Trace Operator on von Koch’s Snowflake
Springer Science and Business Media LLC - Trang 1-26 - 2024
Tóm tắt
We study properties of the boundary trace operator on the Sobolev space
$$W^1_1(\Omega )$$
. Using the density result by Koskela and Zhang (Arch. Ration. Mech. Anal. 222(1), 1-14 2016), we define a surjective operator
$$Tr: W^1_1(\Omega _K)\rightarrow X(\Omega _K)$$
, where
$$\Omega _K$$
is von Koch’s snowflake and
$$X(\Omega _K)$$
is a trace space with the quotient norm. Since
$$\Omega _K$$
is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý
(2017) that there exists a right inverse to Tr, i.e. a linear operator
$$S: X(\Omega _K) \rightarrow W^1_1(\Omega _K)$$
such that
$$Tr \circ S= Id_{X(\Omega _K)}$$
. In this paper we provide a different, purely combinatorial proof based on geometrical structure of von Koch’s snowflake. Moreover we identify the isomorphism class of the trace space as
$$\ell _1$$
. As an additional consequence of our approach we obtain a simple proof of the Peetre’s theorem (Special Issue 2, 277-282 1979) about non-existence of the right inverse for domain
$$\Omega $$
with regular boundary, which explains Banach space geometry cause for this phenomenon.
Tài liệu tham khảo
Bonk, M., Saksman, E., Soto, T.: Triebel-Lizorkin spaces on metric spaces via hyperbolic fillings. Indiana Univ. Math. J. 67(4), 1625–1663 (2018)
Brudnyi, A., Brudnyi, Y.: Methods of geometric analysis in extension and trace problems. vol. 1, volume 102 of Monographs in Mathematics. Birkhäuser/Springer Basel AG, Basel (2012)
Buckley, S., Koskela, P.: Sobolev-Poincaré implies John. Math. Res. Lett. 2(5), 577–593 (1995)
Carleson, L.: On the support of harmonic measure for sets of Cantor type. Ann. Acad. Sci. Fenn. Ser. A I Math. 10, 113–123 (1985)
Ciesielski, Z.: On the isomorphisms of the spaces \(H_{\alpha }\) and \(m\). Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 8, 217–222 (1960)
Derezinski, M.: Isomorphic properties of function space bv on simply connected planar sets. Master’s thesis, Uniwersity of Warsaw (2013)
Derezinski, M., Nazarov, F., Wojciechowski, M.: Isomorphic properties of function space bv on simply connected planar sets. in preparation
Gagliardo, E.: Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in \(n\) variabili. Rendiconti del Seminario Matematico della Università di Padova. The Mathematical Journal of the University of Padova 27, 284–305 (1957)
Gmeineder, F., Raiţă, B., Van Schaftingen, J.: On limiting trace inequalities for vectorial differential operators. Indiana Univ. Math. J. 70(5), 2133–2176 (2021)
Hajłasz, P., Martio, O.: Traces of Sobolev functions on fractal type sets and characterization of extension domains. J. Funct. Anal. 143(1), 221–246 (1997)
Jonsson, A., Wallin, H.: Function spaces on subsets of \({\bf R}^n\). Math. Rep., 2(1), xiv+221 (1984)
Koskela, P., Rajala, T., Zhang, Y.R.-Y.: A density problem for sobolev spaces on gromov hyperbolic domains. Nonlinear Anal: Theory Method Appl 154, 189–209 (2017)
Koskela, P., Zhang, Y.R.-Y.: A density problem for Sobolev spaces on planar domains. Arch. Ration. Mech. Anal. 222(1), 1–14 (2016)
Lahti, P., Li, X., Wang, Z.: Traces of Newton-Sobolev, Hajlasz-Sobolev, and BV functions on metric spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 22(3), 1353–1383 (2021)
Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces. I. Springer-Verlag, Berlin-New York, 1977. Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92
Lukáš Malý: Trace and extension theorems for sobolev-type functions in metric spaces (2017)
Meyer, Y.: Wavelets and operators, vol. 37 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1992). Translated from the 1990 French original by D. H. Salinger
Peetre, J.: A counterexample connected with Gagliardo’s trace theorem. Commentationes Mathematicae. Special Issue 2, 277–282 (1979). Special issue dedicated to Władysław Orlicz on the occasion of his seventy-fifth birthday
Pełczyński, A.: Projections in certain Banach spaces. Studia Math. 19, 209–228 (1960)
Pełczyński, A., Wojciechowski, M.: Spaces of functions with bounded variation and Sobolev spaces without local unconditional structure. Journal für die Reine und Angewandte Mathematik. [Crelle’s Journal], 558, 109–157 (2003)
Roginskaya, M., Wojciechowski, M.: Bounded approximation property for Sobolev spaces on simply-connected planar domains. arXiv:1401.7131
Weaver, N.: Lipschitz algebras. World Scientific Publishing Co., Inc, River Edge, NJ (1999)