Related papers: The inclusion relation between Sobolev and modulat…
The aim of this work is to study the first order Dirac-Sobolev spaces in $L^p$ norm on an open subset of ${\mathbb R}^3$ to clarify its relationship with the corresponding Sobolev spaces. It is shown that for $1< p <\infty$, they coincide,…
We study higher complex Sobolev spaces and their corresponding functional capacities. In particular, we prove the Moser-Trudinger inequality for these spaces and discuss some relationships between these spaces and the complex…
We prove a characterization of the Sobolev spaces $H^\alpha$ on the unit sphere $\mathbb{S}^{d-1}$, where the smoothness index $\alpha$ is any positive real number and $d\geq 2$. This characterization does not use differentiation and it is…
We obtain new multilinear multiplier theorems for symbols of restricted smoothness which lie locally in certain Sobolev spaces. We provide applications concerning the boundedness of the commutators of Calder\'on and…
We consider the situation when an elliptic problem in a subdomain $\Omega_1$ of an $n$-dimensional bounded domain $\Omega$ is coupled via inhomogeneous canonical transmission conditions to a parabolic problem in $\Omega\setminus\Omega_1$.…
This article pertains to interpolation of Sobolev functions at shrinking lattices $h\mathbb{Z}^d$ from $L_p$ shift-invariant spaces associated with cardinal functions related to general multiquadrics,…
It is shown that the modulation spaces $M_{p}^{w}$ can be characterized by the approximation behavior of their elements using Local Fourier bases. In analogy to the Local Fourier bases, we show that the modulation spaces can also be…
Fractional Sobolev spaces $\widehat{H}^s(\mathbb{R})$ have been playing important roles in analysis of many mathematical subjects. In this work, we re-consider fractional Sobolev spaces under the perspective of fractional operators and…
Morrey--Sobolev inequalities are established for functions in weighted Sobolev spaces on the $n$-dimensional half-space, where the weight is a power of the distance to the boundary, as well as for Sobolev spaces on the $n$-dimensional…
We study relations between the variational Sobolev 1-capacity and versions of variational BV-capacity in a complete metric space equipped with a doubling measure and supporting a weak $(1,1)$-Poincar\'e inequality. We prove the equality of…
We obtain fundamental imbeddings for the fractional Sobolev space with variable exponent that is a generalization of well-known fractional Sobolev spaces. As an application, we obtain a-priori bounds and multiplicity of solutions to some…
In this paper, we prove Poincar\'e and Sobolev inequalities for differential forms in $L^1(\mathbb R^n)$. The singular integral estimates that it is possible to use for $L^p$, $p>1$, are replaced here with inequalities which go back to…
The dual purpose of this article is to establish bilinear Poincare-type estimates associated to an approximation of the identity and to explore the connections between bilinear pseudo-differential operators and bilinear potential-type…
A factorization formula for certain automorphisms of a Poisson algebra associated to a quiver is proved, which involves framed versions of moduli spaces of quiver representations. This factorization formula is related to wall-crossing…
In the context of Sobolev spaces with variable exponents, Poincar\'e--Wirtinger inequalities are possible as soon as Luxemburg norms are considered. On the other hand, modular versions of the inequalities in the expected form…
We study the $L^p$-theory for the Schr\"odinger operator $\mathcal L_a$ with inverse-square potential $a|x|^{-2}$. Our main result describes when $L^p$-based Sobolev spaces defined in terms of the operator $(\mathcal L_a)^{s/2}$ agree with…
We develop a theory of BV and Sobolev Spaces via integration by parts formula in abstract metric spaces; the role of vector fields is played by Weaver's metric derivations. The definition hereby given is shown to be equivalent to many…
In the setting of the multidimensional Mellin analysis we introduce moduli of continuity and use them to define Besov-Mellin spaces. We prove that Besov-Mellin spaces are the interpolation spaces (in the sense of J.Peetre) between two…
Recently Gigli developed a Sobolev calculus on non-smooth spaces using module theory. In this paper it is shown that his theory fits nicely into the theory of differentiability spaces initiated by Cheeger, Keith and others. A relaxation…
We investigate the properties of a class of weighted vector-valued $L_p$-spaces and the corresponding (an)isotropic Sobolev-Slobodetskii spaces. These spaces arise naturally in the context of maximal $L_p$-regularity for parabolic…