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## An endpoint estimate for maximal multilinear singular integral operators

### Analysis in Theory and Applications (2007-12-01) 23: 307-314 , December 01, 2007

A weak type endpoint estimate for the maximal multilinear singular integral operator
$$T_A^* f(x) = \mathop {\sup }\limits_{\varepsilon > 0} \left| {\int_{\left| {x - y} \right| > \varepsilon } {\frac{{\Omega (x - y)}}{{\left| {x - y} \right|^{n + 1} }}(A(x) - A(y) - \nabla A(y)(x - y))f(y)dy} } \right|$$
is established, where Θ is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, and *A* has derivatives of order one in BMO(ℝ^{n}). A regularity condition on Θ which implies an *L*log*L* type estimate of *T*_{A}^{*}
is given.

## Maximal characterizations of Herz type Hardy spaces on homogeneous groups

### Analysis in Theory and Applications (2005-12-01) 21: 301-310 , December 01, 2005

In this paper, the authors establish some characterizations of Herz-type Hardy spaces HK
_{q}^{α,p}
(G) and HK
_{q}^{α,p}
(G), where 1<q<∞, Q(1−1/q)≤α<∞, 0<p<∞ and G denotes a graded homogeneous Lie group.

## Bounds for commutators of multilinear fractional integral operators with homogeneous kernels

### Analysis in Theory and Applications (2011-06-01) 27: 181-186 , June 01, 2011

We will show bounds for commutators of multilinear fractional integral operators with some homogeneous kernels.

## On the order of summability of the Fourier inversion formula

### Analysis in Theory and Applications (2010-03-01) 26: 13-42 , March 01, 2010

In this article we show that the order of the point value, in the sense of Łojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order *k*, then its Fourier series is e.v. Cesàro summable to the distributional point value of order *k*+1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesàro summable of order *k*, then the distribution is the (*k*+1)-th derivative of a locally integrable function, and the distribution has a distributional point value of order *k*+2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.

## Mathematical estimation of physiological disturbances in human dermal parts at extreme conditions: One dimensional steady state case

### Analysis in Theory and Applications (2009-12-12) 25: 325-332 , December 12, 2009

A mathematical model for the thermoregulation in the dermal layers of the human body is proposed. The skin is composed mainly of three layers — epidermis, dermis and subcutaneous tissues. The relative constancy of the body temperature is remarkable because there is a continuous exchange of heat with the external environment as well as within the different compartments of the body. A model describes the distribution of dermal temperature as a function of internal and external parameters, such as temperature of the incoming arterial blood, blood flow, ambient temperature, and heat exchange with the environment. It is shown that substantial changes in human dermal temperature can be accomplished only through changes in the temperature of the incoming arterial blood or substantial suppression of blood flow. Other parameters can lead only to temperature changes near the skin surface.

## ε-weakly Chebyshev subspaces of banach spaces

### Analysis in Theory and Applications (2003-06-01) 19: 130-135 , June 01, 2003

We will define and characterize ε-weakly Chebyshev subspaces of Banach spaces. We will prove that all closed subspaces of a Banach space X are ε-weakly Chebyshev if and only if X is reflexive.

## The nonlinear boundary value problem for a class of integro-differential system

### Analysis in Theory and Applications (2006-09-01) 22: 254-261 , September 01, 2006

In this paper, using the theory of differential inequalities, we study the nonlinear boundary value problem for a class of integro-differential system. Under appropriate assumptions, the existence of solution is proved and the uniformly valid asymptotic expansions for arbitrary n-th order approximation and the estimation of remainder term are obtained simply and conveniently.

## Generalization of the interaction between Haar approximation and polynomial operators to higher order methods

### Analysis in Theory and Applications (2006-12-01) 22: 301-318 , December 01, 2006

In applications it is useful to compute the local average of a function f(u) of an input u from empirical statistics on u. A very simple relation exists when the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.

## BMO spaces associated to generalized parabolic sections

### Analysis in Theory and Applications (2011-03-01) 27: 1-9 , March 01, 2011

Parabolic sections were introduced by Huang[1] to study the parabolic Monge-Ampère equation. In this note, we introduce the generalized parabolic sections *P* and define BMO
_{P}^{q}
spaces related to these sections. We then establish the John-Nirenberg type inequality and verify that all BMO
_{P}^{q}
are equivalent for *q* ≥ 1.

## On Polya-Szegö inequalities

### Analysis in Theory and Applications (2005-12-01) 21: 395-398 , December 01, 2005

This paper gives two new inequalities, which improve two Polya-Szegö’s inequlities.