Open AccessHeat kernel estimates and functional calculi of $-b \Delta$
Author(s) -
Alan McIntosh,
Andrea R. Nahmod
Publication year - 2000
Publication title -
mathematica scandinavica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.553
H-Index - 30
eISSN - 1903-1807
pISSN - 0025-5521
DOI - 10.7146/math.scand.a-14310
Subject(s) - mathematics , bounded function , heat kernel , kernel (algebra) , heat equation , combinatorics , quadratic equation , operator (biology) , function (biology) , measurable function , mathematical analysis , geometry , chemistry , biochemistry , repressor , evolutionary biology , biology , transcription factor , gene
We show that the elliptic operator ${\mathcal L} = - b(x) \Delta$ has a bounded $H^\infty$ functional calculus in $L^p(\boldsymbol R^n), 1 < p < \infty$, where $b$ is a bounded measurable complex-valued function with positive real part. In the process, we prove quadratic estimates for ${\mathcal L}$, and obtain bounds with fast decay and Hölder continuity estimates for $k_t(x,y) b(y)$ and its gradient, where $k_t(x,y)$ is the heat kernel of $-b(x) \Delta$. This implies $L^p$ regularity of solutions to the parabolic equation $\partial_t u + {\mathcal L} u = 0$.