Related papers: Sharp norm estimates for the classical heat equati…

In this note, we prove some new entropy formula for linear heat equation on static Riemannian manifold with nonnegative Ricci curvature. The results are analogies of Cao and Hamilton's entropies for Ricci flow coupled with heat-type…

In this manuscript, we provide local $L^q$-estimates for the gradient of solutions of a class of quasilinear equations whose principal part lacks strong monotonicity. These estimates are used to establish uniform large-scale $L^q$-estimates…

This paper introduces a new method for solving the planar heat equation based on the Lightning Method. The lightning method is a recent development in the numerical solution of linear PDEs which expresses solutions using sums of polynomials…

A representation of the sharp coefficient in a pointwise estimate for the gradient of the generalized Poisson integral of a function $f$ on ${\mathbb R}^n$ is obtained under the assumption that $f$ belongs to $L^p$. The explicit value of…

The heat kernel associated with the setting of the classical Jacobi polynomials is defined by an oscillatory sum which cannot be computed explicitly, in contrast to the situation for the two other classical systems of orthogonal…

We provide an accurate verification method for solutions of heat equations with a superlinear nonlinearity. The verification method numerically proves the existence and local uniqueness of the exact solution in a neighborhood of a…

We consider $L^p$-$L^q$ estimates for the spherical harmonic projection operators and obtain sharp bounds on a certain range of $p$, $q$. As an application, we provide a proof of off-diagonal Carleman estimates for the Laplacian, which…

We find the nodes that minimise divided differences and use them to find the sharp constant in a sublevel set estimate. We also find the sharp constant in the first instance of the van der Corput Lemma using a complex mean value theorem for…

Let $R_{1,2}$ be scalar Riesz transforms on $\mathbb{R}^2$. We prove that the $L^p$ norms of $k$-th powers of the operator $R_2+iR_1$ behave exactly as $|k|^{1-2/p}p$, uniformly in $k\in\mathbb{Z}\backslash\{0\}$, $p\geq2$. This gives a…

We consider the classical Cauchy problem for the linear heat equation and integrable initial data in the Euclidean space $\mathbb{R}^N$. In the case $N=1$ we show that given a weighted $L^p$-space $L_w^p(\mathbb{R})$ with $1 \leq p <…

In this short survey paper, we first recall the log gradient estimates for the heat equation on manifolds by Li-Yau, R. Hamilton and later by Perelman in conjunction with the Ricci flow. Then we will discuss some of their applications and…

The paper contains the proof of $L^p$-weighted norm inequalities for both, martingales square functions and the classical square functions in harmonic analysis of Littlewood-Paley and Lusin. Furthermore, the bounds are completely explicit…

We study the heat equation with a random potential term. The potential is a one-sided stable noise, with positive jumps, which does not depend on time. To avoid singularities, we define the equation in terms of a construction similar to the…

We provide $L^1$ estimates for a class of transport equations containing singular integral operators. While our main application is for a specific problem in General Relativity we believe that the phenomenon which our result illustrates is…

In this paper, we study elliptic gradient estimates for a nonlinear $f$-heat equation, which is related to the gradient Ricci soliton and the weighted log-Sobolev constant of smooth metric measure spaces. Precisely, we obtain Hamilton's and…

We prove certain $L^p$ estimates ($1<p<\infty$) for non-isotropic singular integrals along surfaces of revolution. As an application we obtain $L^p$ boundedness of the singular integrals under a sharp size condition on their kernels.

In this paper we prove the $L^{p}-L^{\acute{p}}$ estimate for the Schr\"{o}dinger equation on the half-line and with homogeneous Dirichlet boundary condition at the origin.

We prove a product estimate that allows to estimate the quadratic first order nonlinearity of the harmonic map flow in the $L^p$ norm. Then the parabolic analogue of Weyl's lemma for the Lapace operator is established. Both results are…

In this short note we present local derivative estimates for heat equations on Riemannian manifolds following the line of W.-X. Shi. As an application we generalize a second derivative estimate of R. Hamilton for heat equations on compact…

Gradient inequalities of the Hamilton type and the Li-Yau type for positive solutions to the heat equation are established from a probabilistic viewpoint, which simplifies the proofs of some results of Sun [{\it Pacific J. Math.}, 253…