A sharp Leibniz rule for $${\mathrm {BV}}$$ BV functions in metric spaces
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A sharp Leibniz rule for BV functions in metric spaces Panu Lahti1 Received: 15 November 2018 / Accepted: 1 November 2019 © Universidad Complutense de Madrid 2019
Abstract We prove a Leibniz rule for BV functions in a complete metric space that is equipped with a doubling measure and supports a Poincaré inequality. Unlike in previous versions of the rule, we do not assume the functions to be locally essentially bounded and the end result does not involve a constant C ≥ 1, and so our result seems to be essentially the best possible. In order to obtain the rule in such generality, we first study the weak* convergence of the variation measure of BV functions, with quasi semicontinuous test functions. Keywords Function of bounded variation · Leibniz rule · Metric measure space · Weak* convergence · Quasi semicontinuity Mathematics Subject Classification 30L99 · 31E05 · 26B30
1 Introduction The Leibniz rule for functions of bounded variation (BV functions) says that if u, v ∈ BV(Rn ) ∩ L ∞ (Rn ), then the variation measures satisfy d D(uv) = u d Dv + v d Du,
(1.1)
where u, v are the so-called precise representatives of u and v; see [31] or [32, Section 4.6.4]. More precisely, this result is proved in the above references with somewhat weaker assumptions; in particular, the boundedness assumption can be weakened to only one of the functions being locally (essentially) bounded. In the past two decades, a theory of BV functions as well as other topics in analysis has been developed in the abstract setting of metric measure spaces. The standard assumptions in this setting are that (X , d, μ) is a complete metric space equipped with
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Panu Lahti [email protected] Institut für Mathematik, Universität Augsburg, Universitätsstr. 14, 86159 Augsburg, Germany
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a Borel regular, doubling outer measure μ, and that X supports a Poincaré inequality. See Sect. 2 for definitions. In this setting, the following Leibniz rule for BV functions was proved in [17]. Proposition 1.1 ([17, Proposition 4.2]) Let u, v ∈ BV(X ) ∩ L ∞ (X ) be nonnegative functions. Then uv ∈ BV(X ) ∩ L ∞ (X ) such that dD(uv) ≤ Cv ∨ dDu + Cu ∨ dDv for some constant C ≥ 1 that depends only on the doubling constant of the measure and the constants in the Poincaré inequality. Note that in metric spaces, one cannot talk about the vector measure Du, only the total variation Du. In the above Leibniz rule, we see that again the functions are assumed to be in L ∞ (X ). Additionally, there is a multiplicative constant C ≥ 1 that arises from the use of a discrete convolution technique in the proof of the Leibniz rule. This is a common technique in metric space analysis, and sometimes the constant C appearing in an end result cannot be removed, see e.g. [13, Remark 4.7, Example 4.8]. On the other hand, for the upper gradients of Newton–Sobolev functions (a generalization of Sobolev functions to metric spaces), one has the Leibniz rule guv ≤ ugv + vgu , which does not involve a constant C. Thus it is natural to ask whether the constan
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