Weakly canceling operators and singular integrals.(English)Zbl 1462.42025
Proc. Steklov Inst. Math.312, 249-260 (2021) and Tr. Mat. Inst. Steklova 312, 259-271 (2021).
Summary: We suggest an elementary harmonic analysis approach to canceling and weakly canceling differential operators, which allows us to extend these notions to the anisotropic setting and replace differential operators with Fourier multiplies with mild smoothness regularity. In this more general setting of anisotropic Fourier multipliers, we prove the inequality \(\|f\|_{L_\infty} \lesssim \|Af\|_{L_1}\) if \(A\) is a weakly canceling operator of order \(d\) and the inequality \(\|f\|_{L_2} \lesssim \|Af\|_{L_1}\) if \(A\) is a canceling operator of order \(d/2\), provided \(f\) is a function of \(d\) variables.
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B15 | Multipliers for harmonic analysis in several variables |
| 47E99 | Ordinary differential operators |
Cite
References:
| [1] | Ayoush, R.; Stolyarov, D. M.; Wojciechowski, M., Sobolev martingales, Rev. Mat. Iberoam., 0, 0 (2020) |
| [2] | Ayoush, R.; Wojciechowski, M., On dimension and regularity of bundle measures (2017) |
| [3] | Besov, O. V.; Il’in, V. P.; Lizorkin, P. I., \(L_p\)-estimates of a certain class of nonisotropic singular integrals, Sov. Math., Dokl., 7, 1065-1069 (1966) ·Zbl 0158.12903 |
| [4] | Besov, O. V.; Il’in, V. P.; Nikol’skii, S. M., Integral Representations of Functions and Embedding Theorems (1996), Moscow: Nauka, Moscow ·Zbl 0352.46023 |
| [5] | Besov, O. V.; Lizorkin, P. I., Singular integral operators and sequences of convolutions in \(L_p\) spaces, Math. USSR, Sb., 2, 1, 57-76 (1967) ·Zbl 0167.41102 ·doi:10.1070/SM1967v002n01ABEH002324 |
| [6] | Bourgain, J.; Brezis, H., On the equation \(\operatorname{div} Y=f\) and application to control of phases, J. Am. Math. Soc., 16, 2, 393-426 (2002) ·Zbl 1075.35006 ·doi:10.1090/S0894-0347-02-00411-3 |
| [7] | Bourgain, J.; Brezis, H., New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc., 9, 2, 277-315 (2007) ·Zbl 1176.35061 ·doi:10.4171/JEMS/80 |
| [8] | Bousquet, P.; Schaftingen, J. Van, Hardy-Sobolev inequalities for vector fields and canceling linear differential operators, Indiana Univ. Math. J., 63, 5, 1419-1445 (2014) ·Zbl 1325.46037 ·doi:10.1512/iumj.2014.63.5395 |
| [9] | Fabes, E. B.; Riviere, N. M., Singular integrals with mixed homogeneity, Stud. Math., 27, 19-38 (1966) ·Zbl 0161.32403 ·doi:10.4064/sm-27-1-19-38 |
| [10] | Kislyakov, S. V., Sobolev imbedding operators and the nonisomorphism of certain Banach spaces, Funct. Anal. Appl., 9, 4, 290-294 (1975) ·Zbl 0329.46036 ·doi:10.1007/BF01075874 |
| [11] | Kislyakov, S. V.; Maksimov, D. V., An embedding theorem with anisotropy for vector fields, J. Math. Sci., 234, 3, 343-349 (2018) ·Zbl 1409.46027 ·doi:10.1007/s10958-018-4010-y |
| [12] | Kislyakov, S. V.; Maksimov, D. V.; Stolyarov, D. M., Differential expressions with mixed homogeneity and spaces of smooth functions they generate in arbitrary dimension, J. Funct. Anal., 269, 10, 3220-3263 (2015) ·Zbl 1348.46037 ·doi:10.1016/j.jfa.2015.09.001 |
| [13] | Maz’ya, V., Bourgain-Brezis type inequality with explicit constants, Interpolation Theory and Applications, 445, 247-252 (2007) ·Zbl 1143.46017 ·doi:10.1090/conm/445/08605 |
| [14] | Raiţă, B., -estimates for constant rank operators (2018) |
| [15] | Raiţă, B., Critical \(\text{L}^p\)-differentiability of \(\text{BV}^{\mathbb{A}} \)-maps and canceling operators, Trans. Am. Math. Soc., 372, 10, 7297-7326 (2019) ·Zbl 1429.26019 ·doi:10.1090/tran/7878 |
| [16] | Raiţă, B.; Skorobogatova, A., Continuity and canceling operators of order \(n\) on \(\mathbb{R}^n\), Calc. Var. Partial Diff. Eqns., 59, 2 (2020) ·Zbl 1443.47046 ·doi:10.1007/s00526-020-01739-z |
| [17] | Schaftingen, J. Van, Estimates for \(L^1\)-vector fields, C. R., Math., Acad. Sci. Paris, 339, 3, 181-186 (2004) ·Zbl 1049.35069 ·doi:10.1016/j.crma.2004.05.013 |
| [18] | Schaftingen, J. Van, Estimates for \(L^1\)-vector fields under higher-order differential conditions, J. Eur. Math. Soc., 10, 4, 867-882 (2008) ·Zbl 1228.46034 ·doi:10.4171/JEMS/133 |
| [19] | Schaftingen, J. Van, Limiting Sobolev inequalities for vector fields and canceling linear differential operators, J. Eur. Math. Soc., 15, 3, 877-921 (2013) ·Zbl 1284.46032 ·doi:10.4171/JEMS/380 |
| [20] | Schaftingen, J. Van, Limiting Bourgain-Brezis estimates for systems of linear differential equations: Theme and variations, J. Fixed Point Theory Appl., 15, 2, 273-297 (2014) ·Zbl 1311.35005 ·doi:10.1007/s11784-014-0177-0 |
| [21] | Stolyarov, D. M., Bilinear embedding theorems for differential operators in \(\mathbb{R}^2\), J. Math. Sci., 209, 5, 792-807 (2015) ·Zbl 1331.35016 ·doi:10.1007/s10958-015-2527-x |
| [22] | Stolyarov, D., Martingale interpretation of weakly cancelling differential operators, J. Math. Sci., 251, 2, 286-290 (2020) ·Zbl 1466.46030 ·doi:10.1007/s10958-020-05089-1 |
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