arXiv subjects
Publications and source records attributed to Tomasz Jakubowski.
Motivated by the spectral theory of relativistic atoms, we prove matching upper and lower bounds for the transition density of Hardy perturbations of subordinated Bessel heat kernels. The analysis is based on suitable supermedian functions, in particular invariant functions.
We give local in time sharp two sided estimates of the heat kernel associated with the relativistic stable operator perturbed by a critical (Hardy) potential.
We provide sharp two-sided estimates of the heat kernel of the Dirichlet fractional Laplacian on the half-line perturbed by the Hardy potential.
We describe the large-time asymptotics of solutions to the heat equation for the fractional Laplacian with added subcritical or even critical Hardy-type potential. The asymptotics is governed by a self-similar solution of the equation, obtained as a normalized limit at the origin of the kernel of the corresponding Feynman-Kac semigroup.
We obtain Grönwall type estimates for the gradient of the harmonic functions for a Lévy operator with order strictly larger than 1 and minimal assumptions of its Lévy measure.
We consider non-local Schrödinger operators $H=-L-V$ in $L^2(\mathbf{R}^d)$, $d \geq 1$, where the kinetic terms $L$ are pseudo-differential operators which are perturbations of the fractional Laplacian by bounded non-local operators and $V$ is the fractional Hardy potential. We prove pointwise estimates of eigenfunctions corresponding to negative eigenvalues and upper finite-time horizon estimates for heat kernels. We also analyze the relation between the matching lower estimates of the heat kernel and the ground state near the origin. Our results cover the relativistic Schrödinger operator with Coulomb potential.
We consider the fractional Burgers equation $ Δ^{α/2} u + b\cdot \nabla (u|u|^{(α-1)/β})$ on ${\mathbf R}^d$, $d\geq2$, with {$α\in (1,2)$ and} $β>1$ and prove the existence of a solution for a large class of initial conditions, which contains functions that do not belong to any $L^p({\mathbf R}^d)$, $1\leq p\leq\infty$. Next, we apply the general results to the initial condition $u_0(x)=M|x|^{-β}$, $1<β<d$, and show the existence of a selfsimilar solution and derive its properties such as smoothness, two-sided estimates, asymptotics and gradient estimates.
We give ground-state representation for the fractional Laplacian with Dirichlet condition on the half-line
For the fractional Laplacian we give Hardy inequality which is optimal in $L^p$ for $1<p<\infty$. As an application, we explicitly characterize the contractivity of the corresponding Feynman-Kac semigroups on $L^p$.
We investigate whether a fundamental solution of the Schrödinger equation $\partial_t u =(Δ+V)\, u$ has local in time sharp Gaussian estimates. We compare that class with the class of $V$ for which local in time plain Gaussian estimates hold. We concentrate on $V$ that have fixed sign and we present certain conclusions for $V$ in the Kato class.
We provide two-sided pointwise estimates and uniform asymptotics of the solutions to the subcritical quasi-geostrophic equation with initial data in $L^{2/(α-1)}(\mathbb{R}^2)$. Furthermore, we give upper bound of similar type for any derivative of the solutions. Initial data in $L^{p}(\mathbb{R}^2)$, $p>2/(α-1)$, are also discussed.
We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schrödinger operator with negative Hardy potential $$Δ^{α/2} -λ|x|^{-α}$$ on $\RR^d$, where $α\in(0,d\wedge 2)$ and $λ>0$. The proof is purely analytical but elementary. In particular, for upper bounds of heat kernel we use the Chapman-Kolmogorov equation and adopt self-improving argument.
We give sharp two-sided estimates of the semigroup generated by the fractional Laplacian plus the Hardy potential on $\mathbb{R}^d$, including the case of the critical constant. We use Davies' method back-to-back with a new method of integral analysis of Duhamel's formula.
In this paper, we provide two-sided estimates and uniform asymptotics for the solution of $d$-dimensional critical fractal Burgers equation $u_t-Δ^{α/2}u+b\cdot \nabla\left(u|u|^q\right)=0$, $α\in(1,2)$, $b\in\mathbb R^d$ for $q = (α-1)/d$ and $u_0 \in L^1(\mathbb R^d)$. We consider also $q > (α-1)/d$ under additional condition $u_0 \in L^\infty(\mathbb R^d)$. In both cases we assume $u_0\geq0$, which implies that the solution is non-negative. The estimates are given in the terms of the function $P_t u_0$, where $P_t$ is the stable semigroup operator.
In this paper, we provide two-sided estimates for the source solution of $d$-dimensional critical fractal Burgers equation $u_t-Δ^{α/2}+b\cdot \nabla\left(u|u|^q\right)=0$, $q=(α-1)/d$, $α\in(1,2)$, $b\in\mathbb{R}^d$, by the density function of the isotropic $α$-stable process.
We prove that the Green function of a generator of isotropic unimodal Lévy processes with the weak lower scaling order bigger than one and the Green function of its gradient perturbations are comparable for bounded smooth open sets if the drift function is from an appropriate Kato class.
We study iterations of integral kernels satisfying a transience-type condition and we prove exponential estimates analogous to Gronwall\rq{}s inequality. As a consequence we obtain estimates of Schrödinger perturbations of integral kernels, including Markovian semigroups.
Let $L=Δ^{α/2}+ b\cdot\nabla$ with $α\in(1,2)$. We prove the Martin representation and the Relative Fatou Theorem for non-negative singular $L$-harmonic functions on ${\mathcal C}^{1,1}$ bounded open sets.