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A new wavelet-based estimator of the conditional density is investigated. The estimator is constructed by combining a special ratio technique and applying a non negative estimator to the density function in the denominator. We used a wavelet shrinkage technique to find an adaptive estimator for this problem. In particular, a block thresholding estimator is proposed, and we prove that it enjoys powerful mean integrated squared error properties over Besov balls. Moreover, it is shown that convergence rates for the mean integrated squared error (MISE) of the adaptive estimator are optimal under some mild assumptions. Finally, a numerical example has been considered to illustrate the performance of the estimator. [ABSTRACT FROM AUTHOR]

This paper addresses the adaptive wavelet estimations for density derivatives by using data-driven methods. Based on the classical linear wavelet estimator of density derivatives, we provide a point-wise estimation under the local Hölder condition firstly. Moreover, we introduce a data-driven wavelet estimator for adaptivity and prove a point-wise oracle inequality, which does not require any assumption on the underlying function. Finally, by using the point-wise oracle inequality, the point-wise estimation under the local Hölder condition and L p -risk ( 1 ≤ p < ∞ ) estimation on Besov spaces are investigated respectively. [ABSTRACT FROM AUTHOR]

We present a shift theorem for solutions of the Poisson equation in a finite planar cone (and hence also on plane polygons) for Dirichlet, Neumann, and mixed boundary conditions. The range in which the shift theorem holds depends on the angle of the cone. For the right endpoint of the range, the shift theorem is described in terms of Besov spaces rather than Sobolev spaces. [ABSTRACT FROM AUTHOR]

A nonlinear shallow water wave equation containing the famous Degasperis-Procesi and Fornberg-Whitham equations is investigated. The well-posedness of short time solutions is established to illustrate that the solution map of the equation is continuous in the critical Besov space B ∞ , 1 1 (R) . Using the methods to construct high and low frequency functions, we prove that the solution map of the equation is non-uniform continuous dependence on the initial data in B ∞ , 1 1 (R) . [ABSTRACT FROM AUTHOR]

Let Ω be a bounded non-smooth domain in R n that satisfies the measure density condition. In this paper, the authors study the interrelations of three basic types of Besov spaces B p , q s (Ω) , B ˚ p , q s (Ω) and B ~ p , q s (Ω) on Ω , which are defined, respectively, via the restriction, completion and supporting conditions with p , q ∈ [ 1 , ∞) and s ∈ (0 , 1) . The authors prove that B p , q s (Ω) = B ˚ p , q s (Ω) = B ~ p , q s (Ω) , if Ω supports a fractional Besov–Hardy inequality, where the latter is proved under certain conditions on fractional Besov capacity or Aikawa's dimension of the boundary of Ω . [ABSTRACT FROM AUTHOR]

This article considers wavelet estimations for density functions in Besov spaces based on negatively dependent biased random samples. More precisely, wavelet estimators are defined and their L p (1 ≤ p < ∞) convergence rates are provided when the parameter μ is known and unknown, respectively. [ABSTRACT FROM AUTHOR]

This paper establishes non-uniform continuity of the data-to-solution map in the periodic case for the two-component Fornberg-Whitham system in Besov spaces Bs p,r (T) x B s−1 p,r (T) for s > max{2 + 1 p, 5 2 }. In particular, when p = 2 and r = 2, this proves the non-uniform dependence on initial data for the system in Sobolev spaces Hs (T) × H s−1 (T) for s > 5/2 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 3D incompressible magnetohydrodynamic equations and establish several logarithmically improved regularity criteria in the homogeneous Besov space. These criteria are based on certain assumptions made in the velocity field and pressure, respectively. The first result solely based on the velocity field indicates that the velocity field plays a dominant role in determining the regularity of weak solutions, when compared with the magnetic field. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a class of Besov spaces via the heat semigroup on Carnot groups, and provide some basic properties about these spaces. Especially, we reveal the relationship between the Besov space and the known Sobolev space on Carnot groups. Then some Sobolev type inequalities for Besov spaces are established. Finally, we introduce the Besov capacity associated with Besov spaces and investigate some measure theories and geometric properties for this capacity. Furthermore, several applications of the Besov capacity are also obtained. [ABSTRACT FROM AUTHOR]

This paper establishes non-uniform continuity of the data-to-solution map in the periodic case for the two-component Fornberg-Whitham system in Besov spaces $ B^s_{p, r}(\mathbb{T}) \times B^{s-1}_{p, r}(\mathbb{T}) $ for $ s > \max\{2+\frac{1}{p}, \frac{5}{2}\} $. In particular, when $ p = 2 $ and $ r = 2 $, this proves the non-uniform dependence on initial data for the system in Sobolev spaces $ H^s(\mathbb{T})\times H^{s-1}(\mathbb{T}) $ for $ s > \frac{5}{2} $.

We study a class of almost diagonal matrices compatible with the mixed-norm α-modulation spaces Mp,qs,α(ℝn), α ∈ [0, 1], introduced recently by Cleanthous and Georgiadis. The class of almost diagonal matrices is shown to be closed under matrix multiplication and we connect the theory to the continuous case by identifying a suitable notion of molecules for the mixed-norm α-modulation spaces. It is shown that the “change of frame” matrix for a pair of time-frequency frames for the mixed-norm α-modulation consisting of suitable molecules is almost diagonal. As examples of applications, we use the almost diagonal matrices to construct compactly supported frames for the mixed-norm α-modulation spaces, and to obtain a straightforward boundedness result for Fourier multipliers on the mixed-norm α-modulation spaces. [ABSTRACT FROM AUTHOR]

Let k ≥ 1 , 0 ≤ α < d and M b , α k be the k-th order fractional maximal commutator. When α = 0 , we denote M b , α k = M b k . We show that M b , α k is bounded from the first order Sobolev space W 1 , p 1 (R d) to W 1 , p (R d) where 1 < p 1 , p 2 , p < ∞ , 1 / p = 1 / p 1 + k / p 2 - α / d . We also prove that if 0 < s < 1 , 1 < p 1 , p 2 , p , q < ∞ and 1 / p = 1 / p 1 + k / p 2 , then M b k is bounded and continuous from the fractional Sobolev space W s , p 1 (R d) to W s , p (R d) if b ∈ W s , p 2 (R d) , from the inhomogeneous Triebel–Lizorkin space F s p 1 , q (R d) to F s p , q (R d) if b ∈ F s p 2 , q (R d) and from the inhomogeneous Besov space B s p 1 , q (R d) to B s p , q (R d) if b ∈ B s p 2 , q (R d) . It should be pointed out that the main ingredients of proving the above results are some refined and complex difference estimates of higher order maximal commutators as well as some characterizations of the Sobolev spaces, Triebel–Lizorkin spaces and Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for the Camassa–Holm equation on the real line. By presenting a new construction of initial data, we show that the solution map in the smaller space B p , 1 1 ∩ C 0 , 1 with p ∈ (2 , ∞ ] is discontinuous at origin. More precisely, the initial data in B p , 1 1 ∩ C 0 , 1 can guarantee that the Camassa–Holm equation has a unique local solution in W 1 , p ∩ C 0 , 1 , however, this solution is instable and can have an inflation in B p , 1 1 ∩ C 0 , 1 . [ABSTRACT FROM AUTHOR]

The integral with respect to a multidimensional stochastic measure, assuming only its σ-additivity in probability, is studied. The continuity and differentiability of realizations of the integral are established. [ABSTRACT FROM AUTHOR]

The aim of this study was to establish a new regularity criterion for local-in-time smooth solutions for the 3D nematic liquid crystal flows in terms of the horizontal gradient of two horizontal velocity components in the framework of the homogeneous Besov Ḃ∞,∞−1 space and one directional derivative of molecular orientations in the framework of the homogeneous Morrey–Campanato Ṁ2,3 r space. More precisely, we show that the smooth solution (u,d) can be extended beyond T, provided that ∫0T(∥∇ hũ(⋅,t)∥Ḃ∞,∞−12 + ∥∂ 3d(⋅,t)∥Ṁ2,3 r2)dt < ∞. [ABSTRACT FROM AUTHOR]

In this article, via certain lower bound conditions on the measures under consideration, the authors fully characterize the Sobolev embeddings for the scales of Hajłasz–Triebel–Lizorkin and Hajłasz–Besov spaces in the general context of quasi-metric measure spaces for an optimal range of the smoothness parameter s. An interesting facet of this work is how the range of s for which the above characterizations of these embeddings hold is intimately linked (in a quantitative manner) to the geometric makeup of the underlying space. Importantly, although the main results in this article are stated in the context of quasi-metric spaces, the authors provide several examples illustrating how this range of s is strictly larger than similar ones currently appearing in the literature, even in the metric setting. Moreover, the authors relate these values of s to the (non)triviality of these function spaces. [ABSTRACT FROM AUTHOR]

We characterize the matrix-weighted Triebel–Lizorkin spaces and Besov spaces by Peetre maximal function and approximation. Using these characterizations, we obtain the boundedness of pseudo-differential operators with symbol in Hörmander's class on matrix weighted Besov and Triebel–Lizorkin spaces. [ABSTRACT FROM AUTHOR]

We investigate an integral operator T t , λ which preserves the Carleson measure for the Möbius invariant Besov space B p on the unit ball of C n . A holomorphic function space W β p , associated with the Carleson measure for B p , is introduced. As applications for the operator T t , λ , we estimate the distance from Bloch-type functions to the space W β p , which extends Jones' formula. Moreover, the bounded small Hankel operators on B p and the atomic decomposition of W β p are characterized. [ABSTRACT FROM AUTHOR]

Let ... and t, r, p ∈ (0, ∞) . In this paper, we introduce the slice Besov-type space ... and the slice Triebel-Lizorkin-type space ..., and establish their φ-transform characterizations in the sense of Frazier and Jawerth. The embedding properties, characterizations via the Peetre maximal function, the Lusin area function, smooth atomic and molecular decompositions of these spaces are also obtained. As applications, we obtain the boundedness on these spaces of Fourier multipliers with symbols satisfying some generalized Hörmander condition. [ABSTRACT FROM AUTHOR]

The integral with respect to a multidimensional stochastic measure, assuming only its σ-additivity in probability, is studied. The continuity and differentiability of realizations of the integral are established.

This article focuses on local and global existence and uniqueness for the strong solution to the Boussinesq system in Rn{{\mathbb{R}}}^{n} (n≥3n\ge 3) with full viscosity in Besov spaces. Under the hypotheses 1

Let p (·) , q (·) and α (·) be variable exponents satisfying some Hölder continuous conditions. In this paper, the authors characterize the variable inhomogeneous Besov space B p (·) , q (·) α (·) (R n) and the variable inhomogeneous Triebel-Lizorkin space F p (·) , q (·) α (·) (R n) in terms of Hajłasz gradient, via establishing the boundedness of a kind of almost diagonal operator A on the corresponding sequence spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Besov space and the Besov capacity and obtain several important capacitary inequalities in a strictly local Dirichlet space, which satisfies the doubling condition and the weak Bakry–Émery condition. It is worth noting that the capacitary inequalities in this paper are proved if the Dirichlet space supports the weak (1 , 2) -Poincaré inequality, which is weaker than the weak (1 , 1) -Poincaré inequality investigated in the previous references. Moreover, we first consider the strong subadditivity and its equality condition for the Besov capacity in metric space. [ABSTRACT FROM AUTHOR]

We introduce the critical Besov space B p on quasicircles and prove the solvability of the jump problem on d-regular quasicircles with B p boundary values for d < p < d d - 1 . As applications, we obtain the B p (1 < p < ∞) boundedness of the Cauchy integral operator and the Faber operator on chord-arc curves. [ABSTRACT FROM AUTHOR]

In this article, we establish a characterization of the set M (B p , ∞ 0 , b (R n)) of all pointwise multipliers of Besov spaces B p , ∞ 0 , b (R n) with only logarithmic smoothness b ∈ R in the special cases p = 1 and p = ∞ . As applications of these two characterizations, we clarify whether or not the three concrete examples, namely characteristic functions of open sets, continuous functions defined by differences, and the functions e i k · x with k ∈ Z n and x ∈ R n , are pointwise multipliers of B 1 , ∞ 0 , b (R n) and B ∞ , ∞ 0 , b (R n) , respectively; furthermore, we obtain the explicit estimates of ‖ e i k · x ‖ M (B 1 , ∞ 0 , b (R n)) and ‖ e i k · x ‖ M (B ∞ , ∞ 0 , b (R n)) . In the case where p ∈ (1 , ∞) , we give some sufficient conditions and some necessary conditions of the pointwise multipliers of B p , ∞ 0 , b (R n) and a complete characterization of M (B p , ∞ 0 , b (R n)) is still open. However, via a different method, we are still able to accurately calculate ‖ e i k · x ‖ M (B p , ∞ 0 , b (R n)) , k ∈ Z n , in this situation. The novelty of this article is that most of the proofs are constructive and these constructions strongly depend on the logarithmic structure of Besov spaces under consideration. [ABSTRACT FROM AUTHOR]

In this work, we are concerned with the main mechanism for possible blow-up criteria of smooth solutions to the 3D incompressible Boussinesq equations. The main results state that the finite-time blowup/global existence of smooth solutions to the Boussinesq equation is controlled by either of the criteria u h ∈ L 2 0 , T ; B ˙ ∞ , ∞ 0 (R 3) or ∇ h u h ∈ L 1 0 , T ; B ˙ ∞ , ∞ 0 R 3 , where u h and ∇ h denote the horizontal components of the velocity field and partial derivative with respect to the horizontal variables, respectively. We present a new simple proof for the regularity of this system without using the higher-order energy law and without any assumptions on the temperature θ. Our results extend the Navier–Stokes equations results in Dong and Zhang (Nonlinear Anal Real World Appl 11:2415–2421, 2010), Dong and Chen (J Math Anal Appl 338:1–10, 2008) and Gala and Ragusa (Electron J Qual Theory Differ Equ, 2016a) to Boussinesq equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a kind of singular integrals which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation ∂ t ⁡ θ + u ⋅ ∇ ⁡ θ + κ ⁢ Λ 2 ⁢ β ⁢ θ = 0 , (x , t) ∈ ℝ 2 × ℝ + , κ > 0 , where u = - ∇ ⊥ ⁡ Λ - 2 + 2 ⁢ α ⁢ θ , α ∈ [ 0 , 1 2 ] and β ∈ (0 , 1 ] . First, we give a relationship between this kind of singular integrals and Calderón–Zygmund singular integral operators and obtain a uniform Besov estimates. As an application, we give the well-posedness of the generalized 2D dissipative quasi-geostrophic (QG) in the critical Besov space. [ABSTRACT FROM AUTHOR]

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]

We consider multilinear pseudo-differential operators with symbols in the multilinear Hörmander class S 0 , 0 . The aim of this paper is to discuss the boundedness of these operators in the settings of Besov spaces. [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.

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]

This paper is devoted to the real analysis methods for the problem of construction of Markov processes with boundary conditions in probability. Analytically, a Markovian particle in a domain of Euclidean space is governed by an integro-differential operator, called Waldenfels operator, in the interior of the domain, and it obeys a boundary condition, called Ventcel' (Wentzell) boundary condition, on the boundary of the domain. Probabilistically, a Markovian particle moves both by continuous paths and by jumps in the state space and it obeys the Ventcel' boundary condition that consists of six terms corresponding to a diffusion along the boundary, an absorption phenomenon, a reflection phenomenon, a sticking (or viscosity) phenomenon and a jump phenomenon on the boundary and an inward jump phenomenon from the boundary. More precisely, we study a class of first order Ventcel' boundary value problems for second order elliptic Waldenfels integro-differential operators. By using the Calderón–Zygmund theory of singular integrals in real analysis, we prove existence and uniqueness theorems in the framework of Sobolev and Besov spaces, which extend earlier theorems due to Bony–Courrège–Priouret to the VMO (vanishing mean oscillation) case. Our proof is based on various maximum principles for second order elliptic Waldenfels operators with discontinuous coefficients in the framework of Sobolev spaces. [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]

In this paper, using inhomogeneous Calderón's reproducing formulas and the space of test functions associated with a para-accretive function, the inhomogeneous Besov and Triebel-Lizorkin spaces are established. As applications, pointwise multiplier theorems are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, on Lie groups of polynomial growth G, we prove the boundedness of multilinear spectral multipliers from the product of Besov spaces B p 1 , q 1 s 1 (G) × B p 2 , q 2 s 2 (G) × ⋯ × B p N , q N s N (G) to Lebesgue spaces L p (G) with p 1 , … , p N , q 1 , … , q N , p ⩾ 1 and s 1 , … , s N ∈ R . Then we prove the boundedness from the product of Triebel–Lizorkin spaces T p 1 , q 1 s 1 (G) × T p 2 , q 2 s 2 (G) × ⋯ × T p N , q N s N (G) to Lebesgue spaces L p (G) with p 1 , … , p N , q 1 , … , q N > 1 , p ⩾ 1 , s 1 , … , s N ∈ R . [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]

We show that the zero smoothness Besov space does not embed into the Lorentz space unless ; here . This answers in the negative a question posed by O. V. Besov. [ABSTRACT FROM AUTHOR]

We study the Riemann–Liouville space of fractional potentials on the half-line and establish its properties such as embeddings in Besov spaces, Liouville classes, and Lizorkin–Triebel spaces. [ABSTRACT FROM AUTHOR]

We discuss various definitions of Sobolev and Besov classes on infinite-dimensional spaces, give a survey of the results on coincidence of some of these classes, and obtain a number of new results. [ABSTRACT FROM AUTHOR]

In this paper, we establish the Schatten class and endpoint weak Schatten class estimates for the commutator of Riesz transforms on weighted L2 spaces. As an application a weighted version for the estimate of the quantised derivative introduced by Alain Connes and studied recently by Lord–McDonald–Sukochev–Zanin and Frank–Sukochev–Zanin is provided. [ABSTRACT FROM AUTHOR]

Our main purpose is to establish Gagliardo-Nirenberg-type inequalities using fractional homogeneous Sobolev spaces and homogeneous Besov spaces. In particular, we extend some of the results obtained by the authors in previous studies.

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 obtain precise estimates, in terms of the measure of balls, for the Besov capacity of annuli and singletons in complete metric spaces. The spaces are only assumed to be uniformly perfect with respect to the centre of the annuli and equipped with a doubling measure. [ABSTRACT FROM AUTHOR]