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Exponential Contractivity and Propagation of Chaos for Langevin Dynamics of McKean-Vlasov Type with Lévy Noises Potential Anal. (IF 1.1) Pub Date : 2024-03-15
Abstract By the probabilistic coupling approach which combines a new refined basic coupling with the synchronous coupling for Lévy processes, we obtain explicit exponential contraction rates in terms of the standard \(L^1\) -Wasserstein distance for the following Langevin dynamic \((X_t,Y_t)_{t\ge 0}\) of McKean-Vlasov type on \(\mathbb R^{2d}\) : $$\begin{aligned} \left\{ \begin{array}{l} dX_t=Y_t\
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On $$L_{p}-$$ Theory for Integro-Differential Operators with Spatially Dependent Coefficients Potential Anal. (IF 1.1) Pub Date : 2024-03-14 Sutawas Janreung, Tatpon Siripraparat, Chukiat Saksurakan
The parabolic integro-differential Cauchy problem with spatially dependent coefficients is considered in generalized Bessel potential spaces where smoothness is defined by Lévy measures with O-regularly varying profile. The coefficients are assumed to be bounded and Hölder continuous in the spatial variable. Our results can cover interesting classes of Lévy measures that go beyond those comparable
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Large and Moderate Deviations for Empirical Density Fields of Stochastic Seir Epidemics with Vertex-Dependent Transition Rates Potential Anal. (IF 1.1) Pub Date : 2024-03-09 Xiaofeng Xue, Xueting Yin
In this paper, we are concerned with stochastic susceptible-exposed-infected-removed epidemics on complete graphs with vertex-dependent transition rates. Large and moderate deviations of empirical density fields of our models are given. Proofs of our main results utilize exponential martingale strategies. In the proof of the moderate deviation principle, we introduce an iteration approach to check
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Probabilistic Characterization of Weakly Harmonic Maps with Respect to Non-Local Dirichlet Forms Potential Anal. (IF 1.1) Pub Date : 2024-03-09 Fumiya Okazaki
We characterize weakly harmonic maps with respect to non-local Dirichlet forms by Markov processes and martingales. In particular, we can obtain discontinuous martingales on Riemannian manifolds from the image of symmetric stable processes under fractional harmonic maps in a weak sense. Based on this characterization, we also consider the continuity of weakly harmonic maps along the paths of Markov
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Stochastic Generalized Porous Media Equations Over $$\sigma $$ -finite Measure Spaces with Non-continuous Diffusivity Function Potential Anal. (IF 1.1) Pub Date : 2024-03-04 Michael Röckner, Weina Wu, Yingchao Xie
In this paper, we prove that stochastic porous media equations over \(\sigma \)-finite measure spaces \((E,\mathcal {B},\mu )\), driven by time-dependent multiplicative noise, with the Laplacian replaced by a self-adjoint transient Dirichlet operator L and the diffusivity function given by a maximal monotone multi-valued function \(\Psi \) of polynomial growth, have a unique solution. This generalizes
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Trace Operator on von Koch’s Snowflake Potential Anal. (IF 1.1) Pub Date : 2024-02-09 Krystian Kazaniecki, Michał Wojciechowski
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A Fourier Integral Formula for Logarithmic Energy Potential Anal. (IF 1.1) Pub Date : 2024-02-09 L. Frerick, J. Müller, T. Thomaser
A formula which expresses logarithmic energy of Borel measures on \(\mathbb {R}^n\) in terms of the Fourier transforms of the measures is established and some applications are given. In addition, using similar techniques a (known) formula for Riesz energy is reinvented.
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New Universal Inequalities for Eigenvalues of a Clamped Plate Problem Potential Anal. (IF 1.1) Pub Date : 2024-02-08 Yiling Jin, Shiyun Pu, Yuxia Wei, Yue He
In this paper, we study the universal inequalities for eigenvalues of a clamped plate problem, and establish some new universal inequalities that are different from those already present in the literature, such as (Wang and Xia J. Funct. Anal. 245(1), 334-352 2007), (Wang and Xia Calc. Var. Partial Differential 653 Equations 40(1-2), 273-289 2011), (Chen, Zheng, and Lu Pacific J. Math. 255(1), 41-54
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Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations Potential Anal. (IF 1.1) Pub Date : 2024-01-31
Abstract By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics
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A Basic Homogenization Problem for the p-Laplacian in $$\mathbb {R}^d$$ Perforated along a Sphere: $$L^\infty $$ Estimates Potential Anal. (IF 1.1) Pub Date : 2024-01-31 Peter V. Gordon, Fedor Nazarov, Yuval Peres
We consider a boundary value problem for the p-Laplacian, posed in the exterior of small cavities that all have the same p-capacity and are anchored to the unit sphere in \(\mathbb {R}^d\), where \(10\). We show that the problem possesses a critical window characterized by \(\tau :=\lim _{\varepsilon \downarrow 0}\alpha /\alpha _c \in (0,\infty )\), where \(\alpha _c=\varepsilon ^{1/\gamma }\) and
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A Discovery Tour in Random Riemannian Geometry Potential Anal. (IF 1.1) Pub Date : 2024-01-26
Abstract We study random perturbations of a Riemannian manifold \((\textsf{M},\textsf{g})\) by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields \(h^\bullet : \omega \mapsto h^\omega \) will act on the manifold via the conformal transformation \(\textsf{g}\mapsto \textsf{g}^\omega := e^{2h^\omega }\,\textsf{g}\) . Our focus will be on the
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Non-Degeneracy and Infinitely Many Solutions for Critical SchrÖDinger-Maxwell Type Problem Potential Anal. (IF 1.1) Pub Date : 2024-01-25 Yuxia Guo, Yichen Hu, Shaolong Peng
In this paper, we consider the following Schrödinger-Maxwell type equation with critical exponent \(-\Delta u=K(y)\Big (\frac{1}{|x|^{n-2}}*K(x)|u|^{\frac{n+2}{n-2}}\Big )u^{\frac{4}{n-2}},\quad {in}\,\, \mathbb {R}^n, \qquad \text {(0.1)}\) where the function K satisfies the assumption \(\mathcal {F}\), and \(*\) stands for the standard convolution. We first derived the non-degeneracy result for the
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Fractional Derivative Description of the Bloch Space Potential Anal. (IF 1.1) Pub Date : 2024-01-09
Abstract We establish new characterizations of the Bloch space \(\mathcal {B}\) which include descriptions in terms of classical fractional derivatives. Being precise, for an analytic function \(f(z)=\sum _{n=0}^\infty \widehat{f}(n) z^n\) in the unit disc \(\mathbb {D}\) , we define the fractional derivative \( D^{\mu }(f)(z)=\sum \limits _{n=0}^{\infty } \frac{\widehat{f}(n)}{\mu _{2n+1}} z^n \)
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Decay Rate of the Eigenvalues of the Neumann-Poincaré Operator Potential Anal. (IF 1.1) Pub Date : 2023-12-22 Shota Fukushima, Hyeonbae Kang, Yoshihisa Miyanishi
If the boundary of a domain in three dimensions is smooth enough, then the decay rate of the eigenvalues of the Neumann-Poincaré operator is known and it is optimal. In this paper, we deal with domains with less regular boundaries and derive quantitative estimates for the decay rates of the Neumann-Poincaré eigenvalues in terms of the Hölder exponent of the boundary. Estimates in particular show that
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Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds Potential Anal. (IF 1.1) Pub Date : 2023-12-20 Effie Papageorgiou
This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with \(L^1\) initial
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Uniqueness of Conformal Metrics with Constant Q-Curvature on Closed Einstein Manifolds Potential Anal. (IF 1.1) Pub Date : 2023-12-18 Jérôme Vétois
On a smooth, closed Einstein manifold (M, g) of dimension \(n \ge 3\) with positive scalar curvature and not conformally diffeomorphic to the standard sphere, we prove that the only conformal metrics to g with constant Q-curvature of order 4 are the metrics \(\lambda \) g with \(\lambda > 0\) constant.
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Heat Kernel Estimates for Stable-driven SDEs with Distributional Drift Potential Anal. (IF 1.1) Pub Date : 2023-11-24 Mathis Fitoussi
We consider the formal SDE \(\textrm{d} X_t = b(t,X_t)\textrm{d} t + \textrm{d} Z_t, \qquad X_0 = x \in \mathbb {R}^d, (\text {E})\) where \(b\in L^r ([0,T],\mathbb {B}_{p,q}^\beta (\mathbb {R}^d,\mathbb {R}^d))\) is a time-inhomogeneous Besov drift and \(Z_t\) is a symmetric d-dimensional \(\alpha \)-stable process, \(\alpha \in (1,2)\), whose spectral measure is absolutely continuous w.r.t. the Lebesgue
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The Fractional Laplacian with Reflections Potential Anal. (IF 1.1) Pub Date : 2023-11-20 Krzysztof Bogdan, Markus Kunze
Motivated by the notion of isotropic \(\alpha \)-stable Lévy processes confined, by reflections, to a bounded open Lipschitz set \(D\subset \mathbb {R}^d\), we study some related analytical objects. Thus, we construct the corresponding transition semigroup, identify its generator and prove exponential speed of convergence of the semigroup to a unique stationary distribution for large time.
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Some Inequalities Between Ahlfors Regular Conformal Dimension And Spectral Dimensions For Resistance Forms Potential Anal. (IF 1.1) Pub Date : 2023-11-11 Kôhei Sasaya
Quasisymmetric maps are well-studied homeomorphisms between metric spaces preserving annuli, and the Ahlfors regular conformal dimension \(\dim _\textrm{ARC}(X,d)\) of a metric space (X, d) is the infimum over the Hausdorff dimensions of the Ahlfors regular images of the space by quasisymmetric transformations. For a given regular Dirichlet form with the heat kernel, the spectral dimension \(d_s\)
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On the Restriction of a Right Process Outside a Negligible Set Potential Anal. (IF 1.1) Pub Date : 2023-11-09 Liping Li, Michael Röckner
The objective of this paper is to examine the restriction of a right process on a Radon topological space, excluding a negligible set, and investigate whether the restricted object can induce a Markov process with desirable properties. We address this question in three aspects: the induced process necessitates only right continuity; it is a right process, and the semi-Dirichlet form of the induced
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Rank 5 Trivializable Subriemannian Structure on $$\mathbb {S}^7$$ and Subelliptic Heat Kernel Potential Anal. (IF 1.1) Pub Date : 2023-10-28 Wolfram Bauer, Abdellah Laaroussi, Daisuke Tarama
We present an explicit form of the subelliptic heat kernel of the intrinsic sublaplacian \(\Delta _{\textrm{sub}}^5\) induced by a rank 5 trivializable subriemannian structure on the Euclidean seven dimensional sphere \(\mathbb {S}^7\). This completes the heat kernel analysis of trivializable subriemannian structures on \(\mathbb {S}^7\) induced by a Clifford module action on \(\mathbb {R}^8\). As
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Lower Bound for the Green Energy of Point Configurations in Harmonic Manifolds Potential Anal. (IF 1.1) Pub Date : 2023-10-19 Carlos Beltrán, Víctor de la Torre, Fátima Lizarte
In this paper, we get the sharpest known to date lower bounds for the minimal Green energy of the compact harmonic manifolds of any dimension. Our proof generalizes previous ad-hoc arguments for the most basic harmonic manifold, i.e. the sphere, extending it to the general case and remarkably simplifying both the conceptual approach and the computations.
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Extensions of Harmonic Functions of the Complex Plane Slit Along a Line Segment Potential Anal. (IF 1.1) Pub Date : 2023-10-19 Armen Grigoryan, Andrzej Michalski, Dariusz Partyka
Let I be a line segment in the complex plane \(\mathbb C\). We describe a method of constructing a bi-Lipschitz sense-preserving mapping of \(\mathbb C\) onto itself, which is harmonic in \(\mathbb C\setminus I\) and coincides with a given sufficiently regular function \(f:I\rightarrow \mathbb C\). As a result we show that a quasiconformal self-mapping of \(\mathbb C\) which is harmonic in \(\mathbb
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A Convergence Rate for Extended-Source Internal DLA in the Plane Potential Anal. (IF 1.1) Pub Date : 2023-10-16 David Darrow
Internal DLA (IDLA) is an internal aggregation model in which particles perform random walks from the origin, in turn, and stop upon reaching an unoccupied site. Levine and Peres showed that, when particles start instead from fixed multiple-point distributions, the modified IDLA processes have deterministic scaling limits related to a certain obstacle problem. In this paper, we investigate the convergence
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Harmonic Bergman Projectors on Homogeneous Trees Potential Anal. (IF 1.1) Pub Date : 2023-10-13 Filippo De Mari, Matteo Monti, Maria Vallarino
In this paper we investigate some properties of the harmonic Bergman spaces \(\mathcal A^p(\sigma )\) on a q-homogeneous tree, where \(q\ge 2\), \(1\le p<\infty \), and \(\sigma \) is a finite measure on the tree with radial decreasing density, hence nondoubling. These spaces were introduced by J. Cohen, F. Colonna, M. Picardello and D. Singman. When \(p=2\) they are reproducing kernel Hilbert spaces
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The Stochastic Klausmeier System and A Stochastic Schauder-Tychonoff Type Theorem Potential Anal. (IF 1.1) Pub Date : 2023-10-13 Erika Hausenblas, Jonas M. Tölle
On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step
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Asymptotic Behavior for Multi-scale SDEs with Monotonicity Coefficients Driven by Lévy Processes Potential Anal. (IF 1.1) Pub Date : 2023-10-11 Yinghui Shi, Xiaobin Sun, Liqiong Wang, Yingchao Xie
In this paper, we study the asymptotic behavior for multi-scale stochastic differential equations driven by Lévy processes. The optimal strong convergence order 1/2 is obtained by studying the regularity estimates for the solution of Poisson equation with polynomial growth coefficients, and the optimal weak convergence order 1 is got by using the technique of Kolmogorov equation. The main contribution
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Dual Spaces for Weak Martingale Hardy Spaces Associated with Rearrangement-Invariant Spaces Potential Anal. (IF 1.1) Pub Date : 2023-10-11 Xingyan Quan, Niyonkuru Silas, Guangheng Xie
Given a probability space \((\Omega ,\mathcal {F},\mathbb P)\) and a rearrangement-invariant quasi-Banach function space X, the authors of this article first prove the \(\alpha \)-atomic (\(\alpha \in [1,\infty )\)) characterization of weak martingale Hardy spaces \(WH_X(\Omega )\) associated with X via simple atoms. The authors then introduce the generalized weak martingale \(\textrm{BMO}\) spaces
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Schauder Estimates for Nonlocal Equations with Singular Lévy Measures Potential Anal. (IF 1.1) Pub Date : 2023-10-05 Zimo Hao, Zhen Wang, Mingyan Wu
In this paper, we establish Schauder’s estimates for the following non-local equations in \(\mathbb {R}^{d}\) : $$ \partial _t u=\mathscr {L}^{(\alpha )}_{\kappa ,\sigma } u+b\cdot \nabla u+f,\ u(0)=0, $$ where \(\alpha \in (1/2,2)\) and \( b:\mathbb {R}_+\times \mathbb {R}^d\rightarrow \mathbb R\) is an unbounded local \(\beta \)-order Hölder function in x uniformly in t, and \(\mathscr {L}^{(\alpha
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Necessary cancellation conditions for the boundedness of operators on local Hardy spaces Potential Anal. (IF 1.1) Pub Date : 2023-09-07 Galia Dafni, Chun Ho Lau, Tiago Picon, Claudio Vasconcelos
In this work we present necessary cancellation conditions for the continuity of linear operators in \(h^p({\mathbb R^n})\), \(0
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On the Shape of the First Fractional Eigenfunction Potential Anal. (IF 1.1) Pub Date : 2023-09-05 Nicola Abatangelo, Sven Jarohs
We show that the first eigenfunction of the fractional Laplacian \({\left( -\Delta \right) }^{s}\), \(s\in (1/2,1)\), is superharmonic in the unitary ball up to dimension 11. To this aim, we also rely on a computer-assisted step to estimate a rather complicated constant depending on the dimension and the power s.
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Heyde Theorem on Locally Compact Abelian Groups with the Connected Component of Zero of Dimension 1 Potential Anal. (IF 1.1) Pub Date : 2023-09-04 Gennadiy Feldman
Let X be a locally compact Abelian group with the connected component of zero of dimension 1. Let \(\xi _1\) and \(\xi _2\) be independent random variables with values in X with nonvanishing characteristic functions. We prove that if a topological automorphism \(\alpha\) of the group X satisfies the condition \({\text {Ker}(I+\alpha )=\{0\}}\) and the conditional distribution of the linear form \({L_2
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Parabolic Muckenhoupt Weights with Time Lag on Spaces of Homogeneous Type with Monotone Geodesic Property Potential Anal. (IF 1.1) Pub Date : 2023-08-30 Juha Kinnunen, Kim Myyryläinen, Dachun Yang, Chenfeng Zhu
This work discusses parabolic Muckenhoupt weights with time lag on spaces of homogeneous type with an extra monotone geodesic property. The main results include a characterization in terms of weighted norm inequalities for parabolic maximal operators, a reverse Hölder inequality, and a Jones-type factorization result for this class of weights. The connection between the space of parabolic bounded mean
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Two-Sided Boundary Points of Sobolev Extension Domains on Euclidean Spaces Potential Anal. (IF 1.1) Pub Date : 2023-08-29 Miguel García-Bravo, Tapio Rajala, Jyrki Takanen
We prove an estimate on the Hausdorff dimension of the set of two-sided boundary points of general Sobolev extension domains on Euclidean spaces. We also present examples showing lower bounds on possible dimension estimates of this type.
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p-Capacity with Bessel Convolution Potential Anal. (IF 1.1) Pub Date : 2023-08-29 Á. P. Horváth
We define and examine nonlinear potential by Bessel convolution with Bessel kernel. We investigate removable sets with respect to Laplace-Bessel inequality. By studying the maximal and fractional maximal measure, a Wolff type inequality is proved. Finally the relation of B-p capacity and B-Lipschitz mapping, and the B-p capacity and weighted Hausdorff measure and the B-p capacity of Cantor sets are
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Non-Gaussian Measures in Infinite Dimensional Spaces: the Gamma-Grey Noise Potential Anal. (IF 1.1) Pub Date : 2023-08-23 Luisa Beghin, Lorenzo Cristofaro, Janusz Gajda
In the context of non-Gaussian analysis, Schneider [29] introduced grey noise measures, built upon Mittag-Leffler functions; analogously, grey Brownian motion and its generalizations were constructed (see, for example, [6, 7, 9, 27]). In this paper, we construct and study a new non-Gaussian measure, by means of the incomplete-gamma function (exploiting its complete monotonicity). We label this measure
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Connections and Loops Intertwinning Potential Anal. (IF 1.1) Pub Date : 2023-08-22 Yves Le Jan
On a finite graph, we prove that trace of holonomies determine an intertwining relation between merge-and-split generators on collections of geodesic loops ensembles and Casimir operators on moduli of unitary connections. By adding a deformation part to the generator on loops, this result is extended to the Casimir operator modified in order to be self adjoint with respect to Yang-Mills measure.
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A Smooth 1-Parameter Family of Delaunay-Type Domains for an Overdetermined Elliptic Problem in $$\mathbb {S}^n\times \mathbb {R}$$ and $$\mathbb {H}^n\times \mathbb {R}$$ Potential Anal. (IF 1.1) Pub Date : 2023-08-18 Guowei Dai, Filippo Morabito, Pieralberto Sicbaldi
We prove the existence of a non-compact smooth one-parameter family of domains \(\Omega _s\subset \mathbb {M}^n\times \mathbb {R}\), where \(\mathbb {M}^n\) denotes the Riemannian manifold \(\mathbb {S}^n\) or \(\mathbb {H}^n\) (for \(n \ge 2\)), bifurcating from the straight cylinder \(B_1\times \mathbb {R}\) (where \(B_1\) is a geodesic unit ball in \(\mathbb {M}^n\)) such that there exists a positive
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Weighted Poincaré Inequalities and Degenerate Elliptic and Parabolic Problems: An Approach via the Distance Function Potential Anal. (IF 1.1) Pub Date : 2023-08-08 D. D. Monticelli, F. Punzo
We obtain weighted Poincaré inequalities in bounded domains, where the weight is given by a symmetric nonnegative definite matrix, which can degenerate on submanifolds. Furthermore, we investigate uniqueness and nonuniqueness of solutions to degenerate elliptic and parabolic problems, where the diffusion matrix can degenerate on subsets of the boundary of the domain. Both the results are obtained by
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Mittag–Leffler Euler Integrator and Large Deviations for Stochastic Space-Time Fractional Diffusion Equations Potential Anal. (IF 1.1) Pub Date : 2023-08-08 Xinjie Dai, Jialin Hong, Derui Sheng
Stochastic space-time fractional diffusion equations often appear in the modeling of the heat propagation in non-homogeneous medium. In this paper, we firstly investigate the Mittag–Leffler Euler integrator of a class of stochastic space-time fractional diffusion equations, whose super-convergence order is obtained by developing a helpful decomposition way for the time-fractional integral. Here, the
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$$p-$$ Harmonic Functions in the Upper Half-space Potential Anal. (IF 1.1) Pub Date : 2023-08-07 E. Abreu, R. Clemente, J. M. do Ó, E. Medeiros
This paper investigates the existence, nonexistence, and qualitative properties of p-harmonic functions in the upper half-space \(\mathbb {R}^N_+\) \((N\ge 3)\) satisfying nonlinear boundary conditions for \(1
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q-Poincaré Inequalities on Carnot Groups with Filiform Type Lie Algebra Potential Anal. (IF 1.1) Pub Date : 2023-07-31 Marianna Chatzakou, Serena Federico, Boguslaw Zegarlinski
In this paper we prove (global) q- Poincaré inequalities for probability measures on nilpotent Lie groups with filiform Lie algebra of any length. The probability measures under consideration have a density with respect to the Haar measure given as a function of a suitable homogeneous norm.
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Manifolds with Density and the First Steklov Eigenvalue Potential Anal. (IF 1.1) Pub Date : 2023-07-31 Márcio Batista, José I. Santos
There are many interesting eigenvalue problems in a variety of settings; one of them is the well-known Steklov eigenvalue problem. In this work, we are interested in studying some Steklov eigenvalue problems for elliptic operators of second and fourth order using a well-known Reilly formula. Some upper and lower bounds for the first eigenvalue are obtained, and the rigidity case is carefully analyzed
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Extremal of Log-Sobolev Functionals and Li-Yau Estimate on $$\textrm{RCD}^*(K,N)$$ Spaces Potential Anal. (IF 1.1) Pub Date : 2023-07-27 Samuel Drapeau, Liming Yin
In this work, we study the extremal functions of the log-Sobolev functional on compact metric measure spaces satisfying the \(\textrm{RCD}^*(K,N)\) condition for K in \(\mathbb {R}\) and N in \((2,\infty )\). We show the existence, regularity and positivity of non-negative extremal functions. Based on these results, we prove a Li-Yau type estimate for the logarithmic transform of any non-negative extremal
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The $$\textrm{CMO}$$ -Dirichlet Problem for Elliptic Systems in the Upper Half-Space Potential Anal. (IF 1.1) Pub Date : 2023-07-25 Mingming Cao
We prove that for any second-order, homogeneous, \(N \times N\) elliptic system L with constant complex coefficients in \(\mathbb {R}^n\), the Dirichlet problem in \(\mathbb {R}^n_+\) with boundary data in \(\textrm{CMO}(\mathbb {R}^{n-1}, \mathbb {C}^N)\) is well-posed under the assumption that \(d\mu (x', t) := |\nabla u(x)|^2\, t \, dx' dt\) is a strong vanishing Carleson measure in \(\mathbb {R}^n_+\)
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Asymptotics, Trace, and Density Results for Weighted Dirichlet Spaces Defined on the Half-line Potential Anal. (IF 1.1) Pub Date : 2023-07-24 Claudia Capone, Agnieszka Kałamajska
We give an analytic description for the completion of \(C_0^\infty (\mathbb {R}_+)\), where \(\mathbb {R}_+= (0,\infty )\), in Dirichlet space \(D^{1,p}(\mathbb {R}_+, \omega ):= \{ u \, :\mathbb {R}_+\rightarrow {{\mathbb {R}}}: u\ \) is locally absolutely continuous on \(\mathbb {R}_+\, and \, \Vert u^{'}\Vert _{L^p(\mathbb {R}_+, \omega )}<\infty \}\), for given continuous positive weight \(\omega
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Inner Riesz Pseudo-Balayage and its Applications to Minimum Energy Problems with External Fields Potential Anal. (IF 1.1) Pub Date : 2023-07-22 Natalia Zorii
For the Riesz kernel \(\kappa _\alpha (x,y):=|x-y|^{\alpha -n}\) of order \(0<\alpha
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On complete Weingarten Hypersurfaces Under Assumptions on the Gauss-Kronecker Curvature Potential Anal. (IF 1.1) Pub Date : 2023-07-22 Eudes L. de Lima
We provide a classification result for complete rotation Weingarten hypersurfaces of the Euclidean space satisfying \(K = aH_k + b\), for constants \(a, b \in \mathbb {R}\), where K is the Gauss-Kronecker curvature and \(H_k\) stands for the k-th mean curvature of the hypersurface, \(1 \le k < n\). Moreover, we prove that the only hypersurface of the Euclidean space with \(H_r = aH_k + b\), \(1 \le
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On the Upper Rate Functions of Some Time Inhomogeneous Diffusion Processes Potential Anal. (IF 1.1) Pub Date : 2023-07-13 Daehong Kim, Yoichi Oshima
In the present paper, we study an upper escape rate of some time inhomogeneous diffusion process associated with a family of regular and local Dirichlet forms. In particular, by making full use of Gaussian type’s heat kernel estimates, we establish integral tests for an upper rate function of the time inhomogeneous diffusion process with a coefficient that is not necessarily bounded concerning space
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Littlewood-Paley Theorem, Nikolskii Inequality, Besov Spaces, Fourier and Spectral Multipliers on Graded Lie Groups Potential Anal. (IF 1.1) Pub Date : 2023-07-12 Duván Cardona, Michael Ruzhansky
In this paper we investigate Besov spaces on graded Lie groups. We prove a Nikolskii type inequality (or the Reverse Hölder inequality) on graded Lie groups and as consequence we obtain embeddings of Besov spaces. We prove a version of the Littlewood-Paley theorem on graded Lie groups. The results are applied to obtain embedding properties of Besov spaces and multiplier theorems for both spectral and
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Nonlinear McKean-Vlasov Diffusions under the Weak Hörmander Condition with Quantile-Dependent Coefficients Potential Anal. (IF 1.1) Pub Date : 2023-07-10 Yaozhong Hu, Michael A. Kouritzin, Jiayu Zheng
In this paper, the strong existence and uniqueness for a degenerate finite system of quantile-dependent McKean-Vlasov stochastic differential equations are obtained under a weak Hörmander condition. The approach relies on the a priori bounds for the density of the solution to time inhomogeneous diffusions. The time inhomogeneous Feynman-Fac formula is used to construct a contraction map for this degenerate
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Stochastic Doubly Nonlinear PDE: Large Deviation Principles and Existence of Invariant Measure Potential Anal. (IF 1.1) Pub Date : 2023-07-10 Ananta K. Majee
In this paper, we establish large deviation principle for the strong solution of a doubly nonlinear PDE driven by small multiplicative Brownian noise. Motononicity arguments and the weak convergence approach have been exploited in the proof. Moreover, by using certain a-priori estimates and sequentially weakly Feller property of the associated Markov semigroup, we show existence of invariant probability
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Corrigendum to “From $$A_1$$ to $$A_\infty $$ : New Mixed Inequalities for Certain Maximal Operators” Potential Anal. (IF 1.1) Pub Date : 2023-07-10 Fabio Berra
We devote this note to correct an estimate concerning mixed inequalities for the generalized maximal function \(M_\Phi \) given in Berra (Potential Anal. 57(1), 1–27, 2022), when certain properties of the associated Young function \(\Phi \) are assumed. Although the obtained estimates turn out to be slightly different, they are good extensions of mixed inequalities for the classical Hardy-Littlewood
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Boundary Lipschitz Regularity and the Hopf Lemma for Fully Nonlinear Elliptic Equations Potential Anal. (IF 1.1) Pub Date : 2023-07-10 Yuanyuan Lian, Kai Zhang
In this paper, we study the boundary regularity for viscosity solutions of fully nonlinear elliptic equations. We use a unified, simple method to prove that if the domain \(\Omega \) satisfies the exterior \(C^{1,\textrm{Dini}}\) condition at \(x_0\in \partial \Omega \), the solution is Lipschitz continuous at \(x_0\); if \(\Omega \) satisfies the interior \(C^{1,\textrm{Dini}}\) condition at \(x_0\)
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Growth of Harmonic Mappings and Baernstein Type Inequalities Potential Anal. (IF 1.1) Pub Date : 2023-07-08 Suman Das, Anbareeswaran Sairam Kaliraj
Seminal works of Hardy and Littlewood on the growth of analytic functions contain the comparison of the integral means \(M_p(r,f)\), \(M_p(r,f')\), \(M_q(r,f)\). For a complex-valued harmonic function f in the unit disk, using the notation \(\vert \nabla f \vert =(\vert f_z{\vert }^2+\vert f_{\bar{z}}{\vert ^2})^{1/2}\) we explore the relation between \(M_p(r,f)\) and \(M_p(r,\nabla f)\). We show that
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Rate of Convergence in the Smoluchowski-Kramers Approximation for Mean-field Stochastic Differential Equations Potential Anal. (IF 1.1) Pub Date : 2023-07-03 Ta Cong Son, Dung Quang Le, Manh Hong Duong
In this paper we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent stochastic noise. Using techniques from Malliavin calculus, we establish explicit rates of convergence in the zero-mass limit (Smoluchowski-Kramers approximation) in the
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Global Well-Posedness of Solutions to a Class of Double Phase Parabolic Equation With Variable Exponents Potential Anal. (IF 1.1) Pub Date : 2023-06-13 Wen-Shuo Yuan, Bin Ge, Qing-Hai Cao
The main objective of this paper is to study a class of parabolic equation driven by double phase operator with initial-boundary value conditions. As is well known, subcritical hypotheses play an important role in investigating well-posedness result to parabolic and elliptic equations. The highlight of this paper is to overcome the difficulties without subcritical assumption creates by restricting
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Fractional Higher Differentiability for Solutions of Stationary Stokes and Navier-Stokes Systems with Orlicz Growth Potential Anal. (IF 1.1) Pub Date : 2023-05-02 Flavia Giannetti, Antonia Passarelli di Napoli, Christoph Scheven
We consider weak solutions \((u,\pi ):{\Omega }\to \mathbb {R}^{n}\times \mathbb {R}\) to stationary ϕ-Navier-Stokes systems of the type \( \left \{ \begin {array}{ll} -\mathrm {div~} a(x,\mathcal {E} u)+\nabla \pi +[Du]u=f \\ \mathrm {div~} u=0 \end {array} \right . \) in \({\Omega }\subset \mathbb {R}^{n}\), and to the corresponding ϕ-Stokes systems, in which the convective term [Du]u does not appear
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Schatten Classes and Commutator in the Two Weight Setting, I. Hilbert Transform Potential Anal. (IF 1.1) Pub Date : 2023-04-06 Michael Lacey, Ji Li, Brett D. Wick
We characterize the Hilbert–Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten class Sp result for dyadic paraproducts, where \(1< p < \infty \). We discuss the difficulties in extending the dyadic result to the full range of Schatten
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Removable Singularities for Solutions of the Fractional Heat Equation in Time Varying Domains Potential Anal. (IF 1.1) Pub Date : 2023-03-21 Joan Mateu, Laura Prat
In this paper we study removable singularities for solutions of the fractional heat equation in the spacial-time space. We introduce associated capacities and we study some of its metric and geometric properties.