Extrapolation Theory Research Articles

Two‐weight extrapolation on function spaces and applications

AbstractThis paper is devoted to studying the extrapolation theory of Rubio de Francia on general function spaces. We present endpoint extrapolation results including , , and extrapolation in the context of Banach function spaces, and also on modular spaces. We also include several applications that can be easily obtained using extrapolation: local decay estimates for various operators, Coifman–Fefferman inequalities that can be used to show some known sharp inequalities, Muckenhoupt–Wheeden and Sawyer's conjectures are also presented for many operators, which go beyond Calderón–Zygmund operators. Finally, we obtain two‐weight inequalities for Littlewood–Paley operators and Fourier integral operators on weighted Banach function spaces.

Damage of interface between slab track bases and cable-stayed bridge under train load based on fatigue cohesion model

To investigate the fatigue damage of the interface between bases of slab tracks and the beam of a cable-stayed bridge (CSB), this study established a fatigue cohesion model based on the damage extrapolation theory. Through finite element analysis, the fatigue damage of the base interface under moving train loads was analyzed. The results showed that significant deformations occurred on the track near the piers and towers of the CSB, revealing noticeable incongruous deformations between the track and beam. Substantial stress was observed in the base and interface, which was induced by bridge deformations and approached damage criteria. The damage locations primarily occurred in the center and the ends of the base directly above the support. At the piers and towers, the fatigue life of the interface between the slab tracks and the beam was found to be 20.63 years and 45.21 years, respectively. Since the bridge deformation under train load mainly caused interface tensile cracking, the influence of normal bonding strength on interface damage was significant, while the impact of tangential bonding strength on base interface damage was less apparent. Changing the relative position of the bases and supports did not prevent damage to the base end interface completely, but can lead to an increase in fatigue life in the center significantly. Therefore, adopting a rational layout of slab tracks can effectively extend the fatigue life of the base interface. It was suggested to place base joints as close as possible to the supports at the piers and towers.

Extrapolation to mixed Herz spaces and its applications

AbstractIn this paper, we extend the extrapolation theory to mixed Herz spaces and . To prove the main result, we first study the dual spaces of mixed Herz spaces, and then give the boundedness of the Hardy–Littlewood maximal operator on mixed Herz spaces. By using the extrapolation theorems, we obtain the boundedness of many integral operators on mixed Herz spaces. We also give a new characterization of via the boundedness of commutators of some operators on mixed Herz spaces.

Calderón–Zygmund operators, Bochner–Riesz means, and parametric Marcinkiewicz integrals on Hardy–Morrey spaces with variable exponents

We obtain the boundedness of Calderón–Zygmund operators on the entire Hardy–Morrey spaces with variable exponents. We obtain this result by refining the extrapolation theory. This refined extrapolation theory also gives the boundedness of Fourier multipliers, Bochner–Riesz means, and parametric Marcinkiewicz integrals on Hardy–Morrey spaces with variable exponents.

Temperature distribution simulation and thermal properties investigation of hollow slab beam based on meteorological data

Natural environment has a great impact on the performance of bridge structures, such as the ambient temperature, solar radiation, climate extremes and so on. Due to the lack of regional pertinence in the provisions of the existing specifications on the temperature load standard, and the lack of verification of its adaptability of hollow slab beam bridge in the reconstruction and expansion project, it is necessary to study the temperature load of the beam based on the temperature data of the area where the bridge site is located. Based on the measured meteorological data, the cross-section temperature distribution characteristics of the hollow slab beam and the temperature load effect were investigated. The vertical temperature load model of the short and medium-span hollow slab beam was proposed according to the simplified temperature curve. Firstly, based on the annual measured meteorological data, the plane model was established to analyze the temperature distribution characteristics of the hollow slab beam. Secondly, according to the temperature field distribution of the beam cross-section, the solid model of 16 m hollow slab beam was established to analyze the distribution of the temperature load effect. Finally, based on the preliminarily proposed temperature curve, vertical temperature load models were proposed by introducing the theory of extreme value extrapolation. The studies have shown that the temperature of the hollow slab beam is approximately horizontal distributed, and temperature extreme value appear at the location of the beam web. The roof and floor of the hollow slab beam are compressed and the webs are tensioned in the positive temperature difference state, the stress condition is opposite under the negative temperature difference state. The positive and negative temperature load models can meet the extrapolation requirements of the bridge design reference period. This study innovatively proposed a temperature load model based on the temperature curve corresponding to the most unfavorable effect of the beam. The extrapolation idea in the derivation of vehicle load model is introduced as well. And a temperature load model suitable for the extension of hollow slab bridges in Guangdong is established in this study.

Extrapolation on function and modular spaces, and applications

We generalize the extrapolation theory of Rubio de Francia to the context of Banach function spaces and modular spaces. Our results are formulated in terms of some natural weighted estimates for the Hardy-Littlewood maximal function and are stated in measure spaces and for general Muckenhoupt bases. Finally, we give several applications in analysis and partial differential equations.

Fractional integral operators on Orlicz slice Hardy spaces

This paper gives the mapping properties of the fractional integral operators on Orlicz slice Hardy spaces. We use the extrapolation theory for Hardy type spaces to obtain this result. In particular, our result yields the mapping properties of the fractional integral operators on Hardy slice spaces.

Sublinear operators on Herz–Hardy spaces with variable exponents

AbstractIn this paper, we establish the mapping properties of sublinear operators on Herz–Hardy spaces with variable exponents. We obtain these mapping properties by extending the extrapolation theory to Herz–Hardy spaces with variable exponents. In particular, we study the mapping properties of the multiplier operators, the Littlewood–Paley functions and the parametric Marcinkiewicz integrals on the Herz–Hardy spaces with variable exponents.

Study on fracture behavior for sharp V-notched specimens combined with 3D-DIC and caustics

In this paper, the fracture behavior of Three-Point Bending (TPB) beams with V-notch under mode I and mixed mode I-II loading is investigated. For this purpose, first, an experimental system combining three-dimensional digital image correlation (3D-DIC) and a new digital laser caustic is established. Then, based on displacement field for V-notch and the displacement extrapolation theory, a new method (3D-DIC) for measuring Notch Stress Intensity Factors (NSIFs) of V-notches under mixed mode loading is presented and verified. Furthermore, based on the established system, the effect of eccentric loading and the rotation angle of V-notches on the initiation and propagation for V-notch is analysed, and the SIF results for V-notched specimen from initiation to crack arrest are provided. Obtained results demonstrate that SIF increases rapidly at first, and then decreases with the decrease of energy at the initial stage of notch propagation. Under the action of eccentric load, V-notches initiate and propagate toward the load position. In addition, the values of mode I NSIFs (KIV) and the length of failure time of these specimens are ranked as: specimen B (notch rotation angle of 0°) > specimen D (notch rotation angle of −60°) > specimen C (notch rotation angle of 60°). However, the absolute value of mode II NSIFs (KIIV) and the distance of crack arrest in X direction of these specimens are ranked as: specimen C > specimen D > specimen B. Finally, the Maximum Tensile Stress (MTS) criterion suitable for V-notches under mode I or mixed mode loading is proposed, and the validity of the criterion is verified by experimental results.

Extrapolation of compactness on weighted spaces

Let T be a linear operator that, for some p_1\in(1,\infty) , is bounded on L^{p_1}(\tilde w) for all \tilde w\in A_{p_1}(\mathbb{R}^d) and in addition compact on L^{p_1}(w_1) for some w_1\in A_{p_1}(\mathbb{R}^d) . Then T is bounded and compact on L^p(w) for all p\in(1,\infty) and all w\in A_p(\mathbb{R}^d) . This “compact version” of Rubio de Francia's celebrated weighted extrapolation theorem follows from a combination of results in the interpolation and extrapolation theory of weighted spaces on the one hand, and of compact operators on abstract spaces on the other hand. Moreover, generalizations of this extrapolation of compactness are obtained for operators that are bounded from one space to a different one (“off-diagonal estimates”) or only in a limited range of the L^p scale. As applications, we easily recover several recent results on the weighted compactness of commutators of singular integral operators, fractional integrals and pseudo-differential operators, and obtain new results about the weighted compactness of commutators of Bochner–Riesz multipliers.

Singular integrals and sublinear operators on amalgam spaces and Hardy-amalgam spaces

In this paper, we establish the extrapolation theory for the amalgam spaces and the Hardy-amalgam spaces. By using the extrapolation theory, we obtain the mapping properties for the Calderón-Zygmund operators and its commutator, the Carleson operators and establish the Rubio de Francia inequalities for Littlewood-Paley functions of arbitrary intervals to the amalgam spaces. We also obtain the boundedness of the Calder{ó}n-Zygmund operators and the intrinsic square function on the Hardy-amalgam spaces.

Calderón Operator on Local Morrey Spaces with Variable Exponents

In this paper, we establish the boundedness of the Calderón operator on local Morrey spaces with variable exponents. We obtain our result by extending the extrapolation theory of Rubio de Francia to the local Morrey spaces with variable exponents. The exponent functions of the local Morrey spaces with the exponent functions are only required to satisfy the log-Hölder continuity assumption at the origin and infinity only. As special cases of the main result, we have Hardy’s inequalities, the Hilbert inequalities and the boundedness of the Riemann–Liouville and Weyl averaging operators on local Morrey spaces with variable exponents.

Extrapolation to mixed norm spaces and applications

This paper establishes extrapolation theory to mixed norm spaces. By applying this extrapolation theory, we obtain the mapping properties of the Rubio de Francia Littlewood-Paley functions and the geometrical maximal functions on mixed norm spaces. As special cases of these results, we have the mapping properties on the mixed norm Lebesgue spaces with variable exponents and the mixed norm Lorentz spaces.

Extrapolation to weighted Morrey spaces with variable exponents and applications

AbstractThis paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

$C_p$ estimates for rough homogeneous singular integrals and sparse forms

We consider Coifman--Fefferman inequalities for rough homogeneous singular integrals $T_\Omega$ and $C_p$ weights. It was recently shown by Li-P\'erez-Rivera-R\'ios-Roncal that $$ \|T_\Omega \|_{L^p(w)} \le C_{p,T,w} \|Mf\|_{L^p(w)} $$ for every $0 \max\{1,p\}$ without using extrapolation theory. Although the bounds we prove are new even in a qualitative sense, we also give the quantitative bound with respect to the $C_q$ characteristic. Our techniques rely on recent advances in sparse domination theory and we actually prove most of our estimates for sparse forms. Our second goal is to continue the structural analysis of $C_p$ classes. We consider some weak self-improving properties of $C_p$ weights and weak and dyadic $C_p$ classes. We also revisit and generalize a counterexample by Kahanp\a\a and Mejlbro who showed that $C_p \setminus \bigcup_{q > p} C_q \neq \emptyset$. We combine their construction with techniques of Lerner to define an explicit weight class $\widetilde{C}_p$ such that $\bigcup_{q > p} C_q \subsetneq \widetilde{C}_p \subsetneq C_p$ and every $w \in \widetilde{C}_p$ satisfies Muckenhoupt's conjecture. In particular, we give a different, self-contained proof for the fact that the $C_{p+\varepsilon}$ condition is not necessary for the Coifman--Fefferman inequality and our ideas allow us to consider also dimensions higher than $1$.

Asymmetric Attention

We document that the expectations of households, firms, and professional forecasters in standard surveys simultaneously extrapolate from recent events and underreact to new information. Existing models of expectation formation, whether behavioral or rational, cannot account for these observations. We develop a rational theory of extrapolation based on limited attention, which is consistent with this evidence. In particular, we show that limited, asymmetric attention to procyclical variables can explain the coexistence of extrapolation and underreactions. We illustrate these mechanisms in a microfounded macroeconomic model, which generates expectations consistent with the survey data, and show that asymmetric attention increases business cycle fluctuations. (JEL C53, D83, D84, E23, E27, E32)

Extrapolation to product Herz spaces and some applications

Abstract This paper extends the extrapolation theory to product Herz spaces. To prove the main result, we first investigate the dual space of the product Herz space, and then show the boundedness of the strong maximal operator on product Herz spaces. By using this extrapolation theory, we establish the John–Nirenberg inequality, the characterization of little bmo, the Fefferman–Stein vector-valued inequality, the boundedness of the bi-parameter singular integral operator, the strong fractional maximal operator, and the bi-parameter fractional integral operator on product Herz spaces. We also give a new characterization of little bmo via the boundedness of the commutators of some bi-parameter operators on product Herz spaces. Even in the one-parameter setting, some of our results are new.

Singular integral operators and sublinear operators on Hardy local Morrey spaces with variable exponents

This paper establishes the mapping properties of the singular integral operators, the Littlewood-Paley functions and the maximal Bochner-Riesz means on the Hardy local Morrey spaces with variable exponents by using extrapolation theory.

Weighted inequalities for q-functions

Let f be a martingale on an arbitrary atomic probability space equipped with a tree-like structure and let S(f,q) denote the associated q-function. The paper is devoted to weighted Lp-estimates, cp,q,w−1‖S(f,q)‖ Lp(w)≤‖f‖Lp(w)≤Cp,q,w‖S(f,q)‖Lp(w),1≤p<∞, for Muckenhoupt weights. Using the combination of the theory of sparse operators, extrapolation, and Bellman function method, we identify the optimal dependence of the constants cp,q,w and Cp,q,w on the Ap characteristics of the weights involved.

Operators on Orlicz-slice spaces and Orlicz-slice Hardy spaces

We establish the mapping properties of the singular integral operators, the Fourier integral operators and the geometric maximal operators on the Orlicz-slice spaces by using extrapolation. By extending the extrapolation theory to the Hardy Orlicz-slice spaces, we also obtain the mapping properties of the maximal Bochner-Riesz means, the parametric Marcinkiewicz integrals and the multiplier operators on the Hardy Orlicz-slice spaces.