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Hardy–Littlewood and Ulyanov inequalities
About this Title
Yurii Kolomoitsev and Sergey Tikhonov
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 271, Number 1325
ISBNs: 978-1-4704-4758-8 (print); 978-1-4704-6628-2 (online)
DOI: https://doi.org/10.1090/memo/1325
Published electronically: July 6, 2021
Keywords: Moduli of smoothness,
$K$-functionals,
Ulyanov’s inequalities,
Hardy–Litlewood–Nikol’skii’s inequalities,
embedding theorems,
fractional derivatives
Table of Contents
Chapters
- Basic notation
- 1. Introduction
- 2. Auxiliary results
- 3. Polynomial inequalities of Nikol’skii–Stechkin–Boas–types
- 4. Basic properties of fractional moduli of smoothness
- 5. Hardy–Littlewood–Nikol’skii inequalities for trigonometric polynomials
- 6. General form of the Ulyanov inequality for moduli of smoothness, $K$-functionals, and their realizations
- 7. Sharp Ulyanov inequalities for $K$-functionals and realizations
- 8. Sharp Ulyanov inequalities for moduli of smoothness
- 9. Sharp Ulyanov inequalities for realizations of $K$-functionals related to Riesz derivatives and corresponding moduli of smoothness
- 10. Sharp Ulyanov inequality via Marchaud inequality
- 11. Sharp Ulyanov and Kolyada inequalities in Hardy spaces
- 12. $(L_p,L_q)$ inequalities of Ulyanov-type involving derivatives
- 13. Embedding theorems for function spaces
- List of symbols
Abstract
We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness $\omega _\alpha (f,t)_q$ and $\omega _\beta (f,t)_p$ for $0<p<q\le \infty$. A similar problem for the generalized $K$-functionals and their realizations between the couples $(L_p, W_p^\psi )$ and $(L_q, W_q^\varphi )$ is also solved.
The main tool is the new Hardy–Littlewood–Nikol’skii inequalities. More precisely, we obtained the asymptotic behavior of the quantity \begin{equation*} \sup _{T_n} \frac {\Vert \mathcal {D}(\psi )(T_n)\Vert _q}{\Vert \mathcal {D}({\varphi })(T_n)\Vert _p},\qquad 0<p<q\le \infty , \end{equation*} where the supremum is taken over all nontrivial trigonometric polynomials $T_n$ of degree at most $n$ and $\mathcal {D}(\psi ), \mathcal {D}({\varphi })$ are the Weyl-type differentiation operators.
We also prove the Ulyanov and Kolyada-type inequalities in the Hardy spaces. Finally, we apply the obtained estimates to derive new embedding theorems for the Lipschitz and Besov spaces.
- È. S. Belinskiĭ, An application of the Fourier transform to the summability of Fourier series, Sibirsk. Mat. Ž. 18 (1977), no. 3, 497–511, 717 (Russian). MR 0463785
- E. S. Belinsky, Strong summability for the Marcinkiewicz means in the integral metric and related questions, J. Austral. Math. Soc. Ser. A 65 (1998), no. 3, 303–312. MR 1660418
- Edward Belinskii and Elijah Liflyand, Approximation properties in $L_p$, $0<p<1$, Funct. Approx. Comment. Math. 22 (1993), 189–199 (1994). MR 1304373
- Colin Bennett and Robert C. Sharpley, Weak-type inequalities in analysis, Linear spaces and approximation (Proc. Conf., Math. Res. Inst., Oberwolfach, 1977) Lecture Notes in Biomath., vol. 21, Springer, Berlin-New York, 1978, pp. 151–162. MR 501473
- Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
- O. V. Besov, On some conditions for derivatives of periodic functions to belong to $L_p$, Nauchn. Dokl. Vyssh. Shkoly Fiz.-Mat. Nauki 1 (1959), 13–17 (in Russian).
- O. V. Besov, V. P. Ilin, S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorems. Halsted Press, New York-Toronto-London, 1978.
- R. P. Boas, Quelques généralisations d’un théorème de S. Bernstein sur la dérivée d’un polynôme trigonométrique, C. R. Acad. Sci., Paris 227 (1948), 618–619 (in French).
- Jan Boman and Harold S. Shapiro, Comparison theorems for a generalized modulus of continuity, Ark. Mat. 9 (1971), 91–116. MR 340901, DOI 10.1007/BF02383639
- Ju. A. Brudnyĭ, Piecewise polynomial approximation, embedding theorem and rational approximation, Approximation theory (Proc. Internat. Colloq., Inst. Angew. Math. Univ. Bonn, Bonn, 1976) Springer, Berlin, 1976, pp. 73–98. MR 0617764
- V. I. Burenkov, Imbedding and extension theorems for classes of differentiable functions of several variables defined on the entire spaces, Mathematical Analysis 1965 (Russian), Akad. Nauk SSSR Inst. Naučn. Informacii, Moscow, 1966, pp. 71–155 (Russian). MR 0206698
- D. L. Burkholder, R. F. Gundy, and M. L. Silverstein, A maximal function characterization of the class $H^{p}$, Trans. Amer. Math. Soc. 157 (1971), 137–153. MR 274767, DOI 10.1090/S0002-9947-1971-0274767-6
- P. L. Butzer, H. Dyckhoff, E. Görlich, and R. L. Stens, Best trigonometric approximation, fractional order derivatives and Lipschitz classes, Canadian J. Math. 29 (1977), no. 4, 781–793. MR 442566, DOI 10.4153/CJM-1977-081-6
- A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. MR 167830, DOI 10.4064/sm-24-2-113-190
- Ferruccio Colombini and Nicolas Lerner, Hyperbolic operators with non-Lipschitz coefficients, Duke Math. J. 77 (1995), no. 3, 657–698. MR 1324638, DOI 10.1215/S0012-7094-95-07721-7
- Albert Cohen, Numerical analysis of wavelet methods, Studies in Mathematics and its Applications, vol. 32, North-Holland Publishing Co., Amsterdam, 2003. MR 1990555
- F. Dai and Z. Ditzian, Littlewood-Paley theory and a sharp Marchaud inequality, Acta Sci. Math. (Szeged) 71 (2005), no. 1-2, 65–90. MR 2160356
- F. Dai, Z. Ditzian, and S. Tikhonov, Sharp Jackson inequalities, J. Approx. Theory 151 (2008), no. 1, 86–112. MR 2403897, DOI 10.1016/j.jat.2007.04.015
- Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR 1261635
- 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
- Z. Ditzian, On the Marchaud-type inequality, Proc. Amer. Math. Soc. 103 (1988), no. 1, 198–202. MR 938668, DOI 10.1090/S0002-9939-1988-0938668-8
- Z. Ditzian, Fractional derivatives and best approximation, Acta Math. Hungar. 81 (1998), 323–348.
- Z. Ditzian, V. H. Hristov, and K. G. Ivanov, Moduli of smoothness and $K$-functionals in $L_p$, $0<p<1$, Constr. Approx. 11 (1995), no. 1, 67–83. MR 1323964, DOI 10.1007/BF01294339
- Z. Ditzian and K. Runovskii, Realization and smoothness related to the Laplacian, Acta Math. Hungar. 93 (2001), no. 3, 189–223. MR 1925231, DOI 10.1023/A:1013978628115
- Z. Ditzian and S. Tikhonov, Ul′yanov and Nikol′skiĭ-type inequalities, J. Approx. Theory 133 (2005), no. 1, 100–133. MR 2122270, DOI 10.1016/j.jat.2004.12.008
- Z. Ditzian and S. Tikhonov, Moduli of smoothness of functions and their derivatives, Studia Math. 180 (2007), no. 2, 143–160. MR 2314094, DOI 10.4064/sm180-2-4
- Mikhail Dyachenko and Sergey Tikhonov, A Hardy-Littlewood theorem for multiple series, J. Math. Anal. Appl. 339 (2008), no. 1, 503–510. MR 2370669, DOI 10.1016/j.jmaa.2007.06.057
- M. Dyachenko and S. Tikhonov, General monotone sequences and convergence of trigonometric series, Topics in classical analysis and applications in honor of Daniel Waterman, World Sci. Publ., Hackensack, NJ, 2008, pp. 88–101. MR 2569380, DOI 10.1142/9789812834447_{0}006
- D. E. Edmunds and H. Triebel, Function spaces, entropy numbers, differential operators, Cambridge Tracts in Mathematics, vol. 120, Cambridge University Press, Cambridge, 1996. MR 1410258, DOI 10.1017/CBO9780511662201
- John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR 2261424
- José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
- Amiran Gogatishvili, Bohumír Opic, Sergey Tikhonov, and Walter Trebels, Ulyanov-type inequalities between Lorentz-Zygmund spaces, J. Fourier Anal. Appl. 20 (2014), no. 5, 1020–1049. MR 3254611, DOI 10.1007/s00041-014-9343-4
- M. L. Gol′dman, Imbedding theorems for anisotropic Nikol′skiĭ-Besov spaces with moduli of continuity of a general type, Trudy Mat. Inst. Steklov. 170 (1984), 86–104, 275 (Russian). Studies in the theory of differentiable functions of several variables and its applications, X. MR 790329
- M. L. Goldman, A criterion for the embedding of different metrics for isotropic Besov spaces with arbitrary moduli of continuity, Proc. Steklov Inst. Math. 201 (2) (1994), 155–181; translation from Trudy Mat. Inst. Steklov. 201 (1992), 186–218.
- Dmitry Gorbachev and Sergey Tikhonov, Moduli of smoothness and growth properties of Fourier transforms: two-sided estimates, J. Approx. Theory 164 (2012), no. 9, 1283–1312. MR 2948566, DOI 10.1016/j.jat.2012.05.017
- G. H. Hardy and J. E. Littlewood, A convergence criterion for Fourier series, Math. Z. 28 (1928), no. 1, 612–634. MR 1544980, DOI 10.1007/BF01181186
- G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. II, Math. Z. 34 (1932), no. 1, 403–439. MR 1545260, DOI 10.1007/BF01180596
- Dorothee D. Haroske and Hans Triebel, Embeddings of function spaces: a criterion in terms of differences, Complex Var. Elliptic Equ. 56 (2011), no. 10-11, 931–944. MR 2838229, DOI 10.1080/17476933.2010.551203
- Tord Holmstedt, Interpolation of quasi-normed spaces, Math. Scand. 26 (1970), 177–199. MR 415352, DOI 10.7146/math.scand.a-10976
- 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
- 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
- H. Johnen, K. Scherer, On the equivalence of the K-functional and moduli of continuity and some applications, Constructive theory of functions of several variables (Proc. Conf., Math. Res. Inst., Oberwolfach 1976), Lecture Notes in Math., vol. 571, Springer-Verlag, Berlin-Heidelberg 1977, 119–140.
- Yu. S. Kolomoitsev, Description of a class of functions with the condition $\omega _r(f,h)_p\le Mh^{r-1+1/p}$ for $0<p<1$, Vestn. Dnepr. Univ., Ser. Mat. 8 (2003), 31–43 (in Russian).
- Yu. Kolomoitsev, The inequality of Nikol’skii-Stechkin-Boas type with fractional derivatives in $L_p$, $0<p<1$, Tr. Inst. Prikl. Mat. Mekh. 15 (2007), 115–119 (in Russian).
- Yu. S. Kolomoĭtsev, On a class of functions representable as a Fourier integral, Proceedings of the Institute of Applied Mathematics and Mechanics. Volume 25 (Russian), Tr. Inst. Prikl. Mat. Mekh., vol. 25, Nats. Akad. Nauk Ukrainy Inst. Prikl. Mat. Mekh., Donetsk, 2012, pp. 125–132 (Russian, with English, Russian and Ukrainian summaries). MR 3155722
- Yuriĭ S. Kolomoĭtsev, On moduli of smoothness and $K$-functionals of fractional order in Hardy spaces, Ukr. Mat. Visn. 8 (2011), no. 3, 421–446, 462 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 181 (2012), no. 1, 78–97. MR 2906700, DOI 10.1007/s10958-012-0677-7
- Yu. S. Kolomoĭtsev, Multiplicative sufficient conditions for Fourier multipliers, Izv. Ross. Akad. Nauk Ser. Mat. 78 (2014), no. 2, 145–166 (Russian, with Russian summary); English transl., Izv. Math. 78 (2014), no. 2, 354–374. MR 3234820, DOI 10.1070/im2014v078n02abeh002690
- Yu. S. Kolomoitsev, Inequalities for the fractional derivatives of trigonometric polynomials in spaces with integral metrics, Ukrainian Math. J. 67 (2015), no. 1, 45–61. Translation of Ukraïn. Mat. Zh. 67 (2015), no. 1, 42–56. MR 3403604, DOI 10.1007/s11253-015-1064-6
- Yu. Kolomoitsev, S. Tikhonov, Properties of moduli of smoothness in $L_p (\Bbb {R}^d)$. J. Approx. Theory 257 (2020), 105423.
- V. I. Kolyada, On the relations between moduli of continuity in various metrics, Trudy Mat. Inst. Steklov. 181 (1988), 117–136, 270 (Russian). Translated in Proc. Steklov Inst. Math. 1989, no. 4, 127–148; Studies in the theory of differentiable functions of several variables and its applications, XII (Russian). MR 945427
- V. I. Kolyada, Rearrangements of functions, and embedding theorems, Uspekhi Mat. Nauk 44 (1989), no. 5(269), 61–95 (Russian); English transl., Russian Math. Surveys 44 (1989), no. 5, 73–117. MR 1040269, DOI 10.1070/RM1989v044n05ABEH002287
- V. I. Kolyada, On the embedding of Sobolev spaces, Mat. Zametki 54 (1993), no. 3, 48–71, 158 (Russian, with Russian summary); English transl., Math. Notes 54 (1993), no. 3-4, 908–922 (1994). MR 1248284, DOI 10.1007/BF01209556
- V. I. Kolyada, Embeddings of fractional Sobolev spaces and estimates for Fourier transformations, Mat. Sb. 192 (2001), no. 7, 51–72 (Russian, with Russian summary); English transl., Sb. Math. 192 (2001), no. 7-8, 979–1000. MR 1861373, DOI 10.1070/SM2001v192n07ABEH000579
- V. I. Kolyada, Inequalities of Gagliardo-Nirenberg type and estimates for the moduli of continuity, Russian Math. Surveys 60 (6) (2005), 1147–1164; translation from Uspekhi Mat. Nauk 60 (6)(366) (2005), 139–156.
- V. I. Kolyada and F. J. Pérez Lázaro, Inequalities for partial moduli of continuity and partial derivatives, Constr. Approx. 34 (2011), no. 1, 23–59. MR 2796090, DOI 10.1007/s00365-010-9088-5
- Steven G. Krantz, Fractional integration on Hardy spaces, Studia Math. 73 (1982), no. 2, 87–94. MR 667967, DOI 10.4064/sm-73-2-87-94
- V. G. Krotov, On the differentiability of functions from $L^{p}$, $0<p<1$, Mat. Sb. (N.S.) 117(159) (1982), no. 1, 95–113 (Russian). MR 642492
- 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
- O. I. Kuznecova and R. M. Trigub, Two-sided estimations of the approximation of functions by Riesz and Marcinkiewicz means, Dokl. Akad. Nauk SSSR 251 (1980), no. 1, 34–36 (Russian). MR 563172
- E. R. Liflyand, Lebesgue constants of multiple Fourier series, Online J. Anal. Comb. 1 (2006), Art. 5, 112. MR 2249993
- P. I. Lizorkin, Bounds for trigonometrical integrals and an inequality of Bernstein for fractional derivatives, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 109–126 (Russian). MR 0178306
- S. Lu, Four Lectures on Real $H_p$ Spaces. World Scientific, Singapore, 1995.
- M. A. Marchaud, Sur les dérivées et sur les différences des fonctions de variables réelles, NUMDAM, [place of publication not identified], 1927 (French). MR 3532941
- J. Marcinkiewicz, Sur quelques intégrales du type de Dini, Ann. Soc. Polon. Math. 17 (1938), 42–50.
- M. Mateljević and M. Pavlović, An extension of the Hardy-Littlewood inequality, Mat. Vesnik 6(19)(34) (1982), no. 1, 55–61. MR 681659
- Joaquim Martín and Mario Milman, Fractional Sobolev inequalities: symmetrization, isoperimetry and interpolation, Astérisque 366 (2014), x+127 (English, with English and French summaries). MR 3308452
- S. G. Mihlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR (N.S.) 109 (1956), 701–703 (Russian). MR 0080799
- Ferenc Móricz, On double cosine, sine, and Walsh series with monotone coefficients, Proc. Amer. Math. Soc. 109 (1990), no. 2, 417–425. MR 1010803, DOI 10.1090/S0002-9939-1990-1010803-4
- Yu. V. Netrusov, Embedding theorems for the spaces $H^{\omega ,k}_p$ and $H^{s,\omega ,k}_p$, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 159 (1987), no. Chisl. Metody i Voprosy Organiz. Vychisl. 8, 83–102, 177–178 (Russian); English transl., J. Soviet Math. 47 (1989), no. 6, 2881–2895. MR 885078, DOI 10.1007/BF01305217
- S. N. Bernšteĭn, A generalization of an inequality of S. B. Stečkin to entire functions of finite degree, Doklady Akad. Nauk SSSR (N.S.) 60 (1948), 1487–1490 (Russian). MR 0024995
- 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 0048565
- S. M. Nikol’skii, Approximation of functions of several variables and embedding theorems, Springer, New York, 1975.
- E. Nursultanov and S. Tikhonov, A sharp Remez inequality for trigonometric polynomials, Constr. Approx. 38 (2013), no. 1, 101–132. MR 3078275, DOI 10.1007/s00365-012-9172-0
- P. Osval′d, Approximation by splines in the metric $L_{p}$, $0<p<1$, Math. Nachr. 94 (1980), 69–96 (Russian). MR 582521, DOI 10.1002/mana.19800940107
- Jaak Peetre, A remark on Sobolev spaces. The case $0<p<1$, J. Approximation Theory 13 (1975), 218–228. MR 374900, DOI 10.1016/0021-9045(75)90034-9
- Jaak Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Duke University, Mathematics Department, Durham, N.C., 1976. MR 0461123
- Aleksander Pełczyński and MichałWojciechowski, Sobolev spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1361–1423. MR 1999199, DOI 10.1016/S1874-5849(03)80039-6
- M. K. Potapov, B. V. Simonov, and S. Yu. Tikhonov, Relations between moduli of smoothness in different metrics, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 3 (2009), 17–25 (Russian, with English and Russian summaries); English transl., Moscow Univ. Math. Bull. 64 (2009), no. 3, 105–112. MR 2664494, DOI 10.3103/S0027132209030036
- M. K. Potapov, B. V. Simonov, S. Yu. Tikhonov, Fractional Moduli of Smoothness, Max Press, 2016, Moscow, 338 p.
- T. V. Radoslavova, Decrease orders of the $L^{p}$-moduli of continuity $(0<p\leq \infty )$, Anal. Math. 5 (1979), no. 3, 219–234 (English, with Russian summary). MR 549239, DOI 10.1007/BF01908905
- James C. Robinson, Log-Lipschitz embeddings of homogeneous sets with sharp logarithmic exponents and slicing products of balls, Proc. Amer. Math. Soc. 142 (2014), no. 4, 1275–1288. MR 3162249, DOI 10.1090/S0002-9939-2014-11852-1
- K. Runovski, Approximation of families of linear polynomial operators, Disser. of Doctor of Science, Moscow State University, 2010.
- K. Runovski and H.-J. Schmeisser, On some extensions of Bernstein’s inequality for trigonometric polynomials, Funct. Approx. Comment. Math. 29 (2001), 125–142. MR 2135603, DOI 10.7169/facm/1538186723
- K. Runovski and H.-J. Schmeisser, On families of linear polynomial operators generated by Riesz kernels, Eurasian Math. J. 1 (2010), no. 4, 124–139. MR 2905205, DOI 10.1007/s12572-010-0002-y
- K. Runovski and H.-J. Schmeisser, General moduli of smoothness and approximation by families of linear polynomial operators, New perspectives on approximation and sampling theory, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2014, pp. 269–298. MR 3363034
- Konstantin Runovski and Hans-Jürgen Schmeisser, Moduli of smoothness related to the Laplace-operator, J. Fourier Anal. Appl. 21 (2015), no. 3, 449–471. MR 3345363, DOI 10.1007/s00041-014-9373-y
- Konstantin V. Runovski and Hans-Jürgen Schmeisser, Moduli of smoothness related to fractional Riesz-derivatives, Z. Anal. Anwend. 34 (2015), no. 1, 109–125. MR 3300960, DOI 10.4171/ZAA/1531
- Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
- B. V. Simonov and S. Yu. Tikhonov, Embedding theorems in the constructive theory of approximations, Mat. Sb. 199 (2008), no. 9, 107–148 (Russian, with Russian summary); English transl., Sb. Math. 199 (2008), no. 9-10, 1367–1407. MR 2466856, DOI 10.1070/SM2008v199n09ABEH003964
- Boris Simonov and Sergey Tikhonov, Sharp Ul′yanov-type inequalities using fractional smoothness, J. Approx. Theory 162 (2010), no. 9, 1654–1684. MR 2718890, DOI 10.1016/j.jat.2010.04.010
- S. B. Stečkin, A generalization of some inequalities of S. N. Bernšteĭn, Doklady Akad. Nauk SSSR (N.S.) 60 (1948), 1511–1514 (Russian). MR 0024993
- S. B. Steckin, On absolute convergence of Fourier series, Izv. Akad. Nauk SSSR. Ser. Mat. 20 (1956), 385–412 (in Russian).
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- È. A. Storoženko, Imbedding theorems and best approximations, Mat. Sb. (N.S.) 97(139) (1975), no. 2(6), 230–241, 317 (Russian). MR 0404965
- È. A. Storoženko, Approximation of functions of class $H^{p}$, $0<p\leq 1$, Mat. Sb. 105(147) (1978), no. 4, 601–621, 640 (Russian). MR 496597
- È. A. Storoženko and P. Osval′d, Jackson’s theorem in the spaces $L^{p}(\textbf {R}^{k})$, $0<p<1$, Sibirsk. Mat. Ž. 19 (1978), no. 4, 888–901, 956 (Russian). MR 0493121
- È. A. Storoženko, V. G. Krotov, and P. Osval′d, Direct and inverse theorems of Jackson type in the spaces $L^p, 0<p<1$, Mat. Sb. (N.S.) 98(140) (1975), no. 3(11), 395–415, 495 (Russian). MR 0402384
- R. Taberski, Differences, moduli and derivatives of fractional orders, Comment. Math. Prace Mat. 19 (1976/77), no. 2, 389–400. MR 477582
- S. Tikhonov, On moduli of smoothness of fractional order, Real Anal. Exchange 30 (2004/05), no. 2, 507–518. MR 2177415
- Sergey Tikhonov, Weak type inequalities for moduli of smoothness: the case of limit value parameters, J. Fourier Anal. Appl. 16 (2010), no. 4, 590–608. MR 2671173, DOI 10.1007/s00041-009-9101-1
- S. Tikhonov and M. Zeltser, Weak monotonicity concept and its applications, Fourier analysis, Trends Math., Birkhäuser/Springer, Cham, 2014, pp. 357–374. MR 3362029
- A. F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon Press, Oxford, London, New York, Paris, 1963.
- M. F. Tīman, The difference properties of functions of several variables, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 667–676 (Russian). MR 0271282
- E. C. Titchmarsh, A Note on Fourier Transforms, J. London Math. Soc. 2 (1927), no. 3, 148–150. MR 1574410, DOI 10.1112/jlms/s1-2.3.148
- Walter Trebels, Inequalities for moduli of smoothness versus embeddings of function spaces, Arch. Math. (Basel) 94 (2010), no. 2, 155–164. MR 2592762, DOI 10.1007/s00013-009-0078-4
- Hans Triebel, Theory of function spaces, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2010. Reprint of 1983 edition [MR0730762]; Also published in 1983 by Birkhäuser Verlag [MR0781540]. MR 3024598
- Roal′d M. Trigub, Fourier multipliers and $K$-functionals of spaces of smooth functions, Ukr. Mat. Visn. 2 (2005), no. 2, 236–280, 296 (Russian, with English and Russian summaries); English transl., Ukr. Math. Bull. 2 (2005), no. 2, 239–284. MR 2172631
- R. M. Trigub, Summability of trigonometric Fourier series at $d$-points and a generalization of the Abel-Poisson method, Izv. Math. 79 (4) (2015), 838–858; translation from Izv. Ross. Akad. Nauk Ser. Mat. 79 (4) (2015), 205–224.
- Roald M. Trigub and Eduard S. Bellinsky, Fourier analysis and approximation of functions, Kluwer Academic Publishers, Dordrecht, 2004. [Belinsky on front and back cover]. MR 2098384, DOI 10.1007/978-1-4020-2876-2
- P. L. Ul’yanov, The imbedding of certain function classes $H_p^\omega$, Math. USSR-Izv. 2 (3) (1968), 601–637; translation from Izv. Akad. Nauk SSSR Ser. Mat. 3 (1968), 649–686.
- Stephen Wainger, Special trigonometric series in $k$-dimensions, Mem. Amer. Math. Soc. 59 (1965), 102. MR 182838
- 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
- Zhi Xin Liu and Shan Zhen Lu, Applications of Hörmander multiplier theorem to approximation in real Hardy spaces, Harmonic analysis (Tianjin, 1988) Lecture Notes in Math., vol. 1494, Springer, Berlin, 1991, pp. 119–129. MR 1187072, DOI 10.1007/BFb0087763
- Enrique Zuazua, Log-Lipschitz regularity and uniqueness of the flow for a field in $(W^{n/p+1,p}_\textrm {loc}(\Bbb R^n))^n$, C. R. Math. Acad. Sci. Paris 335 (2002), no. 1, 17–22 (English, with English and French summaries). MR 1920427, DOI 10.1016/S1631-073X(02)02426-3
- A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587