On the Regularity of Characteristic Functions of Weakly Exterior Thick Domains

Winfried Sickel

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Theory of Functions of Several Real Variables and Its Applications, Tome 323 (2023), pp. 137-166.

Résumé

Let $E$ be a domain in $\mathbb R^d$. We investigate the regularity of the characteristic function $\mathcal X_E$ depending on the behavior of the $\delta $-neighborhoods of the boundary of $E$. The regularity is measured in terms of the Nikol'skii–Besov and Lizorkin–Triebel spaces.

Détail

Keywords: Lizorkin–Triebel spaces, characterization by differences, snowflake domain, regular domains, weakly exterior thick domains, thick domains, quasiballs.
Mots-clés : Nikol'skii–Besov spaces, Haar wavelet decomposition, $(\varepsilon,\delta)$-domains

Bibliographie

[1] Bechtel S., “The extension problem for fractional Sobolev spaces with a partial vanishing trace condition”, Arch. Math., 117:1 (2021), 79–85 | DOI | MR | Zbl

[2] O. V. Besov, “Extension by zero of functions of several variables”, Math. Notes, 64:3 (1998), 303–315 | DOI | DOI | MR | MR | Zbl

[3] O. V. Besov, V. P. Il'in, and S. M. Nikol'skii, Integral Representations of Functions and Imbedding Theorems, v. 1, 2, 1st ed., J. Wiley Sons, New York, 1978 | MR | MR | Zbl

[4] Bricchi M., “Existence and properties of $h$-sets”, Georgian Math. J., 9:1 (2002), 13–32 | DOI | MR | Zbl

[5] Bricchi M., “Tailored Besov spaces and $h$-sets”, Math. Nachr., 263–264 (2004), 36–52 | DOI | MR | Zbl

[6] Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Grundl. Math. Wiss., 285, Springer, Berlin, 1988 | MR | Zbl

[7] Burenkov V.I., Sobolev spaces on domains, Teubner-Texte Math., 137, Teubner, Stuttgart, 1998 | DOI | MR | Zbl

[8] Caetano A.M., Hewett D.P., Moiola A., “Density results for Sobolev, Besov and Triebel–Lizorkin spaces on rough sets”, J. Funct. Anal., 281:3 (2021), 109019 | DOI | MR | Zbl

[9] Evans E.J., Extension operators and finite elements for fractal boundary value problems, PhD Thesis, Worcester Polytech. Inst., Worcester, MA, 2011

[10] Falconer K., Fractal geometry: Mathematical foundations and applications, J. Wiley Sons, Chichester, 1990 | MR | Zbl

[11] Falconer K., Techniques in fractal geometry, J. Wiley Sons, Chichester, 1997 | MR | Zbl

[12] Faraco D., Rogers K.M., “The Sobolev norm of characteristic functions with applications to the Calderón inverse problem”, Q. J. Math., 64:1 (2013), 133–147 | DOI | MR | Zbl

[13] Fleming W.H., Rishel R., “An integral formula for total gradient variation”, Arch. Math., 11 (1960), 218–222 | DOI | MR | Zbl

[14] Frazier M., Jawerth B., “A discrete transform and decompositions of distribution spaces”, J. Funct. Anal., 93:1 (1990), 34–170 | DOI | MR | Zbl

[15] A. B. Gulisashvili, “Multipliers on Besov spaces”, J. Sov. Math., 31:1 (1985), 2662–2672 | DOI | Zbl | Zbl

[16] A. B. Gulisashvili, “Multipliers in Besov spaces and traces of functions on subsets of Euclidean space”, Sov. Math., Dokl., 31:2 (1985), 332–336 | MR | Zbl

[17] Hajłasz P., Martio O., “Traces of Sobolev functions on fractal type sets and characterization of extension domains”, J. Funct. Anal., 143:1 (1997), 221–246 | DOI | MR | Zbl

[18] Hedberg L.I., Netrusov Yu., An axiomatic approach to function spaces, spectral synthesis, and Luzin approximation, Mem. AMS, 188, no. 882, Amer. Math. Soc., Providence, RI, 2007 | MR

[19] Jaffard S., Meyer Y., Wavelet methods for pointwise regularity and local oscillations of functions, Mem. AMS, 123, no. 587, Amer. Math. Soc., Providence, RI, 1996 | MR

[20] Jawerth B., “Some observations on Besov and Lizorkin–Triebel spaces”, Math. Scand., 40:1 (1977), 94–104 | DOI | MR | Zbl

[21] Jerison D.S., Kenig C.E., “Boundary behavior of harmonic functions in non-tangentially accessible domains”, Adv. Math., 46:1 (1982), 80–147 | DOI | MR | Zbl

[22] Jones P., “Quasiconformal mappings and extendability of functions in Sobolev spaces”, Acta math., 147 (1981), 71–88 | DOI | MR | Zbl

[23] Jonsson A., Wallin H., Function spaces on subsets of $\mathbb R^d$, Math. Rep., 2, Harwood Acad. Publ., London, 1984 | MR

[24] G. A. Kalyabin, “Descriptions of functions in classes of Besov–Lizorkin–Triebel type”, Proc. Steklov Inst. Math., 156 (1983), 89–118 | MR | Zbl | Zbl

[25] Koskela P., Lectures on quasiconformal and quasisymmetric mappings, Jyväskylä Lect. Math., 1, Univ. Jyväskylä, Jyväskylä, 2009

[26] V. G. Maz'ya, Sobolev Spaces, Springer, Berlin, 1985 | MR | MR | Zbl

[27] Nečas J., Direct methods in the theory of elliptic equations, Springer, Berlin, 2012 | MR

[28] S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems, Grundl. Math. Wiss., 205, Springer, Berlin, 1975 | MR | Zbl

[29] Oswald P., Multilevel finite element approximation: Theory and applications, Teubner, Stuttgart, 1994 | MR | Zbl

[30] Runst T., Sickel W., Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, De Gruyter Ser. Nonlinear Anal. Appl., 3, W. de Gruyter, Berlin, 1996 | MR | Zbl

[31] Schneider C., Vybíral J., “Non-smooth atomic decompositions, traces on Lipschitz domains, and pointwise multipliers in function spaces”, J. Funct. Anal., 264:5 (2013), 1197–1237 | DOI | MR | Zbl

[32] Seeger A., “A note on Triebel–Lizorkin spaces”, Approximation and function spaces: Proc. 27th semest. (Warsaw, 1986), Banach Center Publ., 22, PWN–Pol. Sci. Publ., Warszawa, 1989, 391–400 | DOI | MR

[33] Sickel W., “On pointwise multipliers for $F^s_{p,q}(\mathbb R^n)$ in case $\sigma_{p,q}/p$”, Ann. Mat. Pura Appl. Ser. 4, 176 (1999), 209–250 | DOI | MR | Zbl

[34] Sickel W., “Pointwise multipliers of Lizorkin–Triebel spaces”, The Maz'ya anniversary collection, v. 2, Oper. Theory Adv. Appl., 110, Rostock Conf., 1998, Birkhäuser, Basel, 1999, 295–321 | MR | Zbl

[35] Sickel W., “On the regularity of characteristic functions”, Anomalies in partial differential equations, Springer INdAM Ser., 43, Springer, Cham, 2021, 395–441 ; Stein I.M., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | DOI | MR | Zbl

[36] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser., 30, Princeton Univ. Press, Princeton, NJ, 1970 | MR | Zbl

[37] Strichartz R.S., “Wavelets and self-affine tilings”, Constr. Approx., 9:2–3 (1993), 327–346 ; Tribel Kh., Teoriya funktsionalnykh prostranstv, Mir, M., 1986 | DOI | MR | Zbl

[38] H. Triebel, Theory of Function Spaces, Monogr. Math., 78, Birkhäuser, Basel, 1983 | MR | Zbl

[39] Triebel H., Theory of function spaces II, Monogr. Math., 84, Birkhäuser, Basel, 1992 | MR | Zbl

[40] Triebel H., Fractals and spectra related to Fourier analysis and function spaces, Monogr. Math., 91, Birkhäuser, Basel, 1997 | MR | Zbl

[41] Triebel H., The structure of functions, Monogr. Math., 97, Birkhäuser, Basel, 2001 | MR | Zbl

[42] Triebel H., “Non-smooth atoms and pointwise multipliers in function spaces”, Ann. Mat. Pura Appl. Ser. 4, 182:4 (2003), 457–486 | DOI | MR | Zbl

[43] Triebel H., Theory of function spaces III, Monogr. Math., 100, Birkhäuser, Basel, 2006 | MR | Zbl

[44] Triebel H., Function spaces and wavelets on domains, EMS Tracts Math., 7, Eur. Math. Soc., Zürich, 2008 | MR | Zbl

[45] Triebel H., Bases in function spaces, sampling, discrepancy, numerical integration, EMS Tracts Math., 11, Eur. Math. Soc., Zürich, 2010 | MR | Zbl

[46] Zhou Y., “Fractional Sobolev extension and imbedding”, Trans. Amer. Math. Soc., 367:2 (2015), 959–979 | DOI | MR | Zbl