We consider parabolic equations of the form $$ u_{t}-\text{div} \left( |\nabla u|^{p-2}\nabla u+ a(x,t)|\nabla u|^{q-2}\nabla u\right)= 0, a(x,t)\geq 0. $$ In the range \(\frac {2n}{n+1}

We construct a family of steady solutions of the lake model perturbed by some small Coriolis force, that converge to a singular vortex pair. The desingularized solutions are obtained by maximization of the kinetic energy over a class of rearrangements of sign changing functions. The precise localization of the asymptotic singular vortex pair is proved to depend on the depth function and the Coriolis

We introduce and study the unconstrained polarization (or Chebyshev) problem which requires to find an N-point configuration that maximizes the minimum value of its potential over a set A in p-dimensional Euclidean space. This problem is compared to the constrained problem in which the points are required to belong to the set A. We find that for Riesz kernels 1/|x − y|s with s > p − 2 the optimum unconstrained

We show that BMO-solvability implies scale invariant quantitative absolute continuity (specifically, the weak-\(A_{\infty }\) property) of caloric measure with respect to surface measure, for an open set Ω ⊂ ℝn+ 1, assuming as a background hypothesis only that the essential boundary of Ω satisfies an appropriate parabolic version of Ahlfors-David regularity, entailing some backwards in time thickness

In this paper, the main aim is to discuss the existence of the extreme points and support points of families of harmonic Bloch mappings and little harmonic Bloch mappings. First, in terms of the Bloch unit-valued set, we prove a necessary condition for a harmonic Bloch mapping (resp. a little harmonic Bloch mapping) to be an extreme point of the unit ball of the normalized harmonic Bloch spaces (resp

We study a multilinear version of the Hörmander multiplier theorem, namely $$ \Vert T_{\sigma}(f_{1},\dots,f_{n})\Vert_{L^{p}}\lesssim \sup_{k\in\mathbb{Z}}{\Vert \sigma(2^{k}\cdot,\dots,2^{k}\cdot)\widehat{\phi^{(n)}}\Vert_{L^{2}_{(s_{1},\dots,s_{n})}}}\Vert f_{1}\Vert_{H^{p_{1}}}\cdots\Vert f_{n}\Vert_{H^{p_{n}}}. $$ We show that the estimate does not hold in the limiting case \(\min \limits {(s_{1}

The Dirichlet forms related to various infinite systems of interacting Brownian motions are studied. For a given random point field μ, there exist two natural infinite-volume Dirichlet forms \( (\mathcal {E}^{\mathsf {upr}},\mathcal {D}^{\mathsf {upr}})\) and \((\mathcal {E}^{\mathsf {lwr}},\mathcal {D}^{\mathsf {lwr}})\) on L2(S,μ) describing interacting Brownian motions each with unlabeled equilibrium

In this manuscript, we appeal to Potential Theory to provide a sufficient condition for existence of distributional solutions to fractional elliptic problems with non-linear first-order terms and measure data ω: $$ \left\{ \begin{array}{rcll} (-{\Delta})^{s}u&=&|\nabla u|^{q} + \omega \quad \text{in }\mathbb{R}^{n}, s \in (1/2, 1)\\ u & > &0 \quad \text{in } \mathbb{R}^{n}\\ \lim_{|x|\to \infty}u(x)

A classical theorem of Bôcher says a positive harmonic function on a punctured domain \(\mathcal {O}\setminus p\) in Euclidean space can be written as the sum of a constant multiple of the Green function with pole at p and a function harmonic on all of \(\mathcal {O}\). Here we establish such a result in the setting of functions on a Riemannian manifold with rather rough metric tensor.

It is shown in quantitative terms that the maximal Bergman projection $$ {P}^{+}_{\omega}(f)(z)={\int}_{\mathbb{D}} f(\zeta)|{B}^{\omega}_{z}(\zeta)|\omega(\zeta) dA(\zeta), $$ is bounded from \(L^{p}_{\nu }\) to \(L^{p}_{\eta }\) if and only if $$ \underset{0

We prove Wasserstein distance bounds between the probability distributions of stochastic integrals with jumps, based on the integrands appearing in their stochastic integral representations. Our approach does not rely on the Stein equation or on the propagation of convexity property for Markovian semigroups, and makes use instead of forward-backward stochastic calculus arguments. This allows us to

A class of (possibly) degenerate integro-differential equations of parabolic type is considered, which includes the Kolmogorov equations for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces and in Sobolev-Slobodeckij spaces. Generalisations to stochastic integro-differential equations, arising in filtering theory of jump diffusions, will be given

In this paper we establish weighted estimates for the bilinear Bochner-Riesz operator \(\mathcal B^{\alpha }\) at the critical index \(\alpha =n-\frac {1}{2}\) with respect to bilinear weights.

Area minimizing hypersurfaces and, more generally, almost minimizing hypersurfaces frequently occur in geometry, dynamics and physics. A central problem is that a general (almost) minimizing hypersurface H contains a complicated singular set Σ. Nevertheless, we manage to develop a detailed potential theory on H ∖Σ applicable to large classes of linear elliptic second order operators. We even get a

For possibly discontinuous functions including, for instance, Sobolev functions, we present new Blaschke-Privaloff-type criteria for superharmonicity and harmonicity. This opens the way for introduction of a substantial generalization of the Laplace operator. These potential-theoretic considerations lead to a new kind of non-absolutely convergent integral where the integrand may be a highly oscillating

In the present paper, as a continuation of our preceding paper (Ishiwata et al. 2018), we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a view point of discrete geometric analysis developed by Kotani and Sunada. We introduce a one-parameter family of random walks which interpolates between the original non-symmetric random walk

In this paper we establish Sobolev type compact embedding theorems for Hörmander classes of pseudodifferential operators \(OpS^{-\alpha }_{1,\delta }\) on localizable Hardy space. Our work include new optimal boundedness results. As application, we obtain compact embeddings for compactly supported distributions with respect to the space variables in the nonhomogeneous localizable Hardy-Sobolev spaces

The asymptotic log-Harnack inequality is proved for Leray-α model with degenerate type noise using the asymptotic coupling method. In particular, we don’t impose any lower bound assumption for the viscosity constant. As applications, we also derive ergodicity and further asymptotic properties for stochastic 3D Leray-α model.

We study weighted (Lp, Lq)-boundedness properties of Riesz potentials and fractional maximal functions for the Dunkl transform. In particular, we obtain the weighted Hardy–Littlewood–Sobolev type inequality and weighted week (L1, Lq) estimate. We find a sharp constant in the weighted Lp-inequality, generalizing the results of W. Beckner and S. Samko.

We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc. We prove that every such series is summable with respect to unrestricted rectangular partial sums, everywhere except for a set of zero multi-parametric logarithmic capacity. Conversely, given a compact set in the torus of zero capacity, we construct a Fourier series in the class which

Let σ and ω be locally finite Borel measures on ℝd, and let \(p\in (1,\infty )\) and \(q\in (0,\infty )\). We study the two-weight norm inequality \( \lVert T(f\sigma ) \rVert _{L^{q}(\omega )}\leq C \lVert f \rVert _{L^{p}(\sigma )}, \quad \text {for all} f \in L^{p}(\sigma ), \) for both the positive summation operators T = Tλ(⋅σ) and positive maximal operators T = Mλ(⋅σ). Here, for a family {λQ}

We study the sub-Laplacian of the 15-dimensional unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the octonionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of a related sub-Laplacian. As a byproduct we also obtain the spectrum of the sub-Laplacian, the small-time asymptotics

The purpose of this paper is to provide necessary and sufficient conditions of the boundedness for singular integrals on the local Hardy space and its dual. Particularly the singular integrals considered in this paper include the pseudo-differential operators $T_{\sigma }f(x)=\int \limits \sigma (x \xi )e^{2\pi ix\xi }\hat {f}(\xi )d\xi $ with \(\sigma \in S_{1 0}^{0}\). As a consequence our results

We extend the well-known result that any \(f \in W^{1,n}({\Omega }, \mathbb {R}^{n})\), \({\Omega } \subset \mathbb {R}^{n}\) with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces \(W^{s,\frac {n}{s}}({\Omega })\) for any \(s \geq \frac {n}{n+1}\), where the sign condition on the Jacobian is understood in a distributional sense. Along the way we also

In this paper we present a short proof of the following classification Theorem for g −harmonic functions in half-spaces. Assume that u is a nonnegative solution to Δgu = 0 in {xn > 0} that continuously vanishes on the flat boundary {xn = 0}. Then, modulo normalization, u(x) = xn in {xn ≥ 0}. Our proof depends on a recent quantitative version of the Hopf-Oleı̆nik Lemma proven by the authors in Braga

For a Dirichlet form \((\mathcal {E},\mathcal {F})\) on L2(E;m), let \(\mathbb {G}(\mathcal {E})=\{X_{u};u\in \mathcal {F}_{e}\}\) be the Gaussian field indexed by the extended Dirichlet space \(\mathcal {F}_{e}\). We first solve the equilibrium problem for a regular recurrent Dirichlet form \(\mathcal {E}\) of finding for a closed set B a probability measure μB concentrated on B whose recurrent potential

Let Δk be the Dunkl Laplacian relative to a fixed root system \(\mathcal {R}\) in \(\mathbb {R}^{d}\), d ≥ 2, and to a nonnegative multiplicity function k on \(\mathcal {R}\). Our first purpose in this paper is to solve the Δk-Dirichlet problem for annular regions. Secondly, we introduce and study the Δk-Green function of the annulus and we prove that it can be expressed by means of Δk-spherical harmonics

In this paper, we mainly construct a connection between synchronized systems and multi-scale equations, and then use the averaging principle as an intermediate step to obtain synchronization. This strategy solves the synchronization problem of dissipative stochastic differential equations, regardless of the structure of the noise. Moreover, the averaging principle of stationary solutions is also investigated

We study the size of the set of points where the α-divided difference of a function in the Hölder class Λα is bounded below by a fixed positive constant. Our results are obtained from their discrete analogues which can be stated in the language of dyadic martingales. Our main technical result in this setting is a sharp estimate of the Hausdorff measure of the set of points where a dyadic martingale

We establish Leibniz-type rules for bilinear and biparameter Fourier multiplier operators with limited Sobolev regularity. Applications of our result are given including the biparameter Leibniz rules, smoothing properties of bilinear-biparameter fractional integrals, and mapping properties of scattering operators for a system of PDEs.

We consider the stochastic differential equation dXt = A(Xt−)dZt, X0 = x, driven by cylindrical α-stable process Zt in , where α ∈ (0,1) and d ≥ 2. We assume that the determinant of A(x) = (aij(x)) is bounded away from zero, and aij(x) are bounded and Lipschitz continuous. We show that for any fixed γ ∈ (0,α) the semigroup Pt of the process Xt satisfies \(|P_{t} f(x) - P_{t} f(y)| \le c t^{-\gamma

Let D be a non-empty open subset of \(\mathbb {R}^{m}, m\ge 2\), with boundary ∂D, with finite Lebesgue measure |D|, and which satisfies a parabolic Harnack principle. Let K be a compact, non-polar subset of D. We obtain the leading asymptotic behaviour as ε↓ 0 of the \(L^{\infty }\) norm of the torsion function with a Neumann boundary condition on ∂D, and a Dirichlet boundary condition on ∂(εK), in

We establish some exact asymptotic results for a matching problem with respect to a family of beta distributions. Let X1,…,Xn be independent random variables with common distribution the symmetric Jacobi measure \(d\mu (x) = C_{d} (1-x^{2})^{\frac d2 -1} dx\) with dimension d ≥ 1 on [− 1,1], and let \(\mu _{n} = \frac {1}{n} {\sum }_{i = 1}^{n} \delta _{X_{i}}\) be the associated empirical measure

We show that, under a natural scaling, the small-time behavior of the logarithmic derivatives of the subelliptic heat kernel on SU(2) converges to their analogues on the Heisenberg group at time 1. Realizing SU(2) as \(\mathbb {S}^{3}\), we then generalize these results to higher-order odd-dimensional spheres equipped with their natural subRiemannian structure, where the limiting spaces are now the

We prove a Chebyshev transform formula for a notion of (weighted) transfinite diameter that is defined using a generalized notion of polynomial degree. We also generalize Leja points to this setting. As an application of our main formula, we prove that in the unweighted case, these generalized Leja points recover the transfinite diameter.

Carbery proved that if \(u:\mathbb {R}^{n} \rightarrow \mathbb {R}\) is a positive, strictly convex function satisfying \(\det D^{2}u \geq 1\), then we have the estimate $$ \left| \left\{x \in \mathbb{R}^{n}: u(x) \leq s \right\} \right| \lesssim_{n} s^{n/2} $$ and this is optimal. We give a short proof that also implies other results. Our main result is an estimate for the sublevel set of functions

In this paper we use potential theoretic arguments to establish new results concerning the overconvergence of Dirichlet series. Let \({\sum }_{j=0}^{\infty } a_{j}e^{-\lambda _{j}s}\) converge on the half-plane {Re(s) > 0} to a holomorphic function f. Our first result gives sufficient conditions for a subsequence of partial sums of the series to converge at every regular point of f. The second result

For a hyperbolic Brownian motion in the hyperbolic space \(\mathbb {H}^{n}, n\ge 3\), we prove a representation of a Green function and a Poisson kernel for bounded and smooth sets in terms of the corresponding objects for an ordinary Euclidean Brownian motion and a conditional gauge functional. Using this representation we prove bounds for the Green functions and Poisson kernels for smooth sets. In

We show that the fractional Brownian motion (fBM) defined via the Volterra integral representation with Hurst parameter \(H\geq \frac {1}{2}\) is a quasi-surely defined Wiener functional on the classical Wiener space, and we establish the large deviation principle (LDP) for such an fBM with respect to (p,r)-capacity on the classical Wiener space in Malliavin’s sense.

We study the problem of maximizing the expected lifetime of drift diffusion in a bounded domain. More formally, we consider the PDE $$ - {\Delta} u + b(x) \cdot \nabla u = 1 \qquad \text{in}~{\Omega} $$ subject to Dirichlet boundary conditions for \(\|b\|_{L^{\infty }}\) fixed. We show that, in any given C2 −domain Ω, the vector field maximizing the expected lifetime is (nonlinearly) coupled to the

Hunt’s hypothesis (H) and the related Getoor’s conjecture is one of the most important problems in the basic theory of Markov processes. In this paper, we investigate the invariance of Hunt’s hypothesis (H) for Markov processes under two classes of transformations, which are change of measure and subordination. Our first theorem shows that for two standard processes (Xt) and (Yt), if (Xt) satisfies

We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form \(\partial _{t}u-A_{t}u-f=(\dot X_{t}(x) \cdot \nabla + \dot Y_{t}(x))u\) on \([0,T]\times \mathbb {R}^{d}.\) To do so, we introduce a concept of “differential rough driver”, which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces

We obtain some fine gradient estimates near the boundary for solutions to fractional elliptic problems subject to exterior Dirichlet boundary conditions. Our results provide, in particular, the sign of the normal derivative of such solutions near the boundary of the underlying domain.

In this paper, we introduce the finite neighboring type and the finite chain length conditions for a connected self-similar set K. We show that with these two conditions, K is a finitely ramified graph directed (f.r.g.d.) fractal defined by Hambly and Nyberg (Proc. Edinb. Math. Soc. (2) 46(1), 1–34 2003). We give some nontrivial examples and compute the harmonic structures on them explicitly. Furthermore

We prove the equivalence of two different types of capacities in abstract Wiener spaces. This yields a criterion for the Lp-uniqueness of the Ornstein-Uhlenbeck operator and its integer powers defined on suitable algebras of functions vanishing in a neighborhood of a given closed set Σ of zero Gaussian measure. To prove the equivalence we show the Wr,p(B,μ)-boundedness of certain smooth nonlinear truncation

We prove a comparison theorem for super- and sub-solutions with non-vanishing gradients to semilinear PDEs provided a nonlinearity f is Lp function with p > 1. The proof is based on a strong maximum principle for solutions of divergence type elliptic equations with VMO leading coefficients and with lower order coefficients from a Kato class. An application to estimation of periodic water waves profiles

We consider a modified Euler equation on \(\mathbb {R}^{2}\). We prove existence of weak global solutions for bounded (and fast decreasing at infinity) initial conditions and construct Gibbs-type measures on function spaces which are quasi-invariant for the Euler flow. Almost everywhere with respect to such measures (and, in particular, for less regular initial conditions), the flow is shown to be

This paper is concerned with higher Hölder regularity for viscosity solutions to non-translation invariant second order integro-PDEs, compared to Mou (2018). We first obtain C1,α regularity estimates for fully nonlinear integro-PDEs. We then prove the Schauder estimates for solutions if the equation is convex.

We prove sharp version of Riesz-Fejér inequality for functions in harmonic Hardy space \(h^{p}(\mathbb {D})\) on the unit disk \(\mathbb {D}\), for p > 1, thus extending the result from Kayumov et al. (Potential Anal. 52, 105–113, 2020) and resolving the posed conjecture.

This paper explores the limiting weak-type behaviors of certain classical operators in harmonic analysis including maximal operators, singular and fractional integral operators and maximal truncated singular integrals as well as the general convolution operators with weak-type Young’s inequalities. Some optimal limiting weak-type behaviors are given, which essentially improve and extend the previous

The aim of this paper is twofold. Firstly, we derive upper and lower non-Gaussian bounds for the densities of the marginal laws of the solutions to backward stochastic differential equations (BSDEs) driven by fractional Brownian motions. Our arguments consist of utilizing a relationship between fractional BSDEs and quasilinear partial differential equations of mixed type, together with the profound

In this paper, we prove that any W2,1 strong solution to second-order non-divergence form elliptic equations is locally \(W^{2,\infty }\) and piecewise C2 when the leading coefficients and data are of piecewise Dini mean oscillation and the lower-order terms are bounded. Somewhat surprisingly here the interfacial boundaries are only required to be in C1,Dini. We also derive global weak-type (1,1) estimates

We construct a decomposition of the identity operator on a Riemannian manifold M as a sum of smooth orthogonal projections subordinate to an open cover of M. This extends a decomposition on the real line by smooth orthogonal projection due to Coifman and Meyer (C. R. Acad. Sci. Paris, Sér. I Math., 312(3), 259–261 1991) and Auscher, Weiss, Wickerhauser (1992), and a similar decomposition when M is

In the existing language for tensor calculus on RCD spaces, tensor fields are only defined \(\mathfrak {m}\)-a.e.. In this paper we introduce the concept of tensor field defined ‘2-capacity-a.e.’ and discuss in which sense Sobolev vector fields have a 2-capacity-a.e. uniquely defined quasi-continuous representative.

In this note we will show a Calderón–Zygmund decomposition associated with a function \(f\in L^{1}(\mathbb {T}^{\omega })\). The idea relies on an adaptation of a more general result by J. L. Rubio de Francia in the setting of locally compact groups. Some related results about differentiation of integrals on the infinite-dimensional torus are also discussed.

We give sharp estimates for distortion of harmonic mappings u from the unit disc \(\mathbb {U}\) into \(\mathbb {R}^{m}\), at a prescribed point by means of diameter and area of the corresponding surface \(S=u(\mathbb {U})\), and via the generalized length of the boundary of S.

Let Pt be the (Neumann) diffusion semigroup Pt generated by a weighted Laplacian on a complete connected Riemannian manifold M without boundary or with a convex boundary. It is well known that the Bakry-Emery curvature is bounded below by a positive constant ≪> 0 if and only if$$W_{p}(\mu_{1}P_{t}, \mu_{2}P_{t})\leq e^{-\ll t} W_{p} (\mu_{1},\mu_{2}),\ \ t\geq 0, p\geq 1 $$holds for all probability

In this paper we refine the recent sparse domination of the integrated p = 2 matrix weighted dyadic square function by T. Hytonen, S. Petermichl, and A. Volberg to prove a pointwise sparse domination of general matrix weighted dyadic square functions. We then use this to prove sharp two matrix weighted strong type inequalities for matrix weighted dyadic square functions when 1 < p ≤ 2.

Let \(p\in (1, \infty )\), ? ? (0,1) and \(w\in A_{p}({\mathbb C}).\) In this article, the authors obtain a boundedness (resp., compactness) characterization of the Beurling–Ahlfors commutator \([\mathcal B, b]\) on the weighted Morrey space \(L_{w}^{p, \kappa }(\mathbb C)\) via \(\text {BMO}({\mathbb C})\) [resp., \(\text {CMO}({\mathbb C})\)], where \(\mathcal B\) denotes the Beurling–Ahlfors transform

In this paper we study the Dirichlet-to-Neumann map for solutions to mean value formulas on trees. We give two alternative definition of the Dirichlet-to-Neumann map. For the first definition (that involves the product of a “gradient” with a “normal vector”) and for a linear mean value formula on the directed tree (taking into account only the successors of a given node) we obtain that the Dirichlet-to-Neumann