Newtonian Spaces Based on Quasi-Banach Function Lattices

Malý, Lukáš

doi:10.7146/math.scand.a-24188

Cited by 16 publications

(24 citation statements)

References 16 publications

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“…A statement holds for Mod X -a.e. curve γ if the family of exceptional curves Γ e , for which the statement fails, has zero X -modulus, i.e., if there is a Borel function ρ ∈ X such that´γ ρ ds = ∞ for every curve γ ∈ Γ e (see Malý [34,Proposition 4.8]).…”

Section: Preliminariesmentioning

confidence: 99%

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Regularization of Newtonian functions on metric spaces via weak boundedness of maximal operators

Malý

2018

JAMA

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Density of Lipschitz functions in Newtonian spaces based on quasi-Banach function lattices is discussed. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces defined via (weak) upper gradients. Our main focus lies on metric spaces with a doubling measure that support a p-Poincaré inequality. Absolute continuity of the function lattice quasi-norm is shown to be crucial for approximability by (locally) Lipschitz functions. The proof of the density result uses, among others, that a suitable maximal operator is locally weakly bounded. In particular, various sufficient conditions for such boundedness on rearrangement-invariant spaces are established and applied.

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Section: Preliminariesmentioning

confidence: 99%

“…[16,23,39,46]. Most recently, general complete quasi-normed lattices of measurable functions were considered as the base function space in Malý [34,35].…”

Section: Introductionmentioning

confidence: 99%

Regularization of Newtonian functions on metric spaces via weak boundedness of maximal operators

Malý

2018

JAMA

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“…The Dirichlet type seminorms provide a homogeneous counterpart of more familiar Newtonian type norms, see [2], [6] and [12]. Then, as ρ is bounded away from zero, the lengths of γ i are bounded and we can reparametrize these paths to be defined on the same interval with a bounded Lipschitz constant.…”

Section: Estimates Of M -Modulusmentioning

confidence: 99%

“…The M F -modulus has been already studied in an even broader generality by L. Malý [12]. Of course, we need both moduli for comparison purposes.…”

Section: Introductionmentioning

confidence: 99%

Modulus in Banach function spaces

Exnerová

Malý

Мартио³

2017

Arkiv För Matematik

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Abstract. Moduli of path families are widely used to mark curves which may be neglected for some applications. We introduce ordinary and approximation modulus with respect to Banach function spaces. While these moduli lead to the same result in reflexive spaces, we show that there are important path families (like paths tangent to a given set) which can be labeled as negligible by the approximation modulus with respect to the Lorentz L p,1 -space for an appropriate p, in particular, to the ordinary L 1 -space if p=1, but not by the ordinary modulus with respect to the same space.

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“…It is also known that the p-Poincaré inequality implies the Sobolev embedding K 0 (E) → L q (E) for some q > p, and that the embedding Note that a Banach function space is a more restrictive concept requiring some further axioms, see the discussion in Malý [15].…”

Section: Examples Of Gradient Spaces I Sobolev Spacesmentioning

confidence: 99%

An axiomatic approach to gradients with applications to Dirichlet and obstacle problems beyond function spaces

Arnlind

Björn

2016

Nonlinear Analysis

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Abstract. We develop a framework for studying variational problems in Banach spaces with respect to gradient relations, which encompasses many of the notions of generalized gradients that appear in the literature. We stress the fact that our approach is not dependent on function spaces and therefore applies equally well to functions on metric spaces as to operator algebras. In particular, we consider analogues of Dirichlet and obstacle problems, as well as first eigenvalue problems, and formulate conditions for the existence of solutions and their uniqueness. Moreover, we investigate to what extent a lattice structure may be introduced on (ordered) Banach spaces via a norm-minimizing variational problem. A multitude of examples is provided to illustrate the versatility of our approach.

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Newtonian Spaces Based on Quasi-Banach Function Lattices

Cited by 16 publications

References 16 publications

Regularization of Newtonian functions on metric spaces via weak boundedness of maximal operators

Regularization of Newtonian functions on metric spaces via weak boundedness of maximal operators

Modulus in Banach function spaces

An axiomatic approach to gradients with applications to Dirichlet and obstacle problems beyond function spaces

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