May  2012, 11(3): 885-909. doi: 10.3934/cpaa.2012.11.885

Heat equations and the Weighted $\bar\partial$-problem

1. 

Department of Mathematical Sciences, University of Arkansas, SCEN 301, Fayetteville, AR 72701, United States

Received  June 2010 Revised  September 2011 Published  December 2011

The purpose of this article is to establish regularity and pointwise upper bounds for the (relative) fundamental solution of the heat equation associated to the weighted $\bar\partial$-operator in $L^2(C^n)$ for a certain class of weights. The weights depend on a parameter, and we find pointwise bounds for heat kernel, as well as its derivatives in time, space, and the parameter. We also prove cancellation conditions for the heat semigroup. We reduce the $n$-dimensional case to the one-dimensional case, and the estimates in one-dimensional case are achieved by Duhamel's principle and commutator properties of the operators. As an application, we recover estimates of the □$_b$-heat kernel on polynomial models in $C^2$.
Citation: Andrew Raich. Heat equations and the Weighted $\bar\partial$-problem. Communications on Pure & Applied Analysis, 2012, 11 (3) : 885-909. doi: 10.3934/cpaa.2012.11.885
References:
[1]

Bo Berndtsson, Some recent results on estimates for the $\overline\partial$-equation,, In, (1994), 27.

[2]

Bo Berndtsson, $\bar\partial$ and Schrödinger operators,, Math. Z., 221 (1996), 401.

[3]

Albert Boggess and Andrew Raich, Heat kernels, smoothness estimates and exponential decay,, preprint, ().

[4]

Albert Boggess and Andrew Raich, A simplified calculation for the fundamental solution to the heat equation on the Heisenberg group,, Proc. Amer. Math. Soc., 137 (2009), 937.

[5]

Albert Boggess and Andrew Raich, The □$_b$-heat equation on quadric manifolds,, J. Geom. Anal., 21 (2011), 256.

[6]

Michael Christ, Pointwise estimates for the relative fundamental solution of $\overline\partial_b$,, Proc. Am. Math. Soc., 104 (1988), 787.

[7]

Michael Christ, Regularity properties of the $\overline\partial_b$ equation on weakly pseudoconvex CR manifolds of dimension 3,, J. Amer. Math. Soc., 1 (1988), 587.

[8]

Michael Christ, On the $\bar\partial$ equation in weighted $L^2$ norms in $C^1$,, J. Geom. Anal., 1 (1991), 193.

[9]

John Erik Fornæss and Nessim Sibony, On $L^p$ estimates for $\overline\partial$,, In, (1989), 129.

[10]

Siqi Fu and Emil Straube, Semi-classical analysis of Schrödinger operators and compactness in the $\overline\partial$-Neumann problem,, J. Math. Anal. Appl., 271 (2002), 267.

[11]

Siqi Fu and Emil Straube, Correction to: "Semi-classical analysis of Schrödinger operators and compactness in the $\overline\partial$-Neumann problem",, J. Math. Anal. Appl., 280 (2003), 195.

[12]

Ewa Ligocka, On the Forelli-Rudin construction and weighted Bergman projections,, Studia Math., 94 (1989), 257.

[13]

Matei Machedon, Estimates for the parametrix of the Kohn Laplacian on certain domains,, Invent. Math., 91 (1988), 339.

[14]

Matei Machedon, Szegö kernels on pseudoconvex domains with one degenerate eigenvalue,, Ann. of Math., 128 (1988), 619.

[15]

Jeffrey McNeal, Boundary behavior of the Bergman kernel function in $C^2$,, Duke Math. J., 58 (1989), 499.

[16]

Jeffrey McNeal, Convex domains of finite type,, J. Funct. Anal., 108 (1992), 361.

[17]

Alexander Nagel, John-Pierre Rosay, Elias Stein and Stephen Wainger, Estimates for the Bergman and Szegö kernels in $C^2$,, Ann. of Math., 129 (1989), 113.

[18]

Alexander Nagel and Elias Stein, The □$_b$-heat equation on pseudoconvex manifolds of finite type in $C^2$,, Math. Z., 238 (2001), 37.

[19]

Alexander Nagel and Elias Stein, The $\bar\partial_b$-complex on decoupled domains in $C^n$, $n \geq 3$,, Ann. of Math., 164 (2006), 649.

[20]

Alexander Nagel, Elias Stein and Stephen Wainger, Balls and metrics defined by vector fields I: Basic properties,, Acta Math., 155 (1985), 103.

[21]

Andrew Raich, Heat equations in $R\times C$,, J. Funct. Anal., 240 (2006), 1.

[22]

Andrew Raich, One-parameter families of operators in $C$,, J. Geom. Anal., 16 (2006), 353.

[23]

Andrew Raich, Pointwise estimates of relative fundamental solutions for heat equations in $R\times C$,, Math. Z., 256 (2007), 193.

[24]

Brian Street, The □$_b$-heat equation and multipliers via the wave equation,, Math. Z., 263 (2009), 861.

show all references

References:
[1]

Bo Berndtsson, Some recent results on estimates for the $\overline\partial$-equation,, In, (1994), 27.

[2]

Bo Berndtsson, $\bar\partial$ and Schrödinger operators,, Math. Z., 221 (1996), 401.

[3]

Albert Boggess and Andrew Raich, Heat kernels, smoothness estimates and exponential decay,, preprint, ().

[4]

Albert Boggess and Andrew Raich, A simplified calculation for the fundamental solution to the heat equation on the Heisenberg group,, Proc. Amer. Math. Soc., 137 (2009), 937.

[5]

Albert Boggess and Andrew Raich, The □$_b$-heat equation on quadric manifolds,, J. Geom. Anal., 21 (2011), 256.

[6]

Michael Christ, Pointwise estimates for the relative fundamental solution of $\overline\partial_b$,, Proc. Am. Math. Soc., 104 (1988), 787.

[7]

Michael Christ, Regularity properties of the $\overline\partial_b$ equation on weakly pseudoconvex CR manifolds of dimension 3,, J. Amer. Math. Soc., 1 (1988), 587.

[8]

Michael Christ, On the $\bar\partial$ equation in weighted $L^2$ norms in $C^1$,, J. Geom. Anal., 1 (1991), 193.

[9]

John Erik Fornæss and Nessim Sibony, On $L^p$ estimates for $\overline\partial$,, In, (1989), 129.

[10]

Siqi Fu and Emil Straube, Semi-classical analysis of Schrödinger operators and compactness in the $\overline\partial$-Neumann problem,, J. Math. Anal. Appl., 271 (2002), 267.

[11]

Siqi Fu and Emil Straube, Correction to: "Semi-classical analysis of Schrödinger operators and compactness in the $\overline\partial$-Neumann problem",, J. Math. Anal. Appl., 280 (2003), 195.

[12]

Ewa Ligocka, On the Forelli-Rudin construction and weighted Bergman projections,, Studia Math., 94 (1989), 257.

[13]

Matei Machedon, Estimates for the parametrix of the Kohn Laplacian on certain domains,, Invent. Math., 91 (1988), 339.

[14]

Matei Machedon, Szegö kernels on pseudoconvex domains with one degenerate eigenvalue,, Ann. of Math., 128 (1988), 619.

[15]

Jeffrey McNeal, Boundary behavior of the Bergman kernel function in $C^2$,, Duke Math. J., 58 (1989), 499.

[16]

Jeffrey McNeal, Convex domains of finite type,, J. Funct. Anal., 108 (1992), 361.

[17]

Alexander Nagel, John-Pierre Rosay, Elias Stein and Stephen Wainger, Estimates for the Bergman and Szegö kernels in $C^2$,, Ann. of Math., 129 (1989), 113.

[18]

Alexander Nagel and Elias Stein, The □$_b$-heat equation on pseudoconvex manifolds of finite type in $C^2$,, Math. Z., 238 (2001), 37.

[19]

Alexander Nagel and Elias Stein, The $\bar\partial_b$-complex on decoupled domains in $C^n$, $n \geq 3$,, Ann. of Math., 164 (2006), 649.

[20]

Alexander Nagel, Elias Stein and Stephen Wainger, Balls and metrics defined by vector fields I: Basic properties,, Acta Math., 155 (1985), 103.

[21]

Andrew Raich, Heat equations in $R\times C$,, J. Funct. Anal., 240 (2006), 1.

[22]

Andrew Raich, One-parameter families of operators in $C$,, J. Geom. Anal., 16 (2006), 353.

[23]

Andrew Raich, Pointwise estimates of relative fundamental solutions for heat equations in $R\times C$,, Math. Z., 256 (2007), 193.

[24]

Brian Street, The □$_b$-heat equation and multipliers via the wave equation,, Math. Z., 263 (2009), 861.

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