Besov Spaces Research Articles

Path Regularity of the Brownian Motion and the Brownian Sheet

By the work of P. Lévy, the sample paths of the Brownian motion are known to satisfy a certain Hölder regularity condition almost surely. This was later improved by Ciesielski, who studied the regularity of these paths in Besov and Besov-Orlicz spaces. We review these results and propose new function spaces of Besov type, strictly smaller than those of Ciesielski and Lévy, in which the sample paths of the Brownian motion almost surely lie. In the same spirit, we review and extend the work of Kamont, who investigated the same question for the multivariate Brownian sheet and function spaces of dominating mixed smoothness.

Logarithmically improved extension criteria involving the pressure for the Navier–Stokes equations in Rn$\mathbb {R}^{n}$

AbstractWe consider the Cauchy problem of the Navier–Stokes equations in ( and establish several new extension criteria involving the pressure or its gradient. In particular, we improve the previous results by means of the homogeneous Besov space with negative differential orders in the case . Our method is based on the interpolation inequality and the trilinear estimate due to Gérard–Meyer–Oru (Séminaire É. D. P. (1996–1997), Exp. No. IV, 1–8) and Guo–Kučera–Skalák (J. Math. Anal. Appl. 458 (2018) 755–766), respectively.

Boundary value problems with rough boundary data

We consider linear boundary value problems for higher-order parameter-elliptic equations, where the boundary data do not belong to the classical trace spaces. We employ a class of Sobolev spaces of mixed smoothness that admits a generalized boundary trace with values in Besov spaces of negative order. We prove unique solvability for rough boundary data in the half-space and in sufficiently smooth domains. As an application, we show that the operator related to the linearized Cahn–Hilliard equation with dynamic boundary conditions generates a holomorphic semigroup in Lp(R+n)×Lp(Rn−1).

Strong periodic solutions to quasilinear parabolic equations: An approach by the Da Prato–Grisvard theorem

AbstractThis article develops an approach to unique, strong periodic solutions to quasilinear evolution equations by means of the classical Da Prato–Grisvard theorem on maximal ‐regularity in real interpolation spaces. The method is used to show that quasilinear Keller–Segel systems admit a unique, strong ‐periodic solution in a neighborhood of 0 provided the external forces are ‐periodic and satisfy certain smallness conditions. A similar assertion applies to a Nernst–Planck–Poisson type system in electrochemistry. The proof for the quasilinear Keller–Segel systems relies also on a new mixed derivative theorem in real interpolation spaces, that is, Besov spaces, which is of independent interest.

Besov estimates for weak solutions of a class of quasilinear parabolic equations with general growth

AbstractIn this paper, we obtain the local higher fractional differentiability regularity estimates in Besov spaces of weak solutions for the quasilinear parabolic equations with general growth under some proper conditions on , and . Moreover, we would like to point out that the results in the present work improve the known results for such parabolic equations.

Schatten Classes and Commutator in the Two Weight Setting, I. Hilbert Transform

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<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1< p < \\infty $\\end{document}. We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform.

Hyperbolic Anderson Model 2: Strichartz Estimates and Stratonovich Setting

Abstract We study a wave equation in dimension $d\in \{1,2\}$ with a multiplicative space-time Gaussian noise. The existence and uniqueness of the Stratonovich solution is obtained under some conditions imposed on the Gaussian noise. The strategy is to develop some Strichartz-type estimates for the wave kernel in weighted Besov spaces, by which we can prove the well-posedness of an associated Young-type equation. Those Strichartz bounds are of independent interest.

Besov spaces of analytic type: Interpolation, convolution, Fourier multipliers, inclusions

We consider a family of Besov spaces of analytic type on the Šilov boundary N of a homogeneous Siegel domain D, and study their properties concerning convolution, Fourier multipliers, and complex interpolation. In addition, we study how these Besov spaces of analytic type can be compared with the ‘classical’ Besov spaces on N.

Operators Induced by Radial Measures Acting on the Dirichlet Space

Let {mathbb {D}} be the unit disc in the complex plane. Given a positive finite Borel measure mu on the radius [0, 1), we let mu _n denote the n-th moment of mu and we deal with the action on spaces of analytic functions in {mathbb {D}} of the operator of Hibert-type {mathcal {H}}_mu and the operator of Cesàro-type {mathcal {C}}_mu which are defined as follows: If f is holomorphic in {mathbb {D}}, f(z)=sum _{n=0}^infty a_nz^n (zin {mathbb {D}}), then {mathcal {H}}_mu (f) is formally defined by {mathcal {H}}_mu (f)(z) = sum _{n=0}^infty left( sum _{k=0}^infty mu _{n+k}a_kright) z^n (zin {mathbb {D}}) and {mathcal {C}}_mu (f) is defined by mathcal C_mu (f)(z) = sum _{n=0}^infty mu _nleft( sum _{k=0}^na_kright) z^n (zin {mathbb {D}}). These are natural generalizations of the classical Hilbert and Cesàro operators. A good amount of work has been devoted recently to study the action of these operators on distinct spaces of analytic functions in {mathbb {D}}. In this paper we study the action of the operators {mathcal {H}}_mu and {mathcal {C}}_mu on the Dirichlet space {mathcal {D}} and, more generally, on the analytic Besov spaces B^p (1le p<infty ).

Periodic solutions of second‐order degenerate differential equations with infinite delay in Banach spaces

AbstractWe consider the well‐posedness of the second‐order degenerate differential equations with infinite delay on [0, 2π] in Lebesgue–Bochner spaces and periodic Besov spaces , where , and M are closed linear operators in a Banach space X satisfying and the kernels . Using known operator‐valued Fourier multiplier theorems, we are able to give necessary and sufficient conditions for the well‐posedness of (P) in and . These results are applied to examine some concrete examples.

Degree of convergence of Fourier series in Besov spaces

Degree of convergence of Fourier series in Besov spaces

Dilation operators in Besov spaces over local fields

We consider a dilation operator on Besov spaces $$B^s_{r,t}(K)$$ over local fields and estimate an operator norm on such a field for $$s > \sigma _r = \text {max}\big (\frac{1}{r} -1,\,0\big )$$ which depends on the constant k unlike the case of Euclidean spaces. In $${\mathbb {R}}^n,$$ it is independent of constant k, the constant appears for limiting case $$s=0$$ and $$s=\sigma _r.$$ In local fields, the limiting case is still open.

Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions

We study the approximation by tensor networks (TNs) of functions from classical smoothness classes. The considered approximation tool combines a tensorization of functions in $L^p([0,1))$, which allows to identify a univariate function with a multivariate function (or tensor), and the use of tree tensor networks (the tensor train format) for exploiting low-rank structures of multivariate functions. The resulting tool can be interpreted as a feed-forward neural network, with first layers implementing the tensorization, interpreted as a particular featuring step, followed by a sum-product network with sparse architecture. In part I of this work, we presented several approximation classes associated with different measures of complexity of tensor networks and studied their properties. In this work (part II), we show how classical approximation tools, such as polynomials or splines (with fixed or free knots), can be encoded as a tensor network with controlled complexity. We use this to derive direct (Jackson) inequalities for the approximation spaces of tensor networks. This is then utilized to show that Besov spaces are continuously embedded into these approximation spaces. In other words, we show that arbitrary Besov functions can be approximated with optimal or near to optimal rate. We also show that an arbitrary function in the approximation class possesses no Besov smoothness, unless one limits the depth of the tensor network.

Sobolev-Type Embeddings for Neural Network Approximation Spaces

We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated (with error measured in L^p) by ReLU neural networks with an increasing number of coefficients, subject to bounds on the magnitude of the coefficients and the number of hidden layers. We prove embedding theorems between these spaces for different values of p. Furthermore, we derive sharp embeddings of these approximation spaces into Hölder spaces. We find that, analogous to the case of classical function spaces (such as Sobolev spaces, or Besov spaces) it is possible to trade “smoothness” (i.e., approximation rate) for increased integrability. Combined with our earlier results in Grohs and Voigtlaender (Proof of the theory-to-practice gap in deep learning via sampling complexity bounds for neural network approximation spaces, 2021. arXiv preprint arXiv:2104.02746), our embedding theorems imply a somewhat surprising fact related to “learning” functions from a given neural network space based on point samples: if accuracy is measured with respect to the uniform norm, then an optimal “learning” algorithm for reconstructing functions that are well approximable by ReLU neural networks is simply given by piecewise constant interpolation on a tensor product grid.

On New Decomposition Theorems for Mixed-Norm Besov Spaces with Ingredient Modulus of Smoothness

In this paper, we introduce and study the concept of the ingredient modulus of smoothness in component form in Lp→(Rd) and a kind of mixed-norm Sobolev space. We obtain some new properties, inequalities, and auxiliary results in mixed-norm spaces Lp→(Rd). In addition, a new concept of mixed-norm Besov space is presented and a new decomposition theorem for mixed-norm Besov spaces is established.

Ill-posedness for a two-component Novikov system in Besov space

In this paper, we consider the Cauchy problem for a two-component Novikov system on the line. By specially constructed initial data (ρ0,u0) in Bp,∞s−1(R)×Bp,∞s(R) with s>max⁡{2+1p,52} and 1≤p≤∞, we show that any energy bounded solution starting from (ρ0,u0) does not converge back to (ρ0,u0) in the metric of Bp,∞s−1(R)×Bp,∞s(R) as time goes to zero, thus results in discontinuity of the data-to-solution map and ill-posedness.

Formation of singularity of solution to a nonlinear shallow water equation

This paper is mainly concerned with behaviors of solution to the Cauchy problem for a generalized shallow water equation with dispersive term and dissipative term in the Besov space. It is shown that the problem of nonlinear shallow water equation is locally well posed. The H^{1}(mathbb{R}) norm of solution to the problem is bounded under certain assumption on the initial value. Several blow-up criteria of solution are presented. The solution has compact support provided that the initial value has compact support. More specifically, the solution exponentially decays at infinity if the initial value exponentially decays at infinity. Our main new contribution is that the effects of coefficients λ and β on solution are illustrated. To the best of our knowledge, the results in Theorems 1.1–1.7 are new.

Pathwise regularisation of singular interacting particle systems and their mean field limits

We investigate the regularising effect of certain perturbations by noise in singular interacting particle systems under the mean field scaling. In particular, we show that the addition of a suitably irregular path can regularise these dynamics and we recover the McKean–Vlasov limit under very broad assumptions on the interaction kernel; only requiring it to be controlled in a possibly distributional Besov space. In the particle system we include two sources of randomness, a common noise path Z which regularises the dynamics and a family of idiosyncratic noises, which we only assume to converge in mean field scaling to a representative noise in the McKean–Vlasov equation.

Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, II

Continuing previous work, this paper provides maximal characterizations of anisotropic Triebel-Lizorkin spaces dot{textbf{F}}^{alpha }_{p,q} for the endpoint case of p = infty and the full scale of parameters alpha in mathbb {R} and q in (0,infty ]. In particular, a Peetre-type characterization of the anisotropic Besov space dot{textbf{B}}^{alpha }_{infty ,infty } = dot{textbf{F}}^{alpha }_{infty ,infty } is obtained. As a consequence, it is shown that there exist dual molecular frames and Riesz sequences in dot{textbf{F}}^{alpha }_{infty ,q}.

On the Cauchy problem for Keller‐Segel model with nonlinear chemotactic sensitivity and signal secretion in Besov spaces

In this paper, the local well‐posedness (which means that the initial‐to‐solution map is existence, uniqueness and continuous) for the Cauchy problem of the Keller‐Segel model with nonlinear chemotactic sensitivity and signal secretion in Besov spaces with and was established, and the solution of PEKS (Keller‐Segel system with ) converges to the solution of HEKS (Keller‐Segel system with ) as the diffusion coefficient in these Besov spaces was proved. In addition, we show that the solution of this parabolic‐elliptic chemotaxis‐growth system is local well‐posedness in the critical spaces but ill‐posed (not continuous) in Besov spaces . Moreover, we show a further continuity result that the initial‐to‐solution map is Hölder continuous in Besov spaces equipped with weaker topology. Finally, a blow‐up criteria for the solutions of this system in Besov spaces was also obtained.