Operators associated to flat plane curves: \(L^ p\) estimates via dilation methods

DOI10.1215/S0012-7094-89-05930-9zbMath0723.44006MaRDI QIDQ756639

Anthony Carbery, Michael Christ, Stephen Wainger, David K. Watson, James Vance

Publication date: 1989

Published in: Duke Mathematical Journal (Search for Journal in Brave)

zbMATH Keywords

maximal operatorHilbert transform\(L^ p\)-estimates\(L^ p\)-boundednessdilation methods

Mathematics Subject Classification ID

Maximal functions, Littlewood-Paley theory (42B25) Special integral transforms (Legendre, Hilbert, etc.) (44A15)

Related Items (38)

Classes of singular integrals along curves and surfaces ⋮ \(L^p\) bounds of maximal operators along variable planar curves in the Lipschitz regularity ⋮ Maximal functions along convex curves with lacunary directions ⋮ The maximal Hilbert transform along nonconvex curves ⋮ H¹ boundedness of oscillatory singular integrals with degenerate phase functions ⋮ Maximal functions associated to a family of flat curves in lacunary directions ⋮ Estimates for Hilbert transforms along variable general curves ⋮ \(L^2\)-boundedness of Hilbert transforms along variable curves ⋮ \(L^p\)-estimates for the rough singular integrals associated to surfaces ⋮ Weighted estimates for singular integrals via Fourier transform estimates ⋮ Hilbert transforms along convex curves for \(l^q\)-valued functions ⋮ Rough singular integrals associated to surfaces of revolution ⋮ \(\mathcal {A}_{p, {\mathbb {E}}}\) weights, maximal operators, and Hardy spaces associated with a family of general sets ⋮ Hilbert transforms along double variable fractional monomials ⋮ A general nonlinear version of Roth's theorem on the real line ⋮ \(L^ p\)-estimates for operators associated to oscillating plane curves ⋮ \(L^p\) estimates for rough singular integrals associated to some hypersurfaces ⋮ On rough Marcinkiewicz integrals along surfaces ⋮ The Hilbert transform along the extended evil ladder curve ⋮ Boundedness of singular integrals along surfaces on Lebesgue spaces ⋮ Mixed-norm estimates for a class of nonisotropic directional maximal operators and Hilbert transforms ⋮ Multiparameter singular integrals and maximal operators along flat surfaces ⋮ Maximal functions associated to surfaces of revolution on product domains ⋮ \(L^2\) boundedness of Hilbert transforms along variable flat curves ⋮ Hilbert transforms along variable planar curves: Lipschitz regularity ⋮ \(L^p\) boundedness of Carleson \& Hilbert transforms along plane curves with certain curvature constraints ⋮ \(L^p\) bounds for a class of Marcinkiewicz integral operators ⋮ $L^p$-boundedness for the Hilbert transform and maximal operator along a class of nonconvex curves ⋮ MAXIMAL OPERATORS AND HILBERT TRANSFORMS ALONG FLAT CURVES NEAR L¹ ⋮ Oscillatory hyper Hilbert transforms along variable curves ⋮ Singular integral and maximal operators associated to hypersurfaces: \(L^p\) theory ⋮ Unnamed Item ⋮ Singular integrals associated to hypersurfaces: \(L^2\) theory ⋮ \(L^p\)-estimates for singular integrals and maximal operators associated with flat curves on the Heisenberg group ⋮ On the p-independence boundedness property of Calderón-Zygmund theory ⋮ Maximal singular integral operators along surfaces ⋮ Singular Radon transforms along odd curves on the Heisenberg group ⋮ Hilbert transforms and maximal functions along variable flat curves

Cites Work

• Hilbert transforms for convex curves
• Examples of singular maximal functions unbounded on \(L^ p\)
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• Maximal and singular integral operators via Fourier transform estimates
• Hilbert transforms along curves. I: Nilpotent groups
• Maximal functions for convex curves
• Weak type (1,1) bounds for rough operators
• Hilbert transforms along curves. II: A flat case
• On \(L_ p\) multipliers
• Analyse harmonique non-commutative sur certains espaces homogènes. Etude de certaines intégrales singulières. (Non-commutative harmonic analysis on certain homogeneous spaces. Study of certain singular integrals.)
• Singular integrals and multiplier operators
• 𝐿^{𝑝} estimates for maximal functions and Hilbert transforms along flat convex curves in 𝐑²
• The Hilbert Transform for Convex Curves in R n
• A Remark on Singular Calderon-Zygmund Theory
• The Hilbert Transform and Maximal Function for Approximately Homogeneous Curves
• Singular Integrals and Maximal Functions Associated with Highly Monotone Curves
• Differentiation in lacunary directions
• Problems in harmonic analysis related to curvature
• Conditionally Convergent Series of Linear Operators on L^p -Spaces and L^p -Estimates for Pseudodifferential Operators
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