On approach regions for the conjugate Poisson integral and singular integrals

• Autorzy: S. Ferrando , R. L. Jones , K. Reinhold
• Języki publikacji: EN

Abstrakty

EN

Let ũ denote the conjugate Poisson integral of a function $f ∈ L^{p}(ℝ)$. We give conditions on a region Ω so that $lim_{(v,ε)→(0,0)}_{(v,ε)∈Ω} ũ(x+v,ε) = Hf(x)$, the Hilbert transform of f at x, for a.e. x. We also consider more general Calderón-Zygmund singular integrals and give conditions on a set Ω so that $sup_{(v,r)∈Ω} |ʃ_{|t|>r} k(x+v-t)f(t)dt|$ is a bounded operator on $L^p$, 1 < p < ∞, and is weak (1,1).

Słowa kluczowe

EN

cone condition; conjugate Poisson integral; singular integrals; ergodic Hilbert transform

Kategorie tematyczne

• 28D10: One-parameter continuous families of measure-preserving transformations
• 42A50: Conjugate functions, conjugate series, singular integrals
• 42B20: Singular and oscillatory integrals (Calder\'on-Zygmund, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1996
• Tom: 120
• Numer: 2
• Strony: 169-182

Daty

wydano

1996

otrzymano

1995-10-31

poprawiono

1996-02-22

Twórcy

autor

S. Ferrando

• Departamento de Matemáticas, Universidad Nacional de Mar del Plata, Mar del Plata, Bs. As., Argentina ,

autor

R. L. Jones

• Department of Mathematics, DePaul University, 2219 N. Kenmore, Chicago, Illinois 60614, U.S.A.,

autor

K. Reinhold

• Department of Mathematics, University at Albany, SUNY, 1400 Western Ave., Albany, New York 12222, U.S.A.,

Bibliografia

• [1] M. A. Akcoglu and Y. Déniel, Moving weighted averages, Canad. J. Math. 45 (1993), 449-469.
• [2] A. P. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 349-353.
• [3] P. Fatou, Séries trigonométriques et séries de Taylor, Acta Math. 30 (1906), 335-400.
• [4] S. Ferrando, Moving ergodic theorems for superadditive processes, Ph.D. thesis, Univ. of Toronto, 1994.
• [5] A. Nagel and E. M. Stein, On certain maximal functions and approach regions, Adv. Math. 54 (1984), 83-106.
• [6] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970.
• [7] E. M. Stein, Harmonic Analysis, Princeton Univ. Press, Princeton, N.J., 1993.