Estimates for the maximal singular integral in terms of the singular integral: the case of even kernels

Abstract

Let $T$ be a smooth homogeneous Calderón-Zygmund singular integral operator in $\mathbb{R}^n$. In this paper we study the problem of controlling the maximal singular integral $T^{\star}f$ by the singular integral $Tf$. The most basic form of control one may consider is the estimate of the $L^2(\mathbb{R}^n)$ norm of $T^{\star}f$ by a constant times the $L^2(\mathbb{R}^n)$ norm of $Tf$. We show that if $T$ is an even higher order Riesz transform, then one has the stronger pointwise inequality $T^{\star}f(x) \leq C \, M(Tf)(x)$, where $C$ is a constant and $M$ is the Hardy-Littlewood maximal operator. We prove that the $L^2$ estimate of $T^{\star}$ by $T$ is equivalent, for even smooth homogeneous Calderón-Zygmund operators, to the pointwise inequality between $T^{\star}$ and $M(T)$. Our main result characterizes the $L^2$ and pointwise inequalities in terms of an algebraic condition expressed in terms of the kernel $\frac{\Omega(x)}{|x|^n}$ of $T$, where $\Omega$ is an even homogeneous function of degree $0$, of class $C^\infty(S^{n-1})$ and with zero integral on the unit sphere $S^{n-1}$. Let $\Omega= \sum P_j$ be the expansion of $\Omega$ in spherical harmonics $P_j$ of degree $j$. Let $A$ stand for the algebra generated by the identity and the smooth homogeneous Calderón-Zygmund operators. Then our characterizing condition states that $T$ is of the form $R\circ U$, where $U$ is an invertible operator in $A$ and $R$ is a higher order Riesz transform associated with a homogeneous harmonic polynomial $P$ which divides each $P_j$ in the ring of polynomials in $n$~variables with real coefficients.

  • [ACL] N. Aronszajn, T. M. Creese, and L. J. Lipkin, Polyharmonic Functions, New York: The Clarendon Press Oxford University Press, 1983.
    @book {ACL, MRKEY = {0745128},
      AUTHOR = {Aronszajn, Nachman and Creese, Thomas M. and Lipkin, Leonard J.},
      TITLE = {Polyharmonic Functions},
      SERIES = {Oxford Math. Monogr.},
      PUBLISHER = {The Clarendon Press Oxford University Press},
      ADDRESS = {New York},
      YEAR = {1983},
      PAGES = {x+265},
      ISBN = {0-19-853906-1},
      MRCLASS = {31-02 (26E05 31B30 32Axx 35C99)},
      MRNUMBER = {0745128},
      MRREVIEWER = {E. Gerlach},
      ZBLNUMBER = {0514.31001},
      }
  • [BM] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge: Cambridge Univ. Press, 2002, vol. 27.
    @book {BM, MRKEY = {1867882},
      AUTHOR = {Majda, Andrew J. and Bertozzi, Andrea L.},
      TITLE = {Vorticity and Incompressible Flow},
      SERIES = {Cambridge Texts Appl. Math.},
      VOLUME = {27},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2002},
      PAGES = {xii+545},
      ISBN = {0-521-63057-6; 0-521-63948-4},
      MRCLASS = {76-02 (35Q30 35Q35 76B03 76D03 76D05)},
      MRNUMBER = {1867882},
      MRREVIEWER = {Yuxi Zheng},
      ZBLNUMBER = {0983.76001},
      }
  • [CZ] Go to document A. P. Calderón and A. Zygmund, "On a problem of Mihlin," Trans. Amer. Math. Soc., vol. 78, pp. 209-224, 1955.
    @article {CZ, MRKEY = {0068028},
      AUTHOR = {Calder{ó}n, A. P. and Zygmund, A.},
      TITLE = {On a problem of {M}ihlin},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {78},
      YEAR = {1955},
      PAGES = {209--224},
      ISSN = {0002-9947},
      MRCLASS = {42.4X},
      MRNUMBER = {0068028},
      MRREVIEWER = {F. Smithies},
      DOI = {10.2307/1992955},
      ZBLNUMBER = {0065.04104},
      }
  • [Ch] J. Chemin, "Fluides parfaits incompressibles," Astérisque, vol. 230, p. 177, 1995.
    @article {Ch, MRKEY = {1340046},
      AUTHOR = {Chemin, Jean-Yves},
      TITLE = {Fluides parfaits incompressibles},
      JOURNAL = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {230},
      YEAR = {1995},
      PAGES = {177},
      ISSN = {0303-1179},
      MRCLASS = {76C05 (35-02 35Q35 76-02)},
      MRNUMBER = {1340046},
      MRREVIEWER = {Denis Serre},
      ZBLNUMBER = {0829.76003},
      }
  • [DS] G. David and S. Semmes, Singular integrals and rectifiable sets in ${\mathbb R}^n$: Beyond Lipschitz graphs, Paris: Soc. Math. France, 1991, vol. 193.
    @book{DS,
      author={David, G. and Semmes, S.},
      TITLE={Singular integrals and rectifiable sets in ${\mathbb R}^n$: {B}eyond {L}ipschitz graphs},
      SERIES={Astérisque},
      VOLUME={193},
      PUBLISHER={Soc. Math. France},
      ADDRESS={Paris},
      YEAR={1991},
      MRNUMBER={1113517},
      ZBLNUMBER={0743.49018},
      }
  • [Gr] L. Grafakos, Classical and Modern Fourier Analysis, Upper Saddle River, NJ: Pearson Education, 2004.
    @book {Gr, MRKEY = {2449250},
      AUTHOR = {Grafakos, Loukas},
      TITLE = {Classical and Modern {F}ourier Analysis},
      PUBLISHER = {Pearson Education},
      ADDRESS={Upper Saddle River, NJ},
      YEAR = {2004},
      PAGES = {xii+931},
      ISBN = {0-13-035399-X},
      MRCLASS = {42-01},
      MRNUMBER = {2449250},
      ZBLNUMBER = {1148.42001},
      }
  • [GKP] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. A Foundation for Computer Science, Second ed., Reading, MA: Addison-Wesley Publishing Company, 1994.
    @book {GKP, MRKEY = {1397498},
      AUTHOR = {Graham, Ronald L. and Knuth, Donald E. and Patashnik, Oren},
      TITLE = {Concrete Mathematics. A Foundation for Computer Science},
      EDITION = {Second},
      PUBLISHER = {Addison-Wesley Publishing Company},
      ADDRESS = {Reading, MA},
      YEAR = {1994},
      PAGES = {xiv+657},
      ISBN = {0-201-55802-5},
      MRCLASS = {68-01 (00-01 00A05 05-01 68Rxx)},
      MRNUMBER = {1397498},
      MRREVIEWER = {Volker Strehl},
      ZBLNUMBER = {0836.00001},
      }
  • [K] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Boston, MA: Birkhäuser, 1985.
    @book {K, MRKEY = {0789602},
      AUTHOR = {Kunz, Ernst},
      TITLE = {Introduction to Commutative Algebra and Algebraic Geometry},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1985},
      PAGES = {xi+238},
      ISBN = {3-7643-3065-1},
      MRCLASS = {14-01 (13-01)},
      MRNUMBER = {0789602},
      ZBLNUMBER = {0563.13001},
      }
  • [LS] Go to document L. Lorch and P. Szego, "A singular integral whose kernel involves a Bessel function," Duke Math. J., vol. 22, pp. 407-418, 1955.
    @article {LS, MRKEY = {0087774},
      AUTHOR = {Lorch, Lee and Szego, Peter},
      TITLE = {A singular integral whose kernel involves a {B}essel function},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {22},
      YEAR = {1955},
      PAGES = {407--418},
      ISSN = {0012-7094},
      MRCLASS = {33.0X},
      MRNUMBER = {0087774},
      MRREVIEWER = {A. P. Calder{ó}n},
      DOI = {10.1215/S0012-7094-55-02244-4},
      ZBLNUMBER = {0066.05201},
      }
  • [Lo] G. G. Lorentz, Approximation of Functions, Second ed., New York: Chelsea Publishing Co., 1986.
    @book {Lo, MRKEY = {0917270},
      AUTHOR = {Lorentz, G. G.},
      TITLE = {Approximation of Functions},
      EDITION = {Second},
      PUBLISHER = {Chelsea Publishing Co.},
      ADDRESS = {New York},
      YEAR = {1986},
      PAGES = {x+188},
      ISBN = {0-8284-0322-8},
      MRCLASS = {41-01},
      MRNUMBER = {0917270},
      ZBLNUMBER = {0643.41001},
      }
  • [LZ] Go to document R. Lyons and K. Zumbrun, "Homogeneous partial derivatives of radial functions," Proc. Amer. Math. Soc., vol. 121, iss. 1, pp. 315-316, 1994.
    @article {LZ, MRKEY = {1227524},
      AUTHOR = {Lyons, Russell and Zumbrun, Kevin},
      TITLE = {Homogeneous partial derivatives of radial functions},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {121},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {315--316},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {26B05 (31B05 35A99)},
      MRNUMBER = {1227524},
      MRREVIEWER = {J. Luke{š}},
      DOI = {10.2307/2160399},
      ZBLNUMBER = {0815.26006},
      }
  • [MNOV] Go to document J. Mateu, Y. Netrusov, J. Orobitg, and J. Verdera, "BMO and Lipschitz approximation by solutions of elliptic equations," Ann. Inst. Fourier $($Grenoble$)$, vol. 46, iss. 4, pp. 1057-1081, 1996.
    @article {MNOV, MRKEY = {1415957},
      AUTHOR = {Mateu, Joan and Netrusov, Y. and Orobitg, J. and Verdera, J.},
      TITLE = {B{MO} and {L}ipschitz approximation by solutions of elliptic equations},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {46},
      YEAR = {1996},
      NUMBER = {4},
      PAGES = {1057--1081},
      ISSN = {0373-0956},
      CODEN = {AIFUA7},
      MRCLASS = {41A30 (31C15 35J30)},
      MRNUMBER = {1415957},
      MRREVIEWER = {Juan Carlos Fari{ñ}a Gil},
      URL = {http://www.numdam.org/item?id=AIF_1996__46_4_1057_0},
      ZBLNUMBER = {0853.31007},
      }
  • [MO] Go to document J. Mateu and J. Orobitg, "Lipschitz approximation by harmonic functions and some applications to spectral synthesis," Indiana Univ. Math. J., vol. 39, iss. 3, pp. 703-736, 1990.
    @article {MO, MRKEY = {1078735},
      AUTHOR = {Mateu, Joan and Orobitg, Joan},
      TITLE = {Lipschitz approximation by harmonic functions and some applications to spectral synthesis},
      JOURNAL = {Indiana Univ. Math. J.},
      FJOURNAL = {Indiana University Mathematics Journal},
      VOLUME = {39},
      YEAR = {1990},
      NUMBER = {3},
      PAGES = {703--736},
      ISSN = {0022-2518},
      CODEN = {IUMJAB},
      MRCLASS = {46E15 (31B05)},
      MRNUMBER = {1078735},
      DOI = {10.1512/iumj.1990.39.39035},
      ZBLNUMBER = {0768.46006},
      }
  • [MOV] Go to document J. Mateu, J. Orobitg, and J. Verdera, "Extra cancellation of even Calderón-Zygmund operators and quasiconformal mappings," J. Math. Pures Appl., vol. 91, iss. 4, pp. 402-431, 2009.
    @article {MOV, MRKEY = {2518005},
      AUTHOR = {Mateu, Joan and Orobitg, Joan and Verdera, Joan},
      TITLE = {Extra cancellation of even {C}alderón-{Z}ygmund operators and quasiconformal mappings},
      JOURNAL = {J. Math. Pures Appl.},
      FJOURNAL = {Journal de Mathématiques Pures et Appliquées. Neuvième Série},
      VOLUME = {91},
      YEAR = {2009},
      NUMBER = {4},
      PAGES = {402--431},
      ISSN = {0021-7824},
      CODEN = {JMPAAM},
      MRCLASS = {42B20 (30C62)},
      MRNUMBER = {2518005},
      MRREVIEWER = {Caroline P. Sweezy},
      DOI = {10.1016/j.matpur.2009.01.010},
      ZBLNUMBER = {1179.30017},
      }
  • [MOPV] Go to document J. Mateu, J. Orobitg, C. Pérez, and J. Verdera, "New estimates for the maximal singular integral," Int. Math. Res. Not., vol. 2010, iss. 19, pp. 3658-3722, 2010.
    @article {MOPV, MRKEY = {2725509},
      AUTHOR = {Mateu, Joan and Orobitg, Joan and P{é}rez, Carlos and Verdera, Joan},
      TITLE = {New estimates for the maximal singular integral},
      JOURNAL = {Int. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices. IMRN},
      YEAR = {2010},
      NUMBER = {19},
      PAGES = {3658--3722},
      ISSN = {1073-7928},
      MRCLASS = {42B35 (42B25)},
      MRNUMBER = {2725509},
      VOLUME = {2010},
      ZBLNUMBER = {1208.42005},
      DOI = {10.1093/imrn/rnq017},
      URL={http://rmi.rsme.es/index.php?option=com_docman&task=doc_details&gid=13 1&Itemid=91&lang=en},
      }
  • [MPV] Go to document J. Mateu, L. Prat, and J. Verdera, "The capacity associated to signed Riesz kernels, and Wolff potentials," J. Reine Angew. Math., vol. 578, pp. 201-223, 2005.
    @article {MPV, MRKEY = {2113895},
      AUTHOR = {Mateu, Joan and Prat, Laura and Verdera, Joan},
      TITLE = {The capacity associated to signed {R}iesz kernels, and {W}olff potentials},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {578},
      YEAR = {2005},
      PAGES = {201--223},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {31B15 (31C45)},
      MRNUMBER = {2113895},
      MRREVIEWER = {Jana Bj{ö}rn},
      DOI = {10.1515/crll.2005.2005.578.201},
      ZBLNUMBER = {1086.31005},
      }
  • [MV1] J. Mateu and J. Verdera, "BMO harmonic approximation in the plane and spectral synthesis for Hardy-Sobolev spaces," Rev. Mat. Iberoamericana, vol. 4, iss. 2, pp. 291-318, 1988.
    @article {MV1, MRKEY = {1028743},
      AUTHOR = {Mateu, Joan and Verdera, Joan},
      TITLE = {B{MO} harmonic approximation in the plane and spectral synthesis for {H}ardy-{S}obolev spaces},
      JOURNAL = {Rev. Mat. Iberoamericana},
      FJOURNAL = {Revista Matemática Iberoamericana},
      VOLUME = {4},
      YEAR = {1988},
      NUMBER = {2},
      PAGES = {291--318},
      ISSN = {0213-2230},
      MRCLASS = {42B30 (46E30)},
      MRNUMBER = {1028743},
      MRREVIEWER = {Weixing Zheng},
      ZBLNUMBER = {0702.31001},
      }
  • [MV2] Go to document J. Mateu and J. Verdera, "$L^p$ and weak $L^1$ estimates for the maximal Riesz transform and the maximal Beurling transform," Math. Res. Lett., vol. 13, iss. 5-6, pp. 957-966, 2006.
    @article {MV2, MRKEY = {2280788},
      AUTHOR = {Mateu, Joan and Verdera, Joan},
      TITLE = {{$L\sp p$} and weak {$L\sp 1$} estimates for the maximal {R}iesz transform and the maximal {B}eurling transform},
      JOURNAL = {Math. Res. Lett.},
      FJOURNAL = {Mathematical Research Letters},
      VOLUME = {13},
      YEAR = {2006},
      NUMBER = {5-6},
      PAGES = {957--966},
      ISSN = {1073-2780},
      MRCLASS = {42B25},
      MRNUMBER = {2280788},
      MRREVIEWER = {P. K. Ratnakumar},
      ZBLNUMBER = {1134.42322},
      URL={http://www.mrlonline.org/mrl/2006-013-006/2006-013-006-010.html},
      }
  • [MaV] Go to document P. Mattila and J. Verdera, "Convergence of singular integrals with general measures," J. Eur. Math. Soc. $($JEMS$)$, vol. 11, iss. 2, pp. 257-271, 2009.
    @article {MaV, MRKEY = {2486933},
      AUTHOR = {Mattila, Pertti and Verdera, Joan},
      TITLE = {Convergence of singular integrals with general measures},
      JOURNAL = {J. Eur. Math. Soc. $($JEMS$)$},
      FJOURNAL = {Journal of the European Mathematical Society (JEMS)},
      VOLUME = {11},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {257--271},
      ISSN = {1435-9855},
      MRCLASS = {42B20},
      MRNUMBER = {2486933},
      DOI = {10.4171/JEMS/149},
      ZBLNUMBER = {1163.42005},
      }
  • [RS] Go to document F. Ricci and E. M. Stein, "Harmonic analysis on nilpotent groups and singular integrals. I. Oscillatory integrals," J. Funct. Anal., vol. 73, iss. 1, pp. 179-194, 1987.
    @article {RS, MRKEY = {0890662},
      AUTHOR = {Ricci, Fulvio and Stein, E. M.},
      TITLE = {Harmonic analysis on nilpotent groups and singular integrals. {I}. {O}scillatory integrals},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {73},
      YEAR = {1987},
      NUMBER = {1},
      PAGES = {179--194},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {42B20 (22E30 43A80 58G15)},
      MRNUMBER = {0890662},
      MRREVIEWER = {Detlef H. M{ü}ller},
      DOI = {10.1016/0022-1236(87)90064-4},
      ZBLNUMBER={0622.42010},
      }
  • [St] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton, N.J.: Princeton Univ. Press, 1970, vol. 30.
    @book {St, MRKEY = {0290095},
      AUTHOR = {Stein, Elias M.},
      TITLE = {Singular Integrals and Differentiability Properties of Functions},
      SERIES = {Princeton Math. Series},
      VOLUME={30},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, N.J.},
      YEAR = {1970},
      PAGES = {xiv+290},
      MRCLASS = {46.38 (26.00)},
      MRNUMBER = {0290095},
      MRREVIEWER = {R. E. Edwards},
      ZBLNUMBER = {0207.13501},
      }
  • [SW] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton Univ. Press, 1971, vol. 32.
    @book {SW, MRKEY = {0304972},
      AUTHOR = {Stein, Elias M. and Weiss, Guido},
      TITLE = {Introduction to {F}ourier Analysis on {E}uclidean Spaces},
      SERIES = {Princeton Math. Series},
      VOLUME={32},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, N.J.},
      YEAR = {1971},
      PAGES = {x+297},
      MRCLASS = {42A92 (31B99 32A99 46F99 47G05)},
      MRNUMBER = {0304972},
      MRREVIEWER = {Edwin Hewitt},
      ZBLNUMBER = {0232.42007},
      }
  • [Ve1] Go to document J. Verdera, "$C^m$ approximation by solutions of elliptic equations, and Calderón-Zygmund operators," Duke Math. J., vol. 55, iss. 1, pp. 157-187, 1987.
    @article {Ve1, MRKEY = {0883668},
      AUTHOR = {Verdera, Joan},
      TITLE = {{$C\sp m$} approximation by solutions of elliptic equations, and {C}alderón-{Z}ygmund operators},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {55},
      YEAR = {1987},
      NUMBER = {1},
      PAGES = {157--187},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {35A35 (35J30)},
      MRNUMBER = {0883668},
      MRREVIEWER = {V. S. Rabinovich},
      DOI = {10.1215/S0012-7094-87-05509-8},
      ZBLNUMBER = {0654.35007},
      }
  • [Ve2] J. Verdera, "$L^2$ boundedness of the Cauchy integral and Menger curvature," in Harmonic Analysis and Boundary Value Problems, Providence, RI: Amer. Math. Soc., 2001, vol. 277, pp. 139-158.
    @incollection {Ve2, MRKEY = {1840432},
      AUTHOR = {Verdera, Joan},
      TITLE = {{$L\sp 2$} boundedness of the {C}auchy integral and {M}enger curvature},
      BOOKTITLE = {Harmonic Analysis and Boundary Value Problems},
      VENUE={{F}ayetteville, {AR},
      2000},
      SERIES = {Contemp. Math.},
      VOLUME = {277},
      PAGES = {139--158},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2001},
      MRCLASS = {30E20 (30E25)},
      MRNUMBER = {1840432},
      MRREVIEWER = {N. V. Rao},
      ZBLNUMBER = {1002.42011},
      }

Authors

Joan Mateu

Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra (Barcelona)
Catalonia

Joan Orobitg

Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra (Barcelona)
Catalonia

Joan Verdera

Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra (Barcelona)
Catalonia