On the Uniform Boundedness of a Class of Hypersingular Integral Operators on the Hardy Space
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Abstract
For a class of hypersingular integral operators, we establish optimal uniform bounds for their norms on the Hardy space $H^1(\R)$. Our results extend the classical result of Fefferman-Stein for the phase function $1/y$ to phase functions of the form $1/P(y)$ where $P$ is an arbitrary real polynomial. It is revealed that the presence and absence of a constant term in $P$ play a crucial role in the outcome.
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References
H. Al-Qassem, L. Cheng and Y. Pan, Uniform boundedness of oscillatory singular integrals with rational phases,
Forum Math., 37(6) (2025), 1665--1672.
R. Coifman, A real variable characterization of $H^p$,
Studia Math. 51 (1974), 269--274.
D. Fan and Y. Pan, Boundedness of certain oscillatory singular integrals,
Studia Math. 114 (1995), 105--116.
C. Fefferman, Inequalities for strongly singular convolution operators,
Acta Math. 124 (1970), 9--36.
C. Fefferman and E.M. Stein, Hardy spaces of several variables,
Acta. Math. 129 (1972), 137--193.
L. Grafakos, Classical and Modern Fourier Analysis,
Pearson Education, Inc., 2004.
P. Sj"olin, Convolution with oscillating kernels on $H^p$ spaces, J. London Math. Soc. 23 (1981), 442--454.
E.M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and
Oscillatory Integrals, Princeton Univ. Press,
Princeton, NJ, 1993.