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Hardy and Hardy-Sobolev Spaces on Strongly Lipschitz Domains and Some Applications

Xiaming Chen, Renjin Jiang, Dachun Yang (2016)

Analysis and Geometry in Metric Spaces

Let Ω ⊂ Rn be a strongly Lipschitz domain. In this article, the authors study Hardy spaces, Hpr (Ω)and Hpz (Ω), and Hardy-Sobolev spaces, H1,pr (Ω) and H1,pz,0 (Ω) on , for p ∈ ( n/n+1, 1]. The authors establish grand maximal function characterizations of these spaces. As applications, the authors obtain some div-curl lemmas in these settings and, when is a bounded Lipschitz domain, the authors prove that the divergence equation div u = f for f ∈ Hpz (Ω) is solvable in H1,pz,0 (Ω) with suitable...

Hardy inequalities in function spaces

Hans Triebel (1999)

Mathematica Bohemica

Let Ω be a bounded C ∞ domain in ℝ n . The paper deals with inequalities of Hardy type related to the function spaces B p q s ( Ω ) and F p q s ( Ω ) .

Hardy-type inequalities related to degenerate elliptic differential operators

Lorenzo D’Ambrosio (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We prove some Hardy-type inequalities related to quasilinear second-order degenerate elliptic differential operators L p u : = - ∇ L * ( ∇ L u p - 2 ∇ L u ) . If φ is a positive weight such that - L p φ ≥ 0 , then the Hardy-type inequalityholds. We find an explicit value of the constant involved, which, in most cases, results optimal. As particular case we derive Hardy inequalities for subelliptic operators on Carnot Groups.

Harmonic analysis of the space BV.

Albert Cohen, Wolfgang Dahmen, Ingrid Daubechies, Ronald DeVore (2003)

Revista Matemática Iberoamericana

We establish new results on the space BV of functions with bounded variation. While it is well known that this space admits no unconditional basis, we show that it is almost characterized by wavelet expansions in the following sense: if a function f is in BV, its coefficient sequence in a BV normalized wavelet basis satisfies a class of weak-l1 type estimates. These weak estimates can be employed to prove many interesting results. We use them to identify the interpolation spaces between BV and Sobolev...

Hessian determinants as elements of dual Sobolev spaces

Teresa Radice (2014)

Studia Mathematica

In this short note we present new integral formulas for the Hessian determinant. We use them for new definitions of Hessian under minimal regularity assumptions. The Hessian becomes a continuous linear functional on a Sobolev space.

High order representation formulas and embedding theorems on stratified groups and generalizations

Guozhen Lu, Richard Wheeden (2000)

Studia Mathematica

We derive various integral representation formulas for a function minus a polynomial in terms of vector field gradients of the function of appropriately high order. Our results hold in the general setting of metric spaces, including those associated with Carnot-Carathéodory vector fields, under the assumption that a suitable L 1 to L 1 Poincaré inequality holds. Of particular interest are the representation formulas in Euclidean space and stratified groups, where polynomials exist and L 1 to L 1 Poincaré...

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