A weak estimate of Calder�n-Zygmund operators

1994 ◽  
Vol 20 (4) ◽  
pp. 383-394
Author(s):  
Dashan Fan
Keyword(s):  
Weak Estimate

Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group

2021 ◽  
Vol 19 (1) ◽  
pp. 316-328
Author(s):  
Yangkendi Deng ◽  
Xingsong Zhang ◽  
Dunyan Yan ◽  
Mingquan Wei
Keyword(s):  
Heisenberg Group ◽  
Weak Type ◽  
Acta Math ◽  
Hardy Operator ◽  
Sharp Constant ◽  
Hausdorff Operator ◽  
Weak Estimate ◽  
Hausdorff Operators

Abstract This paper is devoted to the weak and strong estimates for the linear and multilinear fractional Hausdorff operators on the Heisenberg group H n {{\mathbb{H}}}^{n} . A sharp strong estimate for T Φ m {T}_{\Phi }^{m} is obtained. As an application, we derive the sharp constant for the product Hardy operator on H n {{\mathbb{H}}}^{n} . Some weak-type ( p , q ) \left(p,q) ( 1 ≤ p ≤ ∞ ) \left(1\le p\le \infty ) estimates for T Φ , β {T}_{\Phi ,\beta } are also obtained. As applications, we calculate some sharp weak constants for the fractional Hausdorff operator on the Heisenberg group. Besides, we give an explicit weak estimate for T Φ , β → m {T}_{\Phi ,\overrightarrow{\beta }}^{m} under some mild assumptions on Φ \Phi . We extend the results of Guo et al. [Hausdorff operators on the Heisenberg group, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 11, 1703–1714] to the fractional setting.

scholarly journals A weak estimate of the fractal dimension of the Mandelbrot boundary

1989 ◽  
Vol 136 (7-8) ◽  
pp. 358-362 ◽  
Author(s):  
Bruce Elenbogen ◽  
Thomas Kaeding
Keyword(s):  
Fractal Dimension ◽  
Weak Estimate

scholarly journals The Gradient Superconvergence of Bilinear Finite Volume Element for Elliptic Problems

2016 ◽  
Vol 9 (4) ◽  
pp. 579-594
Author(s):  
Tie Zhang ◽  
Lixin Tang
Keyword(s):  
Finite Volume ◽  
Bilinear Form ◽  
Elliptic Problems ◽  
Volume Element ◽  
Average Gradient ◽  
Weak Estimate ◽  
Finite Volume Element ◽  
Image Position

AbstractWe study the gradient superconvergence of bilinear finite volume element (FVE) solving the elliptic problems. First, a superclose weak estimate is established for the bilinear form of the FVE method. Then, we prove that the gradient approximation of the FVE solution has the superconvergence property:where denotes the average gradient on elements containing point P and S is the set of optimal stress points composed of the mesh points, the midpoints of edges and the centers of elements.

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