ABSTRACT
We prove a lower bound estimate for Hajłasz-Besov capacity in metric spaces in terms of Netrusov-Hausdorff content. We also prove a similar estimate for Hajłasz-Triebel-Lizorkin capacity in terms of Hausdoroff content. These results are improvements of the earlier results obtained by Nuutinen in 2016 and the first author in 2020.
Funding statement: N. K. would like to thank BITS Pilani (BITS/GAU/RIG/2020/H0749) and DST-SERB (SRG/2021/000118), and D. M. would like to thank BITS Pilani (ID no. 2020PHXF0038H) for the financial support.
Funding statement: The authors are also thankful to Prof. Tord Sjödin for asking the question answered in this paper.
REFERENCES
[1] ADAMS, D. R.: The classification problem for the capacities associated with the Besov and Triebel-Lizorkin spaces, Banach Center Publ. 22 (1989), 9–24.10.4064/-22-1-9-24Search in Google Scholar
[2] ADAMS, D. R.: Besov capacity redux, J. Math. Sci. (N.Y.) 162(3) (2009), 307–318.10.1007/s10958-009-9639-0Search in Google Scholar
[3] ADAMS, D. R.—HEDBERG, L. I.: Function Spaces and Potential Theory. Grundlehren der Math. Wiss., Springer-Verlag, Berlin, 1996.10.1007/978-3-662-03282-4Search in Google Scholar
[4] ADAMS, D. R.—HURRI-SYRJÄNEN, R.: Besov functions and vanishing exponential integrability, Illinois J. Math. 47(4) (2003), 1137–1150.10.1215/ijm/1258138095Search in Google Scholar
[5] ADAMS, D. R.—XIAO, J.: Strong type estimates for homogeneous Besov capacities, Math. Ann. 325(4) (2003), 695–709.10.1007/s00208-002-0396-3Search in Google Scholar
[6] BOURDON, M.: Une caractérisation algébrique des homéomorphismes quasi-Möbius, Ann. Acad. Sci. Fenn. Math. 32(1) (2007), 235–250.Search in Google Scholar
[7] COSTEA, S.: Besov capacity and Hausdorff measures in metric measure spaces, Publ. Mat. 53(1) (2009), 141–178.10.5565/PUBLMAT_53109_07Search in Google Scholar
[8] DORRONSORO, J. R.: On the differentiability of LipschitzBesov functions, Trans. Amer. Math. Soc. 303(1) (1987), 229–240.10.1090/S0002-9947-1987-0896019-5Search in Google Scholar
[9] FEDERER, H.: Geometric Measure Theory. Grundlehren Math. Wiss. 153, Springer-Verlag New York Inc., New York, 1969.Search in Google Scholar
[10] FUJII, N.: A condition for a two-weight norm inequality for singular integral operators, Studia Math. 98(3) (1991), 175–190.10.4064/sm-98-3-175-190Search in Google Scholar
[11] GOGATISHVILI, A.—KOSKELA, P.—ZHOU, Y.: Characterizations of Besov and Triebel-Lizorkin spaces on metric measure spaces, Forum Math. 25(4) (2013), 787–819.Search in Google Scholar
[12] HAJŁASZ, P.: Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403–415.10.1007/BF00275475Search in Google Scholar
[13] HEDBERG, L. I.—NETRUSOV, Y.: An axiomatic approach to function spaces, spectral synthesis, and Luzin approximation, Mem. Amer. Math. Soc. 188(882) (2007), 97 pp.10.1090/memo/0882Search in Google Scholar
[14] HEIKKINEN, T.—IHNATSYEVA, L.—TUOMINEN, H.: Measure density and extension of Besov and Triebel-Lizorkin functions, J. Fourier Anal. Appl. 22(2) (2016), 334–382.10.1007/s00041-015-9419-9Search in Google Scholar
[15] HEIKKINEN, T.—KOSKELA, P.—TUOMINEN, H.: Approximation and quasicontinuity of Besov and Triebel-Lizorkin functions, Trans. Amer. Math. Soc. 369(5) (2017), 3547–3573.10.1090/tran/6886Search in Google Scholar
[16] HEIKKINEN, T.—TUOMINEN, H.: Approximation by Holder functions in Besov and TriebelLizorkin spaces, Constr. Approx. 44(3) (2016), 455–482.10.1007/s00365-016-9322-xSearch in Google Scholar
[17] JAWERTH, B.—PÉREZ, C.—WELLAND, G.: The positive cone in TriebelLizorkin spaces and the relation among potential and maximal operators, In: Harmonic Analysis and Partial Differential Equations (Boca Raton, FL, 1988), Contemp. Math. 107, Amer. Math. Soc., Providence, RI, 1990, pp. 71–91.10.1090/conm/107/1066471Search in Google Scholar
[18] JAWERTH, B.—TORCHINSKY, A.: Local sharp maximal functions, J. Approx. Theory 43(3) (1985), 231–270.10.1016/0021-9045(85)90102-9Search in Google Scholar
[19] KARAK, N.: Generalized Lebesgue points for Sobolev functions, Czechoslovak Math. J. 67(1) (2017), 143–150.10.21136/CMJ.2017.0405-15Search in Google Scholar
[20] KARAK, N.: Triebel-Lizorkin capacity and Hausdorff measure in metric spaces, Math. Slovaca 70(3) (2020), 617–624.10.1515/ms-2017-0376Search in Google Scholar
[21] KINNUNEN, J.—LATVALA, V.: Lebesgue points for Sobolev functions on metric spaces, Rev. Mat. Iberoam. 18 (2002), 685–700.10.4171/RMI/332Search in Google Scholar
[22] KINNUNEN, J.—MARTIO, O.: The Sobolev capacity on metric spaces, Ann. Acad. Sci. Fenn. Math. 21 (1996), 367–382.Search in Google Scholar
[23] KOSKELA, P.—YANG, D.—ZHOU, Y.: Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal mappings, Adv. Math. 226(4) (2011), 3579–3621.10.1016/j.aim.2010.10.020Search in Google Scholar
[24] LERNER, A. K.: A pointwise estimate for the local sharp maximal function with applications to singular integrals, Bull. Lond. Math. Soc. 42(5) (2010), 843–856.10.1112/blms/bdq042Search in Google Scholar
[25] LERNER, A. K.—PÉREZ, C.: Self-improving properties of generalized Poincaré type inequalities through rearrangements, Math. Scand. 97(2) (2005), 217–234.10.7146/math.scand.a-14973Search in Google Scholar
[26] LI, Z.—YANG, D.—YUAN, W.: Lebesgue points of Besov and Triebel-Lizorkin spaces with generalized smoothness, Mathematics 9 (2021), Art. No. 2724.10.3390/math9212724Search in Google Scholar
[27] MOCANU, M.: Lebesgue points for OrliczSobolev functions on metric measure spaces, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 57 (2011), 175–186.10.2478/v10157-011-0012-5Search in Google Scholar
[28] MILMAN, M.—XIAO, J.: The ∞-Besov capacity problem, Math. Nachr. 290(17-18) (2017), 2961–2976.10.1002/mana.201600500Search in Google Scholar
[29] NETRUSOV, Y.: Sets of singularities of functions in spaces of Besov and LizorkinTriebel type, Tr. Mat. Inst. Steklova 187 (1989), 162-177 (in Russian); English transl.: Proc. Steklov Inst. Math. (1990), 185–203.Search in Google Scholar
[30] NETRUSOV, Y.: Metric estimates for the capacities of sets in Besov spaces, Tr. Mat. Inst. Steklov 190 (1989), 159-185 (in Russian); English transl.: Proc. Steklov Inst. Math. 1 (1992), 167–192.Search in Google Scholar
[31] NETRUSOV, Y.: Estimates of capacities associated with Besov spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 201 (1992), 124-156 (in Russian); English transl.: J. Math. Sci. 78(2) (1996), 199–217.Search in Google Scholar
[32] NUUTINEN, J.: The Besov capacity in metric spaces, Ann. Polon. Math. 117(1) (2016), 59–78.10.4064/ap3843-4-2016Search in Google Scholar
[33] PROKHOROVICH, M. A.: Capacities and Lebesgue points for HajlaszSobolev fractional classes on metric measure spaces, Vestsī Nats. Akad. Navuk Belarusī Ser. Fīz. Mat. Navuk 124 (2006), 19–23.Search in Google Scholar
[34] POHELHUIS, J.—TORCHINSKY, A.: Medians, continuity, and vanishing oscillation, Studia Math. 213(3) (2012), 227–242.10.4064/sm213-3-3Search in Google Scholar
[35] ROGERS, C. A.: Hausdroff Measures, Cambridge Mathematical Library, Cambridhe University Press, Cambridge, Reprint of the 1970 original, With a foreword by Falconer, K. J., 1998.Search in Google Scholar
[36] STOCKE, B. M.: Differentiability properties of Bessel potentials and Besov functions, Ark. Mat. 22(2) (1984), 269–286.10.1007/BF02384383Search in Google Scholar
[37] YANG, D.: New characterizations of Hajlasz-Sobolev spaces on metric spaces, Sci. China Ser. A Math. 46 (2003), 675–689.10.1360/02ys0343Search in Google Scholar
[38] ZHOU, Y.: Fractional Sobolev extension and imbedding, Trans. Amer. Math. Soc. 367(2) (2015), 959–979.10.1090/S0002-9947-2014-06088-1Search in Google Scholar
[39] ZIEMER, W. P.: Weakly Differentiable Functions. Grad. Texts in Math. 120, Springer-Verlag, New York, 1989.10.1007/978-1-4612-1015-3Search in Google Scholar
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Articles in the same Issue
- A Note on Special Subsets of the Rudin-Frolík Order for Regulars
- The 2-Class Group of Certain Families of Imaginary Triquadratic Fields
- The Deranged Bell Numbers
- On Index and Monogenity of Certain Number Fields Defined by Trinomials
- The k-Generalized Lucas Numbers Close to a Power of 2
- Shifted Power of a Polynomial with Integral Roots
- Further Insights into the Mysteries of the Values of Zeta Functions at Integers
- Memoryless Properties on Time Scales
- A Study of the Higher-Order Schwarzian Derivatives of Hirotaka Tamanoi
- Besov and Triebel-Lizorkin Capacity in Metric Spaces
- Oscillation of Odd Order Linear Differential Equations with Deviating Arguments with Dominating Delay Part
- An Elliptic Type Inclusion Problem on the Heisenberg Lie Group
- Existence Result for a Double Phase Problem Involving the (p(x), q(x))-Laplacian Operator
- A New Series Space Derived by Absolute Generalized Nörlund Means
- Examples of Weinstein Domains in the Complement of Smoothed Total Toric Divisors
- The Uniform Effros Property and Local Homogeneity
- Limit Theorems for Weighted Sums of Asymptotically Negatively Associated Random Variables Under Some General Conditions
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