### Citations

3796 | Elliptic Partial Differential Equations of Second Order - Gilbarg, Trudinger - 1998 |

2332 |
Singular Integrals and Differentiability Properties of Functions
- Stein
- 1971
(Show Context)
Citation Context ...ties. However, it may be very difficult to find estimates for "weighted" Hausdorff measures. 104 Tero Kilpelainen 2.13. Bessel potentials and capacity. By Calder'on's well known theorem [C] =-=(see also [S]-=-) the Sobolev space H 1;p (R n ; dx) is equivalent to the space of Bessel potentials G 1sf , f 2 L p (R n ; dx) . Recall that the Bessel kernel G 1 is the function whose Fourier transform is G 1 (x) =... |

142 |
The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations
- Fabes, Kenig, et al.
- 1982
(Show Context)
Citation Context ... (R n ) with respect to the norm k'k 1;p;w = `Z \Omega j'j w(x) dx ' 1=p + `Z \Omega jr'j w(x) dx ' 1=p : This definition is useful when one studies degenerate elliptic partial differential equations =-=[FKS]-=-, [HKM]. Another approach, for example followed by Kufner [K], is to define the weighted Sobolev space W 1;p w) as the class of functions u such that both u and its distributional gradient ru belong t... |

121 |
Real-Variable Methods in Harmonic Analysis,
- Torchinsky
- 1986
(Show Context)
Citation Context ... that w 2 A q whenever qsp . A more intriguing fact is that there exists an " ? 0 such that w 2 A p\Gamma" , as well. For these and other properties of A p -weights we refer to the monograph=-=s [GCRF], [T]-=-, and [HKM]. As an example, we have that the function w(x) = jxj fl is an A p -weight if and only if \Gamman ! fl ! n (p \Gamma 1) . Moreover, positive superharmonic functions in R n , and more genera... |

114 | Weighted Sobolev Spaces,” - Kufner - 1985 |

92 |
Weakly differentiable functions, Graduate texts in mathematics, 120,
- Ziemer
- 1989
(Show Context)
Citation Context ...ubject Classification: Primary 46E35; Secondary 31C15. 96 Tero Kilpelainen means of (1; p)-capacity: unweighted Sobolev functions possess Lebesgue points except on a set of (1; p)-capacity zero [MK], =-=[Z]-=-. That the corresponding refinement can also be made in the weighted theory seems to belong to the folklore, but we have not been able to find these results in the existing literature. In the present ... |

55 |
Martio O.: Nonlinear potential theory of degenerate elliptic equations
- Heinonen, Kilpeläinen
- 1993
(Show Context)
Citation Context ...position of the behavior of functions in weighted Sobolev spaces and this leads us to use a concept of capacity. The motivation arises from the theory of partial differential equations, see e.g. [F], =-=[HKM]-=-. Most of the results we present are probably not new but according to our knowledge they have not yet appeared in printed form. We define the weighted Sobolev space H 1;p w) to be the completion of C... |

23 | Extension theorems on weighted Sobolev spaces, - Chua - 1992 |

16 |
Capacity methods in the theory of partial differential equations
- Frehse
- 1982
(Show Context)
Citation Context ...nt exposition of the behavior of functions in weighted Sobolev spaces and this leads us to use a concept of capacity. The motivation arises from the theory of partial differential equations, see e.g. =-=[F]-=-, [HKM]. Most of the results we present are probably not new but according to our knowledge they have not yet appeared in printed form. We define the weighted Sobolev space H 1;p w) to be the completi... |

10 | Sobolev spaces and pseudodifferential operators with smooth symbols - Miller, Weighted - 1982 |

8 |
A nonlinear potential theory
- Maz′ya, Havin
- 1974
(Show Context)
Citation Context ...tics Subject Classification: Primary 46E35; Secondary 31C15. 96 Tero Kilpelainen means of (1; p)-capacity: unweighted Sobolev functions possess Lebesgue points except on a set of (1; p)-capacity zero =-=[MK]-=-, [Z]. That the corresponding refinement can also be made in the weighted theory seems to belong to the folklore, but we have not been able to find these results in the existing literature. In the pre... |

7 | Quasi topologies and rational approximation, - Bagby - 1972 |

4 |
Les espaces du type de Beppo Levi.
- Deny, Lions
- 1953
(Show Context)
Citation Context ...t W 1;p (\Omega\Gamma w) be the set of all functions u 2 L p(\Omega\Gamma w) whose distributional gradientsru belongs to L p(\Omega\Gamma w) . A well known theorem of Meyers and Serrin [MS] (see also =-=[DL]-=-) states that in the unweighted case these two definitions result in the same function spaces. We extend this result to the weighted situation. 2.5. Theorem. H 1;p (\Omega\Gamma w) = W 1;p w) . Proof.... |

3 |
Hausdorff measures, capacities, and Sobolev spaces with weights
- Nieminen
- 1991
(Show Context)
Citation Context ...the Hausdorff dimension of E can be derived from known results for unweighted capacities (cf. [HKM, 2.26, 2.27]). But the sets E with 0 2 E cause troubles. A way out of this difficulty is proposed in =-=[N], where the author c-=-onsiders "weighted" Hausdorff measures and their connection with capacities. However, it may be very difficult to find estimates for "weighted" Hausdorff measures. 104 Tero Kilpela... |

2 |
Lebesgue spaces of differentiable functions and distributions
- on
- 1961
(Show Context)
Citation Context ...on with capacities. However, it may be very difficult to find estimates for "weighted" Hausdorff measures. 104 Tero Kilpelainen 2.13. Bessel potentials and capacity. By Calder'on's well know=-=n theorem [C]-=- (see also [S]) the Sobolev space H 1;p (R n ; dx) is equivalent to the space of Bessel potentials G 1sf , f 2 L p (R n ; dx) . Recall that the Bessel kernel G 1 is the function whose Fourier transfor... |

2 |
de Francia: Weighted norm inequalities and related topics
- ia--Cuerva, J, et al.
- 1985
(Show Context)
Citation Context ...;ps1 and that w 2 A q whenever qsp . A more intriguing fact is that there exists an " ? 0 such that w 2 A p\Gamma" , as well. For these and other properties of A p -weights we refer to the m=-=onographs [GCRF]-=-, [T], and [HKM]. As an example, we have that the function w(x) = jxj fl is an A p -weight if and only if \Gamman ! fl ! n (p \Gamma 1) . Moreover, positive superharmonic functions in R n , and more g... |