The Luzin theorem for higher-order derivatives

Francos, Greg

doi:10.1307/mmj/1347040255

Cited by 10 publications

(5 citation statements)

References 3 publications

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“…On the other hand, generalizing Whitney's result [30] to higher orders of differentiability, Isakov [22] and Liu and Tai [25] independently established that a function f : R n → R has the Lusin property of class C k if and only if f is approximately differentiable of order k almost everywhere (and if and only if f has an approximate (k − 1)-Taylor polynomial at almost every point). See also [20,30] for related results.…”

Section: Introduction and Main Resultsmentioning

confidence: 95%

Lusin-type properties of convex functions and convex bodies

Azagra¹,

Hajłasz²

2020

Preprint

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We prove that if f : R n → R is convex and A ⊂ R n has finite measure, then for any ε > 0 there is a convex function g :As an application we deduce that if W ⊂ R n is a compact convex body then, for every ε > 0, there exists a convex bodyfor some linear function ℓ. A consequence of this result is that, if S is the boundary of some convex set with nonempty interior (not necessarily bounded) in R n and S does not contain any line, then for every ε > 0 there exists a convex hypersurface S ε of class C 1,1 loc such that H n−1 (S \ S ε ) < ε.

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Section: Introduction and Main Resultsmentioning

confidence: 95%

Lusin-type properties of convex functions and convex bodies

Azagra¹,

Hajłasz²

2020

Preprint

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“…Then, by the same arguments as in Section 5 of [10] with Theorem 4.1 in place of [10,Theorem 4.1], we obtain the following well known result (cf. [2,14,16]).…”

Section: Alberti's Theoremmentioning

confidence: 99%

Some Results About the Structure of Primitivity Domains for Linear Partial Differential Operators with Constant Coefficients

Delladio

2022

Mediterr. J. Math.

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Let G(D) be a linear partial differential operator on $${\mathbb {R}}^n$$ R n , with constant coefficients. Moreover let $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n be open and $$F\in L^1_{\text {loc}} (\Omega , {\mathbb {C}}^N)$$ F ∈ L loc 1 ( Ω , C N ) . Then any set of the form $$\begin{aligned} A_{f,F}:= \{ x\in \Omega \, \vert \, (G(D)f)(x)=F(x)\}, \text { with }f\in W^{g,1}_{\text {loc}}(\Omega , {\mathbb {C}}^k) \end{aligned}$$ A f , F : = { x ∈ Ω | ( G ( D ) f ) ( x ) = F ( x ) } , with f ∈ W loc g , 1 ( Ω , C k ) is said to be a G-primitivity domain of F. We provide some results about the structure of G-primitivity domains of F at the points of the (suitably defined) G-nonintegrability set of F. A Lusin type theorem for G(D) is also provided. Finally, we give applications to the Maxwell type system and to the multivariate Cauchy-Riemann system.

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“…In addition, we remark brie y that the results of Alberti and Moonens-Pfe er have been generalized to higher order derivatives on Euclidean space in the work of Francos [10] and Hajłasz-Mirra [11], though we do not pursue those lines here.…”

Section: Here C > Is a Constant That Depends Only On Kmentioning

confidence: 99%

Lusin-type Theorems for Cheeger Derivatives on Metric Measure Spaces

David

2015

Analysis and Geometry in Metric Spaces

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A theorem of Lusin states that every Borel function on R is equal almost everywhere to the derivative of a continuous function. This result was later generalized to R n in works of Alberti and Moonens-Pfe er.In this note, we prove direct analogs of these results on a large class of metric measure spaces, those with doubling measures and Poincaré inequalities, which admit a form of di erentiation by a famous theorem of Cheeger.

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The Luzin theorem for higher-order derivatives

Cited by 10 publications

References 3 publications

Lusin-type properties of convex functions and convex bodies

Lusin-type properties of convex functions and convex bodies

Some Results About the Structure of Primitivity Domains for Linear Partial Differential Operators with Constant Coefficients

Lusin-type Theorems for Cheeger Derivatives on Metric Measure Spaces

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