A. P. Calderón :
“On the behaviour of harmonic functions at the boundary Trans. Am. Math. Soc.
68 : 1
(January 1950 ),
pp. 47–54 .
This was one of three published articles combined to make up the author’s 1950 PhD thesis .
MR
0032863 Zbl
0035.18901 article
BibTeX
@article {key0032863m,
AUTHOR = {Calder\'on, A. P.},
TITLE = {On the behaviour of harmonic functions
at the boundary},
JOURNAL = {Trans. Am. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {68},
NUMBER = {1},
MONTH = {January},
YEAR = {1950},
PAGES = {47--54},
DOI = {10.2307/1990537},
NOTE = {This was one of three published articles
combined to make up the author's 1950
PhD thesis. MR:0032863. Zbl:0035.18901.},
ISSN = {0002-9947},
}
A. P. Calderón :
“On a theorem of Marcinkiewicz and Zygmund Trans. Am. Math. Soc.
68 : 1
(January 1950 ),
pp. 55–61 .
This was one of three published articles combined to make up the author’s 1950 PhD thesis .
MR
0032864 Zbl
0035.18903 article
Abstract
BibTeX
The purpose of the present paper is to prove the following result:
Let \( F(P) \) , \( P=(x_1,x_2,\dots,x_n) \) , be a function harmonic for \( x_n > 0 \) and such that for every point \( Q \) of a set \( E \) of positive measure on the hyperplane \( x_n=0 \) there exists a region contained in \( x_n > 0 \) limited by a cone with vertex at \( Q \) and a hyperplane \( x_n = 0 \) there exists a region contained in \( x_n > 0 \) limited by a cone with vertex at \( Q \) and a hyperplane \( x_n = \) const. where the function is bounded. Then except for a set of measure zero, the integral
\[ \int \frac{1}{x_n^{n-2}}\operatorname{grad}^2F \,d\omega, \]
extended over any region limited by a cone with vertex at \( Q \in E \) , a hyperplane \( x_n = \) const., and contained in \( x_n > 0 \) , is finite.
@article {key0032864m,
AUTHOR = {Calder\'on, A. P.},
TITLE = {On a theorem of {M}arcinkiewicz and
{Z}ygmund},
JOURNAL = {Trans. Am. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {68},
NUMBER = {1},
MONTH = {January},
YEAR = {1950},
PAGES = {55--61},
DOI = {10.2307/1990538},
NOTE = {This was one of three published articles
combined to make up the author's 1950
PhD thesis. MR:0032864. Zbl:0035.18903.},
ISSN = {0002-9947},
}
A. P. Calderon and A. Zygmund :
“On the existence of certain singular integrals Acta Math.
88
(December 1952 ),
pp. 85–139 .
Dedicated to Professor Marcel Riesz, on the occasion of his 65th birthday.
MR
0052553 Zbl
0047.10201 article
People
BibTeX
@article {key0052553m,
AUTHOR = {Calderon, A. P. and Zygmund, A.},
TITLE = {On the existence of certain singular
integrals},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {88},
MONTH = {December},
YEAR = {1952},
PAGES = {85--139},
DOI = {10.1007/BF02392130},
NOTE = {Dedicated to Professor Marcel Riesz,
on the occasion of his 65th birthday.
MR:0052553. Zbl:0047.10201.},
ISSN = {0001-5962},
}
A. P. Calderón and A. Zygmund :
“On singular integrals Am. J. Math.
78 : 2
(April 1956 ),
pp. 289–309 .
MR
0084633 Zbl
0072.11501 article
Abstract
People
BibTeX
In earlier work [1952] we considered certain singular integrals arising in various problems of Analysis and studied some of their properties. Here we present a new approach to such integrals. Unlike the method used in [1952] it is based on the theory of Hilbert transforms of functions of one variable, but otherwise it is simpler and yields most results obtained previously, under far less restrictive assumptions. Unfortunately some important cases (\( f\in L \) for instance) seem to be beyond its scope. We have been unable to decide whether the corresponding theorems as presented in [1952] can be likewise strengthened.
@article {key0084633m,
AUTHOR = {Calder\'on, A. P. and Zygmund, A.},
TITLE = {On singular integrals},
JOURNAL = {Am. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {78},
NUMBER = {2},
MONTH = {April},
YEAR = {1956},
PAGES = {289--309},
DOI = {10.2307/2372517},
NOTE = {MR:0084633. Zbl:0072.11501.},
ISSN = {0002-9327},
}
A. P. Calderón and A. Zygmund :
“Algebras of certain singular operators Am. J. Math.
78 : 2
(April 1956 ),
pp. 310–320 .
MR
0087810 Zbl
0072.11601 article
Abstract
People
BibTeX
@article {key0087810m,
AUTHOR = {Calder\'on, A. P. and Zygmund, A.},
TITLE = {Algebras of certain singular operators},
JOURNAL = {Am. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {78},
NUMBER = {2},
MONTH = {April},
YEAR = {1956},
PAGES = {310--320},
DOI = {10.2307/2372518},
NOTE = {MR:0087810. Zbl:0072.11601.},
ISSN = {0002-9327},
}
A.-P. Calderón and A. Zygmund :
“Singular integral operators and differential equations Am. J. Math.
79 : 4
(October 1957 ),
pp. 901–921 .
MR
0100768 Zbl
0081.33502 article
Abstract
People
BibTeX
@article {key0100768m,
AUTHOR = {Calder\'on, A.-P. and Zygmund, A.},
TITLE = {Singular integral operators and differential
equations},
JOURNAL = {Am. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {79},
NUMBER = {4},
MONTH = {October},
YEAR = {1957},
PAGES = {901--921},
DOI = {10.2307/2372441},
NOTE = {MR:0100768. Zbl:0081.33502.},
ISSN = {0002-9327},
}
A. P. Calderón :
“Uniqueness in the Cauchy problem for partial differential equations Am. J. Math.
80 : 1
(January 1958 ),
pp. 16–36 .
MR
0104925 Zbl
0080.30302 article
BibTeX
@article {key0104925m,
AUTHOR = {Calder\'on, A. P.},
TITLE = {Uniqueness in the {C}auchy problem for
partial differential equations},
JOURNAL = {Am. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {80},
NUMBER = {1},
MONTH = {January},
YEAR = {1958},
PAGES = {16--36},
DOI = {10.2307/2372819},
NOTE = {MR:0104925. Zbl:0080.30302.},
ISSN = {0002-9327},
}
A. P. Calderón :
“Existence and uniqueness theorems for systems of partial differential equations ,”
pp. 147–195
in
Fluid dynamics and applied mathematics
(College Park, MD, 28–29 April 1961 ).
Edited by J. B. Diaz and S. Bai .
Gordon and Breach (New York ),
1962 .
MR
0156078 Zbl
0147.08202 incollection
People
BibTeX
@incollection {key0156078m,
AUTHOR = {Calder\'on, A. P.},
TITLE = {Existence and uniqueness theorems for
systems of partial differential equations},
BOOKTITLE = {Fluid dynamics and applied mathematics},
EDITOR = {Diaz, Joaquin B. and Bai, Shiyi},
PUBLISHER = {Gordon and Breach},
ADDRESS = {New York},
YEAR = {1962},
PAGES = {147--195},
NOTE = {(College Park, MD, 28--29 April 1961).
MR:0156078. Zbl:0147.08202.},
}
A. P. Calderón :
“Boundary value problems for elliptic equations ,”
pp. 303–304
in
Outlines of the joint Soviet–American symposium on partial differential equations
(Novosibirsk, August 1963 ).
Akademiia Nauk SSSR Sibirskoe Otdelenie (Moscow ),
1963 .
MR
0203254 incollection
BibTeX
@incollection {key0203254m,
AUTHOR = {Calder\'on, A. P.},
TITLE = {Boundary value problems for elliptic
equations},
BOOKTITLE = {Outlines of the joint {S}oviet--{A}merican
symposium on partial differential equations},
PUBLISHER = {Akademiia Nauk SSSR Sibirskoe Otdelenie},
ADDRESS = {Moscow},
YEAR = {1963},
PAGES = {303--304},
NOTE = {(Novosibirsk, August 1963). MR:0203254.},
}
A. P. Calderón :
“Commutators of singular integral operators Proc. Natl. Acad. Sci. U.S.A.
53 : 5
(May 1965 ),
pp. 1092–1099 .
MR
0177312 Zbl
0151.16901 article
Abstract
BibTeX
Let
\[ A(f) = \lim_{\varepsilon \to 0} \int_{|x-y| > \varepsilon}k(x-y)f(y)\,dy, \]
where \( x \) , \( y \) are points in \( n \) -dimensional Euclidean space \( R^n \) and \( k(x) \) is a homogeneous function of degree \( -n \) with mean value zero on \( |x| = 1 \) , and let
\[ B(f) = b(x)f(x) .\]
It is well known [Calderón and Zygmund 1957] that if \( k \) and \( b \) are sufficiently smooth and \( b \) is bounded, then
\[ (AB - BA)(\partial/\partial x_j) \quad\text{and}\quad (\partial/\partial x_j)(AB - BA) \]
are bounded operators in \( L^p \) , \( 1 < p < \infty \) . The purpose of the present note is to extend and strengthen the preceding result and establish some related facts of independent interest.
@article {key0177312m,
AUTHOR = {Calder\'on, A. P.},
TITLE = {Commutators of singular integral operators},
JOURNAL = {Proc. Natl. Acad. Sci. U.S.A.},
FJOURNAL = {Proceedings of the National Academy
of Sciences of the United States of
America},
VOLUME = {53},
NUMBER = {5},
MONTH = {May},
YEAR = {1965},
PAGES = {1092--1099},
DOI = {10.1073/pnas.53.5.1092},
NOTE = {MR:0177312. Zbl:0151.16901.},
ISSN = {0027-8424},
}
A. P. Calderón and R. Vaillancourt :
“A class of bounded pseudo-differential operators Proc. Natl. Acad. Sci. U.S.A.
69 : 5
(May 1972 ),
pp. 1185–1187 .
MR
0298480 Zbl
0244.35074 article
Abstract
People
BibTeX
Pseudo-differential operators of order \( -M \) and type \( \rho \) , \( \delta_1 \) , \( \delta_2 \) are shown to be bounded in \( L^2 \) provided that
\[ 0 \leq \rho \leq \delta_1 < 1, \quad 0 \leq \rho \leq \delta_2 < 1, \quad\text{and}\quad \frac{M}{n} \geq \frac{1}{2}(\delta_1 + \delta_2) - \rho .\]
@article {key0298480m,
AUTHOR = {Calder\'on, Alberto P. and Vaillancourt,
R{\'e}mi},
TITLE = {A class of bounded pseudo-differential
operators},
JOURNAL = {Proc. Natl. Acad. Sci. U.S.A.},
FJOURNAL = {Proceedings of the National Academy
of Sciences of the United States of
America},
VOLUME = {69},
NUMBER = {5},
MONTH = {May},
YEAR = {1972},
PAGES = {1185--1187},
DOI = {10.1073/pnas.69.5.1185},
NOTE = {MR:0298480. Zbl:0244.35074.},
ISSN = {0027-8424},
}
A. P. Calderón :
“Cauchy integrals on Lipschitz curves and related operators Proc. Natl. Acad. Sci. U.S.A.
74 : 4
(April 1977 ),
pp. 1324–1327 .
MR
0466568 Zbl
0373.44003 article
Abstract
BibTeX
In this note, we establish certain properties of the Cauchy integral on Lipschitz curves and prove the \( L^p \) -boundedness of some related operators. In particular, we obtain the recent results of R. R. Coifman and Y. Meyer [1976] on the continuity of the so-called commutator operators.
@article {key0466568m,
AUTHOR = {Calder\'on, A. P.},
TITLE = {Cauchy integrals on {L}ipschitz curves
and related operators},
JOURNAL = {Proc. Natl. Acad. Sci. U.S.A.},
FJOURNAL = {Proceedings of the National Academy
of Sciences of the United States of
America},
VOLUME = {74},
NUMBER = {4},
MONTH = {April},
YEAR = {1977},
PAGES = {1324--1327},
DOI = {10.1073/pnas.74.4.1324},
NOTE = {MR:0466568. Zbl:0373.44003.},
ISSN = {0027-8424},
}
