Journal of the Mathematical Society of Japan

Weighted inequalities for holomorphic functional calculi of operators with heat kernel bounds

Xuan Thinh DUONG and Lixin YAN

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Let 𝒳 be a space of homogeneous type. Assume that L has a bounded holomorphic functional calculus on L 2 ( Ω ) and L generates a semigroup with suitable upper bounds on its heat kernels where Ω is a measurable subset of 𝒳 . For appropriate bounded holomorphic functions b , we can define the operators b ( L ) on L p ( Ω ) , 1 p . We establish conditions on positive weight functions u , v such that for each p , 1 < p < , there exists a constant c p such that Ω | b ( L ) f ( x ) | p u ( x ) d μ ( x ) c p | | b | | p Ω | f ( x ) | p v ( x ) d μ ( x ) for all f L p ( v d μ ) .

Applications include two-weight L p inequalities for Schrödinger operators with non-negative potentials on R n and divergence form operators on irregular domains of R n .

Article information

J. Math. Soc. Japan, Volume 57, Number 4 (2005), 1129-1152.

First available in Project Euclid: 14 June 2006

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Zentralblatt MATH identifier

Primary: 42B25: Maximal functions, Littlewood-Paley theory 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 47B38: Operators on function spaces (general)

holomorphic functional calculus space of homogeneous type singular integral operator weights semigroup kernel elliptic operator


DUONG, Xuan Thinh; YAN, Lixin. Weighted inequalities for holomorphic functional calculi of operators with heat kernel bounds. J. Math. Soc. Japan 57 (2005), no. 4, 1129--1152. doi:10.2969/jmsj/1150287306.

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