Journal of the Mathematical Society of Japan

Littlewood-Paley theory for variable exponent Lebesgue spaces and Gagliardo-Nirenberg inequality for Riesz potentials

Yoshihiro MIZUTA, Eiichi NAKAI, Yoshihiro SAWANO, and Tetsu SHIMOMURA

Full-text: Open access


Our aim in this paper is to prove the Gagliardo-Nirenberg inequality for Riesz potentials of functions in variable exponent Lebesgue spaces, which are called Musielak-Orlicz spaces with respect to $\Phi(x,t)=t^{p(x)}(\log(c_0+t))^{q(x)}$ for $t$ > 0 and $x \in {\mathbb R}^n$, via the Littlewood-Paley decomposition.

Article information

J. Math. Soc. Japan, Volume 65, Number 2 (2013), 633-670.

First available in Project Euclid: 25 April 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 31B15: Potentials and capacities, extremal length 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Gagliardo-Nirenberg inequality Riesz potentials variable exponents Musielak-Orlicz spaces Littlewood-Paley theory


MIZUTA, Yoshihiro; NAKAI, Eiichi; SAWANO, Yoshihiro; SHIMOMURA, Tetsu. Littlewood-Paley theory for variable exponent Lebesgue spaces and Gagliardo-Nirenberg inequality for Riesz potentials. J. Math. Soc. Japan 65 (2013), no. 2, 633--670. doi:10.2969/jmsj/06520633.

Export citation


  • E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164 (2002), 213–259.
  • E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacean system, J. Reine Angew. Math., 584 (2005), 117–148.
  • D. Cruz-Uribe and A. Fiorenza, $L\log L$ results for the maximal operator in variable $L^{p}$ spaces, Trans. Amer. Math. Soc., 361 (2009), 2631–2647.
  • D. Cruz-Uribe, A. Fiorenza, J. M. Martell and C. Pérez, The boundedness of classical operators on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math., 31 (2006), 239–264.
  • D. Cruz-Uribe, A. Fiorenza and C. J. Neugebauer, The maximal function on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math., 28 (2003), 223–238; 29 (2004), 247–249.
  • G. Dai, Infinitely many solutions for a $p(x)$-Laplacian equation in $\mathbb R\sp N$, Nonlinear Anal., 71 (2009), 1133–1139.
  • L. Diening, Maximal function on generalized Lebesgue spaces $L^{p(\cdot)}$, Math. Inequal. Appl., 7 (2004), 245–253.
  • L. Diening, P. Hästö and A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, FSDONA 2004 Proceedings (Drabek and Rakosnik (eds.); Milovy, Czech Republic, 2004) 38–58.
  • L. Diening and M. R\ružička, Calderón-Zygmund operators on generalized Lebesgue spaces $L^{p(\cdot)}$ and problems related to fluid dynamics, J. Reine Angew. Math., 563 (2003), 197–220.
  • J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math., 29, Amer. Math. Soc., Providence, RI, 2001.
  • X.-L. Fan, Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397–417.
  • M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal., 93 (1990), 34–170.
  • T. Futamura, Y. Mizuta and T. Shimomura, Sobolev embeddings for Riesz potential space of variable exponent, Math. Nachr., 279 (2006), 1463–1473.
  • E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat., 8 (1959), 24–51.
  • J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud., 116, North-Holland Publishing Co., Amsterdam, 1985.
  • P. Harjulehto, P. Hästö and V. Latvala, Sobolev embeddings in metric measure spaces with variable dimension, Math. Z., 254 (2006), 591–609.
  • P. Harjulehto, P. Hästö, U. V. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551–4574.
  • P. Hästö, Y. Mizuta, T. Ohno and T. Shimomura, Sobolev inequalities for Orlicz spaces of two variable exponents, Glasg. Math. J., 52 (2010), 227–240.
  • B. Jawerth and A. Torchinsky, Local sharp maximal functions, J. Approx. Theory., 43 (1985), 231–270.
  • V. Kokilashvili and S. Samko, Maximal and fractional operators in weighted $L^{p(x)}$ spaces, Rev. Mat. Iberoamericana, 20 (2004), 493–515.
  • T. Kopaliani and G. Chelidze, Gagliardo-Nirenberg type inequality for variable exponent Lebesgue spaces, J. Math. Anal. Appl., 356 (2009), 232–236.
  • H. Kozono and H. Wadade, Remarks on Gagliardo-Nirenberg type inequality with critical Sobolev space and BMO, Math. Z., 259 (2008), 935–950.
  • O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41 (1991), 592–618.
  • A. K. Lerner, Weighted norm inequalities for the local sharp maximal function, J. Fourier Anal. Appl., 10 (2004), 465–474.
  • F.-Y. Maeda, Y. Mizuta and T. Ohno, Approximate identities and Young type inequalities in variable Lebesgue-Orlicz spaces $L^{p(\cdot)}(\log L)^{q(\cdot)}$, Ann. Acad. Sci. Fenn. Math., 35 (2010), 405–420.
  • V. Maz'ya and T. Shaposhnikova, On pointwise interpolation inequalities for derivatives, Math. Bohem., 124 (1999), 131–148.
  • A. L. Mazzucato, Decomposition of Besov-Morrey spaces, In: Harmonic Analysis at Mount Holyoke, South Hadley, MA, 2001 (Eds. W. Beckner, A. Nagel, A. Seager and H. F. Smith), Contemp. Math., 320, Amer. Math. Soc., Providence, RI, 2003, pp.,279–294.
  • Y. Mizuta, Potential Theory in Euclidean Spaces, GAKUTO Internat. Ser. Math. Sci. Appl., 6, Gakkōtosho, Tokyo, 1996.
  • Y. Mizuta, E. Nakai, T. Ohno and T. Shimomura, Riesz potentials and Sobolev embeddings on Morrey spaces of variable exponents, Complex Var. Elliptic Equ., 56 (2011), 671–695.
  • Y. Mizuta, T. Ohno and T. Shimomura, Sobolev's inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space $L^{p(\cdot)}(\log L)^{q(\cdot)}$, J. Math. Anal. Appl., 345 (2008), 70–85.
  • J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., 1034, Springer-Verlag, Berlin, 1983.
  • E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal., 262 (2012), 3665–3748.
  • L. Nirenberg, On elliptic partial differential equations, Ann. Scuola. Norm. Sup. Pisa (3), 13 (1959), 115–162.
  • L. Nirenberg, An extended interpolation inequality, Ann. Scuola. Norm. Sup. Pisa (3), 20 (1966), 733–737.
  • M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9), 81 (2002), 847–875.
  • M. R\r užička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture notes in Math., 1748, Springer-Verlag, Berlin, 2000.
  • S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral Transforms Spec. Funct., 16 (2005), 461–482.
  • Y. Sawano, A note on Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces, Acta Math. Sin. (Engl. Ser.), 25 (2009), 1223–1242.
  • Y. Sawano, Identification of the image of Morrey spaces by the fractional integral operators, Proc. A. Razmadze Math. Inst., 149 (2009), 87–93.
  • Y. Sawano, S. Sugano and H. Tanaka, Orlicz-Morrey spaces and fractional operators, Potential Anal., 36 (2012), 517–556.
  • E. M. Stein, Functions of exponential type, Ann. of Math. (2), 65 (1957), 582–592.
  • E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser., 30, Princeton University Press, Princeton, NJ, 1970.
  • H. Wadade, Remarks on the Gagliardo-Nirenberg type inequality in the Besov and the Triebel-Lizorkin spaces in the limiting case, J. Fourier Anal. Appl., 15 (2009), 857–870.
  • A. Zang and Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, Nonlinear Anal., 69 (2008), 3629–3636.
  • W. P. Ziemer, Weakly Differentiable Functions, Grad. Text in Math., 120, Springer-Verlag, New York, 1989.