Sharp Function Research Articles

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.

Green function estimates for second order elliptic operators in non-divergence form with Dini continuous coefficients

Two-sided sharp Green function estimates are obtained for second order uniformly elliptic operators in non-divergence form with Dini continuous coefficients in bounded C1,1 domains, which are shown to be comparable to that of the Dirichlet Laplace operator in the domain. The first and second order derivative estimates of the Green functions are also derived. Moreover, boundary Harnack inequality with an explicit boundary decay rate and interior Schauder’s estimates for these differential operators are established, which may be of independent interest.

Necessary and sufficient conditions for boundedness of commutators of maximal function on the $ p $-adic vector spaces

<abstract><p>In this paper, we first show that the $ p $-adic version of maximal function $ \mathcal{M}_{L\log L}^{p} $ is equivalent to the maximal function $ \mathcal{M}^{p}(\mathcal{M}^{p}) $ and that the class of functions for which the maximal commutators and the commutator with the $ p $-adic version of maximal function or the maximal sharp function are bounded on the $ p $-adic vector spaces are characterized and proved to be the same. Moreover, new pointwise estimates for these operators are proved.</p></abstract>

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].

On the best constant in the Lp estimate for the sharp maximal function

Let I be a fixed subinterval of R and let 1≤p<∞ be a fixed exponent. Suppose that f∈Lp(I) is an arbitrary function of integral zero and let f# be the BLO-based sharp maximal function of f. The paper contains the identification of the best constant Cp in the estimate‖f‖Lp≤Cp‖f#‖Lp. The proof rests on the explicit evaluation of the Bellman function associated with the above estimate.

A pointwise estimate for pseudo-differential operators

Let [Formula: see text] be a pseudo-differential operator defined by the symbol [Formula: see text] with [Formula: see text] [Formula: see text]. It is shown that if [Formula: see text], then the operator satisfies the following pointwise estimate: [Formula: see text] for all [Formula: see text] and all Schwartz function [Formula: see text] Here, [Formula: see text] denotes the John-Nirenberg sharp function of [Formula: see text] and [Formula: see text] stands for the Hardy–Littlewood maximal operator.

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.

Description of global EGAM in the maximum of local frequency during current ramp-up discharges in DIII-D

Energetic-particle-induced geodesic acoustic modes, EGAMs (Fu, Phys. Rev. Let., vol. 101, 2008, pp. 185002), driven by neutral beam injection (NBI), have been observed in many DIII-D tokamak experiments (Nazikian et al., Phys. Rev. Lett., vol. 101, 2008, pp. 185001). This mechanism has been theoretically investigated in (Qiu et al., Plasma Phys. Control. Fusion, vol. 52, 2010, pp. 095003), using a sharp energetic particle distribution function, and in (Qu et al., Plasma Phys. Control. Fusion, vol. 59, 2017, pp. 055018), where the dispersion relation and eigenmode behaviour were obtained for the situation of early beam scenario, that is, for times smaller than the beam slowing down time. In this work, we extend these studies determining the eigenmode for beyond the slowing down time, in a scenario with reverse safety factor $q$ profile, where a small concentration of energetic ions can produce an off-axis maximum in the GAM dispersion relation. The characteristics of EGAM are analytically studied with the drift kinetic equation together with the MHD code NOVA. The toroidal energetic ion transit frequency, coupled with the GAM frequency, produces the maximum in the dispersion relation where the eigenmode can be found. The quantitative correspondence of experimental results with the predictions of the proposed model is analysed.

Sharp threshold for the Erdős–Ko–Rado theorem

AbstractFor positive integers and with , the Kneser graph is the graph with vertex set consisting of all ‐sets of , where two ‐sets are adjacent exactly when they are disjoint. The independent sets of are ‐uniform intersecting families, and hence the maximum size independent sets are given by the Erdős–Ko–Rado Theorem. Let be a random spanning subgraph of where each edge is included independently with probability . Bollobás, Narayanan, and Raigorodskii asked for what does have the same independence number as with high probability. For , we prove a hitting time result, which gives a sharp threshold for this problem at . Additionally, completing work of Das and Tran and work of Devlin and Kahn, we determine a sharp threshold function for all .

Underwater acoustic communication based on chirp spread spectrum with cyclic shifted PN sequences

In this presentation, we propose a new underwater acoustic communication technique that combines a binary PN (Pseudo Noise) sequence to which cyclic shift is applied and a CSS (Chirp Spread Spectrum) method that is robust against low Doppler shift. PN sequences are known to have sharp autocorrelation function and cyclic properties. In the case of the previous CSS method, 1-bit per symbol was mapped according to the sign of the slope in the time-frequency domain of the chirp signal, but the proposed method maps 2-bit per symbol according to the cyclic shift direction of PN sequence and the slope of the chirp. As a result, the transmission data rate is doubled. For example, “00” symbol is expressed as an up-chirp and right cyclic shifted PN sequence, and “11” symbol can be expressed as a down-chirp and left cyclic shifted PN sequence. This combination can have excellent autocorrelation properties and Doppler shift immunity. The performance of the proposed method will be compared with that of the previous method by presenting the results of the sea trial. [Work supported by the Agency for Defense Development, South Korea, Grant No. UD200002DD.]

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.

Weighted and endpoint estimates for commutators of bilinear pseudo-differential operators

&lt;abstract&gt;&lt;p&gt;In this paper, by applying the accurate estimates of the Hörmander class, the authors consider the commutators of bilinear pseudo-differential operators and the operation of multiplication by a Lipschitz function. By establishing the pointwise estimates of the corresponding sharp maximal function, the boundedness of the commutators is obtained respectively on the products of weighted Lebesgue spaces and variable exponent Lebesgue spaces with $ \sigma \in\mathcal{B}BS_{1, 1}^{1} $. Moreover, the endpoint estimate of the commutators is also established on $ L^{\infty}\times L^{\infty} $.&lt;/p&gt;&lt;/abstract&gt;

Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators

Abstract In this paper, we establish some sharp maximal function estimates for certain Toeplitz-type operators associated with some fractional integral operators with general kernel. As an application, we obtain the boundedness of the Toeplitz-type operators on the Lebesgue, Morrey and Triebel-Lizorkin spaces. The operators include the Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

L p Smoothness on Weighted Besov–Triebel–Lizorkin Spaces in terms of Sharp Maximal Functions

It is known, in harmonic analysis theory, that maximal operators measure local smoothness of L p functions. These operators are used to study many important problems of function theory such as the embedding theorems of Sobolev type and description of Sobolev space in terms of the metric and measure. We study the Sobolev-type embedding results on weighted Besov–Triebel–Lizorkin spaces via the sharp maximal functions. The purpose of this paper is to study the extent of smoothness on weighted function spaces under the condition M α # f ∈ L p , μ , where μ is a lower doubling measure, M α # f stands for the sharp maximal function of f , and 0 ≤ α ≤ 1 is the degree of smoothness.

Real interpolation of martingale Orlicz Hardy spaces and BMO spaces

In this article, the authors prove that the real interpolation spaces between martingale Orlicz Hardy spaces and martingale BMO spaces are martingale Orlicz–Lorentz Hardy spaces. Using sharp maximal functions, the authors also establish the characterizations of martingale Orlicz Hardy spaces.

An extension of Pratelli’s inequality

Let X, Y be càdlàg martingales and let Y# denote the sharp function of Y. The paper contains the proof of the estimate ‖∫0∞|d〈X,Y〉t|‖1≤2‖〈X〉1∕2Y#‖1 for the total variation between X and Y. The constant 2 is shown to be the best possible. The proof rests on the construction of an appropriate special function, enjoying certain size and concavity requirements.

Further Results on the Rainbow Vertex-Disconnection of Graphs

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of \(G-S\); whereas when x and y are adjacent, \(S+x\) or \(S+y\) is rainbow and x and y belong to different components of \((G-xy)-S\). Such a vertex subset S is called an x–y rainbow vertex-cut of G. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertex-disconnected. In this paper, we obtain bounds of the rainbow vertex-disconnection number of a graph in terms of the minimum degree and maximum degree of the graph. We give a tighter upper bound for the maximum size of a graph G with \(rvd(G)=k\) for \(k\ge \frac{n}{2}\). We then characterize the graphs of order n with rainbow vertex-disconnection number \(n-1\) and obtain the maximum size of a graph G with \(rvd(G)=n-1\). Moreover, we get a sharp threshold function for the property \(rvd(G(n,p))=n\) and prove that almost all graphs G have \(rvd(G)=rvd({\overline{G}})=n\). Finally, we obtain some Nordhaus–Gaddum-type results: \(n-5\le rvd(G)+rvd({\overline{G}})\le 2n\) and \(n-1\le rvd(G)\cdot rvd({\overline{G}})\le n^2\) for the rainbow vertex-disconnection numbers of nontrivial connected graphs G and \({\overline{G}}\) with order \(n\ge 24\).

High-resolution Lamb waves dispersion curves estimation and elastic property inversion

Ultrasonic Lamb waves have been widely used for non-destructive evaluation and testing. However, the inversion from the measured guided signals to the material properties is still a challenging task in terms of multimodal dispersive signal processing and parameter estimation. This paper presents a robust strategy including the high-resolution extraction of the multimodal dispersion curves and model-based elastic property estimation. First, the estimation of signal parameters via rotation invariant technique (ESPRIT) is employed to extract the dispersion curves of the Lamb waves in the plates. Then, the particle swarm optimization (PSO) algorithm is used to retrieve the optimal model parameters by maximizing the objective function built from the dispersion equations. The elastic properties (i.e., two independent constants for the isotropic plate and four constants for the transversely isotropic plate) can thus be determined. Results of the steel, aluminum and composite plates demonstrate that the estimates are in agreement with the references. The root mean squared errors (RMSEs) between the estimated and theoretical dispersion curves calculated by the inversed model parameters for simulation, steel, aluminum and composite experiments are 0.027 rad/mm, 0.032 rad/mm, 0.033 rad/mm and 0.102 rad/mm respectively. The estimated error of thickness is less than 1%. The proposed model-based inversion strategy offers several advantages: (1) the high-resolution estimation of dispersion curves allows the objective function built for the parameter inversion without a peak-finding process. (2) The ESPRIT based dispersion curves extraction strategy offers a sharp objective function in the parameter space. (3) The inverse problem for ultrasonic waveguide characterization is solved using the PSO optimizer which can be implemented with ease and few parameters need to be tuned.

On the Space of Bounded Mean Oscillations

The space of bounded mean oscillations, abbreviated BMO, was first introduced by F. John and L. Nirenberg in 1961 in the context of partial differential equations. Later, C. Fefferman proved that the BMO is the dual space of well-known Hardy space, popularly known as H1 space and became the center of attraction for mathematicians. With the help of BMO space, many mathematical phenomenon can be characterized clearly. In this article, we discuss the connections of function of bounded mean oscillations with weight functions, sharp maximal functions and Carleson measure.

Extensions of the John–Nirenberg theorem and applications

The John–Nirenberg theorem states that functions of bounded mean oscillation are exponentially integrable. In this article we give two extensions of this theorem. The first one relates the dyadic maximal function to the sharp maximal function of Fefferman–Stein, while the second one concerns local weighted mean oscillations, generalizing a result of Muckenhoupt and Wheeden. Applications to the context of generalized Poincaré type inequalities and to the context of the C p C_p class of weights are given. Extensions to the case of polynomial BMO \operatorname {BMO} type spaces are also given.