# American Institute of Mathematical Sciences

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.   Google Scholar [2] Bo Berndtsson, $\bar\partial$ and Schrödinger operators,, Math. Z., 221 (1996), 401.   Google Scholar [3] Albert Boggess and Andrew Raich, Heat kernels, smoothness estimates and exponential decay,, preprint, ().   Google Scholar [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.   Google Scholar [5] Albert Boggess and Andrew Raich, The □$_b$-heat equation on quadric manifolds,, J. Geom. Anal., 21 (2011), 256.   Google Scholar [6] Michael Christ, Pointwise estimates for the relative fundamental solution of $\overline\partial_b$,, Proc. Am. Math. Soc., 104 (1988), 787.   Google Scholar [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.   Google Scholar [8] Michael Christ, On the $\bar\partial$ equation in weighted $L^2$ norms in $C^1$,, J. Geom. Anal., 1 (1991), 193.   Google Scholar [9] John Erik Fornæss and Nessim Sibony, On $L^p$ estimates for $\overline\partial$,, In, (1989), 129.   Google Scholar [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.   Google Scholar [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.   Google Scholar [12] Ewa Ligocka, On the Forelli-Rudin construction and weighted Bergman projections,, Studia Math., 94 (1989), 257.   Google Scholar [13] Matei Machedon, Estimates for the parametrix of the Kohn Laplacian on certain domains,, Invent. Math., 91 (1988), 339.   Google Scholar [14] Matei Machedon, Szegö kernels on pseudoconvex domains with one degenerate eigenvalue,, Ann. of Math., 128 (1988), 619.   Google Scholar [15] Jeffrey McNeal, Boundary behavior of the Bergman kernel function in $C^2$,, Duke Math. J., 58 (1989), 499.   Google Scholar [16] Jeffrey McNeal, Convex domains of finite type,, J. Funct. Anal., 108 (1992), 361.   Google Scholar [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.   Google Scholar [18] Alexander Nagel and Elias Stein, The □$_b$-heat equation on pseudoconvex manifolds of finite type in $C^2$,, Math. Z., 238 (2001), 37.   Google Scholar [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.   Google Scholar [20] Alexander Nagel, Elias Stein and Stephen Wainger, Balls and metrics defined by vector fields I: Basic properties,, Acta Math., 155 (1985), 103.   Google Scholar [21] Andrew Raich, Heat equations in $R\times C$,, J. Funct. Anal., 240 (2006), 1.   Google Scholar [22] Andrew Raich, One-parameter families of operators in $C$,, J. Geom. Anal., 16 (2006), 353.   Google Scholar [23] Andrew Raich, Pointwise estimates of relative fundamental solutions for heat equations in $R\times C$,, Math. Z., 256 (2007), 193.   Google Scholar [24] Brian Street, The □$_b$-heat equation and multipliers via the wave equation,, Math. Z., 263 (2009), 861.   Google Scholar

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##### References:
 [1] Bo Berndtsson, Some recent results on estimates for the $\overline\partial$-equation,, In, (1994), 27.   Google Scholar [2] Bo Berndtsson, $\bar\partial$ and Schrödinger operators,, Math. Z., 221 (1996), 401.   Google Scholar [3] Albert Boggess and Andrew Raich, Heat kernels, smoothness estimates and exponential decay,, preprint, ().   Google Scholar [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.   Google Scholar [5] Albert Boggess and Andrew Raich, The □$_b$-heat equation on quadric manifolds,, J. Geom. Anal., 21 (2011), 256.   Google Scholar [6] Michael Christ, Pointwise estimates for the relative fundamental solution of $\overline\partial_b$,, Proc. Am. Math. Soc., 104 (1988), 787.   Google Scholar [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.   Google Scholar [8] Michael Christ, On the $\bar\partial$ equation in weighted $L^2$ norms in $C^1$,, J. Geom. Anal., 1 (1991), 193.   Google Scholar [9] John Erik Fornæss and Nessim Sibony, On $L^p$ estimates for $\overline\partial$,, In, (1989), 129.   Google Scholar [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.   Google Scholar [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.   Google Scholar [12] Ewa Ligocka, On the Forelli-Rudin construction and weighted Bergman projections,, Studia Math., 94 (1989), 257.   Google Scholar [13] Matei Machedon, Estimates for the parametrix of the Kohn Laplacian on certain domains,, Invent. Math., 91 (1988), 339.   Google Scholar [14] Matei Machedon, Szegö kernels on pseudoconvex domains with one degenerate eigenvalue,, Ann. of Math., 128 (1988), 619.   Google Scholar [15] Jeffrey McNeal, Boundary behavior of the Bergman kernel function in $C^2$,, Duke Math. J., 58 (1989), 499.   Google Scholar [16] Jeffrey McNeal, Convex domains of finite type,, J. Funct. Anal., 108 (1992), 361.   Google Scholar [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.   Google Scholar [18] Alexander Nagel and Elias Stein, The □$_b$-heat equation on pseudoconvex manifolds of finite type in $C^2$,, Math. Z., 238 (2001), 37.   Google Scholar [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.   Google Scholar [20] Alexander Nagel, Elias Stein and Stephen Wainger, Balls and metrics defined by vector fields I: Basic properties,, Acta Math., 155 (1985), 103.   Google Scholar [21] Andrew Raich, Heat equations in $R\times C$,, J. Funct. Anal., 240 (2006), 1.   Google Scholar [22] Andrew Raich, One-parameter families of operators in $C$,, J. Geom. Anal., 16 (2006), 353.   Google Scholar [23] Andrew Raich, Pointwise estimates of relative fundamental solutions for heat equations in $R\times C$,, Math. Z., 256 (2007), 193.   Google Scholar [24] Brian Street, The □$_b$-heat equation and multipliers via the wave equation,, Math. Z., 263 (2009), 861.   Google Scholar
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