# Weighted inequalities for some integral operators with rough kernels

Open Mathematics (2014)

• Volume: 12, Issue: 4, page 636-647
• ISSN: 2391-5455

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## Abstract

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In this paper we study integral operators with kernels $K\left(x,y\right)={k}_{1}\left(x-{A}_{1}y\right)\cdots {k}_{m}\left(x-{A}_{m}y\right),$${k}_{i}\left(x\right)={{\Omega }_{i}\left(x\right)/\phantom{{\Omega }_{i}\left(x\right)\left|x\right|}\phantom{\rule{0.0pt}{0ex}}\left|x\right|}^{n/\phantom{n{q}_{i}}\phantom{\rule{0.0pt}{0ex}}{q}_{i}}$ where Ωi: ℝn → ℝ are homogeneous functions of degree zero, satisfying a size and a Dini condition, A i are certain invertible matrices, and n/q 1 +…+n/q m = n−α, 0 ≤ α < n. We obtain the appropriate weighted L p-L q estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.

## How to cite

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María Riveros, and Marta Urciuolo. "Weighted inequalities for some integral operators with rough kernels." Open Mathematics 12.4 (2014): 636-647. <http://eudml.org/doc/269403>.

@article{MaríaRiveros2014,
abstract = {In this paper we study integral operators with kernels $K(x,y) = k\_1 (x - A\_1 y) \cdots k\_m \left( \{x - A\_m y\} \right),$$k\_i \left( x \right) = \{\{\Omega \_i \left( x \right)\} \mathord \{\left\bad. \{\vphantom\{\{\Omega \_i \left( x \right)\} \{\left| x \right|\}\}\} \right. \hspace\{0.0pt\}\} \{\left| x \right|\}\}^\{\{n \mathord \{\left\bad. \{\vphantom\{n \{q\_i \}\}\} \right. \hspace\{0.0pt\}\} \{q\_i \}\}\}$ where Ωi: ℝn → ℝ are homogeneous functions of degree zero, satisfying a size and a Dini condition, A i are certain invertible matrices, and n/q 1 +…+n/q m = n−α, 0 ≤ α < n. We obtain the appropriate weighted L p-L q estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.},
author = {María Riveros, Marta Urciuolo},
journal = {Open Mathematics},
keywords = {Fractional operators; Calderón-Zygmund operators; BMO; Muckenhoupt weights; fractional operator; Calderón-Zygmund operator; Muckenhoupt weight},
language = {eng},
number = {4},
pages = {636-647},
title = {Weighted inequalities for some integral operators with rough kernels},
url = {http://eudml.org/doc/269403},
volume = {12},
year = {2014},
}

TY - JOUR
AU - María Riveros
AU - Marta Urciuolo
TI - Weighted inequalities for some integral operators with rough kernels
JO - Open Mathematics
PY - 2014
VL - 12
IS - 4
SP - 636
EP - 647
AB - In this paper we study integral operators with kernels $K(x,y) = k_1 (x - A_1 y) \cdots k_m \left( {x - A_m y} \right),$$k_i \left( x \right) = {{\Omega _i \left( x \right)} \mathord {\left\bad. {\vphantom{{\Omega _i \left( x \right)} {\left| x \right|}}} \right. \hspace{0.0pt}} {\left| x \right|}}^{{n \mathord {\left\bad. {\vphantom{n {q_i }}} \right. \hspace{0.0pt}} {q_i }}}$ where Ωi: ℝn → ℝ are homogeneous functions of degree zero, satisfying a size and a Dini condition, A i are certain invertible matrices, and n/q 1 +…+n/q m = n−α, 0 ≤ α < n. We obtain the appropriate weighted L p-L q estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.
LA - eng
KW - Fractional operators; Calderón-Zygmund operators; BMO; Muckenhoupt weights; fractional operator; Calderón-Zygmund operator; Muckenhoupt weight
UR - http://eudml.org/doc/269403
ER -

## References

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8. [8] Kurtz D.S., Wheeden R.L., Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc., 1979, 255, 343–362 http://dx.doi.org/10.1090/S0002-9947-1979-0542885-8 Zbl0427.42004
9. [9] Muckenhoupt B., Wheeden R., Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 1974, 192, 261–274 http://dx.doi.org/10.1090/S0002-9947-1974-0340523-6 Zbl0289.26010
10. [10] Riveros M.S., Urciuolo M., Weighted inequalities for integral operators with some homogeneous kernels, Czechoslovak Math. J., 2005, 55(130)(2), 423–432 http://dx.doi.org/10.1007/s10587-005-0032-y Zbl1081.42018
11. [11] Riveros M.S., Urciuolo M., Weighted inequalities for fractional type operators with some homogeneous kernels, Acta Math. Sin. (Engl. Ser.), 2013, 29(3), 449–460 http://dx.doi.org/10.1007/s10114-013-1639-9 Zbl1260.42011
12. [12] Rocha P., Urciuolo M., On the H p-L q boundedness of some fractional integral operators, Czechoslovak Math. J., 2012, 62(137)(3), 625–635 http://dx.doi.org/10.1007/s10587-012-0054-1 Zbl1265.42046
13. [13] Watson D.K., Weighted estimates for singular integrals via Fourier transform estimates, Duke Math. J., 1990, 60(2), 389–399 http://dx.doi.org/10.1215/S0012-7094-90-06015-6 Zbl0711.42025

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