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In this article we study the boundedness of classical singular integral operator over local fields with rough kernel. We are relaxing the smoothness condition on kernel by block spaces and found the boundedness of truncated singular integral operator on different function spaces such as Lebesgue spaces, Besov spaces and Triebel–Lizorkin spaces. Unlike Euclidean space, our bound of truncated singular integral operator depends on the constant which leads the unboundedness of singular integral operator over local fields. [ABSTRACT FROM AUTHOR]

In the current study, we define Besov‐type spaces related to the Lebedev‐Skalskaya (LS‐) transform. We obtain Parseval's relation for continuous LS‐wavelet transform in L2(ℝ+;dx)$$ {L}^2\left({\mathrm{\mathbb{R}}}^{+}; dx\right) $$ space, and then using it, we extend the notion of continuous LS‐wavelet transform on Lp(ℝ+;dx)$$ {L}^p\left({\mathrm{\mathbb{R}}}^{+}; dx\right) $$ space and, in the end, derive continuity of LS‐wavelet transform on Besov‐type spaces and characterize the Besov LS‐space using LS‐wavelet coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the Cauchy problem for the Fornberg–Whitham equation is not locally well-posed in B p , r s (R) with (s , p , r) ∈ (1 , 1 + 1 p) × [ 2 , ∞) × [ 1 , ∞ ] or (s , p , r) ∈ { 1 } × [ 2 , ∞) × [ 1 , 2 ] by showing norm inflation phenomena of the solution for some special initial data. [ABSTRACT FROM AUTHOR]

For the famous Camassa-Holm equation, the well-posedness in C ([ 0 , T ] ; B p , 1 1 + 1 p (R)) with 1 ≤ p ≤ 2 and the ill-posedness in C ([ 0 , T ] ; B ∞ , 1 1 (R)) had been studied in [15,16,23]. That is to say, it left an open problem in the critical case C ([ 0 , T ] ; B p , 1 1 + 1 p (R)) with 2 < p < + ∞ proposed by Danchin in [15,16]. In this paper, we solve this problem and obtain the local well-posedness for the Camassa-Holm equation in critical Besov spaces C ([ 0 , T ] ; B p , 1 1 + 1 p (R)) with 1 ≤ p < + ∞. It is worth mentioning that our method is suitable for many Camassa-Holm type equations, such as the Novikov equation and the two-component Camassa-Holm system, and can also improve their index of the local well-posedness. [ABSTRACT FROM AUTHOR]

The research presented in this paper is a continuation of the recent studies of new classes of integral operators on the unit disc 픻, which are called Hausdorff–Berezin operators. We consider general operators of Hausdorff–Berezin type constructed with an arbitrary positive Radon measure on the unit disc within the framework of weighted spaces with the so‐called Möbius weights. This is a fairly wide class of weights in the unit disc which includes classical radial weights with singularities or zeroes at the boundary. Sufficient conditions for the boundedness of the Hausdorff–Berezin operators in these spaces are obtained. In the case of nonnegative kernels, a boundedness criterion is obtained in Lp(픻). Approximation properties of operators of Hausdorff–Berezin type are also studied. We also discuss some applications. [ABSTRACT FROM AUTHOR]

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 L p (R + n) × L p (R n − 1). [ABSTRACT FROM AUTHOR]

In this paper, we analyze the convergence problems of function g of Fourier series in Besov and generalized Zygmund norms using generalized Nörlund-Matrix ( N p , q A ) means of Fourier series. Convergence results are also compared by means of applications. [ABSTRACT FROM AUTHOR]

For a solution u to the Navier-Stokes equations in spatial dimension n ≥ 3 which blows up at a finite time T > 0, we prove the blowup estimate || u(t)||. Bsp+εp,q (Rn) ≥,ε,(pv qv 2) (T-t)-ε/2 for all ε Є [1, 2) and p, q Є [1, n/2-ε), where sp := 1 + n/p is the scaling-critical regularity, is the cutoff function used to define the Littlewood--Paley projections. For ε = 2, we prove the same type of estimate but only for q = 1: || u(t)||. Bsp+εp,q (Rn) ≥,ε,(pv qv 2)) (T-t)-1 for all p Є [1,∞). Under the additional restriction that p, q Є [1, 2] and n = 3, these blowup estimates are implied by those first proved by Robinson, Sadowski, and Silva [J. Math. Phys., 53 (2012), 115618] for p = q = 2 in the case ε Є (1, 2), and by McCormick et al. [SIAM J. Math. Anal., 48 (2016), pp. 2119--2132] for p = 2 in the cases (ε, q) =(1, 2) and (ε, q) =(2, 1). [ABSTRACT FROM AUTHOR]

An Euler type hyperbolic-parabolic system of chemotactic aggregation describing vascular network formation is investigated in the critical regularity setting. For initial data near a constant equilibrium state, the global well-posedness of the classical solution to the Cauchy problem with general pressure laws is proved in critical hybrid Besov spaces, and qualitative regularity estimates uniform with respect to the relaxation parameter are established. Then, the optimal timedecay rates of the global solution are analyzed under an additional regularity assumption on the initial data. Furthermore, the relaxation limit of the hyperbolic-parabolic system toward a parabolic-elliptic Keller-Segel model is justified rigorously. It is shown that as the relaxation parameter tends to zero, the solutions of the hyperbolic-parabolic chemotaxis system converge to the solutions of the Keller--Segel model at an explicit rate of convergence. Our approach relies on the introduction of new effective unknowns in low frequencies and the construction of a Lyapunov functional in the spirit of that of Beauchard and Zuazua [Arch. Ration. Mech. Anal., 199 (2011), pp. 177-227] to treat the high frequencies. [ABSTRACT FROM AUTHOR]

In the present paper, we define Besov type spaces associated with the Kontorovich–Lebedev transform. We widen the concept of continuous Kontorovich–Lebedev wavelet transform on L p ( R + ; x − 1 d x) space and derive continuity of Kontorovich–Lebedev wavelet transform on Besov type spaces and lastly characterize the Besov Kontorovich–Lebedev space by using Kontorovich–Lebedev wavelet coefficients. [ABSTRACT FROM AUTHOR]

Measuring and testing the dependency between multiple random functions is often an important task in functional data analysis. In the literature, a model-based method relies on a model which is subject to the risk of model misspecification, while a model-free method only provides a correlation measure which is inadequate to test independence. In this paper, we adopt the Hilbert–Schmidt Independence Criterion (HSIC) to measure the dependency between two random functions. We develop a two-step procedure by first pre-smoothing each function based on its discrete and noisy measurements and then applying the HSIC to recovered functions. To ensure the compatibility between the two steps such that the effect of the pre-smoothing error on the subsequent HSIC is asymptotically negligible when the data are densely measured, we propose a new wavelet thresholding method for pre-smoothing and to use Besov-norm-induced kernels for HSIC. We also provide the corresponding asymptotic analysis. The superior numerical performance of the proposed method over existing ones is demonstrated in a simulation study. Moreover, in a magnetoencephalography (MEG) data application, the functional connectivity patterns identified by the proposed method are more anatomically interpretable than those by existing methods. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to show that over a large class of bounded domains Ω⊂C², for 1

We study approximation classes for adaptive time-stepping finite element methods for time-dependent partial differential equations. We measure the approximation error in |$L_2([0,T)\times \varOmega)$| and consider the approximation with discontinuous finite elements in time and continuous finite elements in space, of any degree. As a by-product we define anisotropic Besov spaces for Banach-space-valued functions on an interval and derive some embeddings, as well as Jackson- and Whitney-type estimates. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the study of viscous compressible liquid-gas two-phase flow model in multi-dimensional spaces $ d\geq2 $. We investigate the global existence of strong solutions to the Cauchy problem in the $ L^p $ critical regularity framework. In comparison with previous results, the high-frequency of dissipative variable $ (\widetilde{P},\widetilde{u}) $ may be bounded in the critical space $ \dot{B}_{p,1}^{d/p}\times\dot{B}_{p,1}^{d/p-1}(p\geq2) $ and the high-frequency of non-dissipative variable $ \widetilde{\rho} $ to be bounded in the $ L^2 $-type Besov space $ \dot{B}_{2,1}^{d/2} $. Moreover, a Lyapunov-type energy argument can be developed, which leads to the time-decay estimates of solutions without additional smallness assumptions. [ABSTRACT FROM AUTHOR]

Considered herein is the Cauchy problem of the two-component Novikov system with peakons. Based on the local well-posedness results for this problem, it is shown that the solution map z 0 ↦ z (t) of this problem on the line is not uniformly continuous in Besov spaces (B 2 , r s (R)) 2 with s > 3 / 2 , 2 ≤ r ≤ ∞ through the method of approximate solutions. Next, in the periodic case, the non-uniform continuity of the solution map in Besov spaces (B p , r s (T)) 2 and Hölder spaces (C q (T)) 2 with s > max { 3 / 2 , 1 + 1 / p } , q > 3 / 2 , q ∈ R + ∖ N , 1 ≤ p , r ≤ ∞ is discussed. Finally, the Hölder continuity of this solution map in Sobolev spaces is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we mainly focus on the compressible viscoelastic flows of Oldroyd type with the general pressure law, with one of the non-Newtonian fluids exhibiting the elastic behavior. For the viscoelastic flows of Oldroyd type with the general pressure law, P ′ ( ρ ̄) + α > 0 , with α > 0 being the elasticity coefficient of the fluid, we prove the global existence and uniqueness of the strong solution in the critical Besov spaces when the initial data u ⃗ 0 and the low frequency part of ρ0, τ0 are small enough compared to the viscosity coefficients. In particular, when the viscosity is large, the part of the initial data can be large. The proof we display here does not need any compatible conditions. In addition, we also obtain the optimal decay rates of the solution in the Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove Noetherian solvability and obtain the index formula for a singular integral equation with a Carleman shift in Besov spaces. We also obtain the existence of the solution of the singular integral equation as well as their explicit representations in Besov spaces. [ABSTRACT FROM AUTHOR]

Let H = −Δ + ∣x∣2 be a Hermite operator, where Δ is the Laplacian on ℝn. In this paper, we give a new characterization of the Besov space associated with Hermite operator H by the generalized derivatives. As an application, we get some regularities for the Hermite operator H. [ABSTRACT FROM AUTHOR]

We consider a Keller–Segel model coupled to the incompressible Navier–Stokes system in 3-dimensional case. We prove that the system has a unique local solution when (u0, n0, c0) ∈ Φ011 × Φ012 × Φ013, where Φ011 × Φ012 × Φ013 is a subspace of bmo − 1 (R 3) × B ˙ p , ∞ − 2 + 3 / p (R 3) × ( B ˙ q , ∞ 3 / q (R 3) ∩ L ∞ (R 3)) . Furthermore, we obtain that the system exists a unique global solution for any small initial data (u 0 , n 0 , c 0) ∈ BMO − 1 (R 3) × B ˙ p , ∞ − 2 + 3 / p (R 3) × ( B ˙ q , ∞ 3 / q (R 3) ∩ L ∞ (R 3)) . For the difference between these spaces and known ones, our results may be regarded as a new existence theorem on the system. [ABSTRACT FROM AUTHOR]

This article develops an approach to unique, strong periodic solutions to quasilinear evolution equations by means of the classical Da Prato–Grisvard theorem on maximal Lp$L^p$‐regularity in real interpolation spaces. The method is used to show that quasilinear Keller–Segel systems admit a unique, strong T$T$‐periodic solution in a neighborhood of 0 provided the external forces are T$T$‐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. [ABSTRACT FROM AUTHOR]

In this paper we study the transport equation in generalized Campanato spaces Ls The critical case is particularly interesting, and is applied to the local well-posedness problem for the incompressible Euler equations in a space close to the Lipschitz space in our companion paper [Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 2, 201-241]. In the critical case s Dq DN D1, we have the embeddings B1 1;1.Rn/,!L1 1.p;1/.Rn/,!C0;1.Rn/, where B11; 1.Rn/and C0;1.Rn/are the Besov and Lipschitz spaces, respectively. For f0 2L1 1.p;1/.Rn/, v 2L1.0;T IL1 1.p;1/.Rn///and g 2L1.0;T IL1 1.p;1/.Rn///, we prove the existence and uniqueness of solutions to the transport equation in L1.0; T IL1 1.p;1/.Rn//such that Similar results for the other cases are also proved. [ABSTRACT FROM AUTHOR]

This current paper provides a data-driven wavelet estimator for deconvolution density model. Moreover, we investigate the totally adaptive estimations with moderately ill-posed noises over L p risk on Besov spaces B r , q s (R) . Compared with the traditional adaptive wavelet estimators, the estimation for the case of 0 < s ≤ 1 r is considered. On the other hand, the convergence rate in the region of 1 ≤ p ≤ 2 s r + (2 β + 1) r s r + 2 β + 1 is improved than that for not necessarily compactly supported density estimations. [ABSTRACT FROM AUTHOR]

By constructing a series of perturbation functions through localization in the Fourier domain and using a symmetric form of the system, we show that the data-to-solution map for the Euler–Poincaré equations is nowhere uniformly continuous in B p , r s (R d) with s > max { 1 + d 2 , 3 2 } and (p , r) ∈ (1 , ∞) × [ 1 , ∞) . This improves our previous result (Li et al. in Nonlinear Anal RWA 63:103420, 2022) which shows the data-to-solution map for the Euler–Poincaré equations is non-uniformly continuous on a bounded subset of B p , r s (R d) near the origin. [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. [ABSTRACT FROM AUTHOR]

Consider the following time-dependent stable-like operator with drift: L t φ (x) = ∫ R d [ φ (x + z) − φ (x) − z (α) ⋅ ∇ φ (x) ] σ (t , x , z) ν α (d z) + b (t , x) ⋅ ∇ φ (x) , where d ⩾ 1 , ν α is an α -stable type Lévy measure with α ∈ (0 , 1 ] and z (α) = 1 α = 1 1 | z | ⩽ 1 z , σ is a real-valued Borel function on R + × R d × R d and b is an R d -valued Borel function on R + × R d. By using the Littlewood-Paley theory, we establish the well-posedness for the martingale problem associated with L t under the sharp balance condition α + β ⩾ 1 , where β is the Hölder index of b with respect to x. Moreover, we also study a class of stochastic differential equations driven by Markov processes with generators of the form L t. We prove the pathwise uniqueness of strong solutions for such equations when the coefficients are in certain Besov spaces. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem of the Navier–Stokes equations in Rn$\mathbb {R}^n$ (n≥3)$n\ge 3)$ 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 n=3$n=3$. 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. [ABSTRACT FROM AUTHOR]

In 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 a,A$a, A$, and F$\mathbf {F}$. Moreover, we would like to point out that the results in the present work improve the known results for such parabolic equations. [ABSTRACT FROM AUTHOR]

Frisch-Parisi conjecture claims the existence of Baire function spaces in which Baire typical functions share the same multifractal behavior, prescribed in advance, and obey a multifractal formalism. In this paper, we introduce a family B of heterogeneous Besov spaces, which generalize the standard Besov spaces - they are obtained by replacing the Lebesgue measure (which plays a key role in the definition of the standard Besov spaces) by multifractal Radon measures belonging to some class constructed in the companion paper [1]. We find a characterization of the elements of B in terms of wavelet coefficients, and then describe the multifractal properties (singularity spectrum, validity of the multifractal formalism) of their Baire typical functions. This allows us to fully solve the Frisch-Parisi conjecture inside B . [ABSTRACT FROM AUTHOR]

In this work and its companion [1] , we construct Baire function spaces in which typical elements share the same prescribed multifractal behavior and obey a multifractal formalism, providing a solution to the so-called Frisch-Parisi conjecture for functions, an inverse problem raised by S. Jaffard. In this first part, a family E d of almost-doubling fully supported capacities on R d with prescribed singularity spectra is constructed. With each μ ∈ E d we associate a Baire function space B μ (R d) (a generalisation of Hölder-Zygmund spaces) in which typical functions share the same singularity spectrum as μ. This yields a partial solution to the conjecture. In [1] , we introduce and study a family B = { B q μ , p (R d) } μ ∈ E d , (p , q) ∈ [ 1 , + ∞ ] 2 of heterogeneous Besov spaces that contains { B μ (R d) } μ ∈ E d and generalises in a natural direction the family of standard Besov spaces, and we solve the inverse problem exhaustively inside B . [ABSTRACT FROM AUTHOR]

In this paper, we first investigate the instantaneous radius of space analyticity for the solutions of 3D Navier–Stokes system with initial data in the Besov spaces B ˙ p , q s (R 3) for p ∈ ] 1 , ∞ [ , q ∈ [ 1 , ∞ ] and s ∈ [ - 1 + 3 p , 3 p [. Then for initial data u 0 ∈ L p (R 3) with p in ]3, 6[, we prove that 3D Navier–Stokes system has a unique solution u = u L + v with and v ∈ L ~ T ∞ ( B ˙ p , p 2 1 - 3 p ) ∩ L ~ T 1 ( B ˙ p , p 2 3 - 3 p ) for some positive time T. Furthermore, we derive an explicit lower bound for the radius of space analyticity of v, which in particular extends the corresponding results in Hu and Zhang (Chin Ann Math Ser B 43:749–772, 2022) with initial data in L p (R 3) for p ∈ [ 3 , 18 / 5 [. [ABSTRACT FROM AUTHOR]

We present various results concerning the two-weight Hardy inequality on infinite trees. Our main aim is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization in terms of a reverse Hölder inequality for trace measures, and one based on the well-known Muckenhoupt-Wheeden-Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so-called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so-called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy inequality, characterize the compactness of the Hardy operator, provide a list of open problems, and suggest some possible lines of future research. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Cauchy problem for a two-component Camassa-Holm system with high order nonlinearity, which is a multi-component extension of the Fokas-Olver-Rosenau-Qiao equation. Firstly, we state the local well-posedness for the Cauchy problem of the system in the framework of Sobolev-Besov spaces. Then, we establish the precise blow-up mechanism for the strong solutions by means of the transport equation theory. As is well-known, the H 1 -norm conservation law of the velocity component is crucial to study blow-up phenomena of the single Camassa-Holm type equation. However, it is no longer available for our nonlinear coupled system. To overcome this difficulty, we derive some suitable conservation laws by sufficiently exploiting the fine structure of the system. Based on which, we finally construct several new blow-up strong solutions with certain initial profiles in finite time. [ABSTRACT FROM AUTHOR]

We study the diffusive relaxation limit of the Jin-Xin system toward viscous conservation laws in the multi-dimensional setting. For initial data being small perturbations of a constant state in suitable homogeneous Besov norms, we prove the global well-posedness of strong solutions satisfying uniform estimates with respect to the relaxation parameter. Then, we justify the strong relaxation limit and exhibit an explicit convergence rate of the process. Our proof is based on an adaptation of the techniques developed in [12,13] to be able to deal with additional low-order nonlinear terms. [ABSTRACT FROM AUTHOR]

We consider the well‐posedness of the second‐order degenerate differential equations (Mu′)′(t)+Λu′(t)+∫−∞ta(t−s)u′(s)ds=Au(t)+∫−∞tb(t−s)Bu(s)ds+f(t)(P)$$\begin{eqnarray*} &&\hspace*{18pc} (Mu^{\prime })^{\prime }(t) +\Lambda u^{\prime }(t)+\int _{-\infty }^t a(t-s)u^{\prime }(s)ds\\ &&\hspace*{18pc}\quad = Au(t)+ \int _{-\infty }^t b(t-s) Bu(s) ds + f(t)\qquad\qquad\qquad \mathrm{{(P)}} \end{eqnarray*}$$with infinite delay on [0, 2π] in Lebesgue–Bochner spaces Lp(T;X)$L^p(\mathbb {T}; X)$ and periodic Besov spaces Bp,qs(T;X)$B_{p,q}^s (\mathbb {T}; X)$, where A,B,Λ$A,\ B,\ \Lambda$, and M are closed linear operators in a Banach space X satisfying D(A)∩D(B)⊂D(M)∩D(Λ)$D(A)\cap D(B)\subset D(M) \cap D(\Lambda)$ and the kernels a,b∈L1(R+)$ a, b\in L^1(\mathbb {R}_+)$. Using known operator‐valued Fourier multiplier theorems, we are able to give necessary and sufficient conditions for the well‐posedness of (P) in Lp(T;X)$L^p(\mathbb {T}; X)$ and Bp,qs$B_{p,q}^s$(T;X)$(\mathbb {T}; X)$. These results are applied to examine some concrete examples. [ABSTRACT FROM AUTHOR]

We verify that the p-integrable Teichmüller space Tp admits the canonical complex Banach manifold structure for any p ≽ 1. Moreover, we characterize a quasisymmetric homeomorphism corresponding to an element of Tp in terms of the p-Besov space for any p > 1. [ABSTRACT FROM AUTHOR]

We extend the Hairer reconstruction theorem for distributions due to Caravenna and Zambotti [EMS Surv. Math. Sci. 7 (2020), 207–251] to general function spaces satisfying a translation and scaling condition. This includes Besov type spaces with exponents below 1 and Triebel–Lizorkin type spaces. [ABSTRACT FROM AUTHOR]

We provide a convenient framework for the study of the well-posedness of a variety of abstract (integro)differential equations in general Banach function spaces. It allows us to extend and complement the known theory on the maximal regularity of such equations. More precisely, by methods of harmonic analysis, we identify large classes of Banach spaces which are invariant with respect to distributional Fourier multipliers. Such classes include general vector-valued Banach function spaces Φ and/or the scales of Besov and Triebel–Lizorkin spaces defined by Φ . We apply this result to the study of the well-posedness and maximal regularity property of abstract second-order integrodifferential equations, which model various types of elliptic and parabolic problems arising in different areas of applied mathematics. [ABSTRACT FROM AUTHOR]

This paper studies a two-component b-family system, which includes the two-component Camassa-Holm system and the two-component Degasperis-Procesi system as special case. It is shown that the solution map of this system is not uniformly continuous on the initial data in Besov spaces B p , r s - 1 (R) × B p , r s (R) with s > max { 1 + 1 p , 3 2 } , 1 ≤ p , r < ∞ . Our result covers and extends the previous non-uniform continuity in Sobolev spaces H s - 1 (R) × H s (R) for s > 5 2 to Besov spaces (Nonlinear Anal., 2014, 111: 1-14). Compared with the generalized rotation b-family system considered by Holmes et al. (Z. Angew. Math. Mech., 2021), our non-uniform continuity is established in a broader range of Besov spaces. [ABSTRACT FROM AUTHOR]

We introduce and study new modules and spaces of generalized functions that are related to the classical Besov spaces. Various Schwartz distribution spaces are naturally embedded into our new generalized function spaces. We obtain precise criteria for detecting Besov regularity of distributions. [ABSTRACT FROM AUTHOR]

In this paper, we establish the block decomposition for Herz spaces associated with ball Banach function spaces. By using this decomposition, we obtain the boundedness for a class of sublinear operators on Herz spaces associated with ball Banach function spaces. [ABSTRACT FROM AUTHOR]

This paper investigates the nonparametric linear wavelet-based estimators of multivariate regression functions. Under mild conditions, we establish the asymptotic normality under the weak dependence, which incorporates mixing and association concepts. This framework applies to numerous classes of intriguing statistical processes, primarily Gaussian sequences and, more generally, Bernoulli shifts. We give an application for the confidence interval. [ABSTRACT FROM AUTHOR]

This paper is concerned with proving some embeddings of the form As p1,q1 (Rn, | · |α) · Ar p2,q2 (Rn, | · |α),! As p,q1 (Rn, | · |α). The different embeddings obtained here are under certain restrictions on the parameters. Here As*** p,q(Rn, | · |α) stands for either the Besov space Bs p,q(Rn, | · |α) or the Triebel-Lizorkin space Fs p,q(Rn, | · |α). These spaces unify and generalize the classical Lebesgue spaces of power weights, Sobolev spaces of power weights, Besov spaces and Triebel-Lizorkin spaces. Almost all our assumptions on s, r, p1, p2, q1 and q2 are necessary. An application to the continuity of pseudodifferential operators with non-regular symbols on Triebel-Lizorkin spaces of power weights is given. [ABSTRACT FROM AUTHOR]

The paper deals with the problem under which conditions for the parameters s 1 , s 2 ∈ R , 1 ≤ p , q 1 , q 2 ≤ ∞ the Fourier transform F is a nuclear mapping from A p , q 1 s 1 (R n) into A p , q 2 s 2 (R n) , where A ∈ { B , F } stands for a space of Besov or Triebel–Lizorkin type, and n ∈ N . It extends the recent paper ‘Mapping properties of Fourier transforms’ (Triebel in Z Anal Anwend 41(1/2):133–152, , 2022) by the third-named author, where the compactness of F acting in the same type of spaces was studied. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of the d‐dimensional magneto‐micropolar equations (d=2$$ d=2 $$ or d=3$$ d=3 $$) with general fractional dissipation. The aim of this paper is to obtain the existence and uniqueness of solutions in the weakest possible inhomogeneous Besov spaces. Using the technical tools of Litttlewood–Paley decomposition and Besov spaces theory, we obtain the local existence in the functional setting of inhomogeneous Besov spaces. Furthermore, such solutions are unique only in 2D case. [ABSTRACT FROM AUTHOR]

We consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space Rn$\mathbb {R}^n$ for n≥3$n \ge 3$. It is clarified that if w is small in Ḃp*,q′−1+np*$\dot{B}^{-1+\frac{n}{p_\ast }}_{p_\ast , q^{\prime }}$ for 1≤p*0$s > 0$. [ABSTRACT FROM AUTHOR]

We here aim at proving the global existence and uniqueness of solutions to the inhomogeneous incompressible Navier-Stokes system in the case where the initial density ρ 0 is discontinuous and the initial velocity u 0 has critical regularity. Assuming that ρ 0 is close to a positive constant, we obtain global existence and uniqueness in the two-dimensional case whenever the initial velocity u 0 belongs to the critical homogeneous Besov space B ˙ p , 1 - 1 + 2 / p (R 2) (1 < p < 2) and, in the three-dimensional case, if u 0 is small in B ˙ p , 1 - 1 + 3 / p (R 3) (1 < p < 3) . Next, still in a critical functional framework, we establish a uniqueness statement that is valid in the case of large variations of density with, possibly, vacuum. Interestingly, our result implies that the Fujita-Kato type solutions constructed by Zhang (Adv Math 363:107007, 2020) are unique. Our work relies on interpolation results, time weighted estimates and maximal regularity estimates in Lorentz spaces (with respect to the time variable) for the evolutionary Stokes system. [ABSTRACT FROM AUTHOR]

We obtain new equivalent quasinorms of the Besov spaces of variable smoothness and integrability. Our main tools are the continuous version of the Calderón reproducing formula, maximal inequalities, and the variable-exponent technique. However, allowing the parameters to vary from point to point leads to additional difficulties which, in general, can be overcome by imposing regularity assumptions on these exponents. [ABSTRACT FROM AUTHOR]

Considered herein is the Cauchy problem of the two-component Novikov system. In the periodic case, we first constructed an approximate solution sequence that possesses the nonuniform dependence property; then, by applying the energy methods, we managed to prove that the difference between the approximate and actual solution is negligible, thus succeeding in proving the nonuniform dependence result in both supercritical Besov spaces B p , r s (T) × B p , r s (T) with s > max { 3 2 , 1 + 1 p } , 1 ≤ p ≤ ∞ , 1 ≤ r < ∞ and critical Besov space B 2 , 1 3 2 (T) × B 2 , 1 3 2 (T) . In the non-periodic case, we constructed two sequences of initial data with high and low-frequency terms by analyzing the inner structure of the system under investigation in detail, and we proved that the distance between the two corresponding solution sequences is lower-bounded by time t, but converges to zero at initial time. This implies that the solution map is not uniformly continuous both in supercritical Besov spaces B p , r s (R) × B p , r s (R) with s > max { 3 2 , 1 + 1 p } , 1 ≤ p ≤ ∞ , 1 ≤ r < ∞ and critical Besov spaces B p , 1 1 + 1 p (R) × B p , 1 1 + 1 p (R) with 1 ≤ p ≤ 2 . The proof of nonuniform dependence is based on approximate solutions and Littlewood–Paley decomposition theory. These approaches are widely applicable in the study of continuous properties for shallow water equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of the multidimensional compressible Navier--Stokes--Euler system for two-phase flow motion, which consists of the isentropic compressible Navier--Stokes equations and the isothermal compressible Euler equations coupled with each other through a relaxation drag force. We first establish the local existence and uniqueness of the strong solution for general initial data in a critical homogeneous Besov space, and then prove the global existence of the solution if the initial data are a small perturbation of the equilibrium state. Moreover, under the additional condition that the low-frequency part of the initial perturbation also belongs to another Besov space with lower regularity, we obtain the optimal time-decay rates of the global solution toward the equilibrium state. These results imply that the relaxation drag force and the viscosity dissipation affect the regularity properties and long time behaviors of solutions for the compressible Navier--Stokes--Euler system. [ABSTRACT FROM AUTHOR]