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We consider the general framework of a metric measure space satisfying the doubling volume property, associated with a non-negative self-adjoint operator, whose heat kernel enjoys standard Gaussian localization. We study the dual spaces of the classical and nonclassical Besov and Triebel-Lizorkin spaces on this setting. Our results generalize those on Euclidean spaces and are new on several setups of independent interest; the sphere, the ball, more general Riemannian manifolds and other settings. [ABSTRACT FROM AUTHOR]

Gaussian upper and lower bounds and Hölder continuity are established for the heat kernel associated to the prolate spheroidal wave functions (PSWFs) of order zero. These results are obtained by application of a general perturbation principle using the fact that the PSWF operator is a perturbation of the Legendre operator. Consequently, the Gaussian bounds and Hölder inequality for the PSWF heat kernel follow from the ones in the Legendre case. As an application of the general perturbation principle, we also establish Gaussian bounds for the heat kernels associated to generalized univariate PSWFs and PSWFs on the unit ball in R d . Further, we develop the related to the PSWFs of order zero smooth functional calculus, which provides the necessary groundwork in developing the theory of Besov and Triebel–Lizorkin spaces associated to the PSWFs. One of our main results on Besov and Triebel–Lizorkin spaces associated to the PSWFs asserts that they are the same as the Besov and Triebel–Lizorkin spaces generated by the Legendre operator. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we consider traces at initial times for functions with mixed time‐space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity problems for fractional evolution equations and stochastic evolution equations, where uniform trace estimates on the half‐line are shown. [ABSTRACT FROM AUTHOR]

We study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix A into Sobolev spaces, with a focus on the influence of A on the embedding behavior. For a large range of parameters, we derive sharp characterizations of embeddings. [ABSTRACT FROM AUTHOR]

We consider the stationary Navier–Stokes equations in the two‐dimensional torus T2$\mathbb {T}^2$. For any ε>0$\varepsilon >0$, we show the existence, uniqueness, and continuous dependence of solutions in homogeneous toroidal Besov spaces Ḃp+ε,q−1+2p(T2)$\dot{B}^{-1+\frac{2}{p}}_{p+\varepsilon , q}(\mathbb {T}^2)$ for given small external forces in Ḃp+ε,q−3+2p(T2)$\dot{B}^{-3+\frac{2}{p}}_{p+\varepsilon , q}(\mathbb {T}^2)$ when 1≤p<2$1\le p <2$. These spaces become closer to the scaling invariant ones if the difference ε becomes smaller. This well‐posedness is proved by using the embedding property and the para‐product estimate in homogeneous Besov spaces. In addition, for the case (p,q)∈({2}×(2,∞])∪((2,∞]×[1,∞])$(p,q)\in (\lbrace 2\rbrace \times (2,\infty ])\cup ((2,\infty ]\times [1,\infty ])$, we can show the ill‐posedness, even in the scaling invariant spaces. Actually in such cases of p and q, we can prove that ill‐posedness by showing the discontinuity of a certain solution map from Ḃp,q−3+2p(T2)$\dot{B}^{-3+\frac{2}{p}}_{p ,q}(\mathbb {T}^2)$ to Ḃp,q−1+2p(T2)$\dot{B}^{-1+\frac{2}{p}}_{p, q}(\mathbb {T}^2)$. [ABSTRACT FROM AUTHOR]

In this paper, we give a new construction of u 0 ∈ B p , ∞ σ such that the corresponding solution to the hyperbolic Keller-Segel model starting from u 0 is discontinuous at t = 0 in the metric of B p , ∞ σ (R d) with d ≥ 1 and 1 ≤ p ≤ ∞ , which implies the ill-posedness for this equation in B p , ∞ σ . Our result generalizes the recent work in Zhang et al. (J Differ Equ 334:451-489, 2022) where the case d = 1 and p = 2 was considered. [ABSTRACT FROM AUTHOR]

In this paper, we show that the Bergman functions on the Siegel upper half-space enjoy the following uniqueness property: if f\in A_t^p(\mathcal {U}) and \mathcal {L}^{\alpha } f\equiv 0 for some nonnegative multi-index \alpha, then f\equiv 0, where \mathcal {L}^{\alpha }≔(\mathcal {L}_1)^{\alpha _1} \cdots (\mathcal {L}_n)^{\alpha _n} with \mathcal {L}_j = \frac {\partial }{\partial z_j} + 2i \bar {z}_j \frac {\partial }{\partial z_n} for j=1,\ldots, n-1 and \mathcal {L}_n = \frac {\partial }{\partial z_n}. As a consequence, we obtain a new integral representation for the Bergman functions on the Siegel upper half-space. In the end, as an application, we derive a result that relates the Bergman norm to a "derivative norm", which suggests an alternative definition of the Bloch space and a notion of the Besov spaces over the Siegel upper half-space. [ABSTRACT FROM AUTHOR]

In this paper we introduce Besov-type spaces with variable smoothness and integrability. We show that these spaces are characterized by the φ-transforms in appropriate sequence spaces and we obtain atomic decompositions for these spaces. Moreover the Sobolev embeddings for these function spaces are obtained. [ABSTRACT FROM AUTHOR]

Using operator-valued Fourier multiplier theorems on B p , q s (R ; X) (resp. F p , q s (R ; X) ), weighted Besov spaces B p , q s , ω (R ; X) (resp. Triebel–Lizorkin spaces F p , q s , ω (R ; X) ) on the real line and the Carleman transform, we characterize the well-posedness of differential equations u ′ (t) = A u (t) + f (t) and u ′ ′ (t) + γ u ′ (t) = A u (t) + f (t) on B p , q s (R ; X) (resp. F p , q s (R ; X) ), where γ ∈ R , 1 ≤ p , q ≤ ∞ (resp. 1 ≤ p < ∞ ), s > 0 and A is a closed linear operator defined on a Banach space X. [ABSTRACT FROM AUTHOR]

We consider a special approach to investigate a mixed boundary value problem (BVP) for the Lamé system of elasticity in the case of three-dimensional bounded domain Ω ⊂ R 3 , when the boundary surface S = ∂ Ω is divided into two disjoint parts, S D and S N , where the Dirichlet and Neumann type boundary conditions are prescribed respectively for the displacement vector and stress vector. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of linear combination of the single layer and double layer potentials with densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations which do not contain neither extensions of the Dirichlet or Neumann data, nor the Steklov–Poincaré type operator. Moreover, the right hand sides of the resulting pseudodifferential system are vectors coinciding with the Dirichlet and Neumann data of the problem under consideration. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L 2 -based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space [ W 2 1 (Ω) ] 3 and representability of solutions in the form of linear combination of the single layer and double layer potentials with densities supported respectively on the Dirichlet and Neumann parts of the boundary. Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that the operator is invertible in the L p -based Besov spaces with 4 3 < p < 4 , which under appropriate boundary data implies C α -Hölder continuity of the solution to the mixed BVP in the closed domain Ω ‾ with α = 1 2 − ε , where ε > 0 is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

We propose alternatives to Bayesian prior distributions that are frequently used in the study of inverse problems. Our aim is to construct priors that have similar good edge-preserving properties as total variation or Mumford-Shah priors but correspond to well-defined infinite-dimensional random variables, and can be approximated by finite-dimensional random variables. We introduce a new wavelet-based model, where the non-zero coefficients are chosen in a systematic way so that prior draws have certain fractal behaviour. We show that realisations of this new prior take values in Besov spaces and have singularities only on a small set $ \tau $ with a certain Hausdorff dimension. We also introduce an efficient algorithm for calculating the MAP estimator, arising from the the new prior, in the denoising problem. [ABSTRACT FROM AUTHOR]

New local smoothing estimates in Besov spaces adapted to the half-wave group are proved via ℓ 2 -decoupling. We apply these estimates to obtain new well-posedness results for the cubic nonlinear wave equation in two dimensions. The results are compared to new well-posedness results in L p -based Sobolev spaces. [ABSTRACT FROM AUTHOR]

This paper considers wavelet estimations of a multivariate density function based on stratified size-biased and strong mixing data. We provide upper bounds of the mean integrated squared error for linear and nonlinear wavelet estimators in Besov space B p , q s (R d). It is shown that the linear estimator achieves the optimal convergence rate in the case of p ≥ 2. Moreover, the convergence rate of nonlinear estimator coincides with the optimal convergence rate up to a ln N factor for p ∈ [ 1 , + ∞). In addition, the nonlinear wavelet estimator is adaptive. [ABSTRACT FROM AUTHOR]

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

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 (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. [ABSTRACT FROM AUTHOR]

The global existence and convergence rates of solutions to the three-dimensional compressible Navier-Stokes equations without heat conductivity have been obtained since Duan and Ma [Indiana Univ. Math. J., 5 (2008), pp. 2299-2319]. However, as far as we know, how to construct the global small solutions to the compressible Navier-Stokes equations without heat conductivity in R² is still an open problem. In the present paper, we give a positive answer in Rn (n=2,3) with the initial data restricted in the Besov spaces. [ABSTRACT FROM AUTHOR]

We establish the higher fractional differentiability of bounded minimizers to a class of obstacle problems with non-standard growth conditions of the form min ⁡ { ∫ Ω F ⁢ (x , D ⁢ w) ⁢ 푑 x : w ∈ 풦 ψ ⁢ (Ω) } , where Ω is a bounded open set of ℝ n , n ≥ 2 , the function ψ ∈ W 1 , p ⁢ (Ω) is a fixed function called obstacle, and 풦 ψ ⁢ (Ω) := { w ∈ W 1 , p ⁢ (Ω) : w ≥ ψ ⁢ a.e. in ⁢ Ω } is the class of admissible functions. If the obstacle ψ is locally bounded, we prove that the gradient of solution inherits some fractional differentiability property, assuming that both the gradient of the obstacle and the mapping x ↦ D ξ ⁢ F ⁢ (x , ξ) belong to some suitable Besov space. The main novelty is that the Besov type regularity on the partial map x ↦ D ξ ⁢ F ⁢ (x , ξ) is not related to the dimension n. [ABSTRACT FROM AUTHOR]

In this paper, we aim to solving the open question left in [Nie, Yuan: Nonlinear Anal 196 (2020); J. Math. Anal. Appl 505 (2022) and Xiao, Fei: J. Math. Anal. Appl 514 (2022)]. We prove that a multidimensional chemotaxis system is ill-posedness in B ˙ 2 d , r - 3 2 × ( B ˙ 2 d , r - 1 2 ) d when 1 ≤ r < d due to the lack of continuity of the solution. [ABSTRACT FROM AUTHOR]

We study the relation between the composition operators Cϕ with closed range on the weighted Bloch spaces and Cϕ with closed range on the weighted Dirichlet spaces Dαp. In particular, we study the boundedness from below of composition operators between Lpa and Lqa, between Lpa and Hardy space, and between Lpa and Besov space. [ABSTRACT FROM AUTHOR]

Although Strichartz estimates have been widely applied for the nonlinear dispersive equations like Schr o ¨ dinger equation and wave equation in Besov spaces, it is rare in nonlinear kinetic type equations such as Vlasov-Poisson equation. In this paper, we are concerned about the dispersive estimates for the kinetic transport equation in Besov spaces, which could be applied to establish Strichartz estimates in Besov space and further may be exploited to investigate the solutions for Vlasov-Poisson equation and Boltzmann equation in Besov space. [ABSTRACT FROM AUTHOR]

This paper is concerned with the generalized singular integral operator with rough kernel and the approximation problem for the generalized surface quasi-geostrophic equation. For the generalized singular integral operator, we obtain uniform L p − L q estimates with respect to a parameter β. From this one can cover the L p -boundedness of the Calderón-Zygmund operator with rough kernel by letting β → 0. We applied this estimate to study the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation. Local well-posedness in the Besov space B p , q s and some limit behaviour of the solutions are obtained. Our results improve the previous ones by Yu-Zheng-Jiu in 2019 and by Yu-Jiu-Li in 2021. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the regularity criteria of axisymmetric weak solutions to the three‐dimensional (3D) incompressible magnetohydrodynamics (MHD) equations with nonzero swirl component. By making use of techniques of the Littlewood–Paley decomposition, we show that weak solutions to the 3D axisymmetric MHD equations become regular if the swirl component of vorticity satisfies that wθeθ∈L1(0,T;Ḃ∞,∞0)$w_{\theta }e_{\theta }\in L^{1}\big (0,T;\dot{B}_{\infty ,\infty }^{0}\big)$, which partially gives a positive answer to the marginal case for the regularity of MHD equations. [ABSTRACT FROM AUTHOR]

The global existence of solutions to a damped MHD system in Besov spaces is obtained. This research is motivated by the open question [Wu et al., J. Nonlinear Sci. 25 (2015), 157–192, Remark 5.1] on the possibility of extension to the existence of global small solutions from a potential Banach space to a Besov space. The present result positively confirms the open question. What is more, the initial data are not required to be uniformly small. [ABSTRACT FROM AUTHOR]

The aim of this article is to extend previous works about the asymptotics of an ill-prepared fast rotating, highly stratified incompressible Navier–Stokes system. Thanks to improved Strichartz and a priori estimates, we are able not only to cover a case which was unanswered in our previous work (allowing bigger ill-prepared initial data) but also to improve the convergence rates and reduce some assumptions on the initial data. In passing we also widen the range of some parameters and compare two methods to obtain dispersive estimates. [ABSTRACT FROM AUTHOR]

Rate distortion theory is concerned with optimally encoding signals from a given signal class S using a budget of R bits, as R → ∞ . We say that S can be compressed at rates if we can achieve an error of at most O (R - s) for encoding the given signal class; the supremal compression rate is denoted by s ∗ (S) . Given a fixed coding scheme, there usually are some elements of S that are compressed at a higher rate than s ∗ (S) by the given coding scheme; in this paper, we study the size of this set of signals. We show that for certain "nice" signal classes S , a phase transition occurs: We construct a probability measure P on S such that for every coding scheme C and any s > s ∗ (S) , the set of signals encoded with error O (R - s) by C forms a P -null-set. In particular, our results apply to all unit balls in Besov and Sobolev spaces that embed compactly into L 2 (Ω) for a bounded Lipschitz domain Ω . As an application, we show that several existing sharpness results concerning function approximation using deep neural networks are in fact generically sharp. In addition, we provide quantitative and non-asymptotic bounds on the probability that a random f ∈ S can be encoded to within accuracy ε using R bits. This result is subsequently applied to the problem of approximately representing f ∈ S to within accuracy ε by a (quantized) neural network with at most W nonzero weights. We show that for any s > s ∗ (S) there are constants c, C such that, no matter what kind of "learning" procedure is used to produce such a network, the probability of success is bounded from above by min { 1 , 2 C · W ⌈ log 2 (1 + W) ⌉ 2 - c · ε - 1 / s } . [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for the generalized Camassa–Holm equation proposed by Hakkaev and Kirchev (Commun Partial Differ Equ 30:761–781, 2005). We prove that the solution map of the generalized Camassa–Holm equation is not uniformly continuous on the initial data in Besov spaces. Our result includes the present work Li et al. (Differ Equ 269:8686–8700, 2020) on Camassa–Holm equation with Q = 1 and extends the previous non-uniform continuity in Sobolev spaces Mi and Mu (Monatsh Math 176:423–457, 2015) to Besov spaces. [ABSTRACT FROM AUTHOR]

For any p ∈ (0 , 1 ] and s ∈ R , the authors prove two types of characterizations of the pointwise multiplier space M (B p , p s (R n)) of the Besov space B p , p s (R n). One type is based on wavelet analysis and is an extension of a well-known argument of Yves Meyer. The other type works with Fourier analytic terms. As an application of the above two types of characterizations, the authors further obtain a characterization of bounded functions in the uniform space B p , p , unif s , τ (R n) via Haar wavelets in the critical index τ = 1 p − s n. [ABSTRACT FROM AUTHOR]

We study the manifold n -widths of Sobolev and Besov spaces with error measured in the L p -norm. The manifold widths measure how efficiently these spaces can be approximated by continuous non-linear parametric methods. Existing upper and lower bounds only match when the smoothness index q satisfies q ≤ p or 1 ≤ p ≤ 2. We close this gap, obtaining sharp bounds for all 1 ≤ p , q ≤ ∞ for which a compact embedding holds. In the process, we determine the exact value of the manifold widths of finite dimensional ℓ q M -balls in the ℓ p -norm when p ≤ q. Although this result is not new, we provide a new proof and apply it to lower bounding the manifold widths of Sobolev and Besov spaces. Our results show that the Bernstein widths, which are typically used to lower bound the manifold widths, decay asymptotically faster than the manifold widths in many cases. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem of the drift–diffusion system arising from semiconductor model. We prove that if a certain nonlinear function of the initial data is small enough, in a Besov type space, then there is a global solution to this drift–diffusion system. We also provide an example of initial data satisfying that nonlinear smallness condition, but whose norm be chosen arbitrarily large in B ̇ ∞ , ∞ − 2 (R d). [ABSTRACT FROM AUTHOR]

The low-frequency L 1 assumption has been extensively applied to the large-time asymptotics of solutions to the compressible Navier-Stokes equations and incompressible Navier-Stokes equations since the classical efforts due to Kawashima, Matsumura, Nishida, Ponce, Schonbek and Wiegner. In this paper, we establish a sharp decay characterization for the compressible Navier-Stokes equations in the critical L p framework. Precisely, it is proved that the Besov space B ˙ 2 , ∞ σ 1 -boundedness condition (with d 2 − 2 d p ≤ σ 1 < d 2 − 1) of the low-frequency part of initial perturbation is not only sufficient, but also necessary to achieve those upper bounds of time-decay estimates. Furthermore, we show that the upper and lower bounds of time-decay estimates hold if and only if the low-frequency part of initial perturbation belongs to a nontrivial subset of B ˙ 2 , ∞ σ 1 . To the best of our knowledge, our work is the first one addressing the inverse problem for the large-time asymptotics of compressible viscous fluids. [ABSTRACT FROM AUTHOR]

In this paper, we study the global regularity problem for the 2D Rayleigh–Bénard equations with logarithmic supercritical dissipation. By exploiting a combined quantity of the system, the technique of Littlewood-Paley decomposition and Besov spaces, and some commutator estimates, we establish the global regularity of a strong solution to this equations in the Sobolev space H s (R 2) for s ≥ 2. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 3D inhomogeneous incompressible magnetohydrodynamic (MHD) system. By the assumption of the smallness of initial velocity and magnetic fluids in the critical Besov space, the local and global well‐posedness of 3D inhomogeneous incompressible equations is obtained. It improves some previous results of MHD equations by generalizing the range of exponent p$$ p $$ in Besov spaces B˙p,13/p−1$$ {\dot{B}}_{p,1}^{3/p-1} $$ with 1

We are concerned with the global existence and decay rates of large solutions for the Poisson–Nernst–Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin–Lerner‐type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species that is small enough. Moreover, the large solution is obtained for initial densities belonging to the low regularity Besov spaces with different regularity and integral indices, which indicates more specific coupling relations between the difference and the summation of negatively and positively charged densities. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain the higher order fractional differentiability of weak solution pairs for the following system modeling the stationary motion of electro-rheological fluids { − div ((1 + | D u | 2) p (x) − 2 2 D u) + [ ∇ u ] u + ∇ π = − div F in Ω div u = 0 in Ω in Besov space, where F is Besov regular and the non-standard growth exponent p (x) is Besov-Orlicz regular. The main approach to prove this result is to establish a Besov regular priori estimate of weak solution pairs for its Stokes setting without convective term [ ∇ u ] u , and then treat the convective term as a part of nonhomogeneous term in the general systems. [ABSTRACT FROM AUTHOR]

We show the local well-posedness for the Keller-Segel-Navier-Stokes system with initial data in the scaling invariant Besov spaces, where the solution exists globally in time if the initial data is sufficiently small. We also reveal that the solution belongs to the Lorentz spaces in time direction, while the solution is smooth in space and time. Moreover, we obtain the maximal regularity estimates of solutions under the certain conditions. We further show that the solution has the additional regularities if the initial data has higher regularities. This result implies that global solutions decay as the limit t → ∞ in the same norm of the space of the initial data. Our results on the Lorentz regularity estimates are based on the strategy by Kozono-Shimizu (2019) [26]. [ABSTRACT FROM AUTHOR]

This paper establishes the global well-posedness issue for the full viscous MHD equations in the axisymmetric setting. Global solutions are obtained in critical Besov spaces uniformly to the viscosity when the resistivity is fixed in the spirit of [Abidi H, Hmidi T, Keraani S. On the global well-posedness for the axisymmetric Euler equations. Math. Ann. 2010;347:15–41.], [Hassainia Z. On the global well-posedness of the 3D axisymmetric resistive MHD equations. Ann. Henri Poincaré. 2022;23:2877-2917], [Hmidi T, Zerguine M. Inviscid limit axisymmetric Navier–Stokes system. Differential and Integral Equations. 2009;22(11–12):1223–1246.]. Furthermore, strong convergence in the resolution spaces with a rate of convergence is also studied. [ABSTRACT FROM AUTHOR]

Let Ω = [0;1]d be the unit cube in ℝd. We study the problem of how efficiently, in terms of the number of parameters, deep neural networks with the ReLU activation function can approximate functions in the Sobolev spaces Ws(Lq(Ω)) and Besov spaces Bsr (Lq(Ω)), with error measured in the Lp(Ω) norm. This problem is important when studying the application of neural networks in a variety of fields, including scientific computing and signal processing, and has previously been solved only when p = q = ∞. Our contribution is to provide a complete solution for all 1 ≤ p, q ≤ ∞ and s > 0 for which the corresponding Sobolev or Besov space compactly embeds into Lp. The key technical tool is a novel bit-extraction technique which gives an optimal encoding of sparse vectors. This enables us to obtain sharp upper bounds in the non-linear regime where p > q. We also provide a novel method for deriving Lp-approximation lower bounds based upon VC-dimension when p < ∞. Our results show that very deep ReLU networks significantly outperform classical methods of approximation in terms of the number of parameters, but that this comes at the cost of parameters which are not encodable. [ABSTRACT FROM AUTHOR]

In the present paper, we show an improved lower bound for the lifespan of the solutions to the ideal MHD equations in the case of space dimension d = 2. In particular, for small initial magnetic fields b 0 of size (say) ε > 0 , the lifespan T ε > 0 of the corresponding solution goes to +∞ in the limit ε → 0 +. Such a result does not follow from standard quasi-linear hyperbolic theory. For proving it, three are the crucial ingredients: first of all, to work in endpoint Besov spaces B ∞ , r s , under the condition s > 1 and r ∈ [ 1 , + ∞ ] or s = r = 1 ; moreover, to use the Elsässer formulation of the ideal MHD, recasted in its vorticity formulation; finally, to take advantage of the special structure of the non-linear terms. We also rigorously establish the equivalence between the original formulation of the ideal MHD and its Elsässer formulation for a large class of weak solutions. The construction of explicit counterexamples shows the sharpness of our assumptions. Related non-uniqueness issues are discussed as well. [ABSTRACT FROM AUTHOR]

We are concerned with a system of equations governing the evolution of isothermal, viscous, and compressible fluids of the Korteweg type, which is used to describe a two-phase liquid–vapor mixture. It is found that there is a "regularity-gain" dissipative structure of linearized systems in case of zero sound speed P ′ (ρ *) = 0 , in comparison with the classical compressible Navier–Stokes equations. First, we establish the global-in-time existence of strong solutions in hybrid Besov spaces by using Banach's fixed point theorem. Furthermore, we prove that the global solutions with critical regularity are Gevrey analytic in fact. Secondly, based on Gevrey's estimates, we obtain uniform bounds on the growth of the analyticity radius of solutions in negative Besov spaces, which lead to the optimal time-decay estimates of solutions and their derivatives of arbitrary order. [ABSTRACT FROM AUTHOR]

The topic of modulus of smoothness still gets the interest of many researchers due to its applicable usage in different fields, especially for function approximation. In this paper, we define a new modulus of smoothness of weighted type. The properties of our modulus are studied. These properties can be easily used in different fields, in particular, the functions in the Besov spaces Bpr when 0 < p < 1. [ABSTRACT FROM AUTHOR]

This paper gives a further investigation on the regularity criteria for three-dimensional micropolar equations in Besov spaces. More precisely, it is proved that the weak solution (μ, ω) is regular if the velocity u satisfies. [ABSTRACT FROM AUTHOR]

This article is devoted to the long-standing problem on the smoothness of sub-Riemannian geodesics. We prove that the derivatives of sub-Riemannian geodesics are always Lp-Hölder continuous. Additionally, this result has several interesting implications. These include (i) the decay of Fourier coefficients on abnormal controls, (ii) the rate at which they can be approximated by smooth functions, (iii) a generalization of the Poincaré inequality, and (iv) a compact embedding of the set of shortest paths into the space of Bessel potentials. [ABSTRACT FROM AUTHOR]

This paper focuses on the higher order fractional differentiability of weak solution pairs to the following nonlinear stationary Stokes system { div A (x , D u) − ∇ π = div F , in Ω , div u = 0 , in Ω. In terms of the difference quotient method, our first result reveals that if F ∈ Bp,q,locβ(Ω,ℝn)for p = 2 and 1 ≤ q ≤ 2 n n − 2 β , then such extra Besov regularity can transfer to the symmetric gradient Du and its pressure π with no losses under a suitable fractional differentiability assumption on x ↦ A (x , ξ) . Furthermore, when the vector field A (x , D u) is simplified to the full gradient ∇u, we improve the aforementioned Besov regularity for all integrability exponents p and q by establishing a new Campanato-type decay estimates for (∇u, π). [ABSTRACT FROM AUTHOR]

In this paper, we consider the global existence, regularizing decay rate and asymptotic behavior of mild solutions to Cauchy problem of fractional drift diffusion system with power-law nonlinearity. Using the properties of fractional heat semigroup and the classical estimates of fractional heat kernel, we first prove the global-in-time existence and uniqueness of the mild solutions in the frame of mixed time-space Besov space with multi-linear continuous mappings. Then, we show the asymptotic behavior and regularizing-decay rate estimates of the solution to equations with power-law nonlinearity by the method of multi-linear operator and the classical Hardy-Littlewood-Sobolev inequality. [ABSTRACT FROM AUTHOR]

Consider the non-stationary Navier-Stokes equation in the upper half space of R n. We prove the decay estimates of the strong solution and its derivatives in the setting of Besov and Triebel–Lizorkin spaces. This is the first time that the decay estimates of solutions to the non-stationary Navier-Stokes equations on Besov and Triebel-Lizorkin spaces are established. Our results not only extend the known results from Hardy spaces to Besov and Triebel-Lizorkin spaces but also imply new estimates on Sobolev spaces and give better decay estimates on Hardy spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special approach to the investigation of a mixed boundary value problem (BVP) for the Laplace equation in the case of a three-dimensional bounded domain Ω ⊂ ℝ 3 , when the boundary surface S = ∂ ⁡ Ω is divided into two disjoint parts S D and S N where the Dirichlet—Neumann-type boundary conditions are prescribed, respectively. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with the densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L 2 -based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space W 2 1 ⁢ (Ω) . Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that it is invertible in the L p -based Besov spaces, which under appropriate boundary data implies C α -Hölder continuity of the solution to the mixed BVP in the closed domain Ω ¯ with α = 1 2 - ε , where ε > 0 is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

We extend Barker's weak-strong uniqueness results for the Navier-Stokes equations and consider a criterion involving Besov spaces and weighted Lebesgue spaces. [ABSTRACT FROM AUTHOR]

A family of Besov-Morrey type spaces in the open unit disc are introduced in this paper. The boundedness of the embedding from Besov-Morrey type spaces to a class tent spaces is studied. As an application, the boundedness, compactness and essential norm of the Volterra integral operator from Besov-Morrey type spaces to a general function space are investigated. [ABSTRACT FROM AUTHOR]

This paper studies the existence and uniqueness of local weak solutions to the d-dimensional (d ≥ 2 ) fractional micropolar Rayleigh-Bénard convection system without thermal diffusion. When the fractional dissipation index 1 ≤ α < 1 + d 4 , any initial data (u 0 , ω 0) ∈ B 2 , 1 1 + d 2 − 2 α (R d) and θ 0 ∈ B 2 , 1 1 + d 2 − α (R d) yield a local unique weak solution. [ABSTRACT FROM AUTHOR]