Establishes Calderón-Zygmund inequalities on evolving Riemannian manifolds.
problem Calderón-Zygmund inequalities on evolving Riemannian manifolds.
method Establishes various Calderón-Zygmund inequalities on evolving Riemannian manifolds with bounded curvature.
result Provides concrete applications of established inequalities.
The Calderón--Zygmund estimate fails on certain open manifolds.
problem The Lp-Calderón--Zygmund estimate fails on some open manifolds. method Construction of a specific Riemannian open manifold where the estimate is false.
result The Calderón--Zygmund estimate is invalid on the constructed manifold.
Proves inequality for maps from curvature bounds.
problem Deduce Lp bounds on second fundamental form. method Proves nonlinear Calderón-Zygmund inequality.
result Establishes bounds on maps between manifolds.
The paper addresses geometric analysis on non-compact Riemannian manifolds, proving Calderón-Zygmund inequalities.
problem Proving Calderón-Zygmund inequalities on non-compact Riemannian manifolds without positive injectivity radius.
method Probabilistic tools, Hessian formulas, and Bismut type representations for heat semigroups.
result The paper proves the Calderón-Zygmund inequality for 1<p<2 under a lower Ricci curvature bound, and for p>2 under additional curvature conditions. New example shows non-compact manifolds can lack Lp-Calderón-Zygmund inequalities.
problem Exploring Lp-Calderón-Zygmund inequalities on non-compact manifolds. method Developed a concrete example using local deformations of metrics.
result Found a non-compact manifold without Lp-Calderón-Zygmund inequalities. The study shows that close hypersurfaces have uniformly bounded inequalities.
problem Bounding inequalities for close hypersurfaces.
method Analyzing families of smooth hypersurfaces close to a fixed one.
result Uniformly bounded constants in Sobolev, Gagliardo-Nirenberg, and geometric Calderón-Zygmund inequalities.
We introduce the concept of Calderón-Zygmund inequalities on Riemannian manifolds. For 1<p<∞, these are inequalities of the form ∥Hess(u)∥Lp≤C1∥u∥Lp+C2∥Δu∥Lp, valid a priori for all smooth functions $…
New method solves Beltrami equation using Hodge star.
problem Solving the Beltrami equation.
method Using Hodge star operator and elliptic PDE theory.
result Essentially unique homeomorphic solution found.
The study bounds Riesz transforms on manifolds with controlled curvature.
problem Bounding Riesz transforms on manifolds with controlled curvature.
method Established Lp-boundedness of local covariant Riesz transforms for differential forms. result Calderón-Zygmund estimates for manifolds with bounded Riemannian curvature.
Survey of global Calderón-Zygmund inequalities on Riemannian manifolds.
problem Validity and failure of W2,p regularity for Poisson equation solutions. method Various geometric conditions and methods to obtain Lp-Hessian estimates. result Integral inequality may fail even with lower sectional curvature bound.
Estimates on symmetric spaces derived from Euclidean results.
problem Proving estimates on symmetric spaces of non-compact type.
method Duality estimate between vector fields and critical Sobolev space solutions.
result Sharp Calderon-Zygmund estimate for Poisson's equation solutions.
Study Lp boundedness of Riesz transform on differential forms for certain manifolds.
problem Investigate Lp-boundedness of the covariant Riesz transform on differential forms. method Analyze Lp-boundedness on weighted Riemannian manifolds under curvature-dimension and lower bound conditions. result Derive Calderón-Zygmund inequality for 1<p≤2 under curvature-dimension condition. The paper examines Lp gradient and Riesz transform estimates under Ricci lower bounds.
problem Investigating Lp estimates for solutions of the Poisson equation under Ricci lower bounds. method Analyzes Lp estimates for gradient and Riesz transforms under Ricci lower bounds, providing counterexamples and bounds. result Valid Lp estimates for gradient and Riesz transforms under Ricci lower bounds, with conditions on injectivity radius and curvature. Density of smooth functions in Sobolev space on manifolds with curvature bounds.
problem Density of Cc∞ in Wk,p on manifolds with curvature bounds. method Gradient regularity lemma, construction of counterexamples.
result Existence of manifolds where density in Wk,p does not hold. The paper proves boundedness of a Riesz transform on weighted manifolds.
problem Establishing \(L^p\)-boundedness of the covariant Riesz transform on differential forms.
method Heat-kernel criterion, volume doubling, heat kernel estimates, curvature control, gradient bounds.
result The covariant Riesz transform is \(L^p\)-bounded for \(p>2\) on weighted Riemannian manifolds.
The paper examines functional properties on manifolds with very negative curvature.
problem Functional properties on manifolds with very negative curvature.
method New Hardy-type inequalities and first and second order inequalities.
result Functional properties typically hold in manifolds with polynomially growing negative curvature.
Sharp Sobolev theory for scalar elliptic equations on minimal regular manifolds.
problem Well-posedness and regularity for scalar elliptic equations on manifolds of minimal regularity.
method Localization and flat domain techniques combined with Calderón–Zygmund theory and Fredholm alternative.
result Sharp Lp-based Sobolev regularity for scalar elliptic problems on manifolds of minimal regularity. We give an overview of the generalized Calderón-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted estimates for such operators and their commutators with $\BMO$ functions. Lp−Lq of…
Constructs distance-like functions on manifolds with controlled curvature to study Sobolev spaces.
problem Density of smooth functions in Sobolev spaces on manifolds with unbounded geometry.
method Constructs distance-like functions with controlled derivatives on manifolds with specific curvature properties.
result Density of smooth compactly supported functions in Sobolev spaces Wk,p on manifolds with possibly unbounded geometry. In the category of metrics with conical singularities along a smooth divisor with angle in (0,2π), we show that locally defined weak solutions (C1,1−solutions) to the Kähler-Einstein equations actually possess maximum regularity, which means the metrics are actually Hölder continuous in the singular polar coord…
Study on smoothness of solutions to nonlinear equations on Riemannian manifolds.
problem Smoothness of solutions to nonlinear equations with Neumann boundary conditions on Riemannian manifolds.
method Integral refinement of Bochner's identity.
result Semilinear Calderón-Zygmund type results on Sobolev regularity.
We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised by Manfredi & Mingione (Math. Ann. 2007) where only dimension dependent bounds for the growth exponent …
Study on harmonic functions in RCD spaces, focusing on singular points and vanishing gradients.
problem Behavior of harmonic functions at singular points of RCD spaces.
method Analysis of tangent cones and modulus of continuity.
result Gradient of harmonic functions vanishes at certain singular points.
The paper proves density of smooth functions in Sobolev spaces on certain manifolds.
problem Density of smooth functions in Sobolev spaces on manifolds with unbounded geometry.
method Distance-like function with bounded gradient and mild growth of Hessian, proving density results.
result Smooth compactly supported functions are dense in W2,p on the considered manifolds. Estimates for covariant derivatives and Riesz transforms on differential forms.
problem Bounding covariant derivatives and Riesz transforms on differential forms.
method Use Bismut derivative formula to prove heat kernel bounds and Riesz transform boundedness.
result Formulate and prove conjecture on boundedness of covariant local Riesz-transforms in L^p.