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GENERALIZED BESSEL AND RIESZ POTENTIALS ON METRIC MEASURE SPACES
"... Abstract. We introduce generalized Bessel and Riesz potentials on metric measure spaces and the corresponding potential spaces. Estimates of the Bessel and Riesz kernels are given which reflect the intrinsic structure of the spaces. Finally, we state the relationship between Bessel (or Riesz) operat ..."
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Abstract. We introduce generalized Bessel and Riesz potentials on metric measure spaces and the corresponding potential spaces. Estimates of the Bessel and Riesz kernels are given which reflect the intrinsic structure of the spaces. Finally, we state the relationship between Bessel (or Riesz) operators and subordinate semigroups. Contents
OBTAINING UPPER BOUNDS OF HEAT KERNELS FROM LOWER
"... Abstract. We show that a neardiagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an ondiagonal upper bound. If in addition the Dirichlet form is local and regular then we obtain a full offdiagonal upper bound of the heat kernel prov ..."
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Cited by 4 (0 self)
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Abstract. We show that a neardiagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an ondiagonal upper bound. If in addition the Dirichlet form is local and regular then we obtain a full offdiagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a neardiagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel. Contents
Potential spaces on fractals
, 2004
"... Abstract. We introduce potential spaces on fractal metric spaces, investigate their embedding theorems, and derive various Besov spaces. Our starting point is that there exists a local, stochastically complete heat kernel satisfying a twosided estimate on the fractal considered. The results of this ..."
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Abstract. We introduce potential spaces on fractal metric spaces, investigate their embedding theorems, and derive various Besov spaces. Our starting point is that there exists a local, stochastically complete heat kernel satisfying a twosided estimate on the fractal considered. The results of this paper are among the important consequences of the heat kernel on the fractal. 1.
Running Head: Nonlocal Dirichlet Forms
"... Abstract. We use an elementary method to obtain Nashtype inequalities for nonlocal Dirichlet forms on dsets. We obtain twoside estimates for the corresponding heat kernels if the walk dimensions of heat kernels are less than 2; these estimates are obtained by combining probabilistic and analytic ..."
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Abstract. We use an elementary method to obtain Nashtype inequalities for nonlocal Dirichlet forms on dsets. We obtain twoside estimates for the corresponding heat kernels if the walk dimensions of heat kernels are less than 2; these estimates are obtained by combining probabilistic and analytic methods. Our arguments partly simplify those in
HEAT KERNELS AND NONLINEAR PDES ON FRACTALS
"... Abstract. We use the heat kernel to study two distinct types of nonlinear partial differential equations on fractals, namely, inhomogeneous semilinear parabolic equations with source term and nonlinear elliptic equations. Contents ..."
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Abstract. We use the heat kernel to study two distinct types of nonlinear partial differential equations on fractals, namely, inhomogeneous semilinear parabolic equations with source term and nonlinear elliptic equations. Contents
Obtaining Upper . . . From Lower Bounds
, 2007
"... We show that a neardiagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an ondiagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full offdiagonal upper bound of the heat kernel provided the ..."
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We show that a neardiagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an ondiagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full offdiagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a neardiagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel.