Hardy-Littlewood-Sobolev inequality for $p=1$

D. M. Stolyarov

Sbornik. Mathematics, Tome 213 (2022) no. 6, pp. 844-889

Résumé

Let $\mathcal{W}$ be a closed dilation and translation invariant subspace of the space of $\mathbb{R}^\ell$-valued Schwartz distributions in $d$ variables. We show that if the space $\mathcal{W}$ does not contain distributions of the type $a\otimes \delta_0$, $\delta_0$ being the Dirac delta, then the inequality $\|\operatorname{I}_\alpha [f]\|_{L_{d/(d-\alpha),1}}\lesssim \|f\|_{L_1}$ holds true for functions $f\in\mathcal{W}\cap L_1$ with a uniform constant; here $\operatorname{I}_\alpha$ is the Riesz potential of order $\alpha$ and $L_{p,1}$ is the Lorentz space. As particular cases, this result implies the inequality $\|\nabla^{m-1} f\|_{L_{d/(d-1),1}} \lesssim \|A f\|_{L_1}$, where $A$ is a cancelling elliptic differential operator of order $m$, and the inequality $\|\operatorname{I}_\alpha f\|_{L_{d/(d-\alpha),1}} \lesssim \|f\|_{L_1}$, where $f$ is a divergence free vector field. Bibliography: 59 titles.

Détail

Keywords: Hardy-Littlewood-Sobolev inequality, Bourgain-Brezis inequalities, cancelling differential operators.

Bibliographie

[1] D. R. Adams and L. I. Hedberg, Function spaces and potential theory, Grundlehren Math. Wiss., 314, Springer-Verlag, Berlin, 1996, xii+366 pp. | DOI | MR | Zbl

[2] A. Alvino, “Sulla diseguaglianza di Sobolev in spazi di Lorentz”, Boll. Un. Mat. Ital. A (5), 14:1 (1977), 148–156 | MR | Zbl

[3] A. Arroyo-Rabasa, G. De Philippis, J. Hirsch and F. Rindler, “Dimensional estimates and rectifiability for measures satisfying linear PDE constraints”, Geom. Funct. Anal., 29:3 (2019), 639–658 | DOI | MR | Zbl

[4] R. Ayoush, D. M. Stolyarov and M. Wojciechowski, “Sobolev martingales”, Rev. Mat. Iberoam., 37:4 (2021), 1225–1246 | DOI | MR | Zbl

[5] R. Ayoush and M. Wojciechowski, On dimension and regularity of bundle measures, arXiv: 1708.01458

[6] J. Bennett, A. Carbery and T. Tao, “On the multilinear restriction and Kakeya conjectures”, Acta Math., 196:2 (2006), 261–302 | DOI | MR | Zbl

[7] O. V. Besov and V. P. Il'in, “An embedding theorem for a limiting exponent”, Mat. Zametki, 6:2 (1969), 129–138 ; English transl. in Math. Notes, 6:2 (1969), 537–542 | MR | Zbl | DOI

[8] O. V. Besov, V. P. Il'in and S. M. Nikol'skii, Integral representations of functions and imbedding theorems, 2nd ed., rev. and compl., Nauka, Moscow, 1996, 480 pp. ; English transl. of 1st ed., v. I, II, Scripta Series in Mathematics, V. H. Winston Sons, Washington, DC; Halsted Press [John Wiley Sons], New York–Toronto, ON–London, 1978, 1979, viii+345 pp., viii+311 pp. | MR | Zbl | MR | MR | Zbl

[9] J. Bourgain, A Hardy inequality in Sobolev spaces, Vrije Univ., Brussels, 1981

[10] J. Bourgain and H. Brezis, “On the equation $\operatorname{div} Y = f$ and application to control of phases”, J. Amer. Math. Soc., 16:2 (2003), 393–426 | DOI | MR | Zbl

[11] J. Bourgain and H. Brezis, “New estimates for the Laplacian, the div–curl, and related Hodge systems”, C. R. Math. Acad. Sci. Paris, 338:7 (2004), 539–543 | DOI | MR | Zbl

[12] J. Bourgain and H. Brezis, “New estimates for elliptic equations and Hodge type systems”, J. Eur. Math. Soc. (JEMS), 9:2 (2007), 277–315 | DOI | MR | Zbl

[13] J. Bourgain, H. Brezis and P. Mironescu, “$H^{1/2}$ maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation”, Publ. Math. Inst. Hautes Études Sci., 99 (2004), 1–115 | DOI | MR | Zbl

[14] P. Bousquet and J. Van Schaftingen, “Hardy-Sobolev inequalities for vector fields and canceling linear differential operators”, Indiana Univ. Math. J., 63:5 (2014), 1419–1445 | DOI | MR | Zbl

[15] S. Chanillo, J. Van Schaftingen and Po-Lam Yung, “Bourgain-Brezis inequalities on symmetric spaces of non-compact type”, J. Funct. Anal., 273:4 (2017), 1504–1547 | DOI | MR | Zbl

[16] E. Gagliardo, “Ulteriori proprieta di alcune classi di funzioni in piu variabili”, Ricerche Mat., 8 (1959), 24–51 | MR | Zbl

[17] F. Gmeineder, B. Raita and J. Van Schaftingen, “On limiting trace inequalities for vectorial differential operators”, Indiana Univ. Math. J., 70:5 (2021), 2133–2176 | DOI | MR | Zbl

[18] L. Grafakos, Modern Fourier analysis, 2nd ed., Grad. Texts in Math., 250, Springer, New York, 2009, xvi+504 pp. | DOI | MR | Zbl

[19] G. H. Hardy and J. E. Littlewood, “Some new properties of Fourier constants”, Math. Ann., 97:1 (1927), 159–209 | DOI | MR | Zbl

[20] F. Hernandez and D. Spector, Fractional integration and optimal estimates for elliptic systems, arXiv: 2008.05639

[21] S. Janson, “Characterizations of $H^1$ by singular integral transforms on martingales and $R^n$”, Math. Scand., 41:1 (1977), 140–152 | DOI | MR | Zbl

[22] S. V. Kislyakov, “Sobolev imbedding operators and the nonisomorphism of certain Banach spaces”, Funktsional. Anal. i Prilozhen., 9:4 (1975), 22–27 ; English transl. in Funct. Anal. Appl., 9:4 (1975), 290–294 | MR | Zbl | DOI

[23] S. V. Kislyakov and D. V. Maksimov, “An embedding theorem with anisotropy for vector fields”, Investigations on linear operators and function theory. Part 45, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 456, St Petersburg Department of the Steklov Mathematical Institute, St Petersburg, 2017, 114–124 ; English transl. in J. Math. Sci. (N.Y.), 234:3 (2018), 343–349 | MR | Zbl | DOI

[24] S. V. Kislyakov, D. V. Maximov and D. M. Stolyarov, “Differential expressions with mixed homogeneity and spaces of smooth functions they generate in arbitrary dimension”, J. Funct. Anal., 269:10 (2015), 3220–3263 | DOI | MR | Zbl

[25] V. I. Kolyada, “On an embedding of Sobolev spaces”, Mat. Zametki, 54:3 (1993), 48–71 ; English transl. in Math. Notes, 54:3 (1993), 908–922 | MR | Zbl | DOI

[26] V. I. Kolyada, “Embedding theorems for Sobolev and Hardy-Sobolev spaces and estimates of Fourier transforms”, Ann. Mat. Pura Appl. (4), 198:2 (2019), 615–637 | DOI | MR | Zbl

[27] L. Lanzani and E. M. Stein, “A note on div curl inequalities”, Math. Res. Lett., 12:1 (2005), 57–61 | DOI | MR | Zbl

[28] J. Lindenstrauss and A. Pełczyński, “Absolutely summing operators in $\mathscr L_p$-spaces and their applications”, Studia Math., 29:3 (1968), 275–326 | DOI | MR | Zbl

[29] V. Maz'ya, “Bourgain-Brezis type inequality with explicit constants”, Interpolation theory and applications, Contemp. Math., 445, Amer. Math. Soc., Providence, RI, 2007, 247–252 | DOI | MR | Zbl

[30] V. Maz'ya, “Estimates for differential operators of vector analysis involving $L^1$-norm”, J. Eur. Math. Soc. (JEMS), 12:1 (2010), 221–240 | DOI | MR | Zbl

[31] V. Maz'ya, Sobolev spaces, Leningrad University Publishing House, Leningrad, 1985, 416 pp. ; English transl., V. Maz'ya, Sobolev spaces, With applications to elliptic partial differential equations, 2nd rev. and augm. ed., Grundlehren Math. Wiss., 342, Springer, Heidelberg, 2011, xxviii+866 pp. | MR | Zbl | DOI | MR | Zbl

[32] L. Nirenberg, “On elliptic partial differential equations”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 13:2 (1959), 115–162 | MR | Zbl

[33] J. Peetre, New thoughts on Besov spaces, Duke Univ. Math. Ser., 1, Math. Department, Duke Univ., Durham, NC, 1976, vi+305 pp. | MR | Zbl

[34] A. Pełczyński and M. Wojciechowski, “Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm”, Studia Math., 107:1 (1993), 61–100 | MR | Zbl

[35] S. Poornima, “An embedding theorem for the Sobolev space $W^{1,1}$”, Bull. Sci. Math. (2), 107:3 (1983), 253–259 | MR | Zbl

[36] D. Preiss, “Geometry of measures in $\mathbf R^n$: distribution, rectifiability, and densities”, Ann. of Math. (2), 125:3 (1987), 537–643 | DOI | MR | Zbl

[37] B. Raiţă, $L^1$-estimates for constant rank operators, arXiv: 1811.10057

[38] M. Roginskaya and M. Wojciechowski, “Singularity of vector valued measures in terms of Fourier transform”, J. Fourier Anal. Appl., 12:2 (2006), 213–223 | DOI | MR | Zbl

[39] J. Van Schaftingen, “Estimates for $L^1$-vector fields”, C. R. Math. Acad. Sci. Paris, 339:3 (2004), 181–186 | DOI | MR | Zbl

[40] J. Van Schaftingen, “A simple proof of an inequality of Bourgain, Brezis and Mironescu”, C. R. Math. Acad. Sci. Paris, 338:1 (2004), 23–26 | DOI | MR | Zbl

[41] J. Van Schaftingen, “Limiting fractional and Lorentz space estimates of differential forms”, Proc. Amer. Math. Soc., 138:1 (2010), 235–240 | DOI | MR | Zbl

[42] J. Van Schaftingen, “Limiting Sobolev inequalities for vector fields and canceling linear differential operators”, J. Eur. Math. Soc. (JEMS), 15:3 (2013), 877–921 | DOI | MR | Zbl

[43] J. Van Schaftingen, “Limiting Bourgain-Brezis estimates for systems of linear differential equations: theme and variations”, J. Fixed Point Theory Appl., 15:2 (2014), 273–297 | DOI | MR | Zbl

[44] S. K. Smirnov, “Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currents”, Algebra i Analiz, 5:4 (1993), 206–238 ; English transl. in St. Petersburg Math. J., 5:4 (1994), 841–867 | MR | Zbl

[45] S. L. Sobolev, “On a theorem of functional analysis”, Mat. Sb., 4(46):3 (1938), 471–497 ; English transl. in Amer. Math. Soc. Transl. Ser. 2, 34, Amer. Math. Soc., Providence, RI, 1963, 39–68 | Zbl | DOI

[46] V. A. Solonnikov, “Inequalities for functions of the classes $\vec W_{p}(R^n)$”, Boundary-value problems of mathematical physics and related problems of function theory. Part 6, Zap. Nauchn. Sem. LOMI, 27, Nauka, Leningrad. Otdel., Leningrad, 1972, 194–210 ; English transl. in J. Soviet Math., 3 (1975), 549–564 | MR | Zbl | DOI

[47] D. Spector, “New directions in harmonic analysis on $L^1$”, Nonlinear Anal., 192 (2020), 111685, 20 pp. | DOI | MR | Zbl

[48] D. Spector, “An optimal Sobolev embedding for $L^1$”, J. Funct. Anal., 279:3 (2020), 108559, 26 pp. | DOI | MR | Zbl

[49] D. Spector and J. Van Schaftingen, “Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo's lemma”, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 30:3 (2019), 413–436 | DOI | MR | Zbl

[50] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Ser., 30, Princeton Univ. Press, Princeton, NJ, 1970, xiv+290 pp. | MR | Zbl

[51] D. Stolyarov, Dimension estimates for vectorial measures with restricted spectrum, arXiv: 2010.14961

[52] D. Stolyarov, Hardy-Littlewood-Sobolev inequality for $p=1$, arXiv: 2010.05297

[53] D. M. Stolyarov, “Weakly canceling operators and singular integrals”, Tr. Mat. Inst. Steklova, 312, Function Spaces, Approximation Theory, and Related Problems of Analysis. (2021), 259–271 ; English transl. in Proc. Steklov Inst. Math., 312 (2021), 249–260 | DOI | MR | Zbl | DOI

[54] D. M. Stolyarov and M. Wojciechowski, “Dimension of gradient measures”, C. R. Math. Acad. Sci. Paris, 352:10 (2014), 791–795 | DOI | MR | Zbl

[55] M. J. Strauss, “Variations of Korn's and Sobolev's equalities”, Partial differential equations (Univ. California, Berkeley, CA 1971), Proc. Sympos. Pure Math., 23, Amer. Math. Soc., Providence, RI, 1973, 207–214 | DOI | MR | Zbl

[56] T. Tao, Uchiyama's constructive proof of the Fefferman-Stein decomposition, 2007 https://terrytao.wordpress.com/2007/02/23/

[57] T. Tao, Symmetric functions in a fractional number of variables, and the multilinear Kakeya conjecture, 2019 https://terrytao.wordpress.com/2019/06/

[58] L. Tartar, “Imbedding theorems of Sobolev spaces into Lorentz spaces”, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1:3 (1998), 479–500 | MR | Zbl

[59] A. Uchiyama, “A constructive proof of the Fefferman-Stein decomposition of $\operatorname{BMO}(\mathbf R^n)$”, Acta Math., 148 (1982), 215–241 | DOI | MR | Zbl