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We introduce a decoupling method on the Wiener space to define a wide class of anisotropic Besov spaces. The decoupling method is based on a general distributional approach and not restricted to the Wiener space. The class of Besov spaces we introduce contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that characterize quantitative properties of directional derivatives in the Malliavin sense without computing or accessing these Malliavin derivatives explicitly. Regarding BSDEs, we deduce regularity properties of the solution processes from the Besov regularity of the initial data, in particular upper bounds for their Among other tools, we use methods from harmonic analysis. As a by-product, we improve the asymptotic behaviour of the multiplicative constant in a generalized Fefferman inequality and verify the optimality of the bound we established.

This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable (1) Mapping properties for the double and single layer potentials, as well as the Newton potential; (2) Extrapolation-type solvability results: the fact that solvability of the Dirichlet or Neumann boundary value problem at any given (3) Well-posedness for the non-homogeneous boundary value problems. In particular, we prove well-posedness of the non-homogeneous Dirichlet problem with data in Besov spaces for operators with real, not necessarily symmetric coefficients.

This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative d-torus \\mathbb{T}^d_\\theta (with \\theta a skew symmetric real d\\times d-matrix). These spaces share many properties with their classical counterparts. The authors prove, among other basic properties, the lifting theorem for all these spaces and a Poincar type inequality for Sobolev spaces.

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.

We provide a full characterization in terms of the six parameters involved the boundedness of all standard weighted integral operators induced by harmonic Bergman-Besov kernels acting between different Lebesgue classes with standard weights on the unit ball of ℝⁿ. These operators in some sense generalize the harmonic Bergman-Besov projections. To obtain the necessity conditions, we use a technique that heavily depends on the precise inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball. This fruitful technique is new. It has been used first with holomorphic Bergman-Besov kernels by Kaptanoğlu and Üreyen. Methods of the sufficiency proofs we employ are Schur tests or Hölder or Minkowski type inequalities which also make use of estimates of Forelli-Rudin type integrals.

In the recent years, the so-called Morrey smoothness spaces attracted a lot of interest. They can (also) be understood as generalisations of the classical spaces A (sk p,q/s }(ℝ n ) with A ∈{ B,F } in ℝ n , where the parameters satisfy s ∈ ℝ (smoothness), 0 < p ⩽ ∞ (integrability) and 0 < q ⩽ ∞ (summability). In the case of Morrey smoothness spaces, additional parameters are involved. In our opinion, among the various approaches at least two scales enjoy special attention, also in view of applications: the scales A u , p , q s ( R n ) with A ∈ { N , E } and u ⩾ p , and A p,q s,τ (ℝ n ) with A ∈ { B, F } and τ ⩾ 0. We reorganise these two prominent types of Morrey smoothness spaces by adding to ( s,p, q ) the so-called slope parameter ϱ , preferably (but not exclusively) with − n ⩽ ϱ < 0. It comes out that ∣ ϱ ∣ replaces n , and min(∣ ϱ ∣, 1) replaces 1 in slopes of (broken) lines in the ( 1 p , s )-diagram characterising distinguished properties of the spaces A p,q s (ℝ n ) and their Morrey counterparts. Special attention will be paid to low-slope spaces with −1 < ϱ < 0, where the corresponding properties are quite often independent of n ∈ ℕ. Our aim is two-fold. On the one hand, we reformulate some assertions already available in the literature (many of which are quite recent). On the other hand, we establish on this basis new properties, a few of which become visible only in the context of the offered new approach, governed, now, by the four parameters ( s, p, q, ϱ ).

Classical and nonclassical Besov and Triebel-Lizorkin spaces with complete range of indices are developed in the general setting of Dirichlet space with a doubling measure and local scale-invariant Poincaré inequality. This leads to a heat kernel with small time Gaussian bounds and Hölder continuity, which play a central role in this article. Frames with band limited elements of sub-exponential space localization are developed, and frame and heat kernel characterizations of Besov and Triebel-Lizorkin spaces are established. This theory, in particular, allows the development of Besov and Triebel-Lizorkin spaces and their frame and heat kernel characterization in the context of Lie groups, Riemannian manifolds, and other settings.

It is known, since the seminal work [T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998)], that the solution map associated to a controlled differential equation is locally Lipschitz continuous in qq-variation, resp., 1/q1/q-Hölder-type metrics on the space of rough paths, for any regularity 1/q∈(0,1]1/q \\in (0,1]. We extend this to a new class of Besov–Nikolskii-type metrics, with arbitrary regularity 1/q∈(0,1]1/q\\in (0,1] and integrability p∈[q,∞]p\\in [ q,\\infty ], where the case p∈{q,∞}p\\in \\{ q,\\infty \\} corresponds to the known cases. Interestingly, the result is obtained as a consequence of known qq-variation rough path estimates.

The main focus of this article is to investigate the generalized fractional drift-diffusion system with small initial data in Besov-Morrey spaces. Our goal is to establish the global well-posedness and asymptotic stability of mild solutions for this system. The results obtained in this study have broad applicability in the modeling of various types of fractional parabolic systems. In other words, the techniques developed here can be useful in studying similar types of systems in the future.