On fractional operators with more than one singularity
DOI:
https://doi.org/10.33044/revuma.4364Abstract
Let $0\leq \alpha < n$, $m\in \mathbb{N}$ and let $T_{\alpha,m}$ be an integral operator given by a kernel of the form \[K(x,y)=k_1(x-A_1y)k_2(x-A_2y)\dots k_m(x-A_my),\] where $A_i$ are invertible matrices and each $k_i$ satisfies a fractional size and a generalized fractional Hörmander condition that depends on $\alpha$. In this survey, written in honour to Eleonor Harboure, we collect several results about boundedness in different spaces of the operator $T_{\alpha,m}$, obtained along the last 35 years by several members of the Analysis Group of FAMAF, UNC.
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N. Accomazzo, J. C. Martínez-Perales, and I. P. Rivera-Ríos, On Bloom-type estimates for iterated commutators of fractional integrals, Indiana Univ. Math. J. 69 no. 4 (2020), 1207–1230. DOI MR Zbl
E. I. Berezhnoi, Two-weighted estimations for the Hardy-Littlewood maximal function in ideal Banach spaces, Proc. Amer. Math. Soc. 127 no. 1 (1999), 79–87. DOI MR Zbl
A. L. Bernardis, M. Lorente, and M. S. Riveros, Weighted inequalities for fractional integral operators with kernel satisfying Hörmander type conditions, Math. Inequal. Appl. 14 no. 4 (2011), 881–895. DOI MR Zbl
C. Capone, D. Cruz-Uribe, and A. Fiorenza, The fractional maximal operator and fractional integrals on variable $L^p$ spaces, Rev. Mat. Iberoam. 23 no. 3 (2007), 743–770. DOI MR Zbl
D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, The maximal function on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math. 28 no. 1 (2003), 223–238. MR Zbl
D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg, 2013. DOI MR Zbl
L. Diening, P. Harjulehto, P. Hästö, and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics 2017, Springer, Heidelberg, 2011. DOI MR Zbl
E. V. Ferreyra and G. J. Flores, Weighted estimates for integral operators on local $BMO$ type spaces, Math. Nachr. 288 no. 8-9 (2015), 905–916. DOI MR Zbl
G. Ibañez Firnkorn and L. A. Vallejos, Boundedness of commutators of integral operators of fractional type and $M_{α, L^rlog L}$ maximal operator in variable Lebesgue spaces, J. Geom. Anal. 33 no. 11 (2023), Article no. 354, 16 pp. DOI MR
G. H. Ibañez Firnkorn and M. S. Riveros, Certain fractional type operators with Hörmander conditions, Ann. Acad. Sci. Fenn. Math. 43 no. 2 (2018), 913–929. MR Zbl
G. H. Ibañez Firnkorn and M. S. Riveros, Commutators of certain fractional type operators with Hörmander conditions, one-weighted and two-weighted inequalities, Math. Inequal. Appl. 23 no. 4 (2020), 1361–1389. DOI MR Zbl
G. H. Ibañez Firnkorn, M. S. Riveros, and R. E. Vidal, Necessary condition on weights for maximal and integral operators with rough kernels, Studia Math. 263 no. 3 (2022), 293–321. DOI MR Zbl
T. Godoy, L. Saal, and M. Urciuolo, About certain singular kernels $K(x,y)=K_1(x-y)K_2(x+y)$, Math. Scand. 74 no. 1 (1994), 98–110. DOI MR Zbl
T. Godoy and M. Urciuolo, About the $L^p$-boundedness of some integral operators, Rev. Un. Mat. Argentina 38 no. 3-4 (1993), 192–195. MR Zbl
T. Godoy and M. Urciuolo, About the $L^p$-boundedness of integral operators with kernels of the form $K_1(x-y)K_2(x+y)$, Math. Scand. 78 no. 1 (1996), 84–92. DOI MR Zbl
T. Godoy and M. Urciuolo, On certain integral operators of fractional type, Acta Math. Hungar. 82 no. 1-2 (1999), 99–105. DOI MR Zbl
A. K. Lerner, On some questions related to the maximal operator on variable $L^p$ spaces, Trans. Amer. Math. Soc. 362 no. 8 (2010), 4229–4242. DOI MR Zbl
A. K. Lerner, On pointwise estimates involving sparse operators, New York J. Math. 22 (2016), 341–349. MR Zbl Available at https://www.emis.de/journals/NYJM/j/2016/22-15.html.
K. Moen, Sharp one-weight and two-weight bounds for maximal operators, Studia Math. 194 no. 2 (2009), 163–180. DOI MR Zbl
B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. DOI MR Zbl
B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274. DOI MR Zbl
A. Nekvinda, Hardy-Littlewood maximal operator on $L^{p(x)}(ℝ)$, Math. Inequal. Appl. 7 no. 2 (2004), 255–265. DOI MR Zbl
F. Ricci and P. Sjögren, Two-parameter maximal functions in the Heisenberg group, Math. Z. 199 no. 4 (1988), 565–575. DOI MR Zbl
M. S. Riveros and M. Urciuolo, Weighted inequalities for integral operators with some homogeneous kernels, Czechoslovak Math. J. 55 no. 2 (2005), 423–432. DOI MR Zbl
M. S. Riveros and M. Urciuolo, Weighted inequalities for fractional type operators with some homogeneous kernels, Acta Math. Sin. (Engl. Ser.) 29 no. 3 (2013), 449–460. DOI MR Zbl
M. S. Riveros and M. Urciuolo, Weighted inequalities for some integral operators with rough kernels, Cent. Eur. J. Math. 12 no. 4 (2014), 636–647. DOI MR Zbl
P. Rocha and M. Urciuolo, On the $H^p$-$L^q$ boundedness of some fractional integral operators, Czechoslovak Math. J. 62 no. 3 (2012), 625–635. DOI MR Zbl
P. Rocha and M. Urciuolo, On the $H^p$-$L^p$-boundedness of some integral operators, Georgian Math. J. 18 no. 4 (2011), 801–808. DOI MR Zbl
P. Rocha and M. Urciuolo, About integral operators of fractional type on variable $L^p$ spaces, Georgian Math. J. 20 no. 4 (2013), 805–816. DOI MR Zbl
E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 no. 1 (1982), 1–11. DOI MR Zbl
M. Urciuolo and L. Vallejos, Integral operators with rough kernels in variable Lebesgue spaces, Acta Math. Hungar. 162 no. 1 (2020), 105–116. DOI MR Zbl
M. Urciuolo and L. Vallejos, $L^{p(⋅)}$-$L^{q(⋅)}$ boundedness of some integral operators obtained by extrapolation techniques, Georgian Math. J. 27 no. 3 (2020), 479–484. DOI MR Zbl
M. S. Urciuolo and L. A. Vallejos, A generalization of the boundedness of certain integral operators in variable Lebesgue spaces, J. Math. Inequal. 14 no. 2 (2020), 547–557. DOI MR Zbl
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