Abstract
The paper characterizes the kernel functions on ℝn with the property that the associated convolution operators are controlled by certain maximal operators.
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References
L. Ahlfors, A. Beurling, Conformal Invariants and Function Theoretic Null Sets. Acta Math. 83 (1950), 101–129.
H. Alexander, Projections of Polynomial Hulls. J. Funct. Anal. 13 (1973), 13–19.
H. Alexander, On the Area of the Spectrum of an Element of a Uniform Algebra. In Complex Approximation Proceedings, Quebec, July 3–8, 1978, Birkhäuser, 1980, 3–12.
W. Beckner, Geometric Inequalities in Fourier Analysis. In Essays on Fourier Analysis in Honor of Elias M. Stein, Princeton University Press, Princeton, 1995, 36–68.
A. Browder, Introduction to Function Algebras. Benjamin, New York, 1969.
F. Browder, Approximation by Solutions of Partial Differential Equations. Amer. J. Math. 84 (1962), 134–160.
F. Browder, Functional Analysis and Partial Differential Equations. II, Math. Ann. 145 (1961), 81–226.
F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis. Pitman Research Notes in Mathematics Series, 76, 1982.
T.W. Gamelin, Uniform Algebras. Prentice Hall, 1969.
J.E. Gilbert, and M.A.M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge Studies in Advanced Mathematics, 26, Cambridge University Press, 1991.
B. Gustafsson, and D. Khavinson, On Approximation by Harmonic Vector Fields. Houston J. Math. 20 (1994), 75–92.
K. Gürlebeck, and W. Sprössig, Quaternionic and Clifford Calculus for Physicists and Engineers. John Wiley & Sons, New York, 1997.
L. Hedberg, On Certain Convolution Inequalities. Proc. Amer. Math. Soc. 36 (1972), 505–510.
L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol I: Distribution Theory and Fourier Analysis. Springer-Verlag, Berlin, 1983.
D. Khavinson, On Uniform Approximation by Harmonic Functions. Mich. Math. J. 34 (1987), 465–473.
D. Khavinson, Duality and Uniform Approximation by Solutions of Elliptic Equations. Operator Theory: Advances and Applications 35 (1988), Birkhäuser Verlag, Basel, 129–141.
E.H. Lieb, Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities. Annals of Math. 118 (1983), 349–379.
M. Martin, Higher-Dimensional Ahlfors-Beurling Inequalities. Proc. Amer. Math. Soc. 126 (1998), 2863–2871.
M. Martin, Convolution and Maximal Operator Inequalities in Clifford Analysis. In Clifford Algebras and Their Applications in Mathematical Physics, Vol. 2: Clifford Analysis, Progress in Physics 9, Birkhäuser Verlag, Basel, 2000, 95–113.
M. Martin, Uniform Approximation by Closed Forms in Several Complex Variables. Preprint 2002.
M. Martin, and P. Szeptycki, Sharp Inequalities for Convolution Operators with Homogeneous Kernels and Applications. Indiana Univ. Math. 46 (1997), 975–988.
M. Mitrea, Singular Integrals, Hardy Spaces, and Clifford Wavelets. Lecture Notes in Mathematics, 1575, Springer-Verlag, Heidelberg, 1994.
M. Putinar, Extreme Hyponormal Operators. Operator Theory: Advances and Applications 28 (1988), 249–265.
J. Ryan, Dirac Operators, Conformal Transformations and Aspects of Classical Harmonic Analysis. Journal of Lie Theory 8 (1998), 67–82.
E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, NJ, 1970.
E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton, NJ, 1993.
N.N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations. Akademie Verlag, Berlin, 1995.
B.M. Weinstock, Uniform Approximations by Solutions of Elliptic Equations. Proc. Amer. Math. Soc. 41 (1973), 513–517.
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Martin, M., Szeptycki, P. (2004). Integral Transforms Controlled by Maximal Functions. In: Gaşpar, D., Timotin, D., Zsidó, L., Gohberg, I., Vasilescu, FH. (eds) Recent Advances in Operator Theory, Operator Algebras, and their Applications. Operator Theory: Advances and Applications, vol 153. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7314-8_10
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DOI: https://doi.org/10.1007/3-7643-7314-8_10
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