Besov Spaces Research Articles

Global solutions to the rotating Navier–Stokes equations with large data in the critical Fourier–Besov spaces

AbstractWe consider the initial value problem for the 3D incompressible Navier–Stokes equations with the Coriolis force. The aim of this paper is to prove the existence of a unique global solution with arbitrarily large initial data in the scaling critical Fourier–Besov spaces (, ), provided that the size of the Coriolis parameter is sufficiently large. Moreover, if the initial data additionally belong to the scaling sub‐critical spaces, we obtain an explicit relationship between the initial data and the Coriolis force, which ensures the existence of a unique global solution.

Inhomogeneous Besov and Triebel-Lizorkin spaces associated with a para-accretive function and their applications

Inhomogeneous Besov and Triebel-Lizorkin spaces associated with a para-accretive function and their applications

On Embedding of Besov Spaces of Zero Smoothness into Lorentz Spaces

We show that the zero smoothness Besov space \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{p,q}^{0,1}$$\\end{document} does not embed into the Lorentz space \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_{p,q}$$\\end{document} unless \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=q$$\\end{document}; here \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p,q\\in (1,\\infty)$$\\end{document}. This answers in the negative a question posed by O. V. Besov.

Bourgain–Morrey Spaces Mixed with Structure of Besov Spaces

Bourgain–Morrey Spaces Mixed with Structure of Besov Spaces

Besov wavefront set

We develop a notion of wavefront set aimed at characterizing in Fourier space the directions along which a distribution behaves or not as an element of a specific Besov space. Subsequently we prove an alternative, albeit equivalent characterization of such wavefront set using the language of pseudodifferential operators. Both formulations are used to prove the main underlying structural properties. Among these we highlight the individuation of a sufficient criterion to multiply distributions with a prescribed Besov wavefront set which encompasses and generalizes the classical Young’s theorem. At last, as an application of this new framework we prove a theorem of propagation of singularities for a large class of hyperbolic operators.

Besov Spaces, Schatten Classes and Weighted Versions of the Quantised Derivative

Besov Spaces, Schatten Classes and Weighted Versions of the Quantised Derivative

Well‐posedness of degenerate fractional differential equations with finite delay in complex Banach spaces

AbstractWe study the well‐posedness of the degenerate fractional differential equations with finite delay: on Lebesgue–Bochner spaces and periodic Besov spaces , where and are closed linear operators in a complex Banach space satisfying , and are fixed, when , and the delay operator is a bounded linear operator from (resp. ) into . Using known operator‐valued Fourier multiplier theorems on and , we completely characterize the ‐well‐posedness and the ‐well‐posedness of above equations. We also give concrete examples that our abstract results may be applied.

On Mixed Fractional Lifting Oscillation Spaces

We introduce hyperbolic oscillation spaces and mixed fractional lifting oscillation spaces expressed in terms of hyperbolic wavelet leaders of multivariate signals on Rd, with d≥2. Contrary to Besov spaces and fractional Sobolev spaces with dominating mixed smoothness, the new spaces take into account the geometric disposition of the hyperbolic wavelet coefficients at each scale (j1,⋯,jd), and are therefore suitable for a multifractal analysis of rectangular regularity. We prove that hyperbolic oscillation spaces are closely related to hyperbolic variation spaces, and consequently do not almost depend on the chosen hyperbolic wavelet basis. Therefore, the so-called rectangular multifractal analysis, related to hyperbolic oscillation spaces, is somehow ‘robust’, i.e., does not change if the analyzing wavelets were changed. We also study optimal relationships between hyperbolic and mixed fractional lifting oscillation spaces and Besov spaces with dominating mixed smoothness. In particular, we show that, for some indices, hyperbolic and mixed fractional lifting oscillation spaces are not always sharply imbedded between Besov spaces or fractional Sobolev spaces with dominating mixed smoothness, and thus are new spaces of a really different nature.

Well-posedness for the one-dimensional inviscid Cattaneo-Christov system

This paper is devoted to studying the inviscid compressible Cattaneo-Christov system in one-dimensional space. The iterative method will be used to establish the local well-posedness of this system for large data in critical Besov spaces based on the L2 framework. Moreover, by using the renormalized energy method, one can prove the global existence of a strong solution when the initial perturbation around a constant state is sufficiently small.

On the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations

In this paper, we are concerned with the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations, which is a generalization of the Camassa-Holm equation and the Fokas-Olver-Rosenau-Qiao equation. We explore how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we established the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, a sufficient condition on initial data that leads to the finite time blow-up of the second-order derivative of the solutions is described in detail.

Cyclicity in the Drury-Arveson space and other weighted Besov spaces

Let H \mathcal {H} be a space of analytic functions on the unit ball B d {\mathbb {B}_d} in C d \mathbb {C}^d with multiplier algebra M u l t ( H ) \mathrm {Mult}(\mathcal {H}) . A function f ∈ H f\in \mathcal {H} is called cyclic if the set [ f ] [f] , the closure of { φ f : φ ∈ M u l t ( H ) } \{\varphi f: \varphi \in \mathrm {Mult}(\mathcal {H})\} , equals H \mathcal {H} . For multipliers we also consider a weakened form of the cyclicity concept. Namely for n ∈ N 0 n\in \mathbb {N}_0 we consider the classes C n ( H ) = { φ ∈ M u l t ( H ) : φ ≠ 0 , [ φ n ] = [ φ n + 1 ] } . \begin{equation*} \mathcal {C}_n(\mathcal {H})=\{\varphi \in \mathrm {Mult}(\mathcal {H}):\varphi \ne 0, [\varphi ^n]=[\varphi ^{n+1}]\}. \end{equation*} Many of our results hold for N N :th order radially weighted Besov spaces on B d {\mathbb {B}_d} , H = B ω N \mathcal {H}= B^N_\omega , but we describe our results only for the Drury-Arveson space H d 2 H^2_d here. Letting C s t a b l e [ z ] \mathbb {C}_{stable}[z] denote the stable polynomials for B d {\mathbb {B}_d} , i.e. the d d -variable complex polynomials without zeros in B d {\mathbb {B}_d} , we show that a m p ; if d is odd, then C s t a b l e [ z ] ⊆ C d − 1 2 ( H d 2 ) , and a m p ; if d is even, then C s t a b l e [ z ] ⊆ C d 2 − 1 ( H d 2 ) . \begin{align*} &\text { if } d \text { is odd, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d-1}{2}}(H^2_d), \text { and }\\ &\text { if } d \text { is even, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d}{2}-1}(H^2_d). \end{align*} For d = 2 d=2 and d = 4 d=4 these inclusions are the best possible, but in general we can only show that if 0 ≤ n ≤ d 4 − 1 0\le n\le \frac {d}{4}-1 , then C s t a b l e [ z ] ⊈ C n ( H d 2 ) \mathbb {C}_{stable}[z]\nsubseteq \mathcal {C}_n(H^2_d) . For functions other than polynomials we show that if f , g ∈ H d 2 f,g\in H^2_d such that f / g ∈ H ∞ f/g\in H^\infty and f f is cyclic, then g g is cyclic. We use this to prove that if f , g f,g extend to be analytic in a neighborhood of B d ¯ \overline {{\mathbb {B}_d}} , have no zeros in B d {\mathbb {B}_d} , and the same zero sets on the boundary, then f f is cyclic in ∈ H d 2 \in H^2_d if and only if g g is. Furthermore, if the boundary zero set of f ∈ H d 2 ∩ C ( B d ¯ ) f\in H^2_d\cap C(\overline {{\mathbb {B}_d}}) embeds a cube of real dimension ≥ 3 \ge 3 , then f f is not cyclic in the Drury-Arveson space.

Analysis on continuity of the solution map for the Whitham equation in Besov spaces

AbstractThis paper is devoted to studying the continuity of the solution map for the Cauchy problem of the Whitham equation. First, the continuity dependence of solution is established in in the sense of Hadamard. Next, by constructing approximate solutions, we show that the data‐to‐solution map is not uniformly continuous in Besov spaces () on the periodic case and on the line. The crucial technical tool used to prove this result is Lemma 5.1, which generalizes the result of Lemma 3 in [23].

Well‐posedness of stochastic heat equation with distributional drift and skew stochastic heat equation

AbstractWe study stochastic reaction–diffusion equation where is a generalized function in the Besov space , and is a space‐time white noise on . We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever , and . This class includes equations with being measures, in particular, which corresponds to the skewed stochastic heat equation. For , we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalize the results of Gyöngy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in Lê (2020).

The Tb theorem for some inhomogeneous Besov and Triebel-Lizorkin spaces over space of homogeneous type

In this paper, the authors give some new characterizations of the inhomogeneous Besov spaces and inhomogeneous Triebel-Lizorkin spaces respectively over space of homogeneous type, and establish the Tb theorems for the boundedness of the Calderón-Zygmund singular integral operator with non-convolution kernel on these new inhomogeneous Besov and Triebel-Lizorkin spaces.

Dahlberg degeneracy for homogeneous Besov and Triebel–Lizorkin spaces

AbstractWe consider the composition operators acting on the real‐valued homogeneous Besov or Triebel–Lizorkin spaces, realized as dilation invariant subspaces of , denoted as . If and , then any function acting by composition on is necessarily linear. The above conditions are optimal: (i) in case , (Besov space), (Triebel–Lizorkin space), is a quasi‐Banach algebra for the pointwise product, (ii) in case , , , any function such that is a finite measure, and , acts by composition on .

A remark on the multi-dimensional compressible Euler system with damping in the 𝐿^{𝑝} critical Besov spaces

We study the global-in-time well-posedness and relaxation limit of the compressible Euler system with damping in L p L^p -type critical Besov spaces. In comparison with the results obtained by Crin-Barat and Danchin [Pure Appl. Anal. 4 (2022), pp. 85–125; Math. Ann. 386 (2023), pp. 2159–2206], the more general pressure law satisfying P ′ ( ρ ¯ ) > 0 P’(\bar {\rho })>0 is allowed. To achieve it, a new composition estimate is established in the L 2 L^2 - L p L^p hybrid Besov spaces with explicit dependence on the threshold between high frequencies and low frequencies.

Local solvability for real-analytic involutive structures of tube type of corank one in Besov and Triebel-Lizorkin spaces

This study shows the local solvability, in degree one, for the differential complex associated to a real-analytic, involutive structure of tube type of corank one in mixed spaces involving Besov and Triebel-Lizorkin spaces.

Unique local weak solutions of the non-resistive MHD equations in homogeneous Besov space

ABSTRACT In this paper, the local existence and uniqueness of weak solutions to a d-dimensional non-resistive MHD equations in homogeneous Besov spaces are studied. Specifically we obtain the local existence of a weak solution ( u , b ) of the non-resistive MHD equations for the initial data u 0 ∈ B ˙ p , 1 d p − 1 ( R d ) and b 0 ∈ B ˙ p , 1 d p ( R d ) with 1 ≤ p ≤ ∞ , and the uniqueness of the weak solution when 1 ≤ p ≤ 2 d . Compared with the previous results for the non-resistive MHD equations, in the local existence part, the range of p extends to 1 ≤ p ≤ ∞ from 1 ≤ p ≤ 2 d , but the uniqueness of the solution requires 1 ≤ p ≤ 2 d yet.

Characterizations of pointwise multipliers of Besov spaces in endpoint cases with an application to the duality principle

Let p,q∈(0,∞], s∈R, and M(Bp,qs(Rn)) denote the pointwise multiplier space of the Besov space Bp,qs(Rn). In this article, the authors first establish the characterizations of both M(B1,∞s(Rn)) with s∈R∖{0} and M(B∞,1s(Rn)) with s∈(−∞,0]. Then, as an application, the authors give a corrected proof of the well-known duality principle for pointwise multiplier spaces of Besov spaces, namely the formulaM(Bp,qs(Rn))=M(Bp′,q′−s(Rn)), where p,q∈[1,∞], s∈R, and 1/a+1/a′=1 for any a∈[1,∞], and, moreover, the authors also show that this duality principle is sharp in some sense. The proofs of all these results essentially depend on the duality theorem of Besov spaces themselves, some elaborate estimates of paraproducts as well as the relation between M(Bp,qs(Rn)) and the auxiliary multiplier space M(B˜p,qs(Rn)), where B˜p,qs(Rn) denotes the completion of the Schwartz function space S(Rn) in Bp,qs(Rn).

Remarks on Chemin's space of homogeneous distributions

AbstractThis paper focuses on Chemin's space of homogeneous distributions, which was introduced to serve as a basis for the realizations of subcritical homogeneous Besov spaces. We will discuss how this construction fails in multiple ways for supercritical spaces. In particular, we study its intersection with various Banach spaces X, namely supercritical homogeneous Besov spaces and the Lebesgue space . For each X, we investigate whether the intersection is dense in X. If it is not, then we study its closure and prove that the quotient is not separable and that C is not complemented in X.