Best constants for some operators associated with the Fourier and Hilbert transforms

• Autorzy: B. Hollenbeck , N. J. Kalton , I. E. Verbitsky
• Języki publikacji: EN

Abstrakty

EN

We determine the norm in $L^{p}(ℝ₊)$, 1 < p < ∞, of the operator $I - ℱ_{s}ℱ_{c}$, where $ℱ_{c}$ and $ℱ_{s}$ are respectively the cosine and sine Fourier transforms on the positive real axis, and I is the identity operator. This solves a problem posed in 1984 by M. S. Birman [Bir] which originated in scattering theory for unbounded obstacles in the plane.
We also obtain the $L^{p}$-norms of the operators aI + bH, where H is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real a,b. Best constants in other related inequalities are found.
In a more general framework, we present an alternative proof of the important theorem of Cole relating best constant inequalities involving the Hilbert transform and the existence of subharmonic minorants, which extends to several variables and plurisubharmonic minorants.

Kategorie tematyczne

• 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2003
• Tom: 157
• Numer: 3
• Strony: 237-278

Daty

wydano

2003

Twórcy

autor

B. Hollenbeck

• Department of Mathematics, Emporia State University, Emporia, KS 66801, U.S.A.

autor

N. J. Kalton

• Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.

autor

I. E. Verbitsky

• Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.

Identyfikatory

DOI

10.4064/sm157-3-2