Pointwise Regularity Research Articles

On the pointwise regularity of the Multifractional Brownian Motion and some extensions

We study the pointwise regularity of the Multifractional Brownian Motion and, in particular, we obtain the existence of so-called slow points of the process, that is points which exhibit a slow oscillation instead of the a.e. regularity. This result entails that a non self-similar process can also exhibit such a behavior. We also consider various extensions with the aim of imposing weaker regularity assumptions on the Hurst function without altering the regularity of the process.

On [formula omitted] rectangular multifractal multivariate functions

Multifractal analysis has traditionally been based on the pointwise Hölder regularity for locally bounded functions. However, recently an extension to the rectangular pointwise regularity for multivariate locally functions has generated interest. The purpose of this paper is twofold. Firstly, we characterize in hyperbolic wavelet bases the rectangular pointwise Lipschitz regularity for multivariate functions which are non necessarily locally bounded but are locally in Lp. Secondly, we perform applications for both random and deterministic anisotropic self-affine tools for some models of textures in images and flows in turbulence. Actually, we show that fractional anisotropic Wiener field are multivariate monofractal in Lp. We then construct self-affine cascade functions that are multivariate multifractal in Lp.

Fundamental properties of Cauchy–Szegő projection on quaternionic Siegel upper half space and applications

Abstract We investigate the Cauchy–Szegő projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy–Szegő kernel and prove that the Cauchy–Szegő kernel is nonzero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy–Szegő projection on every atom on the quaternionic Heisenberg group, which is used to give an atomic decomposition of regular Hardy space H p {H^{p}} on quaternionic Siegel upper half space for 2 3 < p ≤ 1 {\frac{2}{3}<p\leq 1} . Moreover, we establish the characterisation of singular values of the commutator of Cauchy–Szegő projection based on the kernel estimates. The quaternionic structure (lack of commutativity) is encoded in the symmetry groups of regular functions and the associated partial differential equations.

Boundary pointwise regularity and applications to the regularity of free boundaries

Boundary pointwise regularity and applications to the regularity of free boundaries

Multifractional Hermite processes: Definition and first properties

We define multifractional Hermite processes which generalize and extend both multifractional Brownian motion and Hermite processes. It is done by substituting the Hurst parameter in the definition of Hermite processes as a multiple Wiener–Itô integral by a Hurst function. Then, we study the pointwise regularity of these processes, their local asymptotic self-similarity and some fractal dimensions of their graph. Our results show that the fundamental properties of multifractional Hermite processes are, as desired, governed by the Hurst function. Complements are given in the second order Wiener chaos, using facts from Malliavin calculus.

Generalized Newton-Leibniz formula and the embedding of the Sobolev functions with dominating mixed smoothness into Hölder spaces

<abstract><p>It is well-known that the embedding of the Sobolev space of weakly differentiable functions into Hölder spaces holds if the integrability exponent is higher than the space dimension. In this paper, the embedding of the Sobolev functions into the Hölder spaces is expressed in terms of the minimal weak differentiability requirement independent of the integrability exponent. The proof is based on the generalization of the Newton-Leibniz formula to the $ n $-dimensional rectangle and the inductive application of the Sobolev trace embedding results. The new method is applied to prove the embedding of the Sobolev spaces with dominating mixed smoothness into Hölder spaces. Counterexamples demonstrate that in terms of minimal weak regularity degree the Sobolev spaces with dominating mixed smoothness present the largest class of weakly differentiable functions with the upgrade of pointwise regularity to continuity. Remarkably, it also presents the largest class of weakly differentiable functions where the generalized Newton-Leibniz formula holds.</p></abstract>

Gradient estimates via Riesz potentials and fractional maximal operators for quasilinear elliptic equations with applications

In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type div(A(x,∇u))=div(f)inΩ,andu=gon∂Ω,that related to Riesz potentials. Here, in our setting, Ω⊂Rn, n≥2 is a bounded Reifenberg flat domain (that its boundary is sufficiently flat in sense of Reifenberg) and the small-BMO condition (small bounded mean oscillations) is assumed on the nonlinearity A. Further, the emphasis of the paper is the existence of weak solution to a class of quasilinear elliptic equations containing Riesz potential of the gradient term, as an application of the global point-wise bound. And regarding this study, we also analyze the necessary and sufficient conditions that guarantee the existence of solution to such nonlinear elliptic problems.

A degenerate fully nonlinear free transmission problem with variable exponents

We study degenerate fully nonlinear free transmission problems, where the degeneracy rate varies in the domain. We prove optimal pointwise regularity depending on the degeneracy rate. Our arguments consist of perturbation methods, relating our problem to a homogeneous, fully nonlinear, uniformly elliptic equation.

Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process

We identify three types of pointwise behaviour in the regularity of the (generalized) Rosenblatt process. This extends to a non Gaussian setting previous results known for the (fractional) Brownian motion. On this purpose, fine bounds on the increments of the Rosenblatt process are needed. Our analysis is essentially based on various wavelet methods.

Multifractal Analysis of Rectangular Pointwise Regularity with Hyperbolic Wavelet Bases

Multifractal Analysis of Rectangular Pointwise Regularity with Hyperbolic Wavelet Bases

Pointwise regularity for fractional equations

In this paper, we systematically study the pointwise regularity for distribution solutions of fractional equations. We obtain a series of interior pointwise Ck+[2s+α],2s+α−[2s+α] regularities when 2s+α is not an integer, Ck+2s+α,ln⁡L regularities and Ck+2s+α regularities when 2s+α is an integer for k≥0 under suitable conditions. These pointwise regularities seem to be more essential and characterize the solutions for fractional equations, and our proofs are more direct which can also provide optimal conditions for the above regularities. Furthermore, the classical regularity can be regarded as a consequence of pointwise regularity. The method we developed here can also be applied to many other equations, especially for the Poisson equations.

Generalized spaces of pointwise regularity: toward a general framework for the WLM

In this work we generalize the spaces introduced by Calderón and Zygmund using pointwise conditions emanating from generalized Besov spaces. We give conditions binding the functions belonging to these spaces and their wavelet coefficients. Next, we propose a multifractal formalism based on such spaces which generalizes the so-called wavelet leaders method and show that it is satisfied on a prevalent set.

Forecasting Value-at-Risk in turbulent stock markets via the local regularity of the price process

A new computational approach based on the pointwise regularity exponent of the price time series is proposed to estimate Value at Risk. The forecasts obtained are compared with those of two largely used methodologies: the variance-covariance method and the exponentially weighted moving average method. Our findings show that in two very turbulent periods of financial markets the forecasts obtained using our algorithm decidedly outperform the two benchmarks, providing more accurate estimates in terms of both unconditional coverage and independence and magnitude of losses.

Pointwise gradient bounds for a class of very singular quasilinear elliptic equations

A pointwise gradient bound for weak solutions to Dirichlet problem for quasilinear elliptic equations \begin{document}$ -\mathrm{div}(\mathbb{A}(x,\nabla u)) = \mu $\end{document} is established via Wolff type potentials. It is worthwhile to note that the model case of \begin{document}$ \mathbb{A} $\end{document} here is the non-degenerate \begin{document}$ p $\end{document} -Laplacian operator. The central objective is to extend the pointwise regularity results in [Q.-H. Nguyen, N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equations with measure data, J. Funct. Anal. 278(5) (2020), 108391] to the very singular case \begin{document}$ 1 , where the data \begin{document}$ \mu $\end{document} on right-hand side is assumed belonging to some classes that close to \begin{document}$ L^1 $\end{document} . Moreover, a global pointwise estimate for gradient of weak solutions to such problem is also obtained under the additional assumption that \begin{document}$ \Omega $\end{document} is sufficiently flat in the Reifenberg sense.

Critical 𝐿^{𝑝}-differentiability of 𝐵𝑉^{}𝔸-maps and canceling operators

We give a generalization of Dorronsoro’s theorem on criticalLp\mathrm {L}^p-Taylor expansions forBVk\mathrm {BV}^k-maps onRn\mathbb {R}^n; i.e., we characterize homogeneous linear differential operatorsA\mathbb {A}ofkkth order such thatDk−juD^{k-j}uhasjjth orderLn/(n−j)\mathrm {L}^{n/(n-j)}-Taylor expansion a.e. for allu∈BVlocAu\in \mathrm {BV}^\mathbb {A}_{\operatorname {loc}}(herej=1,…,kj=1,\ldots , k, with an appropriate convention ifj≥nj\geq n). The spaceBVlocA\mathrm {BV}^\mathbb {A}_{\operatorname {loc}}, a single framework coveringBV\mathrm {BV},BD\mathrm {BD}, andBVk\mathrm {BV}^k, consists of those locally integrable mapsuusuch thatAu\mathbb {A} uis a Radon measure onRn\mathbb {R}^n.Forj=1,…,min{k,n−1}j=1,\ldots ,\min \{k, n-1\}, we show that theLp\mathrm {L}^p-differentiability property above is equivalent to Van Schaftingen’s elliptic and canceling condition forA\mathbb {A}. Forj=n,…,kj=n,\ldots , k, ellipticity is necessary, but cancellation is not. To complete the characterization, we determine the class of elliptic operatorsA\mathbb {A}such that the estimate(1)‖Dk−nu‖L∞⩽C‖Au‖L1\begin{align}\tag {1} \|D^{k-n}u\|_{\mathrm {L}^\infty }\leqslant C\|\mathbb {A} u\|_{\mathrm {L}^1} \end{align}holds for all vector fieldsu∈Cc∞u\in \mathrm {C}^\infty _c. Surprisingly, the (computable) condition onA\mathbb {A}such that \eqref{eq:abs} holds is strictly weaker than cancellation.The results onLp\mathrm {L}^p-differentiability can be formulated as sharp pointwise regularity results for overdetermined elliptic systemsAu=μ,\begin{align*} \mathbb {A} u=\mu , \end{align*}whereμ\muis a Radon measure, thereby giving a variant for the limit casep=1p=1of a theorem of Calderón and Zygmund which was not covered before.

A Generalized Multifractal Formalism for the Estimation of Nonconcave Multifractal Spectra

Multifractal analysis has become a powerful signal processing tool that characterizes signals or images via the fluctuations of their pointwise regularity, quantified theoretically by the so-called multifractal spectrum. The practical estimation of the multifractal spectrum fundamentally relies on exploiting the scale dependence of statistical properties of appropriate multiscale quantities, such as wavelet leaders, that can be robustly computed from discrete data. Despite successes of multifractal analysis in various real-world applications, current estimation procedures remain essentially limited to providing concave upper-bound estimates, while there is a priori no reason for the multifractal spectrum to be a concave function. This work addresses this severe practical limitation and proposes a novel formalism for multifractal analysis that enables nonconcave multifractal spectra to be estimated in a stable way. The key contributions reside in the development and theoretical study of a generalized multifractal formalism to assess the multiscale statistics of wavelet leaders, and in devising a practical algorithm that permits this formalism to be applied to real-world data, allowing for the estimation of nonconcave multifractal spectra. Numerical experiments are conducted on several synthetic multifractal processes as well as on a real-world remote-sensing image and demonstrate the benefits of the proposed multifractal formalism over the state of the art.

Pointwise regularity of parameterized affine zipper fractal curves

We study the pointwise regularity of zipper fractal curves generated by affine mappings. Under the assumption of dominated splitting of index-1, we calculate the Hausdorff dimension of the level sets of the pointwise Hölder exponent for a subinterval of the spectrum. We give an equivalent characterization for the existence of regular pointwise Hölder exponent for Lebesgue almost every point. In this case, we extend the multifractal analysis to the full spectrum. In particular, we apply our results for de Rham’s curve.

Pointwise regularity of parameterized affine zipper fractal curves fractal curves

Pointwise regularity of parameterized affine zipper fractal curves fractal curves

A pointwise regularity criterion for axisymmetric Navier–Stokes system

In this paper, we consider the axisymmetric Navier–Stokes equations with non-zero swirl component. It is proved that if rur≥M for some M>−2, then the solution actually is smooth. This extends the result in Pan (2017) [14].

Multifractal analysis of the Brjuno function

The Brjuno function B is a 1-periodic, nowhere locally bounded function, introduced by Yoccoz because it encapsulates a key information concerning analytic small divisor problems in dimension 1. We show that \(T^p_\alpha \) regularity, introduced by Calderon and Zygmund, is the only one which is relevant in order to unfold the pointwise regularity properties of B; we determine its \(T^p_\alpha \) regularity at every point and show that it is directly related to the irrationality exponent \(\tau (x)\): its p-exponent at x is exactly \(1/\tau (x)\). This new example of multifractal function puts into light a new link between dynamical systems and fractal geometry. Finally we also determine the Holder exponent of a primitive of B.