Related papers: Harnack inequality for Bessel operators

In this paper, we consider fully nonlinear integro-differential equations with possibly nonsymmetric kernels. We are able to find different versions of Alexandroff-Backelman-Pucci estimate corresponding to the full class $\cS^{\fL_0}$ of…

Recall that if $(M^n,g)$ satisfies $\mathrm{Ric}\geq 0$, then the Li-Yau Differential Harnack Inequality tells us for each nonnegative $f:M\to \mathbb{R}^+$, with $f_t$ its heat flow, that $\frac{\Delta f_t}{f_t}-\frac{|\nabla…

We establish a Sobolev-type inequality in Lorentz spaces for $\mathcal{L}$-superharmonic functions \[ \|u\|_{L^{\frac{nq}{n-\alpha q},t}(\mathbb{R}^n)} \leq c \left\| \frac{u(x) - u(y)}{|x-y|^{\frac{n}{q}+\alpha}}…

We prove sufficient conditions for Hausdorff convergence of the spectra of sequences of closed operators defined on varying Hilbert spaces and converging in norm-resolvent sense, i.e. $\|J_\varepsilon(1+A_\varepsilon)^{-1} -…

We consider nonnegative solutions $u:\Omega\longrightarrow \mathbb{R}$ of second order hypoelliptic equations \begin{equation*} \mathscr{L} u(x) =\sum_{i,j=1}^n \partial_{x_i} \left(a_{ij}(x)\partial_{x_j} u(x) \right) + \sum_{i=1}^n b_i(x)…

Inverse initial and inverse source problems of a time-fractional differential equation with Bessel operator are considered. Results on existence and uniqueness of solutions to these problems are presented. The solution method is based on…

In this paper, we present some upper bounds for unitarily invariant norms inequalities. Among other inequalities, we show some upper bounds for the Hilbert-Schmidt norm. In particular, we prove \begin{align*} \|f(A)Xg(B)\pm…

We investigate the validity and failure of Liouville theorems and Harnack inequalities for parabolic and elliptic operators with low regularity coefficients. We are particularly interested in operators of the form $\partial_t - \Delta…

It is well known from the work of Caffarelli and Silvestre that the fractional Laplacian $(-\Delta_x)^{\frac{\sigma}{2}}$ for $\sigma \in (0,2)$ can be obtained as a Dirichlet-to-Neumann map through an extension problem to the upper half…

In this paper, several refinements of the Berezin number inequalities are obtained. We generalize inequalities involving powers of the Berezin number for product of two operators acting on a reproducing kernel Hilbert space $\mathcal…

We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R^n,$n \geq 1$, so that the following inequalities hold for all $u \in C_{0}^{\infty}(B)$: $\int_{B}V(x)|\nabla u |^{2}dx…

Several unitarily invariant norm inequalities and numerical radius inequalities for Hilbert space operators are studied. We investigate some necessary and sufficient conditions for the parallelism of two bounded operators. For a finite rank…

We consider the Harnack inequality for harmonic functions with respect to three types of infinite dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack…

We prove Harnack's type inequalities for bounded non-negative solutions of degenerate parabolic equations with $(p,q)$ growth $$ u_{t}-{\rm div}\left(\mid \nabla u \mid^{p-2}\nabla u + a(x,t) \mid \nabla u \mid^{q-2}\nabla u \right)=0,\quad…

Using the properties of geometric mean, we shall show for any $0\le \alpha ,\beta \le 1$, \[f\left( A{{\nabla }_{\alpha }}B \right)\le f\left( \left( A{{\nabla }_{\alpha }}B \right){{\nabla }_{\beta }}A \right){{\sharp}_{\alpha }}f\left(…

We establish the Krylov Safonov Harnack inequalities and Holder estimates for fully nonlinear nonlocal operators of non-divergence form on Riemannian manifolds with nonnegative sectional curvatures. To this end, we first define the nonlocal…

We prove the Harnack inequality for antisymmetric $s$-harmonic functions, and more generally for solutions of fractional equations with zero-th order terms, in a general domain. This may be used in conjunction with the method of moving…

We consider degenerate fully nonlinear parabolic equations, which generalize the p-parabolic equation with $p>2$ to nondivergence form operators. We prove an intrinsic Harnack inequality for nonnegative solutions and a weak Harnack…

Given a homogeneous k-th order differential operator $A (D)$ on $\mathbb{R}^n$ between two finite dimensional spaces, we establish the Hardy inequality $$\int_{\mathbb{R}^n} \frac{\lvert D^{k-1}u\rvert}{\lvert x \rvert} \,\mathrm{d} x \leq…

In this paper we study an extension problem for the Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type and use the solution to prove Hardy-type inequalities for fractional powers of the Laplace-Beltrami operator.…