Sobolev inequality
View article: Trace theory for parabolic boundary value problems with rough boundary conditions
Trace theory for parabolic boundary value problems with rough boundary conditions Open
We characterise the trace spaces arising from intersections of weighted, vector-valued Sobolev spaces, where the weights are powers of the distance to the boundary. These weighted function spaces are particularly suitable for treating boun…
View article: A modified Bakry-Émery $Γ_2$ criterion inequality and the monotonicity of the Tsallis entropy
A modified Bakry-Émery $Γ_2$ criterion inequality and the monotonicity of the Tsallis entropy Open
The Bakry-Émery $Γ_2$ criterion inequality provides a method for establishing the logarithmic Sobolev inequality. We prove a one-parameter family of weighted Bakry-Émery $Γ_2$ criterion inequalities which in the limit case yields the impro…
View article: Yang-Mills Existence and Mass Gap via Information Pressure Theory
Yang-Mills Existence and Mass Gap via Information Pressure Theory Open
Description We construct a four-dimensional Euclidean Yang–Mills theory satisfying the Osterwalder–Schrader axioms with a strictly positive mass gap m ≥ c·Λᵣ > 0, where Λᵣ is the Information–Pressure Theory (IPT) capacity scale. IPT provid…
View article: A modified Bakry-Émery $Γ_2$ criterion inequality and the monotonicity of the Tsallis entropy
A modified Bakry-Émery $Γ_2$ criterion inequality and the monotonicity of the Tsallis entropy Open
The Bakry-Émery $Γ_2$ criterion inequality provides a method for establishing the logarithmic Sobolev inequality. We prove a one-parameter family of weighted Bakry-Émery $Γ_2$ criterion inequalities which in the limit case yields the impro…
View article: Trace theory for parabolic boundary value problems with rough boundary conditions
Trace theory for parabolic boundary value problems with rough boundary conditions Open
We characterise the trace spaces arising from intersections of weighted, vector-valued Sobolev spaces, where the weights are powers of the distance to the boundary. These weighted function spaces are particularly suitable for treating boun…
View article: Yang-Mills Existence and Mass Gap via Information Pressure Theory
Yang-Mills Existence and Mass Gap via Information Pressure Theory Open
Description We construct a four-dimensional Euclidean Yang–Mills theory satisfying the Osterwalder–Schrader axioms with a strictly positive mass gap m ≥ c·Λᵣ > 0, where Λᵣ is the Information–Pressure Theory (IPT) capacity scale. IPT provid…
View article: Best constants in subelliptic fractional Sobolev and Gagliardo-Nirenberg inequalities and ground states on stratified Lie groups
Best constants in subelliptic fractional Sobolev and Gagliardo-Nirenberg inequalities and ground states on stratified Lie groups Open
In this paper, we establish the sharp fractional subelliptic Sobolev inequalities and Gagliardo-Nirenberg inequalities on stratified Lie groups. The best constants are given in terms of a ground state solution of a fractional subelliptic e…
View article: Extended Sobolev scale for vector bundles and its applications
Extended Sobolev scale for vector bundles and its applications Open
We study an extended Sobolev scale for smooth vector bundles over a closed manifold. This scale is built on the base of inner product distribution spaces of generalized smoothness given by an arbitrary positive function OR-varying at infin…
View article: Logarithmic Sobolev Inequalities for Generalised Cauchy Measures
Logarithmic Sobolev Inequalities for Generalised Cauchy Measures Open
View article: Normalized Solutions of Coupled Sobolev Critical Schrodinger Equations with Mass Subcritical Couplings
Normalized Solutions of Coupled Sobolev Critical Schrodinger Equations with Mass Subcritical Couplings Open
We are concerned with qualitative properties of positive solutions to the following coupled Sobolev critical Schrödinger equations [Formula: see text] subject to the mass constraints [Formula: see text] and [Formula: see text], where [Form…
View article: Improved Local Well-Posedness in Sobolev Spaces for Two-Dimensional Compressible Euler Equations
Improved Local Well-Posedness in Sobolev Spaces for Two-Dimensional Compressible Euler Equations Open
We establish the local existence and uniqueness of solutions to the two-dimensional compressible Euler equations with initial velocity $\bv_0$, logarithmic density $ρ_0$, and specific vorticity \(w_0\), which satisfy $(\bv_0, ρ_0, w_0, \na…
View article: Improved Local Well-Posedness in Sobolev Spaces for Two-Dimensional Compressible Euler Equations
Improved Local Well-Posedness in Sobolev Spaces for Two-Dimensional Compressible Euler Equations Open
We establish the local existence and uniqueness of solutions to the two-dimensional compressible Euler equations with initial velocity $\bv_0$, logarithmic density $ρ_0$, and specific vorticity \(w_0\), which satisfy $(\bv_0, ρ_0, w_0, \na…
View article: NSk--PLS--CORE: Poincaré and Log-Sobolev Inequalities for the NSk/ψ Program
NSk--PLS--CORE: Poincaré and Log-Sobolev Inequalities for the NSk/ψ Program Open
This preprint introduces the NSk–PLS–CORE module, a functional-inequalities block in the NSk/ψ program. We develop Poincaré and logarithmic Sobolev inequalities in a setting adapted to NSk: Dirichlet forms associated with a Riemannian metr…
View article: Sobolev embeddings in grand variable Herz-Morrey Besov spaces
Sobolev embeddings in grand variable Herz-Morrey Besov spaces Open
View article: NSk--PLS--CORE: Poincaré and Log-Sobolev Inequalities for the NSk/ψ Program
NSk--PLS--CORE: Poincaré and Log-Sobolev Inequalities for the NSk/ψ Program Open
This preprint introduces the NSk–PLS–CORE module, a functional-inequalities block in the NSk/ψ program. We develop Poincaré and logarithmic Sobolev inequalities in a setting adapted to NSk: Dirichlet forms associated with a Riemannian metr…
View article: Riesz potential estimates under co-canceling constraints
Riesz potential estimates under co-canceling constraints Open
Inequalities for Riesz potentials are well-known to be equivalent to Sobolev inequalities of the same order for domain norms ``far" from $L^1$, but to be weaker otherwise. Recent contributions by Van Schaftingen, by Hernandez, Raiţă and Sp…
View article: Riesz potential estimates under co-canceling constraints
Riesz potential estimates under co-canceling constraints Open
Inequalities for Riesz potentials are well-known to be equivalent to Sobolev inequalities of the same order for domain norms ``far" from $L^1$, but to be weaker otherwise. Recent contributions by Van Schaftingen, by Hernandez, Raiţă and Sp…
View article: Nonlinear Schrodinger-Poisson systems in dimension two: The zero mass case
Nonlinear Schrodinger-Poisson systems in dimension two: The zero mass case Open
View article: Wave Front Sets and Regularity for Fourier Integral Operators with Symbols in Anisotropic Sobolev Spaces
Wave Front Sets and Regularity for Fourier Integral Operators with Symbols in Anisotropic Sobolev Spaces Open
This paper investigates the regularity properties of Fourier integral operators (FIOs) when their symbols belong to anisotropic Sobolev spaces. We establish precise connections between the wave front sets of distributions and the mapping p…
View article: Wave Front Sets and Regularity for Fourier Integral Operators with Symbols in Anisotropic Sobolev Spaces
Wave Front Sets and Regularity for Fourier Integral Operators with Symbols in Anisotropic Sobolev Spaces Open
This paper investigates the regularity properties of Fourier integral operators (FIOs) when their symbols belong to anisotropic Sobolev spaces. We establish precise connections between the wave front sets of distributions and the mapping p…
View article: Uniform Modular Coercivity and a Nonperturbative Proof of the Yang–Mills Mass Gap
Uniform Modular Coercivity and a Nonperturbative Proof of the Yang–Mills Mass Gap Open
We prove the existence of a mass gap for four-dimensional SU(2) Yang–Mills theory by developing a new analytic framework combining modular Dirichlet forms, gauge-adapted op- timal transport, and multiscale logarithmic Sobolev inequalities.…
View article: Optimal Transport on Metric Measure Spaces with Non-Smooth Ricci Bounds
Optimal Transport on Metric Measure Spaces with Non-Smooth Ricci Bounds Open
This paper explores the intricate relationship between optimal transport theory and geometric analysis on metric measure spaces equipped with non-smooth lower Ricci curvature bounds. In classical Riemannian geometry, Ricci curvature plays …
View article: The Six-Stage Architecture of PDE Analysis: A Beginner-Friendly Rigorous Introduction with Applications to Diffusion and Ricci Flow
The Six-Stage Architecture of PDE Analysis: A Beginner-Friendly Rigorous Introduction with Applications to Diffusion and Ricci Flow Open
The Six-Stage Architecture of PDE Analysis: A Beginner-Friendly Rigorous Introduction with Applications to Diffusion and Ricci Flow presents a unified, intuitive, and fully structured framework for understanding the analytical backbone of …
View article: Optimal Transport on Metric Measure Spaces with Non-Smooth Ricci Bounds
Optimal Transport on Metric Measure Spaces with Non-Smooth Ricci Bounds Open
This paper explores the intricate relationship between optimal transport theory and geometric analysis on metric measure spaces equipped with non-smooth lower Ricci curvature bounds. In classical Riemannian geometry, Ricci curvature plays …
View article: Uniform Modular Coercivity and a Nonperturbative Proof of the Yang–Mills Mass Gap
Uniform Modular Coercivity and a Nonperturbative Proof of the Yang–Mills Mass Gap Open
We prove the existence of a mass gap for four-dimensional SU(2) Yang–Mills theory by developing a new analytic framework combining modular Dirichlet forms, gauge-adapted op- timal transport, and multiscale logarithmic Sobolev inequalities.…
View article: The Six-Stage Architecture of PDE Analysis: A Beginner-Friendly Rigorous Introduction with Applications to Diffusion and Ricci Flow
The Six-Stage Architecture of PDE Analysis: A Beginner-Friendly Rigorous Introduction with Applications to Diffusion and Ricci Flow Open
The Six-Stage Architecture of PDE Analysis: A Beginner-Friendly Rigorous Introduction with Applications to Diffusion and Ricci Flow presents a unified, intuitive, and fully structured framework for understanding the analytical backbone of …
View article: Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains
Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains Open
View article: Generalized Sobolev Embeddings on $L^p$ Spaces with Respect to Anisotropic Radon Measures
Generalized Sobolev Embeddings on $L^p$ Spaces with Respect to Anisotropic Radon Measures Open
This paper investigates generalized Sobolev embedding theorems in the context of $L^p$ spaces defined with respect to anisotropic Radon measures. Classical Sobolev embeddings are fundamental in analysis, linking different function spaces a…
View article: Elliptic boundary-value problems in some distribution spaces of generalized smoothness
Elliptic boundary-value problems in some distribution spaces of generalized smoothness Open
We build a solvability theory of elliptic boundary-value problems in normed Sobolev spaces of generalized smoothness for any integrability exponent $$p>1$$ . The smoothness is given by a number parameter and a supplementary function p…
View article: Equi-integrable approximation of Sobolev mappings between manifolds
Equi-integrable approximation of Sobolev mappings between manifolds Open
We show that limits of sequences of smooth maps between compact Riemannian manifolds with equi-integrable $W^{1, p}$-Sobolev energy can always be strongly approximated by smooth maps, giving a counterpart of Hang's density result in $W^{1,…