Heat kernel estimates and functional calculi of $-b \Delta$

• Autores: Alan McIntosh, Andrea R. Nahmod
• Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 87, Nº 2, 2000, págs. 287-319
• Idioma: inglés
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• Resumen

  • 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 $\partial_t u + {\mathcal L} u = 0$.