Show Summary Details
More options …

# Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

IMPACT FACTOR 2018: 3.514
5-year IMPACT FACTOR: 3.524

CiteScore 2018: 3.44

SCImago Journal Rank (SJR) 2018: 1.891
Source Normalized Impact per Paper (SNIP) 2018: 1.808

Mathematical Citation Quotient (MCQ) 2017: 0.98

Online
ISSN
1314-2224
See all formats and pricing
More options …
Volume 21, Issue 3

# The effect of the smoothness of fractional type operators over their commutators with Lipschitz symbols on weighted spaces

Estefanía Dalmasso
• Instituto de Matemática Aplicada del Litoral, UNL, CONICET, FIQ Predio Dr. Alberto Cassano del CCT-CONICET-Santa Fe Colectora Ruta Nacional N o168, Santa Fe, Argentina
• Email
• Other articles by this author:
/ Wilfredo Ramos
Published Online: 2018-07-12 | DOI: https://doi.org/10.1515/fca-2018-0034

## Abstract

We prove boundedness results for integral operators of fractional type and their higher order commutators between weighted spaces, including Lp-Lq, Lp-BMO and Lp-Lipschitz estimates. The kernels of such operators satisfy certain size condition and a Lipschitz type regularity, and the symbol of the commutator belongs to a Lipschitz class. We also deal with commutators of fractional type operators with less regular kernels satisfying a Hörmander’s type inequality. As far as we know, these last results are new even in the unweighted case. Moreover, we give a characterization result involving symbols of the commutators and continuity results for extreme values of p.

MSC 2010: Primary 42B25; Secondary 42B35

Key Words and Phrases: fractional calculus operators; commutators; Lipschitz symbols; weights

## References

• [1]

A. Bernardis, E. Dalmasso, G. Pradolini, Generalized maximal functions and related operators on weighted Musielak-Orlicz spaces. Ann. Acad. Sci. Fenn. Math. 39, No 1 (2014), 23–50; .

• [2]

A. Bernardis, S. Hartzstein, G. Pradolini, Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type. J. Math. Anal. Appl. 322, No 2 (2006), 825–846; .

• [3]

A.L. Bernardis, M. Lorente, M.S. Riveros, Weighted inequalities for fractional integral operators with kernel satisfying Hörmander type conditions. Math. Inequal. Appl. 14, No 4 (2011), 881–895; .

• [4]

M. Bramanti, M.C. Cerutti, Commutators of singular integrals and fractional integrals on homogeneous spaces. In: Harmonic Analysis and Operator Theory 189, Contemp. Math. Series, Amer. Math. Soc., Providence, RI (1995), 81–94; .

• [5]

M. Bramanti, M.C. Cerutti, M. Manfredini, Lp estimates for some ultraparabolic operators with discontinuous coefficients. J. Math. Anal. Appl. 200, No 2 (1996), 332–354, .

• [6]

S. Chanillo, A note on commutators. Indiana Univ. Math. J. 31, No 1 (1982), 7–16; .

• [7]

S. Chanillo, D.K. Watson, R.L. Wheeden, Some integral and maximal operators related to starlike sets. Studia Math. 107, No 3 (1993), 223–255; .

• [8]

F. Chiarenza, M. Frasca, P. Longo, Interior W2,p estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche Mat. 40, No 1 (1991), 149–168.Google Scholar

• [9]

F. Chiarenza, M. Frasca, P. Longo, W2,p-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Amer. Math. Soc. 336, No 2 (1993), 841–853; .

• [10]

D. Cruz-Uribe, A. Fiorenza, Endpoint estimates and weighted norm inequalities for commutators of fractional integrals. Publ. Mat. 47, No 1 (2003), 103–131; .

• [11]

E. Dalmasso, G. Pradolini, Characterizations of the boundedness of generalized fractional maximal functions and related operators in Orlicz spaces. Math. Nach. 290, No 1 (2017), 19–36; .

• [12]

L. Diening, M. Růžička, Calderón-Zygmund operators on generalized Lebesgue spaces Lp(·) and problems related to fluid dynamics. J. Reine Angew. Math. 563 (2003), 197–220; .

• [13]

Y. Ding, S. Lu, Weighted norm inequalities for fractional integral operators with rough kernel. Canad. J. Math. 50, No 1 (1998), 29–39; .

• [14]

Y. Ding, S. Lu, P. Zhang, Weak estimates for commutators of fractional integral operators. Sci. China Ser. A 44, No 7 (2001), 877–888; .

• [15]

J. García-Cuerva, J.L.R. de Francia, Weighted Norm Inequalities and Related Topics. Ser. North-Holland Math. Studies # 116, North-Holland Publ. Co., Amsterdam (1985).Google Scholar

• [16]

J. García-Cuerva, E. Harboure, C. Segovia, J.L. Torrea, Weighted norm inequalities for commutators of strongly singular integrals. Indiana Univ. Math. J. 40, No 4 (1991), 1397–1420; .

• [17]

O. Gorosito, G. Pradolini, O. Salinas, Weighted weak-type estimates for multilinear commutators of fractional integrals on spaces of homogeneous type. Acta Math. Sin. (Engl. Ser.) 23, No 10 (2007), 1813–1826; .

• [18]

E. Harboure, O. Salinas, B. Viviani, Orlicz boundedness for certain classical operators. Colloq. Math. 91, No 2 (2002), 263–282; .

• [19]

E. Harboure, C. Segovia, J.L. Torrea, Boundedness of commutators of fractional and singular integrals for the extreme values of p. Illinois J. Math. 41, No 4 (1997), 676–700.Google Scholar

• [20]

G.H. Hardy, J.E. Littlewood, Some properties of fractional integrals, II. Math. Z. 34, No 1 (1932), 403–439; .

• [21]

F. John, L. Nirenberg, On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14 (1961), 415–426; .

• [22]

A.Y. Karlovich, A.K. Lerner, Commutators of singular integrals on generalized Lp spaces with variable exponent. Publ. Mat. 49, No 1 (2005), 111–125; .

• [23]

D.S. Kurtz, Sharp function estimates for fractional integrals and related operators. J. Austral. Math. Soc. Ser. A 49, No 1 (1990), 129–137.

• [24]

M. Lorente, J.M. Martell, M.S. Riveros, A. de la Torre, Generalized Hörmander’s conditions, commutators and weights. J. Math. Anal. Appl. 342, No 2 (2008), 1399–1425; .

• [25]

Y. Meng, D. Yang, Boundedness of commutators with Lipschitz functions in non-homogeneous spaces. Taiwanese J. Math. 10, No 6 (2006), 1443–1464.

• [26]

B. Muckenhoupt, R. Wheeden, Weighted norm inequalities for fractional integrals. Trans. Amer. Math. Soc. 192 (1974), 261–274.

• [27]

C. Pérez, Endpoint estimates for commutators of singular integral operators. J. Funct. Anal. 128, No 1 (1995), 163–185; .

• [28]

C. Pérez, Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function. J. Fourier Anal. Appl. 3, No 6 (1997), 743–756; .

• [29]

C. Pérez, G. Pradolini, R.H. Torres, R. Trujillo-González, End-point estimates for iterated commutators of multilinear singular integrals. Bull. Lond. Math. Soc. 46, No 1 (2014), 26–42; .

• [30]

G. Pradolini, Two-weighted norm inequalities for the fractional integral operator between Lp and Lipschitz spaces. Comment. Math. (Prace Mat.) 41 (2001), 147–169.Google Scholar

• [31]

G.G. Pradolini, W.A. Ramos, Characterization of Lipschitz functions via the commutators of singular and fractional integral operators in variable Lebesgue spaces. Potential Anal. 46, No 3 (2017), 499–525; .

• [32]

G. Pradolini, O. Salinas, The fractional integral between weighted Orlicz and BMOϕ spaces on spaces of homogeneous type. Comment. Math. Univ. Carolin. 44, No 3 (2003), 469–487.Google Scholar

• [33]

C. Rios, The Lp Dirichlet problem and nondivergence harmonic measure. Trans. Amer. Math. Soc. 355, No 2 (2003), 665–687; .

• [34]

C. Segovia, J.L. Torrea, Weighted inequalities for commutators of fractional and singular integrals. Publ. Mat. 35, No 1 (1991), 209–235; .

• [35]

E.M. Stein, G. Weiss, Fractional integrals on n-dimensional Euclidean space. J. Math. Mech. 7 (1958), 503–514.Google Scholar

Published Online: 2018-07-12

Published in Print: 2018-06-26

Citation Information: Fractional Calculus and Applied Analysis, Volume 21, Issue 3, Pages 628–653, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454,

Export Citation