On a relation between norms of the maximal function and the square function of a martingale

2013 ◽  
Vol 132 (1) ◽  
pp. 13-26
Author(s):  
Masato Kikuchi
Keyword(s):  
Maximal Function ◽  
Square Function

Multi-parameter Carnot-Carath´eodory Geometry

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Singular Integral ◽  
Maximal Function ◽  
Integral Operators ◽  
Product Type ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Square Function ◽  
Frobenius Theorem ◽  
Parameter Theory

This chapter develops the theory of multi-parameter Carnot–Carathéodory geometry, which is needed to study singular integral operators. In the case when the balls are of product type, all of the results are simple variants of results in the single-parameter theory. When the balls are not of product type, these ideas become more difficult. What saves the day is the quantitative Frobenius theorem given in Chapter 2. This can be used to estimate certain integrals, as well as develop an appropriate maximal function and an appropriate Littlewood–Paley square function, all of which are essential to our study of singular integral operators.

scholarly journals Estimates for the Multiplicative Square Function of Solutions to Nondivergence Elliptic Equations

2007 ◽  
Vol 2007 ◽  
pp. 1-13
Author(s):  
Jorge Rivera-Noriega
Keyword(s):  
Elliptic Equation ◽  
Harmonic Function ◽  
Maximal Function ◽  
Lipschitz Domain ◽  
Elliptic Equations ◽  
Borel Measure ◽  
Square Function ◽  
Square Functions ◽  
Nondivergence Elliptic Equation ◽  
Nontangential Maximal Function

We prove distributional inequalities that imply the comparability of theLpnorms of the multiplicative square function ofuand the nontangential maximal function oflogu, whereuis a positive solution of a nondivergence elliptic equation. We also give criteria for singularity and mutual absolute continuity with respect to harmonic measure of any Borel measure defined on a Lipschitz domain based on these distributional inequalities. This extends recent work of M. González and A. Nicolau where the term multiplicative square functions is introduced and where the case whenuis a harmonic function is considered.

Error-aware Design Procedure to Implement Energy-efficient Approximate Squaring Hardware

2020 ◽  
Vol 10 (4) ◽  
pp. 471-477
Author(s):  
Merin Loukrakpam ◽  
Ch. Lison Singh ◽  
Madhuchhanda Choudhury
Keyword(s):  
Energy Saving ◽  
Relative Error ◽  
Energy Efficient ◽  
Piecewise Linear ◽  
Arithmetic Operation ◽  
Design Procedure ◽  
Digital Signal ◽  
Maximum Relative Error ◽  
Square Function ◽  
Efficient Design

Background:: In recent years, there has been a high demand for executing digital signal processing and machine learning applications on energy-constrained devices. Squaring is a vital arithmetic operation used in such applications. Hence, improving the energy efficiency of squaring is crucial. Objective:: In this paper, a novel approximation method based on piecewise linear segmentation of the square function is proposed. Methods: Two-segment, four-segment and eight-segment accurate and energy-efficient 32-bit approximate designs for squaring were implemented using this method. The proposed 2-segment approximate squaring hardware showed 12.5% maximum relative error and delivered up to 55.6% energy saving when compared with state-of-the-art approximate multipliers used for squaring. Results: The proposed 4-segment hardware achieved a maximum relative error of 3.13% with up to 46.5% energy saving. Conclusion:: The proposed 8-segment design emerged as the most accurate squaring hardware with a maximum relative error of 0.78%. The comparison also revealed that the 8-segment design is the most efficient design in terms of error-area-delay-power product.

scholarly journals Some estimates for the commutators of multilinear maximal function on Morrey-type space

2021 ◽  
Vol 19 (1) ◽  
pp. 515-530
Author(s):  
Xiao Yu ◽  
Pu Zhang ◽  
Hongliang Li
Keyword(s):  
Maximal Function ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Bounded Mean Oscillation ◽  
Generalized Morrey Spaces ◽  
Type Space ◽  
Bmo Function ◽  
Mean Oscillation ◽  
Multilinear Maximal Function ◽  
Equivalent Conditions

Abstract In this paper, we study the equivalent conditions for the boundedness of the commutators generated by the multilinear maximal function and the bounded mean oscillation (BMO) function on Morrey space. Moreover, the endpoint estimate for such operators on generalized Morrey spaces is also given.

scholarly journals VAT Efficiency—A Discussion on the VAT System in the European Union

2021 ◽  
Vol 13 (9) ◽  
pp. 4768
Author(s):  
Anna Kowal ◽  
Grzegorz Przekota
Keyword(s):  
European Union ◽  
Tax Evasion ◽  
Value Added ◽  
Square Function ◽  
The European Union ◽  
Tax System ◽  
Aggregated Data ◽  
Tax Gap ◽  
Standard Rate ◽  
Tax Fraud

The effectiveness of the tax system can be analysed in various ways. According to the authors one of manifestations of such effectiveness is resistance to tax evasion. This phenomenon is influenced by multiple factors, with few being the level of VAT rates and the number of rates in force in the country concerned. The aim of the considerations is therefore to analyse how the standard VAT rate as well as the number of rates affect the effectiveness of this tax. The research was based on a literature query in the field of value added tax in the European Union. In addition, the problem of tax evasion was indicated and the aggregated data on the size of the tax gap in the Member States were presented. Then, there are the results of the research for 27 European Union countries for 2011–2019. The efficiency of VAT collection was modelled using square function, determining the significance of the parameters of this function, as well as the value of abscissa, which made it possible to group the countries based on how they maintained the efficiency of VAT collection over the analysed period of time. The final part of the study concentrates on the relationship between the efficiency of tax collection and the amount of both the basic rate and the number of rates. The conclusions of the research are as follows: a tax system with a small number of reduced rates, and preferably with one relatively low standard rate, is the system least susceptible to tax fraud. The research also shows a positive correlation between the value of the basic VAT rate along with the number of preferential rates and the scale of the tax gap, i.e., in countries with a higher standard VAT rate and a greater number of preferential rates, the tax gap is greater. The study will enable further investigation into the strategy of determining the optimal VAT rate and the process of its unification. Proposed changes may contribute to increasing the efficiency of VAT administration in EU countries, reducing the shadow economy, tax fraud and positively influencing economic growth.

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