Sharp gradient estimate for heat kernels on metric measure spaces.
problem Establishing gradient estimates for heat kernels on metric measure spaces.
method Elliptic local Li-Yau gradient estimate for weak solutions of the heat equation.
result Sharp gradient estimate for the logarithm of heat kernels.
Estimates heat kernel gradients on fractal-like cable systems.
problem Bounding gradients of heat kernels on complex fractal structures.
method Pointwise upper estimates for heat kernel gradients.
result Derives Lp-boundedness of quasi-Riesz transforms. The paper provides gradient estimates for heat kernels on manifolds with negative Ricci curvature.
problem Estimating gradients of heat kernels on manifolds with negative Ricci curvature.
method Pointwise and Lp gradient estimates, uniform boundedness results for the heat operator of the Hodge Laplacian. result Uniform boundedness results and gradient estimates for heat kernels and Hodge Laplacian.
Heat kernel estimates on manifolds with mixed boundary conditions.
problem Estimating heat kernels on manifolds with ends and mixed boundary conditions.
method Global harmonic function construction and h-transform technique. result Two-sided heat kernel estimates for Riemannian manifolds with mixed boundary conditions.
In this paper we give Hamilton's Laplacian estimates for the heat equation on complete noncompact manifolds with nonnegative Ricci curvature. As an application, combining Li-Yau's lower and upper bounds of the heat kernel, we give an estimate on Laplacian form of the heat kernel on complete manifolds with nonnegative R…
Extends heat kernel estimates for super Ricci flow.
problem Heat kernel estimates for super Ricci flow.
method Generalizes Bamler-Zhang's geometric analysis to super Ricci flow.
result Obtains Gaussian heat kernel estimates for super Ricci flow.
New heat kernel bounds on manifolds with non-negative Ricci curvature.
problem Establishing new two-sided Gaussian bounds for heat kernels on manifolds.
method Using the non-negative Ricci curvature condition, derive new bounds for the heat kernel.
result Improved two-sided Gaussian bounds for the heat kernel on manifolds with non-negative Ricci curvature.
The article proves a gradient estimate for manifolds with negative Ricci curvature in the Kato class.
problem Estimating the first Betti number of manifolds with negative Ricci curvature.
method Using the heat kernel and Li-Yau type estimates, the first Betti number is bounded by the Kato constant.
result The first Betti number of manifolds with negative Ricci curvature in the Kato class is bounded by the Kato constant.
Sharp gradient estimate on hyperbolic spaces derived from heat kernel.
problem Finding sharp Li-Yau type gradient estimates for positive solutions of heat equations.
method Introduced Li-Yau multiplier set and used recurrence relations of heat kernels on hyperbolic spaces.
result Optimal Li-Yau gradient estimate on hyperbolic spaces.
The study examines heat kernel bounds for manifolds with Ricci curvature in the Kato class.
problem Heat kernel estimates for manifolds with Ricci curvature in the Kato class.
method Kato conditions on the negative part of the Ricci curvature.
result Recent results on heat kernel estimates.
Quantitative Sobolev extensions lead to Neumann heat kernel bounds.
problem Bounding Neumann heat kernels for domains with integral Ricci curvature.
method Quantitative Sobolev extension operators and Neumann heat kernel estimates.
result Uniform bounds on Neumann heat kernels and eigenvalues.
Paper improves heat kernel estimates on Ricci shrinkers.
problem Estimates on heat kernels for Ricci shrinkers.
method Improves estimates from previous work and extends recent progress.
result Theory of $\IF$-convergence holds on Ricci flows induced by Ricci shrinkers.
Sharp heat kernel estimate on graphs proved.
problem Estimating heat kernels on graphs.
method Proved sharp Davies-Gaffney-Grigor'yan lemma.
result Sharp estimate of heat kernels on graphs.
Survey on manifold ends with new heat kernel estimates.
problem Analyzing geometric properties on manifolds with ends.
method Constructing manifolds with ends and analyzing their heat kernel estimates.
result Found manifolds with ends that have different heat kernel estimates.
Study heat kernel on Ricci shrinkers with sharper estimates.
problem Analyze heat kernel in Ricci shrinkers.
method Develop estimates for heat kernel of Ricci flows induced by Ricci shrinkers.
result Improve classical results for Ricci flows induced by Ricci shrinkers.
Heat kernels exist and are Hölder for rough metrics on smooth manifolds.
problem Existence and regularity of heat kernels on rough metrics.
method Local parabolic Harnack estimates for weak solutions in weighted Sobolev spaces.
result Globally continuous heat kernels are Hölder continuous locally.
The paper studies heat kernels on modified manifolds and bounds their properties.
problem Bounding heat kernels on modified Riemannian manifolds.
method Derives upper bounds and gradient estimates for the heat kernel of (M,ildeg). result Establishes upper bounds and gradient estimates for the heat kernel of modified manifolds.
Sharp heat kernel and Green's function estimates on Euclidean volume growth manifolds.
problem Estimating heat kernels and Green's functions on specific Riemannian manifolds.
method Analyzing noncompact Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Sharp Moser-Trudinger inequalities on manifolds with Euclidean volume growth.
The paper studies heat kernels on weighted Riemannian manifolds with curvature bounds.
problem Estimating heat kernels on weighted Riemannian manifolds with lower Ricci curvature bounds.
method Establishing parabolic Harnack inequalities, proving Gaussian bounds for heat kernels, and constructing Li-Yau-type gradient estimates.
result Gaussian upper and lower bounds for the heat kernel, Liouville theorem, uniqueness property, and eigenvalue bounds.
Researchers derived heat kernel expansions for non-compact spaces using Witten deformation.
problem Heat kernel expansions on non-compact spaces, especially for Witten Laplacians.
method Introduced parabolic distance and used it to derive asymptotic expansions.
result Derived an asymptotic expansion of trace of heat kernel for small-time t. The study examines heat kernel bounds on Riemannian manifolds with an end.
problem Estimating heat kernel on Riemannian manifolds with an end.
method Investigates heat kernel estimates of the form pt(x,x)≥cxt−α for large enough t. result Establishes bounds on the form pt(x,x)≥cxt−α for large enough t. Proves upper bounds for heat kernels evolving on manifolds.
problem Bounding heat kernels on evolving manifolds.
method Logarithmic Sobolev inequalities and ultracontractivity estimates.
result Gaussian upper bounds for heat kernels are derived.
One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is Lp bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certai…
Sharp heat kernel estimates on manifolds lead to solutions of the Parabolic Anderson model.
problem Well-posedness and intermittency of solutions to the Parabolic Anderson model on Riemannian manifolds.
method Sharp global heat kernel bounds and geodesic comparison geometry.
result Upper and lower moment bounds for solutions of the Parabolic Anderson model on general compact Riemannian manifolds.
We prove the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups G of H-type: ∣∇Ptf∣≤KPt(∣∇f∣) where Pt is the heat semigroup corresponding to the sublaplacian on G, ∇ is the subelliptic gradient, and K is a constant. This extends a result of H.-…
We study inequalities related to the heat kernel for the hypoelliptic sublaplacian on an H-type Lie group. Specifically, we obtain precise pointwise upper and lower bounds on the heat kernel function itself. We then apply these bounds to derive an estimate on the gradient of solutions of the heat equation, which is kno…
Optimal Poincaré constant estimates on manifolds with ends.
problem Estimating the Poincaré constant on manifolds with ends.
method Heat kernel estimates extended to manifolds with ends, focusing on central balls.
result The Poincaré constant is determined by the second largest end.
In the first part of this paper, we get new Li-Yau type gradient estimates for positive solutions of heat equation on Riemmannian manifolds with Ricci(M)≥−k, k∈R. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci…
Sharp gradient estimates found for heat equation on hyperbolic spaces.
problem Finding sharp gradient estimates for heat equations on hyperbolic spaces.
method Introduced a general form of Li-Yau type gradient estimate and used explicit heat kernel expressions.
result Sharp Li-Yau type gradient estimates obtained for heat equation on hyperbolic spaces.
In this note we give a heat kernel lower bound in term of integral Ricci curvature, extending Cheeger-Yau's estimate.
Following Donaldson's oppenness theorem on deforming a conical Kähler-Einstein metric, we prove a parabolic Schauder-type estimate with respect to conical metrics. As a corollary, we show that the conical Kähler-Ricci Flow exists for short time. The key is to establish the relevant heat kernel estimates, where we use t…
We consider a complete noncompact smooth Riemannian manifold M with a weighted measure and the associated drifting Laplacian. We demonstrate that whenever the q-Bakry-Émery Ricci tensor on M is bounded below, then we can obtain an upper bound estimate for the heat kernel of the drifting Laplacian from the upper b…
This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the Lp norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds …
Study Brownian motions and heat kernel bounds on Kähler and quaternion Kähler manifolds.
problem Understanding Brownian motions and heat kernel bounds on specific geometric manifolds.
method Sharp Laplacian comparison theorems and Cheeger-Yau type lower bounds for heat kernels.
result Sharp Cheeger-Yau type lower bounds for heat kernels and Dirichlet eigenvalues of metric balls.
Develops heat kernel and Green's function estimates for manifolds.
problem Solving Poisson equation on manifolds with Ricci curvature bounds.
method Heat kernel and Green's function estimates for manifolds with positive spectrum.
result Existence and sharp estimates of Poisson equation solutions on manifolds with Ricci curvature bounds.
Study on biharmonic heat equation on manifolds with curvature constraints.
problem Analyzing entire solutions of biharmonic heat equation on manifolds.
method Exponential decay estimates for biharmonic heat kernel under Ricci curvature and noncollapsing conditions. Proving uniqueness criteria for Cauchy problem.
result Conservation law for biharmonic heat kernel and uniform L-infinity estimate for entire solutions.
In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental soluti…
Derives gradient estimates for CR heat equation on pseudo-Hermitian manifolds.
problem Estimating solutions to CR heat equation on complex manifolds.
method Local and global Li-Yau type gradient estimates.
result Gradient estimates and Harnack inequality for positive solutions.
Novel heat flow estimates on ALE manifolds for Schrödinger operators.
problem Estimating heat flows on ALE manifolds with non-trivial L2-kernel. method Combining Fredholm theory for Dirac type operators and heat kernel advances.
result Established Lp−Lq decay estimates for heat flows. The paper extends inequalities to twisted differential forms on Kähler manifolds.
problem Generalizing Sobolev-type inequalities to twisted differential forms.
method Establishing heat kernel estimates for differential forms on Kähler manifolds.
result Proves vanishing theorem and Lq,p-estimates for ∂ˉ-operator. We study new heat kernel estimates for the Neumann heat kernel on a compact manifold with positive Ricci curvature and convex boundary. As a consequence, we obtain new lower bounds for the Neumann eigenvalues which are consistent with Weyl's asymptotics.
In this paper, we first derive a Sobolev inequality along the harmonic-Ricci flow. We then prove a linear parabolic estimate based on the Sobolev inequality and Moser's iteration. As an application, we will obtain an upper bound estimate for the heat kernel under the flow.
Paper reconciles different Ricci flow approaches and proves weak solutions.
problem Proving weak solutions for Ricci flows with singularities.
method Introducing a novel hitting estimate for Brownian motion, compensating for lack of lower heat kernel bounds.
result Every noncollapsed limit of Ricci flows and singular Ricci flows are weak solutions.
The paper studies gradient estimates for heat kernels and harmonic functions in metric measure spaces.
problem Gradient estimates for heat kernels and harmonic functions in metric measure spaces.
method Investigation of properties of harmonic functions, heat kernels, and Riesz transforms in metric measure spaces with a Dirichlet form.
result Equivalence of properties (i), (ii), (iii) for p∈(2,∞) and (i), (ii), (iii), (iv) for p=∞. New method improves Gaussian process regression on complex, sparse point clouds.
problem Traditional Gaussian processes struggle with restricted domains and point clouds.
method Atlas Gaussian Processes (RC-AGPs) combining heat kernel and RBF kernels.
result RC-AGPs outperform existing methods in regression accuracy.
The paper studies heat behavior on curved spaces without radiality assumption.
problem Analyzing heat behavior on curved spaces.
method Examining heat equation solutions on specific Riemannian manifolds.
result Long-time convergence results hold on more general manifolds.
Estimates local Sobolev constants for manifolds with integral Ricci bounds.
problem Extending tools to manifolds with integral Ricci lower bounds.
method Local Sobolev constant estimate for integral Ricci curvature.
result Extension of important tools to manifolds with integral Ricci lower bounds.
In this paper, we study how to get the Ricci expanders from W+-functional through the heat kernel estimate of the conjugate heat equation to the type III singularity of Ricci flow. The Gaussian upper and lower bounds are established for the related heat kernel in accordance to the interesting work of Cao-Zhang for the …