# Trace and extension theorems relating Besov spaces to weighted averaged Sobolev spaces

@article{Barton2016TraceAE, title={Trace and extension theorems relating Besov spaces to weighted averaged Sobolev spaces}, author={Ariel Barton}, journal={arXiv: Functional Analysis}, year={2016} }

There are known trace and extension theorems relating functions in a weighted Sobolev space in a domain U to functions in a Besov space on the boundary bU. We extend these theorems to the case where the Sobolev exponent p is less than one by modifying our Sobolev spaces to consider averages of functions in Whitney balls. Averaged Sobolev spaces are also of interest in the applications in the case where p>1, and so we also provide trace and extension results in that case. Finally, we provide…

## Figures from this paper

## 3 Citations

Layer potentials for general linear elliptic systems

- Mathematics
- 2017

In this paper we construct layer potentials for elliptic differential operators using the Lax-Milgram theorem, without recourse to the fundamental solution; this allows layer potentials to be…

Dirichlet and Neumann boundary values of solutions to higher order elliptic equations

- MathematicsAnnales de l'Institut Fourier
- 2019

We show that if $u$ is a solution to a linear elliptic differential equation of order $2m\geq 2$ in the half-space with $t$-independent coefficients, and if $u$ satisfies certain area integral…

Traces and extensions of certain weighted Sobolev spaces on
$$\mathbb {R}^n$$
R
n
and

- Mathematics
- 2020

The focus of this paper is on Ahlfors Q -regular compact sets $$E\subset \mathbb {R}^n$$ E ⊂ R n such that, for each $$Q-2<\alpha \le 0$$ Q - 2 < α ≤ 0 , the weighted measure $$\mu _{\alpha }$$ μ α…

## References

SHOWING 1-10 OF 68 REFERENCES

Traces of weighted Sobolev spaces. Old and new

- Mathematics
- 2015

Abstract We give short simple proofs of Uspenskiĭ’s results characterizing Besov spaces as trace spaces of weighted Sobolev spaces. We generalize Uspenskiĭ’s results and prove the optimality of these…

Trace theorems for Sobolev-Slobodeckij spaces with or without weights

- Mathematics
- 2007

We prove that the well-known trace theorem for weighted Sobolev spaces holds true under minimal regularity assumptions on the domain. Using this result, we prove the existence of a bounded linear…

Boundary Layers on Sobolev–Besov Spaces and Poisson's Equation for the Laplacian in Lipschitz Domains

- Mathematics
- 1998

We study inhomogeneous boundary value problems for the Laplacian in arbitrary Lipschitz domains with data in Sobolev–Besov spaces. As such, this is a natural continuation of work in [Jerison and…

Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces

- Mathematics
- 2013

This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov…

The dirichlet problem in lipschitz domains for higher order elliptic systems with rough coefficients

- Mathematics
- 2007

We study the Dirichlet problem, in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order with bounded, complex-valued…

Equivalence between Regularity Theorems and Heat Kernel Estimates for Higher Order Elliptic Operators and Systems under Divergence Form

- Mathematics
- 2000

We study the heat kernel of higher order elliptic operators or systems under divergence form on Rn. Ellipticity is in the sense of Garding inequality. We show that for homogeneous operators Gaussian…

The Neumann problem for higher order elliptic equations with symmetric coefficients

- Mathematics
- 2017

In this paper we establish well posedness of the Neumann problem with boundary data in $$L^2$$L2 or the Sobolev space $$\dot{W}^2_{-1}$$W˙-12, in the half space, for linear elliptic differential…

Perturbation of well-posedness and layer potentials for higher-order elliptic systems with rough coefficients

- Mathematics
- 2016

In this paper we study boundary value problems for higher order elliptic differential operators in divergence form. We consider the two closely related topics of inhomogeneous problems and problems…

SHARP ESTIMATES FOR GREEN POTENTIALS ON NON-SMOOTH DOMAINS

- 2004

Given an open, bounded, connected domain Ω ⊂ R, let GD, GN be the solution operators for the Poisson equation for the Laplacian in Ω with homogeneous Dirichlet and Neumann boundary conditions,…

Sharp estimates for green potentials on non-smooth domains

- Mathematics
- 2004

Given an open, bounded, connected domain Ω ⊂ R n , let GD, GN be the solution operators for the Poisson equation for the Laplacian in Ω with homogeneous Dirichlet and Neumann boundary conditions,…