Truncated smooth function spaces

Domínguez, Oscar; Tikhonov, Sergey

doi:10.1090/tran/9259

Truncated smooth function spaces
HTML articles powered by AMS MathViewer

by Oscar Domínguez and Sergey Tikhonov;

Trans. Amer. Math. Soc. 377 (2024), 8877-8934

DOI: https://doi.org/10.1090/tran/9259

Published electronically: September 25, 2024

HTML | PDF | Request permission

Abstract:

We introduce truncated Besov and Triebel–Lizorkin function spaces and study their main properties: embeddings, interpolation, duality, lifting, traces. These new scales allow us to improve several known results in functional analysis and PDE’s. In particular, we obtain a full solution to the trace/extension problem in the critical case as well as sharp Sobolev-type embeddings with critical smoothness parameters.

References

S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Analyse Math. 30 (1976), 1–38. MR 466902, DOI 10.1007/BF02786703
Alexandre Almeida, Wavelet bases in generalized Besov spaces, J. Math. Anal. Appl. 304 (2005), no. 1, 198–211. MR 2124658, DOI 10.1016/j.jmaa.2004.09.017
Sergey V. Astashkin, Konstantin V. Lykov, and Mario Milman, Limiting interpolation spaces via extrapolation, J. Approx. Theory 240 (2019), 16–70. MR 3926051, DOI 10.1016/j.jat.2018.09.007
Fethi Ben Belgacem and Pierre-Emmanuel Jabin, Compactness for nonlinear continuity equations, J. Funct. Anal. 264 (2013), no. 1, 139–168. MR 2995703, DOI 10.1016/j.jfa.2012.10.005
F. Ben Belgacem and P.-E. Jabin, Convergence of numerical approximations to non-linear continuity equations with rough force fields, Arch. Ration. Mech. Anal. 234 (2019), no. 2, 509–547. MR 3995045, DOI 10.1007/s00205-019-01396-3
Colin Bennett and Karl Rudnick, On Lorentz-Zygmund spaces, Dissertationes Math. (Rozprawy Mat.) 175 (1980), 67. MR 576995
Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 482275, DOI 10.1007/978-3-642-66451-9
O. V. Besov, V. P. Il’in, and S. M. Nikol’skiĭ, Integral representations of functions and embedding theorems, Halsted Press, New York-Toronto-London, 1978.
Blanca F. Besoy and Fernando Cobos, Duality for logarithmic interpolation spaces when $0<q<1$ and applications, J. Math. Anal. Appl. 466 (2018), no. 1, 373–399. MR 3818122, DOI 10.1016/j.jmaa.2018.05.082
Haïm Brézis and Stephen Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations 5 (1980), no. 7, 773–789. MR 579997, DOI 10.1080/03605308008820154
Elia Brué and Quoc-Hung Nguyen, On the Sobolev space of functions with derivative of logarithmic order, Adv. Nonlinear Anal. 9 (2020), no. 1, 836–849. MR 3995732, DOI 10.1515/anona-2020-0027
Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York, Inc., New York, 1967. MR 230022, DOI 10.1007/978-3-642-46066-1
António M. Caetano and Walter Farkas, Local growth envelopes of Besov spaces of generalized smoothness, Z. Anal. Anwend. 25 (2006), no. 3, 265–298. MR 2251954, DOI 10.4171/zaa/1289
António M. Caetano and Hans-Gerd Leopold, Local growth envelopes of Triebel-Lizorkin spaces of generalized smoothness, J. Fourier Anal. Appl. 12 (2006), no. 4, 427–445. MR 2256932, DOI 10.1007/s00041-006-6018-9
António M. Caetano and Susana D. Moura, Local growth envelopes of spaces of generalized smoothness: the subcritical case, Math. Nachr. 273 (2004), 43–57. MR 2084956, DOI 10.1002/mana.200310195
António M. Caetano and Susana D. Moura, Local growth envelopes of spaces of generalized smoothness: the critical case, Math. Inequal. Appl. 7 (2004), no. 4, 573–606. MR 2097514, DOI 10.7153/mia-07-58
Dongho Chae, Peter Constantin, and Jiahong Wu, Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal. 202 (2011), no. 1, 35–62. MR 2835862, DOI 10.1007/s00205-011-0411-5
K. M. Chong and N. M. Rice, Equimeasurable rearrangements of functions, Queen’s Papers in Pure and Applied Mathematics, No. 28, Queen’s University, Kingston, ON, 1971. MR 372140
Fernando Cobos and Óscar Domínguez, On Besov spaces of logarithmic smoothness and Lipschitz spaces, J. Math. Anal. Appl. 425 (2015), no. 1, 71–84. MR 3299650, DOI 10.1016/j.jmaa.2014.12.034
Fernando Cobos, Óscar Domínguez, and Hans Triebel, Characterizations of logarithmic Besov spaces in terms of differences, Fourier-analytical decompositions, wavelets and semi-groups, J. Funct. Anal. 270 (2016), no. 12, 4386–4425. MR 3490770, DOI 10.1016/j.jfa.2016.03.007
Fernando Cobos and Alba Segurado, Description of logarithmic interpolation spaces by means of the $J$-functional and applications, J. Funct. Anal. 268 (2015), no. 10, 2906–2945. MR 3331789, DOI 10.1016/j.jfa.2015.03.012
Gianluca Crippa and Camillo De Lellis, Estimates and regularity results for the DiPerna-Lions flow, J. Reine Angew. Math. 616 (2008), 15–46. MR 2369485, DOI 10.1515/CRELLE.2008.016
Ronald A. DeVore, Sherman D. Riemenschneider, and Robert C. Sharpley, Weak interpolation in Banach spaces, J. Functional Analysis 33 (1979), no. 1, 58–94. MR 545385, DOI 10.1016/0022-1236(79)90018-1
Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. MR 2944369, DOI 10.1016/j.bulsci.2011.12.004
Óscar Domínguez, Dorothee D. Haroske, and Sergey Tikhonov, Embeddings and characterizations of Lipschitz spaces, J. Math. Pures Appl. (9) 144 (2020), 69–105 (English, with English and French summaries). MR 4175445, DOI 10.1016/j.matpur.2020.11.004
Óscar Domínguez and Sergey Tikhonov, Function spaces of logarithmic smoothness: embeddings and characterizations, Mem. Amer. Math. Soc. 282 (2023), no. 1393, vii+166. MR 4539365, DOI 10.1090/memo/1393
O. Domínguez and S. Tikhonov, Truncated smooth function spaces, arXiv:2211.01529.
O. Domínguez and S. Tikhonov, Truncated interpolation, Work in progress.
D. E. Edmunds and D. Haroske, Spaces of Lipschitz type, embeddings and entropy numbers, Dissertationes Math. (Rozprawy Mat.) 380 (1999), 43. MR 1677961
David E. Edmunds and Dorothee D. Haroske, Embeddings in spaces of Lipschitz type, entropy and approximation numbers, and applications, J. Approx. Theory 104 (2000), no. 2, 226–271. MR 1761901, DOI 10.1006/jath.2000.3453
W. D. Evans, Bohumír Opic, and Luboš Pick, Real interpolation with logarithmic functors, J. Inequal. Appl. 7 (2002), no. 2, 187–269. MR 1922997, DOI 10.1155/S1025583402000127
Walter Farkas and Hans-Gerd Leopold, Characterisations of function spaces of generalised smoothness, Ann. Mat. Pura Appl. (4) 185 (2006), no. 1, 1–62. MR 2179581, DOI 10.1007/s10231-004-0110-z
Reinhard Farwig and Ryo Kanamaru, Optimality of Serrin type extension criteria to the Navier-Stokes equations, Adv. Nonlinear Anal. 10 (2021), no. 1, 1071–1085. MR 4225513, DOI 10.1515/anona-2020-0130
Jens Franke, On the spaces $\textbf {F}_{pq}^s$ of Triebel-Lizorkin type: pointwise multipliers and spaces on domains, Math. Nachr. 125 (1986), 29–68. MR 847350, DOI 10.1002/mana.19861250104
Michael Frazier and Björn Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), no. 4, 777–799. MR 808825, DOI 10.1512/iumj.1985.34.34041
Emilio Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in $n$ variabili, Rend. Sem. Mat. Univ. Padova 27 (1957), 284–305 (Italian). MR 102739
Amiran Gogatishvili, Bohumír Opic, and Walter Trebels, Limiting reiteration for real interpolation with slowly varying functions, Math. Nachr. 278 (2005), no. 1-2, 86–107. MR 2111802, DOI 10.1002/mana.200310228
Jan Gustavsson, A function parameter in connection with interpolation of Banach spaces, Math. Scand. 42 (1978), no. 2, 289–305. MR 512275, DOI 10.7146/math.scand.a-11754
Kurt Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), no. 1, 77–102. MR 567435, DOI 10.7146/math.scand.a-11827
Dorothee D. Haroske, Envelopes and sharp embeddings of function spaces, Chapman & Hall/CRC Research Notes in Mathematics, vol. 437, Chapman & Hall/CRC, Boca Raton, FL, 2007. MR 2262450
Dorothee D. Haroske and Susana D. Moura, Continuity envelopes of spaces of generalised smoothness, entropy and approximation numbers, J. Approx. Theory 128 (2004), no. 2, 151–174. MR 2068695, DOI 10.1016/j.jat.2004.04.008
Aicke Hinrichs, David Krieg, Erich Novak, and Jan Vybíral, Lower bounds for integration and recovery in $L_2$, J. Complexity 72 (2022), Paper No. 101662, 15. MR 4438175, DOI 10.1016/j.jco.2022.101662
Björn Jawerth, Some observations on Besov and Lizorkin-Triebel spaces, Math. Scand. 40 (1977), no. 1, 94–104. MR 454618, DOI 10.7146/math.scand.a-11678
G. A. Kaljabin and P. I. Lizorkin, Spaces of functions of generalized smoothness, Math. Nachr. 133 (1987), 7–32. MR 912417, DOI 10.1002/mana.19871330102
Ryo Kanamaru, Optimality of logarithmic interpolation inequalities and extension criteria to the Navier-Stokes and Euler equations in Vishik spaces, J. Evol. Equ. 20 (2020), no. 4, 1381–1397. MR 4181952, DOI 10.1007/s00028-020-00559-0
H. Kempka, C. Schneider, and J. Vybiral, Path regularity of the Brownian motion and the Brownian sheet, Constr. Approx. 59 (2024), no. 2, 485–539. MR 4721732, DOI 10.1007/s00365-023-09647-z
Yurii Kolomoitsev and Sergey Tikhonov, Properties of moduli of smoothness in $L_p (\Bbb {R}^d)$, J. Approx. Theory 257 (2020), 105423, 29. MR 4102141, DOI 10.1016/j.jat.2020.105423
Pekka Koskela and Zhuang Wang, Dyadic norm Besov-type spaces as trace spaces on regular trees, Potential Anal. 53 (2020), no. 4, 1317–1346. MR 4159382, DOI 10.1007/s11118-019-09808-5
Alois Kufner and Lars-Erik Persson, Weighted inequalities of Hardy type, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 1982932, DOI 10.1142/5129
Flavien Léger, A new approach to bounds on mixing, Math. Models Methods Appl. Sci. 28 (2018), no. 5, 829–849. MR 3799259, DOI 10.1142/S0218202518500215
Ziwei Li, Winfried Sickel, Dachun Yang, and Wen Yuan, Pointwise multipliers for Besov spaces $B^{0,b}_{p,\infty }(\Bbb R^n)$ with only logarithmic smoothness, Ann. Mat. Pura Appl. (4) 203 (2024), no. 2, 703–763. MR 4715316, DOI 10.1007/s10231-023-01379-y
Liguang Liu, Suqing Wu, Jie Xiao, and Wen Yuan, The logarithmic Sobolev capacity, Adv. Math. 392 (2021), Paper No. 107993, 88. MR 4313960, DOI 10.1016/j.aim.2021.107993
Lukáš Malý, Nageswari Shanmugalingam, and Marie Snipes, Trace and extension theorems for functions of bounded variation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 1, 313–341. MR 3783791
Joaquim Martín, Symmetrization inequalities in the fractional case and Besov embeddings, J. Math. Anal. Appl. 344 (2008), no. 1, 99–123. MR 2416295, DOI 10.1016/j.jmaa.2008.02.028
Vladimir Maz’ya, Sobolev spaces with applications to elliptic partial differential equations, Second, revised and augmented edition, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg, 2011. MR 2777530, DOI 10.1007/978-3-642-15564-2
Alejandro Molero, Mihalis Mourgoglou, Carmelo Puliatti, and Xavier Tolsa, $L^2$-boundedness of gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type, Arch. Ration. Mech. Anal. 247 (2023), no. 3, Paper No. 38, 59. MR 4575398, DOI 10.1007/s00205-023-01852-1
Susana Moura, Function spaces of generalised smoothness, Dissertationes Math. (Rozprawy Mat.) 398 (2001), 88. MR 1876765, DOI 10.4064/dm398-0-1
Susana D. Moura, Júlio S. Neves, and Mariusz Piotrowski, Growth envelopes of anisotropic function spaces, Z. Anal. Anwend. 27 (2008), no. 1, 95–118. MR 2362005, DOI 10.4171/ZAA/1346
Susana D. Moura, Júlio S. Neves, and Mariusz Piotrowski, Continuity envelopes of spaces of generalized smoothness in the critical case, J. Fourier Anal. Appl. 15 (2009), no. 6, 775–795. MR 2570433, DOI 10.1007/s00041-009-9063-3
Susana D. Moura, Júlio S. Neves, and Cornelia Schneider, Optimal embeddings of spaces of generalized smoothness in the critical case, J. Fourier Anal. Appl. 17 (2011), no. 5, 777–800. MR 2838107, DOI 10.1007/s00041-010-9155-0
Susana D. Moura, Júlio S. Neves, and Cornelia Schneider, Spaces of generalized smoothness in the critical case: optimal embeddings, continuity envelopes and approximation numbers, J. Approx. Theory 187 (2014), 82–117. MR 3262928, DOI 10.1016/j.jat.2014.07.010
S. M. Nikol′skiĭ, Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables, Trudy Mat. Inst. Steklov. 38 (1951), 244–278 (Russian). MR 48565
S. M. Nikol′skiĭ, Approximation of functions of several variables and imbedding theorems, Die Grundlehren der mathematischen Wissenschaften, Band 205, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by John M. Danskin, Jr. MR 374877, DOI 10.1007/978-3-642-65711-5
Per Nilsson, Reiteration theorems for real interpolation and approximation spaces, Ann. Mat. Pura Appl. (4) 132 (1982), 291–330 (1983). MR 696048, DOI 10.1007/BF01760986
Takayoshi Ogawa and Yasushi Taniuchi, Remarks on uniqueness and blow-up criterion to the Euler equations in the generalized Besov spaces, J. Korean Math. Soc. 37 (2000), no. 6, 1007–1019. International Conference on Differential Equations and Related Topics (Pusan, 1999). MR 1803285
B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific & Technical, Harlow, 1990. MR 1069756
Jaak Peetre, Remark on the dual of an interpolation space, Math. Scand. 34 (1974), 124–128. MR 372640, DOI 10.7146/math.scand.a-11512
J. Peetre, The trace of Besov spaces—a limiting case, Technical Report, Lund, (1975).
Jaak Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Duke University, Mathematics Department, Durham, NC, 1976. MR 461123
Jaak Peetre, A counterexample connected with Gagliardo’s trace theorem, Comment. Math. Spec. Issue 2 (1979), 277–282. MR 552011
M. K. Potapov, B. V. Simonov, and S. Yu. Tikhonov, Relations for moduli of smoothness in various metrics: functions with restrictions on the Fourier coefficients, Jaen J. Approx. 1 (2009), no. 2, 205–222. MR 2597953
W. Sickel and H. Triebel, Hölder inequalities and sharp embeddings in function spaces of $B^s_{pq}$ and $F^s_{pq}$ type, Z. Anal. Anwendungen 14 (1995), no. 1, 105–140. MR 1327495, DOI 10.4171/ZAA/666
Hans Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503903
Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540, DOI 10.1007/978-3-0346-0416-1
Hans Triebel, Fractals and spectra, Monographs in Mathematics, vol. 91, Birkhäuser Verlag, Basel, 1997. Related to Fourier analysis and function spaces. MR 1484417, DOI 10.1007/978-3-0348-0034-1
Hans Triebel, The structure of functions, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2001. [2012 reprint of the 2001 original] [MR1851996]. MR 3013187
Hans Triebel, Function spaces and wavelets on domains, EMS Tracts in Mathematics, vol. 7, European Mathematical Society (EMS), Zürich, 2008. MR 2455724, DOI 10.4171/019
Hans Triebel, Theory of function spaces IV, Monographs in Mathematics, vol. 107, Birkhäuser/Springer, Cham, [2020] ©2020. MR 4298338, DOI 10.1007/978-3-030-35891-4
Neil S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. MR 216286, DOI 10.1512/iumj.1968.17.17028
Misha Vishik, Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 6, 769–812 (English, with English and French summaries). MR 1717576, DOI 10.1016/S0012-9593(00)87718-6
G. Wilmes, On Riesz-type inequalities and $K$-functionals related to Riesz potentials in $\textbf {R}^{N}$, Numer. Funct. Anal. Optim. 1 (1979), no. 1, 57–77. MR 521687, DOI 10.1080/01630567908816004
V. I. Judovič, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR 138 (1961), 805–808 (Russian). MR 140822
V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat i Mat. Fiz. 3 (1963), 1032–1066 (Russian). MR 158189

AMS Website Down

Transactions of the American Mathematical Society

Truncated smooth function spaces
HTML articles powered by AMS MathViewer

Abstract: