On the regularity of maximal operators supported by submanifolds

Liu, Feng; Wu, Huoxiong

doi:10.1016/j.jmaa.2017.03.058

Cited by 47 publications

(20 citation statements)

References 14 publications

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“…We would like to remark that the main ideas employed in the proofs of Theorems 1.1 and 1.7 are greatly motivated by [29,46], but our methods and techniques are more delicate and complex than those in [29,46]. It should be pointed out that the main ideas in the proofs of Theorems 1.2 and 1.3 are motivated by [44]. Our arguments in the proof of the bounded part in Theorem 1.7 are motivated by [10], but our methods and techniques are somewhat different and direct than those in [10].…”

Section: 2mentioning

confidence: 99%

Regularity and continuity of the multilinear strong maximal operators

Liu

Xue

Yabuta

2020

Journal de Mathématiques Pures et Appliquées

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Let m ≥ 1, in this paper, our object of investigation is the regularity and and continuity properties of the following multilinear strong maximal operatorwhere x ∈ R d and R denotes the family of all rectangles in R d with sides parallel to the axes. When m = 1, denote MR by MR. Then, MR coincides with the classical strong maximal function initially studied by Jessen, Marcinkiewicz and Zygmund. We showed that MR is bounded and continuous from the Sobolev spacesAs a consequence, we further showed that MR is bounded and continuous from the fractional Sobolev spaces W s,p 1 (R d ) × · · · × W s,pm (R d ) to W s,p (R d ) for 0 < s ≤ 1 and 1 < p < ∞. As an application, we obtain a weak type inequality for the Sobolev capacity, which can be used to prove the p-quasicontinuity of MR. In addition, we proved that MR( f ) is approximately differentiable a.e. if f = (f1, . . . , fm) with each fj ∈ L 1 (R d ) being approximately differentiable a.e. The discrete type of the strong maximal operators has also been considered. We showed that this discrete type of the maximal operators enjoys somewhat unexpected regularity properties.Key words and phrases. Multilinear strong maximal operators, discrete multilinear strong maximal operators, Sobolev spaces, regularity, Tribel-Lizorkin spaces and Besov spaces, mixed Lebesgue spaces and Sobolev spaces, Sobolev capacity.

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Section: 2mentioning

confidence: 99%

Regularity and continuity of the multilinear strong maximal operators

Liu

Xue

Yabuta

2020

Journal de Mathématiques Pures et Appliquées

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“…The identification method presented in this paper can combine iteration [37,38] and the data filtering methods to study the identification problems of linear, bilinear and nonlinear systems with different structure and disturbance noise [40][41][42]. Some mathematical skills [43][44][45][46][47][48] and statistical methods [49][50][51][52][53][54][55] can be used to study the performances of parameter estimation algorithms.…”

Section: Discussionmentioning

confidence: 99%

State filtering‐based least squares parameter estimation for bilinear systems using the hierarchical identification principle

Zhang

Ding

et al. 2018

IET Control Theory & Applications

128

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Abstract:In the revised paper, we have highlighted the changes made in Blue FONT. This paper presents a combined parameter and state estimation algorithm for a bilinear system described by its observer canonical state space model based on the hierarchical identification principle. The Kalman filter is known as the best state filter for linear systems, but not applicable for bilinear systems. Thus, a bilinear state observer (BSO) is designed to give the state estimates using the extremum principle. Then a BSO based recursive least squares (BSO-RLS) algorithm is developed. For comparison with the BSO-RLS algorithm, by dividing the system into three fictitious subsystems on basis of the decomposition-coordination principle, a BSO based hierarchical least squares algorithm is proposed to reduce the computation burden. Moreover, a BSO based forgetting factor recursive least squares algorithm is presented to improve the parameter tracking capability. Finally, a numerical example illustrates the effectiveness of the proposed algorithms. IntroductionParameter estimation is a significant part in system identification, and has been widely used in system analysis [1][2][3], system modeling [4][5][6][7], and system control [8,9]. Since many industrial processes are complex and inherently nonlinear, nonlinear system identification has drawn much attention throughout the world [10][11][12]. The bilinear approach for modeling these complex processes are proven to be more precise than any other traditional linear models [13]. However, the nonlinear term existing in the bilinear model brings some challenges for bilinear system identification. During the past decades, much work has been carried out on parameter estimation for bilinear systems. For example, dos Santos et al. presented a subspace identification method for bilinear systems by treating the bilinear term as a second-order white noise process [14]. Larkowski et al. addressed the identification problem of the diagonal bilinear errors-in-variables system and extended the bias compensated least squares technique to bilinear systems [15]. Li et al. applied the polynomial transformation technique to obtain the equivalent input-output representation of the bilinear system and proposed the iterative algorithm for parameter estimation [16][17][18]. The recursive least squares (RLS) approach is known as the most commonly used estimation method among numerous different parameter estimation techniques. Although the RLS method offers a fast convergence rate, there exists several problems such as the increase in the computational burden and the decline in the tracking capability [19,20]. The hierarchical identification principle is applied to decompose a bilinear system into several subsystems Nonlinear filtering techniques have attracted much attention in signal processing [26,27] and have wide applications in many areas [28][29][30][31]. The classical Kalman filter (KF) is recognized as the best linear filter for linear systems under Gaussian noises. However, it is not suitable f...

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“…Later on, Luiro established the continuity of M on

F_{s}^{p, q} ({double-struckR}^{d})

for

0 < s < 1

and

1 < p, q < \infty

. Recently, Liu and Wu extended the above results to the maximal operators associated with polynomial mappings.…”

Section: Introductionmentioning

confidence: 97%

“…This question was investigated by Luiro [28] and later extensions were given in [7,21,29]. More interesting works related to this topic may be found in [2,5,6,8,13,22,[25][26][27].…”

Section: Introductionmentioning

confidence: 99%