Advances in Mathematics, ISSN 0001-8708, 2011, Volume 226, Issue 4, pp. 3579 - 3621

In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces B ˙ p , q s and Triebel–Lizorkin spaces F ˙ p , q s for...

Grand Triebel–Lizorkin space | Fractional Hajłasz gradient | Hajłasz–Besov space | Quasisymmetric mapping | Triebel–Lizorkin space | Hajłasz–Triebel–Lizorkin space | Quasiconformal mapping | Metric measure space | Besov space | Grand Besov space | Secondary | Hajłasz-Besov space | Hajłasz-Triebel-Lizorkin space | Grand Triebel-Lizorkin space | Primary | Triebel-Lizorkin space | Fractional Hajlasz gradient | INEQUALITY | Hajlasz-Besov space | MATHEMATICS | DECOMPOSITIONS | DIMENSION | SOBOLEV FUNCTIONS | Hajlasz-Triebel-Lizorkin space

Grand Triebel–Lizorkin space | Fractional Hajłasz gradient | Hajłasz–Besov space | Quasisymmetric mapping | Triebel–Lizorkin space | Hajłasz–Triebel–Lizorkin space | Quasiconformal mapping | Metric measure space | Besov space | Grand Besov space | Secondary | Hajłasz-Besov space | Hajłasz-Triebel-Lizorkin space | Grand Triebel-Lizorkin space | Primary | Triebel-Lizorkin space | Fractional Hajlasz gradient | INEQUALITY | Hajlasz-Besov space | MATHEMATICS | DECOMPOSITIONS | DIMENSION | SOBOLEV FUNCTIONS | Hajlasz-Triebel-Lizorkin space

Journal Article

Potential Analysis, ISSN 0926-2601, 8/2016, Volume 45, Issue 2, pp. 201 - 227

In this paper we introduce Bessel potentials and the Sobolev potential spaces resulting from them in the context of Ahlfors regular metric spaces. The Bessel...

Geometry | Potential Theory | Functional Analysis | Bessel potential | Probability Theory and Stochastic Processes | Mathematics | 43A85 | Sobolev spaces | Besov spaces | Ahlfors spaces | Fractional derivative | MATHEMATICS | BESOV | HAJLASZ-SOBOLEV SPACES | Mathematics - Classical Analysis and ODEs

Geometry | Potential Theory | Functional Analysis | Bessel potential | Probability Theory and Stochastic Processes | Mathematics | 43A85 | Sobolev spaces | Besov spaces | Ahlfors spaces | Fractional derivative | MATHEMATICS | BESOV | HAJLASZ-SOBOLEV SPACES | Mathematics - Classical Analysis and ODEs

Journal Article

Duke Mathematical Journal, ISSN 0012-7094, 04/2019, Volume 168, Issue 5, pp. 775 - 848

.... We establish that the gradient of any such function is bounded in the interior of the ball by a power of its oscillation...

MATHEMATICS | INEQUALITIES | SET | THEOREM | INTEGRODIFFERENTIAL OPERATORS | EQUATIONS | REGULARITY THEORY | SURFACES | SOBOLEV | rigidity theorems | fractional Sobolev inequalities | nonlocal minimal surfaces | gradient estimates | Matemàtiques i estadística | Equacions diferencials parcials | regularity results | weak Harnack inequalities | Differential equations, Partial | nonlocal minimal graphs | Àrees temàtiques de la UPC

MATHEMATICS | INEQUALITIES | SET | THEOREM | INTEGRODIFFERENTIAL OPERATORS | EQUATIONS | REGULARITY THEORY | SURFACES | SOBOLEV | rigidity theorems | fractional Sobolev inequalities | nonlocal minimal surfaces | gradient estimates | Matemàtiques i estadística | Equacions diferencials parcials | regularity results | weak Harnack inequalities | Differential equations, Partial | nonlocal minimal graphs | Àrees temàtiques de la UPC

Journal Article

Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 6/2018, Volume 57, Issue 3, pp. 1 - 23

...Calc. Var. (2018) 57:74 https://doi.org/10.1007/s00526-018-1357-3 Calculus of Variations Quasilinear equations with natural growth in the gradients in spaces...

Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | 35J92 | 35A01 | Mathematics | 35J66 | SOURCE TERMS | FRACTIONAL INTEGRALS | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | WEIGHTED INEQUALITIES | REGULARITY | POSITIVE SOLUTIONS | ELLIPTIC-EQUATIONS | DOMAINS | OPERATORS

Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | 35J92 | 35A01 | Mathematics | 35J66 | SOURCE TERMS | FRACTIONAL INTEGRALS | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | WEIGHTED INEQUALITIES | REGULARITY | POSITIVE SOLUTIONS | ELLIPTIC-EQUATIONS | DOMAINS | OPERATORS

Journal Article

Transactions of the American Mathematical Society, ISSN 0002-9947, 05/2017, Volume 369, Issue 5, pp. 3547 - 3573

<<<łł exists quasieverywhere and defines a quasicontinuous representative of u. The above limit exists quasieverywhere also for Haj]]>ł<<

Quasicontinuity | Median | Triebel–Lizorkin space | Metric measure space | Besov space | Fractional Sobolev space | CAPACITY | INEQUALITIES | DIFFERENTIABILITY | Triebel-Lizorkin space | metric measure space | EXTENSION | fractional Sobolev space | quasicontinuity | OSCILLATION | MATHEMATICS | median | MAXIMAL FUNCTIONS | SOBOLEV FUNCTIONS | METRIC MEASURE-SPACES | SINGULAR-INTEGRALS

Quasicontinuity | Median | Triebel–Lizorkin space | Metric measure space | Besov space | Fractional Sobolev space | CAPACITY | INEQUALITIES | DIFFERENTIABILITY | Triebel-Lizorkin space | metric measure space | EXTENSION | fractional Sobolev space | quasicontinuity | OSCILLATION | MATHEMATICS | median | MAXIMAL FUNCTIONS | SOBOLEV FUNCTIONS | METRIC MEASURE-SPACES | SINGULAR-INTEGRALS

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 2018, Volume 2018, Issue 1

In this paper we investigate the endpoint regularity of the discrete m -sublinear fractional maximal operator associated with \documentclass[12pt]{minimal}...

42B25 | 46E35 | 26A45 | 39A12 | Sobolev space | Bounded variation | Discrete multisublinear fractional maximal function | Continuity | Research | documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell^{1}$\end{document}ℓ1-balls

42B25 | 46E35 | 26A45 | 39A12 | Sobolev space | Bounded variation | Discrete multisublinear fractional maximal function | Continuity | Research | documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell^{1}$\end{document}ℓ1-balls

Journal Article

Journal of functional analysis, ISSN 0022-1236, 2018, Volume 275, Issue 1, pp. 1 - 44

We investigate a fractional notion of gradient and divergence operator. We generalize the div-curl estimate by Coifman–Lions–Meyer...

Fractional div-curl lemma | Fractional harmonic maps | Fractional divergence | MATHEMATICS | THEOREM | SPACES | SURFACE | 1/2-HARMONIC MAPS | WEAKLY HARMONIC MAPS | COMPACTNESS | H-SYSTEMS | PARTIAL REGULARITY

Fractional div-curl lemma | Fractional harmonic maps | Fractional divergence | MATHEMATICS | THEOREM | SPACES | SURFACE | 1/2-HARMONIC MAPS | WEAKLY HARMONIC MAPS | COMPACTNESS | H-SYSTEMS | PARTIAL REGULARITY

Journal Article

Journal of Fixed Point Theory and Applications, ISSN 1661-7738, 3/2014, Volume 15, Issue 1, pp. 133 - 153

Brezis and Mironescu have announced several years ago that for a compact manifold $${N^n \subset \mathbb{R}^\upsilon}$$ N n ⊂ R υ and for real numbers 0 < s <...

58D15 | fractional Sobolev spaces | simply connectedness | Mathematical Methods in Physics | Sobolev maps | 46E35 | Analysis | 46T20 | Mathematics, general | Mathematics | Strong density | DENSITY | MATHEMATICS | MATHEMATICS, APPLIED | TRACE SPACES | OPERATOR | 2 MANIFOLDS | MAPPINGS | Mathematics - Functional Analysis | Analysis of PDEs

58D15 | fractional Sobolev spaces | simply connectedness | Mathematical Methods in Physics | Sobolev maps | 46E35 | Analysis | 46T20 | Mathematics, general | Mathematics | Strong density | DENSITY | MATHEMATICS | MATHEMATICS, APPLIED | TRACE SPACES | OPERATOR | 2 MANIFOLDS | MAPPINGS | Mathematics - Functional Analysis | Analysis of PDEs

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 05/2017, Volume 154, pp. 241 - 257

In this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit...

Fractional Laplacian | Bounded variation | Non-local energies

Fractional Laplacian | Bounded variation | Non-local energies

Journal Article

Advances in Mathematics, ISSN 0001-8708, 01/2014, Volume 250, pp. 387 - 419

A weighted norm inequality of Muckenhoupt–Wheeden type is obtained for gradients of solutions to a class of quasilinear equations with measure data on Reifenberg flat domains...

Riccati type equations | Weighted norm inequalities | Measure data | Muckenhoupt–Wheeden type inequalities | Quasilinear equations | Capacity | Muckenhoupt-Wheeden type inequalities | EXISTENCE | INEQUALITIES | TERMS | ZYGMUND THEORY | FRACTIONAL INTEGRALS | MATHEMATICS | REGULARITY | NORM | MAPPINGS | DEGENERATE ELLIPTIC-EQUATIONS

Riccati type equations | Weighted norm inequalities | Measure data | Muckenhoupt–Wheeden type inequalities | Quasilinear equations | Capacity | Muckenhoupt-Wheeden type inequalities | EXISTENCE | INEQUALITIES | TERMS | ZYGMUND THEORY | FRACTIONAL INTEGRALS | MATHEMATICS | REGULARITY | NORM | MAPPINGS | DEGENERATE ELLIPTIC-EQUATIONS

Journal Article

中国科学：数学英文版, ISSN 1674-7283, 2017, Volume 60, Issue 8, pp. 1461 - 1476

We investigate the regularity properties of discrete multisublinear fractional maximal operators,both in the centered and uncentered versions.We prove that...

有界变差 | 分数次极大算子 | 非中心 | 离散 | 偏导数 | 点估计 | 正则性 | 分数次极大函数 | bounded variation | 42B25 | 46E35 | 26A45 | 39A12 | discrete multisublinear fractional maximal operator | discrete fractional maximal operator | continuity | Mathematics | Applications of Mathematics | MATHEMATICS | MATHEMATICS, APPLIED | SOBOLEV SPACES | END-POINT REGULARITY | BOUNDEDNESS | OPERATORS

有界变差 | 分数次极大算子 | 非中心 | 离散 | 偏导数 | 点估计 | 正则性 | 分数次极大函数 | bounded variation | 42B25 | 46E35 | 26A45 | 39A12 | discrete multisublinear fractional maximal operator | discrete fractional maximal operator | continuity | Mathematics | Applications of Mathematics | MATHEMATICS | MATHEMATICS, APPLIED | SOBOLEV SPACES | END-POINT REGULARITY | BOUNDEDNESS | OPERATORS

Journal Article

Bulletin of the London Mathematical Society, ISSN 0024-6093, 06/2015, Volume 47, Issue 3, pp. 396 - 406

Abstract The purpose of this short article is to prove some potential estimates that naturally arise in the study of subelliptic Sobolev inequalities for...

FRACTIONAL INTEGRALS | MATHEMATICS | MAXIMAL FUNCTIONS | HORMANDER VECTOR-FIELDS | METRICS | POINCARE INEQUALITIES | REPRESENTATION FORMULAS | OPERATORS

FRACTIONAL INTEGRALS | MATHEMATICS | MAXIMAL FUNCTIONS | HORMANDER VECTOR-FIELDS | METRICS | POINCARE INEQUALITIES | REPRESENTATION FORMULAS | OPERATORS

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 2006, Volume 194, Issue 1, pp. 94 - 114

A typical inequality handled in this article connects the L p -norm of the gradient of a function to a one-dimensional integral of the p -capacitance of the conductor between two level surfaces of the same function...

Multiplicative inequalities | Conductor inequalities | Two-weight integral inequalities | Essential norm | Strong type capacitary inequalities | Weighted Sobolev spaces | Fractional Sobolev spaces | fractional Sobolev spaces | essential norm | MATHEMATICS, APPLIED | CAPACITIES | SPACES | strong type capacitary inequalities | conductor inequalities | WEIGHTS | two-weight integral inequalities | HARDYS INEQUALITY | weighted Sobolev spaces | multiplicative inequalities

Multiplicative inequalities | Conductor inequalities | Two-weight integral inequalities | Essential norm | Strong type capacitary inequalities | Weighted Sobolev spaces | Fractional Sobolev spaces | fractional Sobolev spaces | essential norm | MATHEMATICS, APPLIED | CAPACITIES | SPACES | strong type capacitary inequalities | conductor inequalities | WEIGHTS | two-weight integral inequalities | HARDYS INEQUALITY | weighted Sobolev spaces | multiplicative inequalities

Journal Article

14.
Full Text
Endpoint regularity of discrete multilinear fractional nontangential maximal functions

Advances in Difference Equations, ISSN 1687-1847, 12/2019, Volume 2019, Issue 1, pp. 1 - 18

Given m≥1 $m\geq 1$, 0≤λ≤1 $0\leq \lambda \leq 1$, and a discrete vector-valued function f→=(f1,…,fm) $\vec{f}=(f_{1},\ldots,f_{m})$ with each fj:Zd→R...

Ordinary Differential Equations | Functional Analysis | Bounded variation | Analysis | Discrete multilinear fractional maximal operator | Difference and Functional Equations | Mathematics, general | Mathematics | Continuity | Discrete multilinear fractional nontangential maximal operator | Partial Differential Equations | MATHEMATICS | MATHEMATICS, APPLIED | OPERATOR | Operators (mathematics) | Cubes

Ordinary Differential Equations | Functional Analysis | Bounded variation | Analysis | Discrete multilinear fractional maximal operator | Difference and Functional Equations | Mathematics, general | Mathematics | Continuity | Discrete multilinear fractional nontangential maximal operator | Partial Differential Equations | MATHEMATICS | MATHEMATICS, APPLIED | OPERATOR | Operators (mathematics) | Cubes

Journal Article

Zeitschrift für angewandte Mathematik und Physik, ISSN 0044-2275, 8/2016, Volume 67, Issue 4, pp. 1 - 42

We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear...

35A15 | 35J92 | Exponential attractor | Fractal-like domains | Semilinear reaction–diffusion equation | Nonlocal Robin boundary conditions on non-smooth domains | Theoretical and Applied Mechanics | 35K65 | The Laplace operator | Global attractor | Engineering | Mathematical Methods in Physics | 35B41 | Domains with Hölder cusps | RAMIFIED DOMAINS | REFLECTING DIFFUSIONS | TRACE | MATHEMATICS, APPLIED | PARABOLIC EQUATIONS | LINEAR ELLIPTIC-EQUATIONS | ARBITRARY DOMAINS | LIPSCHITZ-DOMAINS | SIMILAR FRACTAL BOUNDARIES | FRACTIONAL LAPLACIAN | CONSTRUCTION | Domains with Holder cusps | Semilinear reaction-diffusion equation | Applications of mathematics | Infinity | Mathematical analysis | Uniqueness | Boundary conditions | Boundaries | Reaction-diffusion equations | Diffusion

35A15 | 35J92 | Exponential attractor | Fractal-like domains | Semilinear reaction–diffusion equation | Nonlocal Robin boundary conditions on non-smooth domains | Theoretical and Applied Mechanics | 35K65 | The Laplace operator | Global attractor | Engineering | Mathematical Methods in Physics | 35B41 | Domains with Hölder cusps | RAMIFIED DOMAINS | REFLECTING DIFFUSIONS | TRACE | MATHEMATICS, APPLIED | PARABOLIC EQUATIONS | LINEAR ELLIPTIC-EQUATIONS | ARBITRARY DOMAINS | LIPSCHITZ-DOMAINS | SIMILAR FRACTAL BOUNDARIES | FRACTIONAL LAPLACIAN | CONSTRUCTION | Domains with Holder cusps | Semilinear reaction-diffusion equation | Applications of mathematics | Infinity | Mathematical analysis | Uniqueness | Boundary conditions | Boundaries | Reaction-diffusion equations | Diffusion

Journal Article

Communications in partial differential equations, ISSN 0360-5302, 12/2009, Volume 34, Issue 12, pp. 1607 - 1631

Γ-convergence methods are used to prove homogenization results for fractional obstacle problems in periodically perforated domains. The obstacles have random...

Fractional obstacles | Perforated domains | Γ-Convergence | Weighted Sobolev energies

Fractional obstacles | Perforated domains | Γ-Convergence | Weighted Sobolev energies

Journal Article

17.
Full Text
A Fefferman–Phong Type Inequality and Applications to Quasilinear Subelliptic Equations

Potential Analysis, ISSN 0926-2601, 12/1999, Volume 11, Issue 4, pp. 387 - 413

Journal Article

Potential Analysis, ISSN 0926-2601, 3/2017, Volume 46, Issue 3, pp. 403 - 430

We study nonlinear elliptic equations in divergence form div A ( x , Du ) = div G . $\text {div }{\mathcal A}(x,Du)=\text {div } G.$ When A ${\mathcal A}$ has...

35J60 | Probability Theory and Stochastic Processes | Mathematics | Nonlinear elliptic equations | Besov spaces | Geometry | Potential Theory | Functional Analysis | Higher order fractional differentiability | 49N60 | Local well-posedness | 35B65 | 42B37 | MATHEMATICS

35J60 | Probability Theory and Stochastic Processes | Mathematics | Nonlinear elliptic equations | Besov spaces | Geometry | Potential Theory | Functional Analysis | Higher order fractional differentiability | 49N60 | Local well-posedness | 35B65 | 42B37 | MATHEMATICS

Journal Article

Mathematische Nachrichten, ISSN 0025-584X, 08/2020, Volume 293, Issue 8, pp. 1459 - 1480

We establish the higher fractional differentiability of the solutions of a problem in divergence form of the type divA(x,Du)=div|F|p(x)−22FinΩ,u=0on∂Ω.The main...

variable exponents | nonlinear elliptic equations | p(x)‐Laplacian operator | Besov spaces | higher order fractional differentiability

variable exponents | nonlinear elliptic equations | p(x)‐Laplacian operator | Besov spaces | higher order fractional differentiability

Journal Article

Journal of Inequalities and Applications, 12/2018, Volume 2018, Issue 1, pp. 1 - 17

In this paper we investigate the endpoint regularity of the discrete m-sublinear fractional maximal operator associated with ℓ1 $\ell^{1}$-balls, both in the...

26A45 | Sobolev space | Mathematics | Discrete multisublinear fractional maximal function | ℓ 1 $\ell^{1}$ -balls | 42B25 | 46E35 | 39A12 | Bounded variation | Analysis | Mathematics, general | Continuity | Applications of Mathematics

26A45 | Sobolev space | Mathematics | Discrete multisublinear fractional maximal function | ℓ 1 $\ell^{1}$ -balls | 42B25 | 46E35 | 39A12 | Bounded variation | Analysis | Mathematics, general | Continuity | Applications of Mathematics

Journal Article

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