Weighted and vector-valued inequalities for potential operators

Ruiz Blasco, Francisco J.; Torrea Hernández, José L.

doi:10.1090/S0002-9947-1986-0831197-4

Weighted and vector-valued inequalities for potential operators
HTML articles powered by AMS MathViewer

by Francisco J. Ruiz Blasco and José L. Torrea Hernández

Trans. Amer. Math. Soc. 295 (1986), 213-232

DOI: https://doi.org/10.1090/S0002-9947-1986-0831197-4

PDF | Request permission

Abstract:

In this paper we develop some aspect of a general theory parallel to the Calderón-Zygmund theory for operator valued kernels, where the operators considered map functions defined on ${R^n}$ into functions defined on $R_ + ^{n + 1} = {R^n} \times [0,\infty )$. In particular, we apply the obtained results to get vector-valued inequalities for the Poisson integral and fractional integrals. Some weighted norm inequalities are also considered for fractional integrals.

References

C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. MR 284802, DOI 10.2307/2373450
Lars Inge Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505–510. MR 312232, DOI 10.1090/S0002-9939-1972-0312232-4
H. P. Heinig and R. Johnson, Weighted norm inequalities for $L^{r}$-valued integral operators and applications, Math. Nachr. 107 (1982), 161–174. MR 695744, DOI 10.1002/mana.19821070113
Benjamin Muckenhoupt and Richard Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274. MR 340523, DOI 10.1090/S0002-9947-1974-0340523-6
José L. Rubio de Francia, Weighted norm inequalities and vector valued inequalities, Harmonic analysis (Minneapolis, Minn., 1981) Lecture Notes in Math., vol. 908, Springer, Berlin-New York, 1982, pp. 86–101. MR 654181

Calderón- Zygmund theory for operator-valued kernels

Francisco J. Ruiz, A unified approach to Carleson measures and $A_p$ weights, Pacific J. Math. 117 (1985), no. 2, 397–404. MR 779929
Francisco J. Ruiz and José L. Torrea, A unified approach to Carleson measures and $A_p$ weights. II, Pacific J. Math. 120 (1985), no. 1, 189–197. MR 808937
Eric T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 1–11. MR 676801, DOI 10.4064/sm-75-1-1-11
G. V. Welland, Weighted norm inequalities for fractional integrals, Proc. Amer. Math. Soc. 51 (1975), 143–148. MR 369641, DOI 10.1090/S0002-9939-1975-0369641-X
Felipe Zo, A note on approximation of the identity, Studia Math. 55 (1976), no. 2, 111–122. MR 423013, DOI 10.4064/sm-55-2-111-122
Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187
J. Schwartz, A remark on inequalities of Calderon-Zygmund type for vector-valued functions, Comm. Pure Appl. Math. 14 (1961), 785–799. MR 143031, DOI 10.1002/cpa.3160140410