Dyadic John-Nirenberg Spaces: Predual Spaces and Fractional Dyadic Maximal Operators

Abstract

Let

\mathcal{X}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{X}$$\end{document}

represent either ℝⁿ or a cube Q of ℝⁿ with finite edge length. The authors prove that, for any p ∈ (1, ∞), the predual of the dyadic John-Nirenberg space

J{N}_p^{\mathrm{d}}(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$JN_{p}^{{\rm{d}}}({\cal{X}})$$\end{document}

is a dyadic Hardy-type space and then establish some relationships among dyadic John-Nirenberg spaces, dyadic Hardy-type spaces, and their non-dyadic counterparts. Moreover, the authors prove that, for any α ∈ [0, n) and p, q ∈ (1, ∞) with

\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${1 \over q} = {1 \over p} - {\alpha \over n}$$\end{document}

, the fractional dyadic maximal operator

M_{\mathcal{X}}^{\mathrm{d},\alpha }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M_{{\cal{X}}}^{{\rm{d}},\alpha}$$\end{document}

is bounded from

J{N}_p^{\mathrm{d}}(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$JN_{p}^{\rm{d}}(\cal{X})$$\end{document}

to

J{N}_q^{\mathrm{d}}(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$JN_{q}^{\rm{d}}(\cal{X})$$\end{document}

and maps

VJ{N}_p^{\mathrm{d}}(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$VJN_{p}^{\rm{d}}(\cal{X})$$\end{document}

to

VJ{N}_q^{\mathrm{d}}(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$VJN_{q}^{\rm{d}}(\cal{X})$$\end{document}

, where

VJ{N}_p^{\mathrm{d}}(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$VJN_{p}^{\rm{d}}(\cal{X})$$\end{document}

and

VJ{N}_q^{\mathrm{d}}(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$VJN_{q}^{\rm{d}}(\cal{X})$$\end{document}

are, respectively, the vanishing sub-spaces of

J{N}_p^{\mathrm{d}}(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$JN_{p}^{\rm{d}}(\cal{X})$$\end{document}

and

J{N}_q^{\mathrm{d}}(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$JN_{q}^{\rm{d}}(\cal{X})$$\end{document}

. As for the endpoint case p = 1 and α ∈ (0, n), the authors show

M_{\mathcal{X}}^{\mathrm{d},\alpha }:J{N}_1^{\mathrm{d}}(\mathcal{X})\to J{N}_{\frac{n}{n-\alpha }}^{\mathrm{d}}(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M_{\cal{X}}^{{\rm{d,}}\alpha}:JN_1^{\rm{d}}(\cal{X}) \to JN_{{n \over {n - \alpha}}}^{\rm{d}}(\cal{X})$$\end{document}

, which improves the classical weak-type result

M_{\mathcal{X}}^{\mathrm{d},\alpha }:L^1(\mathcal{X})\to L^{\frac{n}{n-\alpha },\mathrm{\infty }}(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M_{\cal{X}}^{{\rm{d}},\alpha}:{L^1}(\cal{X}) \to {L^{{n \over {n - \alpha}},\infty}}(\cal{X})$$\end{document}

because

J{N}_1^{\mathrm{d}}(Q)=L^1(Q)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$JN_{1}^{\rm{d}}(Q)=L^{1}(Q)$$\end{document}

and [inline-graphic not available: see fulltext]. Compared with recent corresponding results of J. Kinnunen and K. Myyryläinen, both the case

\mathcal{X}={\mathbb{R}}^n

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{X}=\mathbb{R}^{n}$$\end{document}

and the mapping result on vanishing subspaces are new. Also, an interesting phenomenon in the endpoint case p = 1 is that

M_{\mathcal{X}}^{\mathrm{d},\alpha }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M_{\cal{X}}^{\rm{d},\alpha}$$\end{document}

maps

J{N}_1^{\mathrm{d}}(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$JN_{1}^{\rm{d}}(\cal{X})$$\end{document}

to the convexification of the dyadic John-Nirenberg space when α = 0, but the convexification disappears when α > 0.

Keywords

Cube / dyadic John-Nirenberg space / fractional dyadic maximal operator / duality / 42B35 / 42B25 / 46E30

References

[1] Aalto D, Berkovits L, Kansanen OE, Yue H. John-Nirenberg lemmas for a doubling measure. Studia Math.. 2011, 204(1): 21-37.

[2] Airta E, Hytönen T, Li K, Martikainen H, Oikari T. Off-diagonal estimates for bicommutators. Int. Math. Res. Not. IMRN. 2022, 2022(23): 18766-18832.

[3] Berkovits L, Kinnunen J, Martell JM. Oscillation estimates, self-improving results and good-λ inequalities. J. Funct. Anal.. 2016, 270(9): 3559-3590.

[4] Betancor J.J., Dalmasso E., Quijano P., Gaussian JN_p spaces. Complex Anal. Oper. Theory, 2025, 19 (5): Paper No. 107, 31 pp.

[5] Blasco O, Espinoza-Villalva C. The norm of the characteristic function of a set in the John-Nirenberg space of exponent p. Math. Methods Appl. Sci.. 2020, 43(16): 9327-9336.

[6] Brudnyi A, Brudnyi Y. On the Banach structure of multivariate BV spaces. Dissertationes Math.. 2020, 548: 1-52.

[7] Campanato S. Su un teorema di interpolazione di G. Stampacchia. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3). 1996, 20: 649-652

[8] Chang D-C, Sadosky C. Functions of bounded mean oscillation. Taiwanese J. Math.. 2006, 10(3): 573-601.

[9] Dafni G, Hytönen T, Korte R, Yue H. The space JN_p: nontriviality and duality. J. Funct. Anal.. 2018, 275(3): 577-603.

[10] Di Plinio F, Hytöonen T, Li K. Sparse bounds for maximal rough singular integrals via the Fourier transform. Ann. Inst. Fourier (Grenoble). 2020, 70(5): 1871-1902.

[11] Domínguez Ó, Milman M. Sparse Brudnyi and John-Nirenberg spaces. C. R. Math. Acad. Sci. Paris. 2021, 359: 1059-1069.

[12] Hurri-Syrjänen R, Marola N, Vähäkangas AV. Aspects of local-to-global results. Bull. Lond. Math. Soc.. 2014, 46(5): 1032-1042.

[13] Hytänen T. The sharp weighted bound for general Calderón-Zygmund operators. Ann. of Math. (2). 2012, 175(3): 1473-1506.

[14] Hytönen T. The L^p-to-L^q boundedness of commutators with applications to the Jacobian operator. J. Math. Pures Appl. (9). 2021, 156: 351-391.

[15] Hytöonen T., Some remarks on convex body domination. J. Math. Anal. Appl., 2024, 529 (2): Paper No. 127565, 26 pp.

[16] Hytöonen T, Rosén A. Causal sparse domination of Beurling maximal regularity operators. J. Anal. Math.. 2023, 150(2): 645-672.

[17] Jia H., Tao J., Yang D., Yuan W., Zhang Y., Boundedness of Calderón-Zygmund operators on special John-Nirenberg-Campanato and Hardy-type spaces via congruent cubes. Anal. Math. Phys., 2022, 12 (1): Paper No. 15, 56 pp.

[18] Jia H, Tao J, Yang D, Yuan W, Zhang Y. Special John-Nirenberg-Campanato spaces via congruent cubes. Sci. China Math.. 2022, 65(2): 359-420.

[19] Jia H, Tao J, Yang D, Yuan W, Zhang Y. Boundedness of fractional integrals on special John-Nirenberg-Campanato and Hardy-type spaces via congruent cubes. Fract. Calc. Appl. Anal.. 2022, 25(6): 2446-2487.

[20] John F, Nirenberg L. On functions of bounded mean oscillation. Comm. Pure Appl. Math.. 1961, 14: 415-426.

[21] Kawasumi R, Nakai E, Shi M. Characterization of the boundedness of generalized fractional integral and maximal operators on Orlicz-Morrey and weak Orlicz-Morrey spaces. Math. Nachr.. 2023, 296(4): 1483-1503.

[22] Kinnunen J, Myyrylöainen K. Dyadic John-Nirenberg space. Proc. Roy. Soc. Edinburgh Sect. A. 2023, 153(1): 1-18.

[23] Korte R., Takala T., The John-Nirenberg space: equality of the vanishing subspaces VJN_P and CJN_P. J. Geom. Anal., 2024, 34 (3): Paper No. 67, 19 pp.

[24] Lerner AK. A simple proof of the A₂ conjecture. Int. Math. Res. Not. IMRN. 2013, 14: 3159-3170.

[25] Marola N, Saari O. Local to global results for spaces of BMO type. Math. Z.. 2016, 282(1–2): 473-484.

[26] Milman M. Marcinkiewicz spaces, Garsia-Rodemich spaces and the scale of John-Nirenberg self improving inequalities. Ann. Acad. Sci. Fenn. Math.. 2016, 41(1): 491-501.

[27] Molla M., John-Nirenberg space on LCA groups. Anal. Math. Phys., 2022, 12 (5): Paper No. 119, 19 pp.

[28] Myyrylöainen K., Median-type John-Nirenberg space in metric measure spaces. J. Geom. Anal., 2022, 32 (4): Paper No. 131, 23 pp.

[29] Myyrylöainen K., Yang D., Parabolic John-Nirenberg spaces with time lag. Math. Z., 2024, 306 (3): Paper No. 46, 20 pp.

[30] Riesz F. Untersuchungen über systeme integrierbarer funktionen. Math. Ann.. 1910, 69(4): 449-497.

[31] Shaabani S., A view from above on JN_p(ℝⁿ). J. Funct. Anal., 2026, 290 (3): Paper No. 111235, 25 pp.

[32] Stampacchia G. The spaces L^(p,λ), N^(p,λ) and interpolation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3). 1965, 19: 443-462

[33] Takala T. Nontrivial examples of JN_p and VJN_p functions. Math. Z.. 2022, 302(2): 1279-1305.

[34] Tao J, Yang D, Yuan W. John–Nirenberg–Campanato spaces. Nonlinear Anal.. 2019, 189: 111584. 36 pp

[35] Tao J., Yang D., Yuan W., A bridge connecting Lebesgue and Morrey spaces via Riesz norms. Banach J. Math. Anal., 2021, 15 (1): Paper No. 20, 29 pp.

[36] Tao J., Yang D., Yuan W., A survey on function spaces of John–Nirenberg type. Mathematics, 2021, 9 (18): Article No. 2264, 57 pp.

[37] Tao J, Yang D, Yuan W. Vanishing John–Nirenberg spaces. Adv. Calc. Var.. 2022, 15(4): 831-861.

[38] Tao J, Yang Z, Yuan W. John–Nirenberg-Q spaces via congruent cubes. Acta Math. Sci. Ser. B (Engl. Ed.). 2023, 432686-718

[39] Zeng Z., Chang D.-C., Tao J., Yang D., Nontriviality of John–Nirenberg–Campanato spaces. Complex Anal. Oper. Theory, 2023, 17 (5): Paper No. 70, 47 pp.

[40] Zhang C. Two weighted estimates for the commutators of multilinear maximal singular integral operators with Dini-type kernels. Acta Math. Sin. (Engl. Ser.). 2023, 39(3): 459-480.

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