Search results for "Mathematical analysis"
showing 10 items of 2409 documents
REGULARITY OF THE FRACTIONAL MAXIMAL FUNCTION
2003
The purpose of this work is to show that the fractional maximal operator has somewhat unexpected regularity properties. The main result shows that the fractional maximal operator maps -spaces boundedly into certain first-order Sobolev spaces. It is also proved that the fractional maximal operator preserves first-order Sobolev spaces. This extends known results for the Hardy–Littlewood maximal operator.
Uniform, Sobolev extension and quasiconformal circle domains
1991
This paper contributes to the theory of uniform domains and Sobolev extension domains. We present new features of these domains and exhibit numerous relations among them. We examine two types of Sobolev extension domains, demonstrate their equivalence for bounded domains and generalize known sufficient geometric conditions for them. We observe that in the plane essentially all of these domains possess the trait that there is a quasiconformal self-homeomorphism of the extended plane which maps a given domain conformally onto a circle domain. We establish a geometric condition enjoyed by these plane domains which characterizes them among all quasicircle domains having no large and no small bo…
Embedding of Sobolev Spaces into Lipschitz Spaces
1989
The main result of the paper is that if Ω is a bounded uniform domain in ℝn and p>n, then the Sobolev space Wl, p(Ω) embeds continously into Cα(Ω), α = 1 - n/p.
An Itô Formula for rough partial differential equations and some applications
2020
AbstractWe investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form $\partial _{t}u-A_{t}u-f=(\dot X_{t}(x) \cdot \nabla + \dot Y_{t}(x))u$ ∂ t u − A t u − f = ( X ̇ t ( x ) ⋅ ∇ + Y ̇ t ( x ) ) u on $[0,T]\times \mathbb {R}^{d}.$ [ 0 , T ] × ℝ d . To do so, we introduce a concept of “differential rough driver”, which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces Wk,p. We also define a natural notion of geometricity in this context, and show how it relates to a product formula for controlled paths. In the case of transport noise (i.e. when Y = 0), we use this framework to prove a…
APPROXIMATION OF BANACH SPACE VALUED NON-ABSOLUTELY INTEGRABLE FUNCTIONS BY STEP FUNCTIONS
2008
AbstractThe approximation of Banach space valued non-absolutely integrable functions by step functions is studied. It is proved that a Henstock integrable function can be approximated by a sequence of step functions in the Alexiewicz norm, while a Henstock–Kurzweil–Pettis and a Denjoy–Khintchine–Pettis integrable function can be only scalarly approximated in the Alexiewicz norm by a sequence of step functions. In case of Henstock–Kurzweil–Pettis and Denjoy–Khintchine–Pettis integrals the full approximation can be done if and only if the range of the integral is norm relatively compact.
Analytic capacity and quasiconformal mappings with $W^{1,2}$ Beltrami coefficient
2008
We show that if $\phi$ is a quasiconformal mapping with compactly supported Beltrami coefficient in the Sobolev space $W^{1,2}$, then $\phi$ preserves sets with vanishing analytic capacity. It then follows that a compact set $E$ is removable for bounded analytic functions if and only if it is removable for bounded quasiregular mappings with compactly supported Beltrami coefficient in $W^{1,2}$.
Holomorphic approximation of ultradifferentiable functions
1981
Introduct ion Let S be a closed subset of some open set in Cn and denote by dT(S) the space of germs of holomorphic functions on (a neighborhood of) S. For a space F(S) of tEvalued (continuous, differentiable etc.) functions on S [containing t~(S)] the problem of holomorphic approximation consists of finding conditions to ensure that the natural mapping Q : e)(S)-~F(S) has dense range with respect to a given topology on F(S). Positive solutions for F = C r, 0_ l . For Q:tP(/3)~O(D)c~C(/3), DCIE n strongly pseudoconvex, proofs were given independently by Henkin [17], Kerzman [21], and Lieb [27], for the case e : (9(/3)~(9(D)c~C~(/3) cf. also [30] and for Sobolev spaces see Bell [3, Sect. 6].…
Sobolev-Poincaré implies John
1995
We establish necessary conditions for the validity of Sobolev-Poincaré type inequalities. We give a geometric characterisation for the validity of this inequality for simply connected plane domains.
Zero viscosity limit of the Oseen equations in a channel
2001
Oseen equations in the channel are considered. We give an explicit solution formula in terms of the inverse heat operators and of projection operators. This solution formula is used for the analysis of the behavior of the Oseen equations in the zero viscosity limit. We prove that the solution of Oseen equations converges in W1,2 to the solution of the linearized Euler equations outside the boundary layer and to the solution of the linearized Prandtl equations inside the boundary layer. © 2001 Society for Industrial and Applied Mathematics.
Southern-Tyrrhenian seismicity in space-time-magnitude domain
2006
An analysis is conducted on a catalogue containing more than 2000 seismic events occurred in the southern Tyrrhenian Sea between 1988 and October 2002, as an attempt to characterise the main seismogenetic processes active in the area in space, time and magnitude domain by means of the parameters of phenomenological laws. We chose to adopt simple phenomenological models, since the low number of data did not allow to use more complex laws. The two main seismogenetic volumes present in the area were considered for the purpose of this work. The first includes a nearly homogeneous distribution of hypocentres in a NW steeply dipping layer as far as a…