Geometric Properties ◽  
Wavelet Approximation ◽  
Multiplier Theorems ◽  
Fourier Multiplier Operators ◽  
Multiplier Operators ◽  
Comprehensive Study

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: The first based on geometric properties of Banach spaces and the second on Littlewood–Paley and Hörmander-type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the [Formula: see text]-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on [Formula: see text], [Formula: see text], [Formula: see text], nonlinear wavelet approximation, etc.

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