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In this note, we consider the Fourier integral operator T ϕ , a ⁢ f ⁢ (x) = ∫ R n e i ⁢ ϕ ⁢ (x , ξ) ⁢ a ⁢ (x , ξ) ⁢ f ^ ⁢ (ξ) ⁢ d ξ with 푎 in the forbidden Hörmander class S ρ , 1 m and ϕ ∈ Φ 2 satisfying the strong non-degeneracy condition. For 0 ≤ ρ ≤ 1 , set m ⁢ (n , ρ , p) = − (n − ρ) ⁢ (1 2 − 1 p) + n p ⁢ (ρ − 1) . When 2 ≤ p ≤ ∞ , 1 ≤ q ≤ ∞ and s > m − m ⁢ (n , ρ , p) , we show that T ϕ , a is bounded from the Besov space B p , q s to B p , q s − m + m ⁢ (n , ρ , p) . This result is a generalization of some theorems proved by Stein, Meyer, Runst and Bourdaud for the pseudo-differential operator T a with a ∈ S 1 , 1 m , and indices 푠 and m ⁢ (n , ρ , p) are sharp in some cases. [ABSTRACT FROM AUTHOR]

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 Fourier multipliers in Besov spaces on graded Lie groups. In particular, we give a number of sufficient conditions for the boundedness of Fourier multipliers in Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper we analyse functions in Besov spaces B q , ∞ 1 / q (R N , R d) , q ∈ (1 , ∞) , and functions in fractional Sobolev spaces W r , q (R N , R d) , r ∈ (0 , 1) , q ∈ [ 1 , ∞) . We prove for Besov functions u ∈ B q , ∞ 1 / q (R N , R d) the summability of the difference between one-sided approximate limits in power q, | u + - u - | q , along the jump set J u of u with respect to Hausdorff measure H N - 1 , and establish the best bound from above on the integral ∫ J u | u + - u - | q d H N - 1 in terms of Besov constants. We show for functions u ∈ B q , ∞ 1 / q (R N , R d) , q ∈ (1 , ∞) that 0.1 lim inf ε → 0 + 1 ε N ∫ B ε (x) | u (z) - u B ε (x) | q d z = 0 for every x outside of a H N - 1 -sigma finite set. For fractional Sobolev functions u ∈ W r , q (R N , R d) we prove that 0.2 lim ε → 0 + 1 ε N ∫ B ε (x) 1 ε N ∫ B ε (x) | u (z) - u (y) | q d z d y = 0 for H N - r q a.e. x, where q ∈ [ 1 , ∞) , r ∈ (0 , 1) and r q ≤ N . We prove for u ∈ W 1 , q (R N) , 1 < q ≤ N , that 0.3 lim ε → 0 + 1 ε N ∫ B ε (x) | u (z) - u B ε (x) | q d z = 0 for H N - q a.e. x ∈ R N . In addition, we prove Lusin-type approximation for fractional Sobolev functions u ∈ W r , q (R N , R d) by Hölder continuous functions in C 0 , r (R N , R d) . [ABSTRACT FROM AUTHOR]

In this paper, by using time-weighted global estimates and the Lagrangian approach, we first investigate the global existence and uniqueness of the solution for the 2D inhomogeneous incompressible asymmetric fluids with the initial (angular) velocity being located in sub-critical Sobolev spaces H s (R 2) (0 < s< 1) and the initial density being bounded from above and below by some positive constants. The global unique solvability of the 2D incompressible inhomogeneous asymmetric fluids with the initial data in the critical Besov space (u 0 , w 0) ∈ B ˙ 2 , 1 0 (R 2) and ρ − 1 − 1 ∈ B ˙ 2 / ε , 1 ε (R 2) is established. In particular, the uniqueness of the solution is also obtained without any more regularity assumptions on the initial density which is an improvement on the recent result of Abidi and Gui (2021) for the 2-D inhomogeneous incompressible Navier-Stokes system. [ABSTRACT FROM AUTHOR]

We study the well-posedness of the fractional differential equations with finite delay: Dαu(t) + BDβu(t) = Au(t) + Fut + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces and periodic Besov spaces , where A and B are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(B), 0 ≤ β < α, α ≥ 1 and F is a bounded linear operator from Lp ([−2π, 0]; X) (resp. into X. We give necessary and sufficient conditions for the Lp -well-posedness and the -well-posedness of above equations by using known operator-valued Fourier multiplier theorems. [ABSTRACT FROM AUTHOR]

We introduce Besov space $ \dot {B}_{p}^{\alpha,q}(\partial \Omega _k) $ B ˙ p α , q (∂ Ω k) and Triebel-Lizorkin space $ \dot {F}^{\alpha,q}_{p}(\partial \Omega _k) $ F ˙ p α , q (∂ Ω k) on a family of model domains $ \partial \Omega _k=\left \{(\mathbf{z},z_{n+1})=(z_1,z_2,\ldots, z_{n+1}):\right. \left. \mbox {Im} (z_{n+1})=\phi (|\mathbf{z}|^2)\right \} $ ∂ Ω k = { (z , z n + 1) = (z 1 , z 2 , ... , z n + 1) : Im (z n + 1) = ϕ (| z | 2) } with $ \phi (x)=x^{k} $ ϕ (x) = x k in $ \mathbf {C}^{n+1} $ C n + 1 which can be considered as a space of homogeneous type in the sense of Coifman RR, Weiss G.[Analyse Harmonique Non-commutative and Certains Espaces Homogǹes. Berlin: Springer; 1971. (Lecture Notes in Math.; 242).], Coifman R, Weiss G. [Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc. 1977;83:569–645.]. We study the sharp estimates on the fundamental solution for the Kohn Laplacian and Cauchy-Szegö projection on $ \partial \Omega _k $ ∂ Ω k in these spaces. [ABSTRACT FROM AUTHOR]

Let \mathcal {H} be a space of analytic functions on the unit ball {\mathbb {B}_d} in \mathbb {C}^d with multiplier algebra \mathrm {Mult}(\mathcal {H}). A function f\in \mathcal {H} is called cyclic if the set [f], the closure of \{\varphi f: \varphi \in \mathrm {Mult}(\mathcal {H})\}, equals \mathcal {H}. For multipliers we also consider a weakened form of the cyclicity concept. Namely for n\in \mathbb {N}_0 we consider the classes \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:th order radially weighted Besov spaces on {\mathbb {B}_d}, \mathcal {H}= B^N_\omega, but we describe our results only for the Drury-Arveson space H^2_d here. Letting \mathbb {C}_{stable}[z] denote the stable polynomials for {\mathbb {B}_d}, i.e. the d-variable complex polynomials without zeros in {\mathbb {B}_d}, we show that \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 and d=4 these inclusions are the best possible, but in general we can only show that if 0\le n\le \frac {d}{4}-1, then \mathbb {C}_{stable}[z]\nsubseteq \mathcal {C}_n(H^2_d). For functions other than polynomials we show that if f,g\in H^2_d such that f/g\in H^\infty and f is cyclic, then g is cyclic. We use this to prove that if f,g extend to be analytic in a neighborhood of \overline {{\mathbb {B}_d}}, have no zeros in {\mathbb {B}_d}, and the same zero sets on the boundary, then f is cyclic in \in H^2_d if and only if g is. Furthermore, if the boundary zero set of f\in H^2_d\cap C(\overline {{\mathbb {B}_d}}) embeds a cube of real dimension \ge 3, then f is not cyclic in the Drury-Arveson space. [ABSTRACT FROM AUTHOR]

This paper addresses the existence and uniqueness of mild solution for fractional Navier–Stokes equation in the homogeneous Besov spaces. The existence and uniqueness of the global mild solution with small initial data is established. The results on existence and uniqueness of the local mild solution for arbitrary initial data are also studied. Meanwhile, we obtain properties about the regularity and decay of mild solutions. These conclusions are mainly based on the fixed point theorems, the implicit function theorem on Banach spaces and the properties of Mittag–Leffler functions. [ABSTRACT FROM AUTHOR]

We study the problem where a one‐dimensional elastic string is immersed in a two‐dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non‐linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time wellposedness for arbitrary initial data in the scaling critical Besov space Ḃ2,13/2(T;R2)$\dot{B}^{3/2}_{2,1}(\mathbb {T}; \mathbb {R}^2)$. We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancelation structure. [ABSTRACT FROM AUTHOR]

In this paper, we focus on Cauchy problem of the non‐isentropic compressible Navier–Stokes–Poisson system with the initial perturbation (ρ0−ρ¯,m0,θ0−θ¯)$$ \left({\rho}&#x0005E;0-\overline{\rho},{\mathbf{m}}&#x0005E;0,{\theta}&#x0005E;0-\overline{\theta}\right) $$ belonging to the space Hl(ℝ3)∩B˙1,∞−s(ℝ3)$$ {H}&#x0005E;l\left({\mathrm{\mathbb{R}}}&#x0005E;3\right)\cap {\dot{B}}_{1,\infty}&#x0005E;{-s}\left({\mathrm{\mathbb{R}}}&#x0005E;3\right) $$, where l⩾4$$ l\geqslant 4 $$ and s∈[0,1]$$ s\in \left[0,1\right] $$. More importantly, we establish the optimal temporal decay rate of the global strong solution, which can be considered as further work. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this work, we consider the magnetohydrodynamics system with the Hall and ion-slip effects in ℝ3. The main result is a sufficient condition for regularity on a time interval [0, T] expressed in terms of the norm of the homogeneous Besov space B ˙ ∞ , ∞ 0 with respect to the pressure and the BMO−norm with respect to the gradient of the magnetic field, respectively ∫ T 0 ‖ ∇ π t ‖ B ˙ ∞ , ∞ 0 2 3 + ‖ ∇ B t ‖ BMO 2 d t < ∞ , which can be regarded as improvement of the result in [3]. [ABSTRACT FROM AUTHOR]

We study sufficient conditions on weight functions under which norm approximations by analytic polynomials are possible. The weights we study include radial, non-radial, and angular weights. [ABSTRACT FROM AUTHOR]

For nonparametric regression estimation, conventional research all focus on isotropic regression function. In this paper, a linear wavelet estimator of anisotropic regression function is constructed, the rate of convergence of this estimator is discussed in anisotropic Besov spaces. More importantly, in order to obtain an adaptive estimator, a regression estimator is proposed with scaling parameter data-driven selection rule. It turns out that our results attain the optimal convergence rate of nonparametric pointwise estimation. [ABSTRACT FROM AUTHOR]

We study the global-in-time well-posedness and relaxation limit of the compressible Euler system with damping in 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'(\bar {\rho })>0 is allowed. To achieve it, a new composition estimate is established in the L^2-L^p hybrid Besov spaces with explicit dependence on the threshold between high frequencies and low frequencies. [ABSTRACT FROM AUTHOR]

This paper focuses on the inviscid limit of the incompressible Boussinesq equations in the same topology as the initial data, and proved that the continuous dependence of the viscous Boussinesq equations uniformly in some Besov spaces with respect to the viscosity. Our results extends the work of Guo et al. (J Funct Anal 276(9):2821–2830, 2019) on Navier–Stokes equations to Boussinesq equations with both stratified limit and earth’s rotation. [ABSTRACT FROM AUTHOR]

The initial data problem to a nonlinear shallow water wave equation in nonhomogeneous Besov space is discussed. Using the decomposition of Littlewood-Paley and the properties of nonhomogeneous Besov space, we establish the well-posedness of short time solutions for the equation in the Besov space. A blow-up criterion of solutions is also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the almost sure convergence, in supremum norm, of the rank-based linear wavelet estimator for the multivariate copula density over Besov classes. Using empirical process tools, we establish a uniform limit law for the deviation of an oracle estimator (which assumes known margins) from its expectation. This enables us to derive strong convergence rates for the rank-based linear estimator. [ABSTRACT FROM AUTHOR]

We are concerned with the global well-posedness of rate type viscoelastic fluids system (1.1)–(1.2). First, we prove the global existence and decay estimate of solution to (1.1)–(1.2) with stress diffusion near a constant state in H N (R 3) (N ≥ 4). We only assume that the H 3 -norm of initial data is small, but the L 2 -norm of the higher order derivatives can be arbitrary large. If the initial data belongs to homogeneous Sobolev or Besov spaces, we obtain the optimal decay rates of the solution and its higher order derivatives. Next, we establish the local existence of smooth solution to (1.1)–(1.2) without stress diffusion and present a blow-up criterion in R 2. From which and the Bony decomposition, we prove the global existence of smooth solution. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem of the 2D viscous shallow water equations in some critical Besov spaces B ˙ p , 1 2 p (R 2) × B ˙ p , q 2 p − 1 (R 2). As is known, this system is locally well-posed for large initial data as well as globally well-posed for small initial data in B ˙ p , 1 2 p (R 2) × B ˙ p , 1 2 p − 1 (R 2) for p < 4 and ill-posed in B ˙ p , 1 2 p (R 2) × B ˙ p , 1 2 p − 1 (R 2) for p > 4. In this paper, we prove that this system is ill-posed for the critical case p = 4 in the sense of "norm inflation". Furthermore, we also show that the system is ill-posed in B ˙ 4 , 1 1 2 (R 2) × B ˙ 4 , q − 1 2 (R 2) for any q ≠ 2. [ABSTRACT FROM AUTHOR]

Let Δ d be the discrete Laplacian defined on Z d by setting Δ d f (n →) = ∑ j = 1 d − [ f (n → + e → j) + f (n → − e → j) − 2 f (n →) ] , n → ∈ Z d , where { e → j : j = 1 , ... , d } is the standard basis for R d. In this paper, we prove weighted mixed norm estimates and end-point estimates for the maximal regularity of the discrete parabolic equation { u t + Δ d u = f , t ∈ [ 0 , T) u (0 , ⋅) = 0 , where T ∈ (0 , ∞). [ABSTRACT FROM AUTHOR]

We consider K‐interpolation spaces involving slowly varying functions, and derive necessary and sufficient conditions for a Holmstedt‐type formula to be held in the limiting case θ0=θ1∈{0,1}$\theta _0=\theta _1\in \lbrace 0,1\rbrace$. We also study the case θ0=θ1∈(0,1)$\theta _0=\theta _1\in (0,1)$. Applications are given to Lorentz–Karamata spaces, generalized gamma spaces, and Besov spaces. [ABSTRACT FROM AUTHOR]

This work deals with well‐posedness and blow‐up in the setting of Lebesgue and Besov spaces to the time‐fractional Keller–Segel model for chemotaxis under homogeneous Neumann boundary conditions in a smooth domain of RN$\mathbb {R}^N$. The KS model consists in a coupled system of partial differential equations. In particular, we also treat the unique continuation of the solution and the persistence of continuous dependence on the initial data for the continued solution. [ABSTRACT FROM AUTHOR]

In this note, we prove scattering for a defocusing nonlinear Schrödinger equation with initial data lying in a critical Besov space. In addition, we obtain polynomial bounds on the scattering size as a function of the critical Besov norm. [ABSTRACT FROM AUTHOR]

This paper is concerned with the estimation of the partial derivatives of a probability density function of directional data on the d-dimensional torus within the local thresholding framework. The estimators here introduced are built by means of the toroidal needlets, a class of wavelets characterised by excellent concentration properties in both the real and the harmonic domains. In particular, we discuss the convergence rates of the Lp-risks for these estimators, investigating their minimax properties and proving their optimality over a scale of Besov spaces, here taken as nonparametric regularity function spaces. [ABSTRACT FROM AUTHOR]

Considered herein is the Cauchy problem for a four‐component Novikov system with peaked solutions. We first investigate the local Gevrey regularity and analyticity of the solutions by a generalized Ovsyannikov theorem. Then, based on the local well‐posedness of this problem, the results with respect to the nonuniformly continuous dependence on initial data of the solutions in Besov spaces B2,15/2(핋)2×B2,13/2(핋)2 and Bp,rs(ℝ)2×Bp,rs−1(ℝ)2(s>max{5/2,2+1/p},1≤p,r≤∞)$$ {\left({B}_{p,r}&amp;#x0005E;s\left(\mathrm{\mathbb{R}}\right)\right)}&amp;#x0005E;2\times {\left({B}_{p,r}&amp;#x0005E;{s-1}\left(\mathrm{\mathbb{R}}\right)\right)}&amp;#x0005E;2\left(s&gt;\max \left\{5/2,2&amp;#x0002B;1/p\right\},1\le p,r\le \infty \right) $$ are established by constructing new approximate solutions and initial data. [ABSTRACT FROM AUTHOR]

This paper investigates the Cauchy problem of a generalized Camassa-Holm equation with quadratic and cubic nonlinearities (alias the mCH-Novikov-CH equation), which is a generalization of some special equations such as the Camassa-Holm (CH) equation, the modified CH (mCH) equation (alias the Fokas-Olver-Rosenau-Qiao equation), the Novikov equation, the CH-mCH equation, the mCH-Novikov equation, and the CH-Novikov equation. We first show the local well-posedness for the strong solutions of the mCH-Novikov-CH equation in Besov spaces by means of the Littlewood-Paley theory and the transport equations theory. Then, the Hölder continuity of the data-to-solution map to this equation are exhibited in some Sobolev spaces. After providing the blow-up criterion and the precise blow-up quantity in light of the Moser-type estimate in the Sobolev spaces, we then trace a portion and the whole of the precise blow-up quantity, respectively, along the characteristics associated with this equation, and obtain two kinds of sufficient conditions on the gradient of the initial data to guarantee the occurance of the wave-breaking phenomenon. Finally, the non-periodic and periodic peakon and multi-peakon solutions for this equation are also explored. [ABSTRACT FROM AUTHOR]

The reconstruction theorem tackles the problem of building a global distribution, on ℝ d or on a manifold, for a given family of sufficiently coherent local approximations. This theorem is a critical tool within Hairer's theory of Regularity Structures. In this paper, we establish a reconstruction theorem in the Besov setting, extending recent results of Caravenna and Zambotti. A Besov reconstruction theorem was first formulated by Hairer and Labbé in the context of regularity structures, exploiting nontrivial results from wavelet analysis. Our calculations follow the more elementary approach of coherent germs due to Caravenna and Zambotti. With this formulation our results are both stated and proved with tools from the theory of distributions without the need of the theory of Regularity Structures. As an application, we present an alternative proof of a (Besov) Young multiplication theorem which does not require the use of para-differential calculus. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

When p > 1 , using as base space classical Lorentz spaces associated to a weight from the Ariño–Muckenhoupt class B p , we will study Gagliardo–Nirenberg inequalities. As a by-product we will also consider Morrey–Sobolev inequalities. These arguments can be generalized to many different frameworks, in particular the proofs are given in the setting of stratified Lie groups. [ABSTRACT FROM AUTHOR]

In this paper we fix and consider to be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure supporting a -Poincaré inequality such that is a uniform domain in its completion . We realize the trace of functions in the Dirichlet–Sobolev space on the boundary as functions in the homogeneous Besov space for suitable ; here, is equipped with a non-atomic Borel regular measure . We show that if satisfies a -codimensional condition with respect to for some , then there is a bounded linear trace operator and a bounded linear extension operator that is a right-inverse of . [ABSTRACT FROM AUTHOR]

This paper is concerned with the logarithmically improved regularity criterion in terms of the middle eigenvalue of the strain tensor to the 3D Boussinesq equations in Besov spaces with negative indices. It is shown that a weak solution is regular on $ (0, T] $ provided that$ \begin{align*} \int_{0}^{T}\frac{\left\Vert \lambda _{2}^{+}(\cdot , t)\right\Vert _{\dot{B} _{\infty , \infty }^{-\delta }}^{\frac{2}{2-\delta }}}{\ln (e+\left\Vert u(\cdot , t)\right\Vert _{\dot{B}_{\infty , \infty }^{-\delta }})}dt<\infty. \end{align*} $for some $ 0<\delta <1 $. As a consequence, this result is some improvements of recent works [11,12] established by Neustupa-Penel and Miller. [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider the Cauchy problem of a weakly dissipative Camassa-Holm equation. We first establish the local well-posedness of equation in Besov spaces B p , r s with s > 1 + 1 p and s = 1 + 1 p , r = 1 , p ∈ [ 1 , ∞). Then, we prove the global existence for small data, and present two blow-up criteria. Finally, we get two blow-up results, which can be used in the proof of the ill-posedness in critical Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new 2‐microlocal Besov and Triebel–Lizorkin spaces of Jaffard type, which contain many classical functions. We establish local theory of these function spaces by the φ‐transform and the wavelet decomposition coefficients. Moreover, we give a characterization of these local spaces by a local polynomial approximation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We introduce hyperbolic oscillation spaces and mixed fractional lifting oscillation spaces expressed in terms of hyperbolic wavelet leaders of multivariate signals on R d , 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 (j 1 , ⋯ , j d) , 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. [ABSTRACT FROM AUTHOR]

In this paper we study the Peskin problem in two dimensions, which describes the dynamics of a one-dimensional closed elastic structure immersed in a steady Stokes flow. We prove the local well-posedness for arbitrary initial configuration in (C²)Ḃ¹ ∞,∞ satisfying the well-stretched condition, and the global well-posedness when the initial configuration is sufficiently close to an equilibrium in .æ. Here (C²)Ḃ¹ ∞,∞ is the closure of C² in the Besov space .æ. The global-in-time solution will converge to an equilibrium exponentially as t A Too. This is the first well-posedness result for the Peskin problem with non-Lipschitz initial data. [ABSTRACT FROM AUTHOR]

We carry on the investigation started in [P. Ambrosio and A. Passarelli di Napoli, Regularity results for a class of widely degenerate parabolic equations, preprint 2022, version 3, https://arxiv.org/abs/2204.05966v3] about the regularity of weak solutions to the strongly degenerate parabolic equation u t - div ⁡ [ (| D ⁢ u | - 1) + p - 1 ⁢ D ⁢ u | D ⁢ u | ] = f in ⁢ Ω T = Ω × (0 , T) , where Ω is a bounded domain in ℝ n for n ≥ 2 , p ≥ 2 and (⋅) + stands for the positive part. Here, we weaken the assumption on the right-hand side, by assuming that f ∈ L loc p ′ ⁢ (0 , T ; B p ′ , ∞ , loc α ⁢ (Ω)) , with α ∈ (0 , 1) and p ′ = p / (p - 1) . This leads us to obtain higher fractional differentiability results for a function of the spatial gradient Du of the solutions. Moreover, we establish the higher summability of Du with respect to the spatial variable. The main novelty of the above equation is that the structure function satisfies standard ellipticity and growth conditions only outside the unit ball centered at the origin. We would like to point out that the main result of this paper can be considered, on the one hand, as the parabolic counterpart of an elliptic result contained in [P. Ambrosio, Besov regularity for a class of singular or degenerate elliptic equations, J. Math. Anal. Appl. 505 2022, 2, Paper No. 125636], and on the other hand as the fractional version of some results established in [P. Ambrosio and A. Passarelli di Napoli, Regularity results for a class of widely degenerate parabolic equations, preprint 2022, version 3, https://arxiv.org/abs/2204.05966v3]. [ABSTRACT FROM AUTHOR]

The present article investigates the two-dimensional Euler-Boussinesq system with critical fractional dissipation and general source term. First, we show that this system admits a global solution of Yudovich type, and as a consequence, we treat the regular vortex patch issue.

We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differential equation (SPDE) (0.1) d u = (a i j (ω , t) u x i x j + f) d t + (σ i k (ω , t) u x i + g k) d w t k , u (0 , x) = u 0 , where { w t k : k = 1 , 2 , ⋯ } is a sequence of independent Brownian motions. The coefficients are merely measurable in (ω , t) and can be unbounded and fully degenerate, that is, coefficients a i j , σ i k merely satisfy (0.2) (α i j (ω , t)) d × d : = (a i j (ω , t) − 1 2 ∑ k = 1 ∞ σ i k (ω , t) σ j k (ω , t)) ≥ 0. In this article, we prove that there exists a unique solution u to (0.1) , and (0.3) ‖ u x x ‖ H p γ (τ , δ) ≤ N (d , p) (‖ u 0 ‖ B p γ + 2 (1 − 1 / p) + ‖ f ‖ H p γ (τ , δ 1 − p) + ‖ g x ‖ H p γ (τ , | σ | p δ 1 − p , l 2) p + ‖ g x ‖ H p γ (τ , δ 1 − p / 2 , l 2)) , where p ≥ 2 , γ ∈ R , τ is an arbitrary stopping time, δ (ω , t) is the smallest eigenvalue of α i j (ω , t) , H p γ (τ , δ) is a weighted stochastic Sobolev space, and B p γ + 2 (1 − 1 / p) is a stochastic Besov space. [ABSTRACT FROM AUTHOR]

We obtain the local existence and uniqueness of solution to the relativistic Vlasov–Poisson system in Besov space, extending the non-relativistic result to the relativistic case. Weaker regularity is required for the initial datum by optimizing the commutator estimates related to the field. Commutator estimates associated with relativistic velocity are proved by dyadic decomposition of the frequency space and the decreasing properties of Bessel potential. [ABSTRACT FROM AUTHOR]

We are concerned with a system of equations in R d (d ≥ 3) governing the evolution of isothermal, viscous and compressible fluids of Korteweg type, that can be used as a phase transition model. In the case of zero sound speed P ′ (ρ ⁎) = 0 , it is found that the linearized system admits the purely parabolic structure, which enables us to establish the global-in-time existence and Gevrey analyticity of strong solutions in hybrid Besov spaces of L p -type. Precisely, if the full viscosity coefficient and capillary coefficient satisfy ν ¯ 2 ≥ 4 κ ¯ , then the acoustic waves are not available in compressible fluids. Consequently, the prior L 2 boundedness on the low frequencies of density and velocity could be improved to the general L p version with 1 ≤ p < d. The proof mainly relies on new nonlinear Besov (-Gevrey) estimates for product and composition of functions. [ABSTRACT FROM AUTHOR]

We study the approximation of univariate functions by combining tensorization of functions with tensor trains (TTs)—a commonly used type of tensor networks (TNs). Lebesgue L p -spaces in one dimension can be identified with tensor product spaces of arbitrary order through tensorization. We use this tensor product structure to define different approximation tools and corresponding approximation spaces of TTs, associated with different measures of complexity. The approximation tools are shown to have (near to) optimal approximation rates for functions with classical Besov smoothness. We then use classical interpolation theory to show that a scale of interpolated smoothness spaces is continuously embedded into the scale of TT approximation spaces and, vice versa, we show that the TT approximation spaces are, in a sense, much larger than smoothness spaces when the depth of the tensor network is not restricted but are embedded into a scale of interpolated smoothness spaces if one restricts the depth. The results of this work can be seen as both an analysis of the approximation spaces of a type of TNs and a study of the expressivity of a particular type of neural networks (NNs)—namely feed-forward sum-product networks with sparse architecture. We point out interesting parallels to recent results on the expressivity of rectifier networks. [ABSTRACT FROM AUTHOR]

We here establish the higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form ∫ Ω ⟨ A ⁢ (x , D ⁢ u) , D ⁢ (φ - u) ⟩ ⁢ d x ≥ 0 for all ⁢ φ ∈ K ψ ⁢ (Ω) , where Ω is a bounded open subset of R n , ψ ∈ W 1 , p ⁢ (Ω) is a fixed function called obstacle and K ψ ⁢ (Ω) = { w ∈ W 1 , p ⁢ (Ω) : w ≥ ψ ⁢ a.e. in ⁢ Ω } is the class of admissible functions. Assuming that the gradient of the obstacle belongs to some suitable Besov space, we are able to prove that some fractional differentiability property transfers to the gradient of the solution. [ABSTRACT FROM AUTHOR]

This paper concerns the study of a non-smooth logistic regression function. The focus is on a high-dimensional binary response case by penalizing the decomposition of the unknown logit regression function on a wavelet basis of functions evaluated on the sampling design. Sample sizes are arbitrary (not necessarily dyadic) and we consider general designs. We study separable wavelet estimators, exploiting sparsity of wavelet decompositions for signals belonging to homogeneous Besov spaces, and using efficient iterative proximal gradient descent algorithms. We also discuss a level by level block wavelet penalization technique, leading to a type of regularization in multiple logistic regression with grouped predictors. Theoretical and numerical properties of the proposed estimators are investigated. A simulation study examines the empirical performance of the proposed procedures, and real data applications demonstrate their effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, we show the weak and strong well-posedness of density dependent stochastic differential equations driven by α -stable processes with α ∈ (1 , 2). The existence part is based on Euler's approximation as Hao et al. (2021), while, the uniqueness is based on the Schauder estimates in Besov spaces for nonlocal Fokker–Planck equations. For the existence, we only assume the drift being continuous in the density variable. For the weak uniqueness, the drift is assumed to be Lipschitz in the density variable, while for the strong uniqueness, we also need to assume the drift being β 0 -order Hölder continuous in the spatial variable, where β 0 ∈ (1 − α / 2 , 1). [ABSTRACT FROM AUTHOR]

Estimating functions with varying degrees of smoothness is a challenging problem in nonparametric function estimation. In this paper, we propose the Lévy adaptive B-Spline regression (LABS) model, an extension of the Lévy adaptive regression kernels (LARK) models, for estimating functions with varying degrees of smoothness. The LABS model is a LARK model with B-spline basis functions as generating kernels. The B-spline basis functions consist of piecewise k-degree polynomials with k -- 1 continuous derivatives, and can systematically express functions with varying degrees of smoothness. By changing the order of the B-spline basis, the LABS model systematically adapts to the smoothness of the functions, for example, jump discontinuities, sharp peaks, and so on. The results of simulation studies and real-data examples show that the proposed model captures smooth areas, jumps, and sharp peaks of functions. The proposed model also perform best in almost all examples. Finally, we provide theoretical results that the mean function for the LABS model belongs to certain Besov spaces, based on the order of the B-spline basis, and that the prior of the model has full support on the Besov spaces. [ABSTRACT FROM AUTHOR]