Boundedness of one-sided fractional integrals in the one-sided Calderón-Hardy spaces

Alejandra Perini

Commentationes Mathematicae Universitatis Carolinae (2011)

  • Volume: 52, Issue: 1, page 57-75
  • ISSN: 0010-2628

Abstract

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In this paper we study the mapping properties of the one-sided fractional integrals in the Calderón-Hardy spaces ℋ q , α p , + ( ω ) for 0 < p ≤ 1 , 0 < α < ∞ and 1 < q < ∞ . Specifically, we show that, for suitable values of p , q , γ , α and s , if ω ∈ A s + (Sawyer’s classes of weights) then the one-sided fractional integral I γ + can be extended to a bounded operator from ℋ q , α p , + ( ω ) to ℋ q , α + γ p , + ( ω ) . The result is a consequence of the pointwise inequality N q , α + γ + I γ + F ; x ≤ C α , γ N q , α + F ; x , where N q , α + ( F ; x ) denotes the Calderón maximal function.

How to cite

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Perini, Alejandra. "Boundedness of one-sided fractional integrals in the one-sided Calderón-Hardy spaces." Commentationes Mathematicae Universitatis Carolinae 52.1 (2011): 57-75. <http://eudml.org/doc/246766>.

@article{Perini2011,
abstract = {In this paper we study the mapping properties of the one-sided fractional integrals in the Calderón-Hardy spaces $\mathcal \{H\}_\{q,\alpha \}^\{p,+\}(\omega )$ for $0< p\le 1$, $0< \alpha < \infty $ and $1< q< \infty $. Specifically, we show that, for suitable values of $p,q,\gamma , \alpha $ and $s$, if $\omega \in A_s^+$ (Sawyer’s classes of weights) then the one-sided fractional integral $I_\{\gamma \}^+$ can be extended to a bounded operator from $\mathcal \{H\}_\{q,\alpha \}^\{p,+\}(\omega )$ to $\mathcal \{H\}_\{q,\alpha + \gamma \}^\{p,+\}(\omega )$. The result is a consequence of the pointwise inequality \[ N\_\{q, \alpha +\gamma \}^+\left( I\_\{\gamma \}^+ F;x\right) \le C\_\{\alpha ,\gamma \} N\_\{q, \alpha \}^+ \left( F;x\right), \] where $N_\{q, \alpha \}^+ (F;x)$ denotes the Calderón maximal function.},
author = {Perini, Alejandra},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {fractional integral; maximal; one-sided Calderón-Hardy; one-sided weights spaces; fractional integral; maximal function; Calderón-Hardy spaces; one-sided Muckenhoupt weights},
language = {eng},
number = {1},
pages = {57-75},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Boundedness of one-sided fractional integrals in the one-sided Calderón-Hardy spaces},
url = {http://eudml.org/doc/246766},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Perini, Alejandra
TI - Boundedness of one-sided fractional integrals in the one-sided Calderón-Hardy spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 1
SP - 57
EP - 75
AB - In this paper we study the mapping properties of the one-sided fractional integrals in the Calderón-Hardy spaces $\mathcal {H}_{q,\alpha }^{p,+}(\omega )$ for $0< p\le 1$, $0< \alpha < \infty $ and $1< q< \infty $. Specifically, we show that, for suitable values of $p,q,\gamma , \alpha $ and $s$, if $\omega \in A_s^+$ (Sawyer’s classes of weights) then the one-sided fractional integral $I_{\gamma }^+$ can be extended to a bounded operator from $\mathcal {H}_{q,\alpha }^{p,+}(\omega )$ to $\mathcal {H}_{q,\alpha + \gamma }^{p,+}(\omega )$. The result is a consequence of the pointwise inequality \[ N_{q, \alpha +\gamma }^+\left( I_{\gamma }^+ F;x\right) \le C_{\alpha ,\gamma } N_{q, \alpha }^+ \left( F;x\right), \] where $N_{q, \alpha }^+ (F;x)$ denotes the Calderón maximal function.
LA - eng
KW - fractional integral; maximal; one-sided Calderón-Hardy; one-sided weights spaces; fractional integral; maximal function; Calderón-Hardy spaces; one-sided Muckenhoupt weights
UR - http://eudml.org/doc/246766
ER -

References

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  9. Ombrosi S., de Rosa L., Boundeness of the Weyl fractional integral on the one-sided weighted Lebesque and Lipchitz spaces, Publ. Mat. 47 (2003), no. 1, 71–102. MR1970895
  10. Ombrosi S., Segovia C., One-sided singular integral operators on Calderón-Hardy spaces, Rev. Un. Mat. Argentina 44 (2003), no. 1, 17–32. Zbl1078.42008MR2051035
  11. de Rosa L., Segovia C., 10.1090/conm/189/02262, Contemp. Math., 189, American Mathematical Society, Providence, RI, 1995, pp. 161–183. DOI10.1090/conm/189/02262
  12. Sawyer E., 10.1090/S0002-9947-1986-0849466-0, Trans. Amer. Math. Soc. 297 (1986), 53–61. Zbl0627.42009MR0849466DOI10.1090/S0002-9947-1986-0849466-0
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