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Non-absolutely Convergent Generalized Laplacian

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Abstract

For possibly discontinuous functions including, for instance, Sobolev functions, we present new Blaschke-Privaloff-type criteria for superharmonicity and harmonicity. This opens the way for introduction of a substantial generalization of the Laplace operator. These potential-theoretic considerations lead to a new kind of non-absolutely convergent integral where the integrand may be a highly oscillating pointwise function or even a distribution-valued function. In turn, this integral gives a precise meaning to some generalized Poisson equations with a wild right hand side.

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Acknowledgments

The problem of sufficiency of Blaschke-Privaloff condition for Sobolev functions was suggested by Antonio Cordoba. We are grateful to him, Joan Verdera and Tony O’Farrell for fruitful discussion and motivation.

We thank the referee for careful reading and useful suggestions.

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  1. Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Jan Malý

  2. Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Ivan Netuka

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  1. Jan Malý
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J. M. has been supported by the grant GA Č R P201/18-07996S of the Czech Science Foundation.

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Malý, J., Netuka, I. Non-absolutely Convergent Generalized Laplacian. Potential Anal 55, 539–562 (2021). https://doi.org/10.1007/s11118-020-09868-y

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