Ulyanov inequalities for the mixed moduli of smoothness in mixed metrics

References

[1] O. V. Besov, V. P. Il’in and S. M. Nikol’skiÄ­, Integral Representations of Functions and Imbedding Theorems, Scripta Ser. Math., V. H. Winston & Sons, Washington, 1978. Search in Google Scholar

[2] Z. Ditzian and S. Tikhonov, Ul’yanov and Nikol’skiÄ­-type inequalities, J. Approx. Theory 133 (2005), no. 1, 100–133. 10.1016/j.jat.2004.12.008Search in Google Scholar

[3] O. Domingues and S. Tikhonov, Embedding of smooth function spaces, extrapolations, and related inequalities, preprint (2020), https://arxiv.org/abs/1909.12818. Search in Google Scholar

[4] P. Glazyrina and S. Tikhonov, Jacobi weights, fractional integration, and sharp Ulyanov inequalities, J. Approx. Theory 195 (2015), 122–140. 10.1016/j.jat.2014.05.005Search in Google Scholar

[5] A. Gogatishvili, B. Opic, S. Tikhonov and W. Trebels, Ulyanov-type inequalities between Lorentz–Zygmund spaces, J. Fourier Anal. Appl. 20 (2014), no. 5, 1020–1049. 10.1007/s00041-014-9343-4Search in Google Scholar

[6] M. L. Gol’dman, Embedding of constructive and structural Lipschitz spaces in symmetric spaces (in Russian), Trudy Mat. Inst. Steklov. 173 (1986), 90–112; translation in Proc Steklov Inst. Math. 173 (1987), no. 4, 93–118. Search in Google Scholar

[7] A. Jumabayeva, Sharp Ul’yanov inequalities for generalized Liouville–Weyl derivatives, Anal. Math. 43 (2017), no. 2, 279–302. 10.1007/s10476-017-0308-0Search in Google Scholar

[8] A. A. Jumabayeva and B. V. Simonov, Transformation of Fourier series by means of general monotone sequences (in Russian), Mat. Zametki 107 (2020), no. 5, 674–692; translation in Math. Notes 107 (2020), no. 5-6, 740–758. Search in Google Scholar

[9] Y. Kolomoitsev and S. Tikhonov, Properties of moduli of smoothness in L p ⁢ ( ℝ d ) , J. Approx. Theory 257 (2020), Article ID 105423. 10.1016/j.jat.2020.105423Search in Google Scholar

[10] Y. Kolomoitsev and S. Tikhonov, Hardy–Littlewood and Ulyanov inequalities, Mem. Amer. Math. Soc. 1325 (2021), 1–118. 10.1090/memo/1325Search in Google Scholar

[11] V. I. Kolyada, On the relations between moduli of continuity in various metrics (in Russian), Trudy Mat. Inst. Steklov. 181 (1988), 117–136; translation in Proc Steklov Inst. Math. 1989, no. 4, 127–148. Search in Google Scholar

[12] M. K. Potapov, Imbedding theorems in a mixed metric (in Russian), Trudy Mat. Inst. Steklov. 156 (1980), 143–156, 263. Search in Google Scholar

[13] M. K. Potapov and B. V. Simonov, Properties of the mixed modulus of smoothness of positive order in a mixed metric (in Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh. (2014), no. 6, 31–40; tranlation in Moscow Univ. Math. Bull. 69 (2014), no. 6, 258–266. Search in Google Scholar

[14] M. K. Potapov and B. V. Simonov, Inequalities for different metrics for trigonometric polynomials (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. (2019), no. 1, 49–62; translation in Russian Math. (Iz. VUZ) 63 (2019), no. 1, 42–54. Search in Google Scholar

[15] M. K. Potapov and B. V. Simonov, Strengthened Ul’yanov’s inequalities for partial moduli of smoothness for functions from spaces with various metrics (in Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh. (2019), no. 3, 26–38; tranlation in Moscow Univ. Math. Bull. 74 (2019), no. 3, 108–120. Search in Google Scholar

[16] M. K. Potapov and B. V. Simonov, Estimates of partial moduli of smoothness in metrics of L p 1 ⁢ ∞ and L ∞ ⁢ p 2 through partial moduli of smoothness in metrics of L p 1 ⁢ p 2 (in Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh. (2020), no. 1, 3–17; tranlation in Moscow Univ. Math. Bull. 75 (2020), no. 1, 1–15. Search in Google Scholar

[17] M. K. Potapov and B. V. Simonov, Refinement of relations between mixed moduli of smoothness in the metrics of L p and L ∞ (in Russian), Mat. Zametki 110 (2021), no. 3, 368–385; translation in Math. Notes 110 (2021), no. 3–4, 347–362. Search in Google Scholar

[18] M. K. Potapov and B. V. Simonov, Refinement of the relations between mixed smoothness moduli in L 1 and L q metrics, Sibirsk. Mat. Zh. 62 (2021), no. 4, 812–829; translation in Sib. Math. J. 62 (2021), no. 4, 661–677. Search in Google Scholar

[19] M. K. Potapov, B. V. Simonov and S. Y. Tikhonov, Relations between mixed moduli of smoothness and embedding theorems for the Nikol’skiÄ­ classes (in Russian), Tr. Mat. Inst. Steklova 269 (2010), 204–214; translation in Proc. Steklov Inst. Math. 269 (2010), no. 1, 197–207. Search in Google Scholar

[20] M. K. Potapov, B. V. Simonov and S. Y. Tikhonov, Mixed moduli of smoothness in L p , 1 < p < ∞ : A survey, Surv. Approx. Theory 8 (2013), 1–57. Search in Google Scholar

[21] M. K. Potapov, B. V. Simonov and S. Y. Tikhonov, Analogues of Ulyanov inequalities for mixed moduli of smoothness, Methods of Fourier Analysis and Approximation Theory, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham (2016), 161–179. 10.1007/978-3-319-27466-9_11Search in Google Scholar

[22] M. K. Potapov, B. V. Simonov and S. Y. Tikhonov, Fractional Moduli of Smoothness, Maks Press, Moscow, 2016. Search in Google Scholar

[23] B. Simonov and S. Tikhonov, Sharp Ul’yanov-type inequalities using fractional smoothness, J. Approx. Theory 162 (2010), no. 9, 1654–1684. 10.1016/j.jat.2010.04.010Search in Google Scholar

[24] S. Tikhonov, Trigonometric series of Nikol’skii classes, Acta Math. Hungar. 114 (2007), no. 1–2, 61–78. 10.1007/s10474-006-0513-ySearch in Google Scholar

[25] S. Tikhonov, Trigonometric series with general monotone coefficients, J. Math. Anal. Appl. 326 (2007), no. 1, 721–735. 10.1016/j.jmaa.2006.02.053Search in Google Scholar

[26] S. Tikhonov, Weak type inequalities for moduli of smoothness: The case of limit value parameters, J. Fourier Anal. Appl. 16 (2010), no. 4, 590–608. 10.1007/s00041-009-9101-1Search in Google Scholar

[27] S. Tikhonov and W. Trebels, Ulyanov-type inequalities and generalized Liouville derivatives, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 1, 205–224. 10.1017/S0308210509001048Search in Google Scholar

[28] W. Trebels, Inequalities for moduli of smoothness versus embeddings of function spaces, Arch. Math. (Basel) 94 (2010), no. 2, 155–164. 10.1007/s00013-009-0078-4Search in Google Scholar

[29] P. L. Ul’janov, The embedding of certain classes H p ω of functions (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 649–686. Search in Google Scholar

[30] A. Zygmund, Trigonometric Series. Vol. I, II, 3rd ed., Cambridge Math. Libr., Cambridge University, Cambridge, 2002. Search in Google Scholar