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The main goals of this paper are: (i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Radon and non-negative. (ii) To employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like

In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.

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.

Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two such spaces, is there an embedding between the two? A decomposition space We establish readily verifiable criteria which ensure the existence of a continuous inclusion (“an embedding”) In a nutshell, in order to apply the embedding results presented in this article, no knowledge of Fourier analysis is required; instead, one only has to study the geometric properties of the involved coverings, so that one can decide the finiteness of certain sequence space norms defined in terms of the coverings. These sufficient criteria are quite sharp: For almost arbitrary coverings and certain ranges of We also prove a The resulting embedding theory is illustrated by applications to

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.

Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let In another article to appear, we will prove that when

In this paper we investigate a class of nonlinear degenerate parabolic equations involving heterogeneous (p,q)-Laplacian operators and subject to Dirichlet boundary conditions. These equations model complex diffusion phenomena with mixed-phase behavior in heterogeneous media. Our aim is to establish existence and uniqueness results for weak solutions under minimal regularity assumptions on the source term f, without requiring any control at infinity. The main difficulties stem from the degeneracy of the operator, the non-standard (p,q)-growth conditions, and the discontinuity of material phases. To overcome these challenges, we develop a variational framework based on Orlicz–Sobolev space theory and employ a generalized version of the Minty–Browder theorem to ensure the surjectivity of the nonlinear operator. Our approach yields new energy estimates, compactness results in non-reflexive settings, and stability under L∞-perturbations of the data. This work provides a rigorous mathematical foundation for analyzing nonlinear diffusion problems in complex and irregular environments.

In this work, we consider refined geometric characterizations of mappings that generate bounded ( p , q )-composition operators on Sobolev spaces. In the case n − 1 < q ≤ p < ∞, geometric characterizations in terms of metric and measure distortions are given with detailed proofs.

This article delves into the exploration of weak solutions’ existence within the context of obstacle problems associated with the following inequality: ∫ Ω V ( z , w , D w ) : D ( v - w ) d z ≥ g , v - w , where v lies in a convex set F O , B . The primary methodology employed in this investigation involves the utilization of Young’s measure theory, complemented by a theorem originating from Kinderlehrer and Stampacchia, specifically tailored for reflexive Orlicz-Sobolev spaces.

In this paper, we study the existence of traces for Sobolev spaces on the hyperbolic filling X of a compact metric space Z equipped with a doubling measure. Given a suitable metric on X , we can regard Z as the boundary of X . After equipping X with a weighted measure μ ρ via the measure on Z and the Euclidean arc length, we give characterizations for the existence of traces for first-order Sobolev spaces.