On finite Hilbert transforms
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- by Kevin F. Clancey PDF
- Trans. Amer. Math. Soc. 212 (1975), 347-354 Request permission
Abstract:
Let E be a bounded measurable subset of the real line. The finite Hilbert transform is the operator ${H_E}$ defined on one of the spaces ${L^p}(E)(1 < p < \infty )$ by \[ {H_E}f(x) = {(\pi i)^{ - 1}}\int _E {f(t){{(t - x)}^{ - 1}}\;dt;} \] here, the singular integral is interpreted as a Cauchy principal value. The main result establishes that for ${H_E}$ to be Fredholm on ${L^p}(E)$, when $p \ne 2$, it is necessary and sufficient that E be equal almost everywhere to a finite union of intervals. The sufficiency of this condition was established in 1960 by H. Widom. In the case where E is not a finite union of intervals and $p < 2$ it is shown that the operator ${H_E}$ has an infinite dimensional null space. The method of proof is constructive.References
- Kevin Clancey, Completely seminormal operators with boundary eigenvalues, Trans. Amer. Math. Soc. 182 (1973), 133–143. MR 341167, DOI 10.1090/S0002-9947-1973-0341167-1
- Adriano M. Garsia, Topics in almost everywhere convergence, Lectures in Advanced Mathematics, No. 4, Markham Publishing Co., Chicago, Ill., 1970. MR 0261253
- I. C. Gohberg and N. Ja. Krupnik, Singular integral operators with piecewise continuous coefficients and their symbols, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 940–964 (Russian). MR 0291893
- R. K. Juberg, Finite Hilbert transforms in $L^{p}$, Bull. Amer. Math. Soc. 78 (1972), 435–438. MR 291894, DOI 10.1090/S0002-9904-1972-12933-1
- Eliahu Shamir, Reduced Hilbert transforms and singular integral equations, J. Analyse Math. 12 (1964), 277–305. MR 165328, DOI 10.1007/BF02807437 E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Clarendon Press, Oxford, 1937.
- F. G. Tricomi, Integral equations, Pure and Applied Mathematics, Vol. V, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. MR 0094665
- Harold Widom, Singular integral equations in $L_{p}$, Trans. Amer. Math. Soc. 97 (1960), 131–160. MR 119064, DOI 10.1090/S0002-9947-1960-0119064-7
- Antoni Zygmund, Intégrales singulières, Lecture Notes in Mathematics, Vol. 204, Springer-Verlag, Berlin-New York, 1971 (French). MR 0467191 —, Trigonometrical series. Vols. I, II, 2nd rev. ed., Cambridge Univ. Press, New York, 1959. MR 21 #6498.
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 212 (1975), 347-354
- MSC: Primary 47G05; Secondary 44A15
- DOI: https://doi.org/10.1090/S0002-9947-1975-0377598-5
- MathSciNet review: 0377598