## Generalized local Fatou theorems and area integrals

HTML articles powered by AMS MathViewer

- by B. A. Mair, Stan Philipp and David Singman
- Trans. Amer. Math. Soc.
**321**(1990), 401-413 - DOI: https://doi.org/10.1090/S0002-9947-1990-0974520-3
- PDF | Request permission

## Abstract:

Let $X$ be a space of homogeneous type and $W$ a subset of $X \times (0,\infty )$. Then, under minimal conditions on $W$, we obtain a relationship between two modes of convergence at the boundary $X$ for functions defined on $W$. This result gives new local Fatou theorems of the Carleson-type for solutions of Laplace, parabolic and Laplace-Beltrami equations as immediate consequences of the classical results. Lusin area integral characterizations for the existence of limits within these more general approach regions are also obtained.## References

- M. Brelot and J. L. Doob,
*Limites angulaires et limites fines*, Ann. Inst. Fourier (Grenoble)**13**(1963), no. fasc. 2, 395–415 (French). MR**196107**
A. P. Calderón, - Lennart Carleson,
*On the existence of boundary values for harmonic functions in several variables*, Ark. Mat.**4**(1962), 393–399 (1962). MR**159013**, DOI 10.1007/BF02591620 - Ronald R. Coifman and Guido Weiss,
*Analyse harmonique non-commutative sur certains espaces homogènes*, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. MR**0499948** - Daryl Geller,
*Some results in $H^{p}$ theory for the Heisenberg group*, Duke Math. J.**47**(1980), no. 2, 365–390. MR**575902** - J. R. Hattemer,
*Boundary behavior of temperatures. I*, Studia Math.**25**(1964/65), 111–155. MR**181838**, DOI 10.4064/sm-25-1-111-155 - Adam Korányi,
*Harmonic functions on Hermitian hyperbolic space*, Trans. Amer. Math. Soc.**135**(1969), 507–516. MR**277747**, DOI 10.1090/S0002-9947-1969-0277747-0 - A. Korányi and R. B. Putz,
*Local Fatou theorem and area theorem for symmetric spaces of rank one*, Trans. Amer. Math. Soc.**224**(1976), no. 1, 157–168. MR**492068**, DOI 10.1090/S0002-9947-1976-0492068-2 - B. A. Mair,
*Fine and parabolic limits for solutions of second-order linear parabolic equations on an infinite slab*, Trans. Amer. Math. Soc.**284**(1984), no. 2, 583–599. MR**743734**, DOI 10.1090/S0002-9947-1984-0743734-7 - B. A. Mair,
*Boundary behavior of positive solutions of the heat equation on a semi-infinite slab*, Trans. Amer. Math. Soc.**295**(1986), no. 2, 687–697. MR**833703**, DOI 10.1090/S0002-9947-1986-0833703-2 - B. A. Mair and David Singman,
*A generalized Fatou theorem*, Trans. Amer. Math. Soc.**300**(1987), no. 2, 705–719. MR**876474**, DOI 10.1090/S0002-9947-1987-0876474-7 - Alexander Nagel and Elias M. Stein,
*On certain maximal functions and approach regions*, Adv. in Math.**54**(1984), no. 1, 83–106. MR**761764**, DOI 10.1016/0001-8708(84)90038-0 - Elias M. Stein,
*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095** - Juan Sueiro,
*On maximal functions and Poisson-Szegő integrals*, Trans. Amer. Math. Soc.**298**(1986), no. 2, 653–669. MR**860386**, DOI 10.1090/S0002-9947-1986-0860386-8 - C. C. Tu,
*Non-tangential limits of a solution of a boundary-value problem for the heat equation*, Math. Systems Theory**3**(1969), 130–138. MR**249840**, DOI 10.1007/BF01746519

*On the behavior of harmonic functions at the boundary*, Trans. Amer. Math. Soc.

**68**(1950), 47-54.

## Bibliographic Information

- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**321**(1990), 401-413 - MSC: Primary 31B25
- DOI: https://doi.org/10.1090/S0002-9947-1990-0974520-3
- MathSciNet review: 974520