Sharp Two-sided Estimates Research Articles

Heat Kernel Estimates of Fractional Schrödinger Operators with Hardy Potential on Half-line

AbstractWe provide sharp two-sided estimates of the heat kernel of the Dirichlet fractional Laplacian on the half-line perturbed by a Hardy potential.

Non-symmetric stable processes: Dirichlet heat kernel, Martin kernel and Yaglom limit

We study a d-dimensional non-symmetric strictly α-stable Lévy process X, whose spherical density is bounded and bounded away from the origin. First, we give sharp two-sided estimates on the transition density of X killed when leaving an arbitrary κ-fat set. We apply these results to get the existence of the Yaglom limit for arbitrary κ-fat cone. In the meantime we also obtain the spacial asymptotics of the survival probability at the vertex of the cone expressed by means of the Martin kernel for Γ and its homogeneity exponent. Our results hold for the dual process X̂, too.

Different degrees of non-compactness for optimal Sobolev embeddings

The structure of non-compactness of optimal Sobolev embeddings of m-th order into the class of Lebesgue spaces and into that of all rearrangement-invariant function spaces is quantitatively studied. Sharp two-sided estimates of Bernstein numbers of such embeddings are obtained. It is shown that, whereas the optimal Sobolev embedding within the class of Lebesgue spaces is finitely strictly singular, the optimal Sobolev embedding in the class of all rearrangement-invariant function spaces is not even strictly singular.

Sharp two-sided Green function estimates for Dirichlet forms degenerate at the boundary

The goal of this paper is to establish Green function estimates for a class of purely discontinuous symmetric Markov processes with jump kernels degenerate at the boundary and critical killing potentials. The jump kernel and the killing potential depend on several parameters. We establish sharp two-sided estimates on the Green functions of these processes for all admissible values of the parameters involved. Depending on the regions where the parameters belong, the estimates on the Green functions are different. In fact, the estimates have three different forms. As applications, we prove that the boundary Harnack principle holds in certain region of the parameters and fails in some other region of the parameters. Together with the main results of our previous paper [Potential Anal., online, 2021], we completely determine the region of the parameters where the boundary Harnack principle holds.

Potential theory of Dirichlet forms degenerate at the boundary: the case of no killing potential

In this paper we consider the Dirichlet form on the half-space \({\mathbb R}^d_+\) defined by the jump kernel \(J(x,y)=|x-y|^{-d-\alpha }{{\mathcal {B}}}(x,y)\), where \({{\mathcal {B}}}(x,y)\) can be degenerate at the boundary. Unlike our previous works [16, 17] where we imposed critical killing, here we assume that the killing potential is identically zero. In case \(\alpha \in (1,2)\) we first show that the corresponding Hunt process has finite lifetime and dies at the boundary. Then, as our main contribution, we prove the boundary Harnack principle and establish sharp two-sided Green function estimates. Our results cover the case of the censored \(\alpha \)-stable process, \(\alpha \in (1,2)\), in the half-space studied in [2].

Heat kernels for reflected diffusions with jumps on inner uniform domains

In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain D D in a length metric space. The length metric is the intrinsic metric of a strongly local Dirichlet form. When D D is an inner uniform domain in the Euclidean space, a prototype for a special case of the processes under consideration is symmetric reflected diffusions with jumps on D D , whose infinitesimal generators are non-local (pseudo-differential) operators L \mathcal {L} on D D of the form \[ L u ( x ) = 1 2 ∑ i , j = 1 d ∂ ∂ x i ( a i j ( x ) ∂ u ( x ) ∂ x j ) + lim ε ↓ 0 ∫ { y ∈ D : ρ D ( y , x ) > ε } ( u ( y ) − u ( x ) ) J ( x , y ) d y \mathcal {L} u(x)\! =\!\frac 12 \!\sum _{i, j=1}^d\! \frac {\partial }{\partial x_i}\! \left (\!\!a_{ij}(x) \frac {\partial u(x)}{\partial x_j}\!\right ) \!+ \lim _{\varepsilon \downarrow 0}\! \int _{\{y\in D: \, \rho _D(y, x)>\varepsilon \}}\!\! (u(y)-u(x)) J(x, y)\, dy \] satisfying “Neumann boundary condition”. Here, ρ D ( x , y ) \rho _D(x,y) is the length metric on D D , A ( x ) = ( a i j ( x ) ) 1 ≤ i , j ≤ d A(x)=(a_{ij}(x))_{1\leq i,j\leq d} is a measurable d × d d\times d matrix-valued function on D D that is uniformly elliptic and bounded, and \[ J ( x , y ) ≔ 1 Φ ( ρ D ( x , y ) ) ∫ [ α 1 , α 2 ] c ( α , x , y ) ρ D ( x , y ) d + α ν ( d α ) , J(x,y)≔\frac {1}{\Phi (\rho _D(x,y))} \int _{[\alpha _1, \alpha _2]} \frac {c(\alpha , x,y)} {\rho _D(x,y)^{d+\alpha }} \,\nu (d\alpha ), \] where ν \nu is a finite measure on [ α 1 , α 2 ] ⊂ ( 0 , 2 ) [\alpha _1, \alpha _2] \subset (0, 2) , Φ \Phi is an increasing function on [ 0 , ∞ ) [ 0, \infty ) with c 1 e c 2 r β ≤ Φ ( r ) ≤ c 3 e c 4 r β c_1e^{c_2r^{\beta }} \le \Phi (r) \le c_3 e^{c_4r^{\beta }} for some β ∈ [ 0 , ∞ ] \beta \in [0,\infty ] , and c ( α , x , y ) c(\alpha , x, y) is a jointly measurable function that is bounded between two positive constants and is symmetric in ( x , y ) (x, y) .

Boundary Harnack principle for diffusion with jumps

Consider the operator Lb=L0+b1⋅∇+Sb2 on Rd, where L0 is a second order differential operator of non-divergence form, the drift function b1 belongs to some Kato class and Sb2f(x)≔∫Rdf(x+z)−f(x)−∇f(x)⋅z1{|z|≤1}b2(x,z)j0(z)dz,f∈Cb2(Rd).Here j0(z) is a nonnegative locally bounded function on Rd∖{0} satisfying that ∫Rd(1∧|z|2)j0(z)dz<∞ and that there are constants β∈(1,2) and c0>0 so that j0(z)≤c0|z|d+βfor|z|≤1,and b2(x,z) is a real-valued bounded function on Rd×Rd. There is conservative Feller process Xb associated with the non-local operator Lb. We derive sharp two-sided Green function estimates of Lb on bounded C1,1 domains, identify the Martin and minimal Martin boundary, and establish the Martin integral representation of Lb-harmonic functions on these domains. The latter in particular reveals how the process Xb exits a bounded C1,1 domain D, or equivalently, the structure of the harmonic measure of Lb on D, which consists of the continuously exiting term and the jump-off term. These results are then used to establish, under some mild conditions, Harnack principle and the boundary Harnack principle with explicit boundary decay rate for the operator Lb on C1,1 open sets.

Heat kernel estimates on spaces with varying dimension

We obtain sharp two-sided heat kernel estimates for some process whose regular Dirichlet form is strongly local on spaces with varying dimension, in which two spaces of general dimension are connected at one point. On these spaces, if the dimensions of the two constituent parts are different, the volume doubling property fails with respect to the measure induced by the associated Lebesgue measures. Thus the parabolic Harnack inequalities fail and the heat kernels do not enjoy Aronson type estimates. Our estimates show that the on-diagonal estimates are independent of the dimensions of the two parts of the space for small time, whereas they depend on their transience or recurrence for large time. These are multidimensional version of a space considered by Z.-Q. Chen and S. Lou (Ann. Probab. 2019), in which a 1-dimensional space and a 2-dimensional space are connected at one point.

Mixed stochastic heat equation with fractional Laplacian and gradient perturbation

We introduce a new stochastic heat equation with a mixed operator, which is a combination of the standard Laplacian, a fractional Laplacian and the gradient operator, driven by an additive Gaussian noise which is white in time and in space. We establish the existence of the solution and we study its behavior with respect to the time variable. In particular, we establish sharp two-sided interesting estimates of its variance increments, which allow us to get some regularity characteristics of the solution sample paths, among many other possible applications.

Heat kernel estimates for subordinate Markov processes and their applications

In this paper, we establish sharp two-sided estimates for transition densities of a large class of subordinate Markov processes. As applications, we show that the parabolic Harnack inequality and Hölder regularity hold for parabolic functions of such processes, and derive sharp two-sided Green function estimates.

Transition densities of spectrally positive Lévy processes

We deduce the asymptotic behavior of transition densities for a large class of spectrally one-sided Lévy processes of unbounded variation satisfying mild condition imposed on the second derivative of the Laplace exponent or, equivalently, on the real part of the characteristic exponent. We also provide sharp two-sided estimates on the density when restricted additionally to processes without Gaussian component.

Heat kernel bounds for a large class of Markov process with singular jump

Let Z=(Z1,…,Zd) be the d-dimensional Lévy processes where Zi’s are independent 1-dimensional Lévy processes with jump kernel Jϕ,1(u,w)=|u−w|−1ϕ(|u−w|)−1 for u,w∈R. Here ϕ is an increasing function with weak scaling condition of order α̲,α¯∈(0,2). Let J(x,y)≍Jϕ(x,y) be the symmetric measurable function where Jϕ(x,y)≔Jϕ,1(xi,yi)if xi≠yi for some i and xj=yj for all j≠i0if xi≠yi for more than one index iCorresponding to the jump kernel J, we show the existence of non-isotropic Markov processes X≔(X1,…,Xd) and obtain sharp two-sided heat kernel estimates for the transition density functions.

Nonlinear convolution integro-differential equation with variable coefficient

Abstract For an integro-differential equation of the convolution type defined on the half-line [0, ∞) with a power nonlinearity and variable coefficient, we use the weight metrics method to prove a global theorem on the existence and uniqueness of a solution in the cone of nonnegative functions in the space C[0, ∞). It is shown that the solution can be found by a successive approximation method of the Picard type; an estimate for the rate of convergence of the approximations is produced. We present sharp two-sided a-priori estimates for the solutions. These estimates enable us to construct a complete metric space which is invariant under the nonlinear convolution operator considered here and to prove that the equation induced by this operator has a unique solution in this space as well as in the class of all non-negative continuous functions vanishing at the origin. Such equations with operators of fractional calculus as the Riemann-Liouville, Erdélyi-Kober, Hadamard fractional integrals arise, in particular, when describing the process of propagation of shock waves in gas-filled pipes, solving the problem about heating a half-infinite body in a nonlinear heat-transfer process, considering models of population genetics, and elsewhere.

Estimates on transition densities of subordinators with jumping density decaying in mixed polynomial orders

In this paper, we discuss estimates on the transition densities of subordinators, which are global in time. We establish sharp two-sided estimates on the transition densities of subordinators whose Lévy measures are absolutely continuous and decaying in mixed polynomial orders. Under a weaker assumption on Lévy measures, we also obtain precise asymptotic behaviors of the transition densities at infinity. Our results cover geometric stable subordinators, Gamma subordinators and much more.

Genuinely sharp heat kernel estimates on compact rank-one symmetric spaces, for Jacobi expansions, on a ball and on a simplex

We prove genuinely sharp two-sided global estimates for heat kernels on all compact rank-one symmetric spaces. This generalizes the authors' recent result obtained for a Euclidean sphere of arbitrary dimension. Furthermore, similar heat kernel bounds are shown in the context of classical Jacobi expansions, on a ball and on a simplex. These results are more precise than the qualitatively sharp Gaussian estimates proved recently by several authors.

Heat Kernel Estimates for Non-symmetric Finite Range Jump Processes

In this paper, we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation $${{\cal S}^b}: = {\overline {\rm{\Delta }} ^{\alpha /2}} + b \cdot \nabla $$ where $${\overline {\rm{\Delta }} ^{\alpha /2}}$$ is the truncated fractional Laplacian, α ∈ (1, 2) and b ∈ K−1 . In the second part, for a more general finite range jump process, we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance ∣x − y∣ in short time.

Asymptotically sharp two-sided estimate for domains of univalence of holomorphic self-maps of a disc with an invariant diameter

Holomorphic self-maps of the unit disc with two fixed diametrically opposite boundary points and an invariant diameter are investigated. Asymptotically sharp estimates for domains of univalence are obtained for functions in such classes, which depend on the product of the angular derivatives at the boundary fixed points. Bibliography: 16 titles.

Sharp Two-Sided Estimate for the Sum of a Sine Series with Convex Slowly Varying Sequence of Coefficients

The sum of a sine series $$g\left({b,x} \right) = \sum\nolimits_{k = 1}^\infty {}$$ bk sin kx with coefficients forming a convex sequence b is known to be positive on the interval (0,π). To estimate its values near zero Telyakovskiĭ used the piecewise-continuous function $$\sigma \left({{\bf{b}},x} \right) = \left({1/m\left(x \right)} \right)\sum\nolimits_{k = 1}^{m\left(x \right) - 1} {{k^2}\left({{b_k} - {b_{k + 1}}} \right),\,\,m\left(x \right) = \left[{\pi /x} \right]}$$ . He showed that in some neighborhood of zero the difference g(b,x) − (bm(x)/2) cot(x/2) can be estimated from both sides two-sided in terms of the function σ(b,x) with absolute constants. In the present paper, sharp values of these constants on the class of convex slowly varying sequences b are found. A sharp two-sided estimate for the sum of a sine series on this class is obtained. Examples that demonstrate good accuracy of the obtained two-sided estimates are given.

Sharp heat kernel estimates for spectral fractional Laplacian perturbed by gradients

Using Duhamel’s formula, we prove sharp two-sided estimates for the spectral fractional Laplacian’s heat kernel with time-dependent gradient perturbation in bounded C1,1 domains. In addition, we obtain a gradient estimate as well as the Holder continuity of the heat kernel’s gradient.

Progressive intrinsic ultracontractivity and heat kernel estimates for non-local Schrödinger operators

We study the long-time asymptotic behaviour of semigroups generated by non-local Schrödinger operators of the form H=−L+V; the free operator L is the generator of a symmetric Lévy process in Rd, d>1 (with non-degenerate jump measure) and V is a sufficiently regular confining potential. We establish sharp two-sided estimates of the corresponding heat kernels for large times and identify a new general regularity property, which we call progressive intrinsic ultracontractivity, to describe the large-time evolution of the corresponding Schrödinger semigroup. We discuss various examples and applications of these estimates, for instance we characterize the heat trace and heat content. Our examples cover a wide range of processes and we have to assume only mild restrictions on the growth, resp. decay, of the potential and the jump intensity of the free process. Our approach is based on a combination of probabilistic and analytic methods; our examples include fractional and quasi-relativistic Schrödinger operators.