Strong differentiability properties of Bessel potentials
Authors:
Daniel J. Deignan and William P. Ziemer
Journal:
Trans. Amer. Math. Soc. 225 (1977), 113122
MSC:
Primary 31B15
MathSciNet review:
0422645
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Abstract: This paper is concerned with the ``strong'' differentiability properties of Bessel potentials of order of functions. Thus, for such a function f, we investigate the size (in the sense of an appropriate capacity) of the set of points x for which there is a polynomial of degree such that where, for example, S is allowed to run through the family of all oriented rectangles containing the origin.
 [A]
David
R. Adams, Maximal operators and
capacity, Proc. Amer. Math. Soc. 34 (1972), 152–156. MR 0350314
(50 #2807), http://dx.doi.org/10.1090/S00029939197203503141
 [BZ]
Thomas
Bagby and William
P. Ziemer, Pointwise differentiability and
absolute continuity, Trans. Amer. Math.
Soc. 191 (1974),
129–148. MR 0344390
(49 #9129), http://dx.doi.org/10.1090/S00029947197403443906
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H. Busemann and W. Feller, Zur differentiation der Lebesgueschen Integrale, Fund. Math. 22 (1934), 226256.
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A.P.
Calderón and A.
Zygmund, Local properties of solutions of elliptic partial
differential equations, Studia Math. 20 (1961),
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(25 #310)
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Daniel J. Deignan, Boundary regularity of weak solutions to a quasilinear parabolic equation, Doctoral Dissertation, Indiana University, 1974.
 [FZ]
Herbert
Federer and William
P. Ziemer, The Lebesgue set of a function whose distribution
derivatives are 𝑝th power summable, Indiana Univ. Math. J.
22 (1972/73), 139–158. MR 0435361
(55 #8321)
 [GZ]
Ronald
Gariepy and William
P. Ziemer, Behavior at the boundary of solutions of quasilinear
elliptic equations, Arch. Rational Mech. Anal. 56
(1974/75), 372–384. MR 0355332
(50 #7807)
 [JMZ]
B. Jessen, J. Marcinkiewicz and A. Zygmund, Note on the differentiability of multiple integrals, Fund. Math. 25 (1935), 217234.
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Norman
G. Meyers, A theory of capacities for potentials of functions in
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255–292 (1971). MR 0277741
(43 #3474)
 [M2]
Norman
G. Meyers, Taylor expansion of Bessel potentials, Indiana
Univ. Math. J. 23 (1973/74), 1043–1049. MR 0348482
(50 #980)
 [M]
Anthony
P. Morse, Perfect blankets, Trans. Amer. Math. Soc. 61 (1947), 418–442. MR 0020618
(8,571h), http://dx.doi.org/10.1090/S00029947194700206180
 [R]
N.
M. Rivière, Singular integrals and multiplier
operators, Ark. Mat. 9 (1971), 243–278. MR 0440268
(55 #13146)
 [S]
S. Saks, Remark on the differentiability of the Lebesgue indefinite integral, Fund. Math. 22 (1934), 257261.
 [ST]
Elias
M. Stein, Singular integrals and differentiability properties of
functions, Princeton Mathematical Series, No. 30, Princeton University
Press, Princeton, N.J., 1970. MR 0290095
(44 #7280)
 [Z1]
A. Zygmund, On the differentiability of multiple integrals, Fund. Math. 23 (1934), 143149.
 [Z2]
, Trigonometric series, 2nd ed., Cambridge Univ. Press, New York, 1959. MR 21 #6498.
 [A]
 David R. Adams, Maximal operators and capacity, Proc. Amer. Math. Soc. 34 (1972), 152156. MR 50 #2807. MR 0350314 (50:2807)
 [BZ]
 T. Bagby and W. P. Ziemer, Pointwise differentiability and absolute continuity, Trans. Amer. Math. Soc. 191 (1974), 129148. MR 49 #9129. MR 0344390 (49:9129)
 [BF]
 H. Busemann and W. Feller, Zur differentiation der Lebesgueschen Integrale, Fund. Math. 22 (1934), 226256.
 [CZ]
 A. P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171225. MR 25 #310. MR 0136849 (25:310)
 [D]
 Daniel J. Deignan, Boundary regularity of weak solutions to a quasilinear parabolic equation, Doctoral Dissertation, Indiana University, 1974.
 [FZ]
 H. Federer and W. P. Ziemer, The Lebesgue set of a function whose distribution derivatives are pth power summable, Indiana Univ. Math. J. 22 (1972), 139158. MR 0435361 (55:8321)
 [GZ]
 Ronald Gariepy and W. P. Ziemer, Behavior at the boundary of solutions of quasilinear elliptic equations, Arch Rational Mech. Anal. 56 (1974/75), 372384. MR 50 #7807. MR 0355332 (50:7807)
 [JMZ]
 B. Jessen, J. Marcinkiewicz and A. Zygmund, Note on the differentiability of multiple integrals, Fund. Math. 25 (1935), 217234.
 [M1]
 N. G. Meyers, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand. 26 (1970), 255292 (1971). MR 43 #3474. MR 0277741 (43:3474)
 [M2]
 , Taylor expansion of Bessel potentials, Indiana Univ. Math J. 23 (1973/74), 10431049. MR 50 #980. MR 0348482 (50:980)
 [M]
 A. P. Morse, Perfect blankets, Trans. Amer. Math. Soc. 61 (1947), 418442. MR 8, 571. MR 0020618 (8:571h)
 [R]
 N. Riviere, Singular integrals and multiplier operators, Ark. Mat. 9 (1971), 243278. MR 0440268 (55:13146)
 [S]
 S. Saks, Remark on the differentiability of the Lebesgue indefinite integral, Fund. Math. 22 (1934), 257261.
 [ST]
 E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1970. MR 44 #7280. MR 0290095 (44:7280)
 [Z1]
 A. Zygmund, On the differentiability of multiple integrals, Fund. Math. 23 (1934), 143149.
 [Z2]
 , Trigonometric series, 2nd ed., Cambridge Univ. Press, New York, 1959. MR 21 #6498.
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DOI:
http://dx.doi.org/10.1090/S00029947197704226457
PII:
S 00029947(1977)04226457
Article copyright:
© Copyright 1977 American Mathematical Society
