Variable Step Mollifiers and Applications
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Integral Equations and Operator Theory
Variable Step Mollifiers and Applications Michael Hinterm¨ uller, Kostas Papafitsoros Carlos N. Rautenberg
and
Abstract. We consider a mollifying operator with variable step that, in contrast to the standard mollification, is able to preserve the boundary values of functions. We prove boundedness of the operator in all basic Lebesgue, Sobolev and BV spaces as well as corresponding approximation results. The results are then applied to extend recently developed theory concerning the density of convex intersections. Mathematics Subject Classification. 46E35, 42B20, 49J40, 26A45. Keywords. Mollification, Integral operators, Boundary values, Density of convex sets.
1. Introduction Consider an open and bounded domain Ω ⊆ RN , and bounded non-negative smooth maps ρ, η : RN → R such that supp(ρ) ⊆ B1 (0) := {x ∈ RN : |x| ≤ 1} with “supp” the support set of a function, and η(x) ≤ dist(x, ∂Ω) for all x ∈ Ω. Then, we define the variable step mollifier T as follows: Definition 1.1. Let f ∈ L1loc (Ω). We define T f (x) for x ∈ Ω, as −1 T f (x) := Mρ ρ(z)f (x − η(x)z) dz, with Mρ := ρ(y) dy , B1 (0)
B1 (0)
(1.1) with B1 (0) := {x ∈ R
N
: |x| < 1}.
Note that if η = 1 and if ρ is a smooth function, then (1.1) reduces to the usual mollifier. In this paper, however, we are interested in η(x) ≤ dist(x, ∂Ω) for all x ∈ Ω, which implies that the domain of integration in the definition of T is always within Ω. To the best of our knowledge, the idea of such mollifiers with variable step was introduced by Burenkov; see [6,7] and references therein. In particular, Burenkov considers mollifiers defined as infinite sums of constant step mollifiers that are localized by means of the partition of unity procedure. 0123456789().: V,-vol
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Further, the operator in Definition 1.1 is mentioned on Remark 26 in [7] but only a few comments are made. We also mention the paper by Shan'kov [21], where this operator is used to prove certain trace theorems for weighted Sobolev spaces. A detailed study of such operators, however, is still lacking in the literature. In particular, we focus on identifying conditions on the pair (ρ, η) to guarantee boundedness of the operator on Lebesgue, Sobolev, and bounded variation (BV) spaces, and the preservation of trace values, respectively. Additionally, we study properties of approximations Tn (where η is replaced by η/n, n ∈ N) of T . Finally, we apply our work to establish results involving the density of convex intersections. For more on applications we refer to [15], and they are also partly recalled at the end of this section, for convenience. In general, ρ denotes a standard smooth function, supported on the unit ball centred at the origin, and η is a non-negative, smooth function in Ω, with the property that its values and all its derivatives, vanish on the boundary of Ω. The existence of such a function follows from a classical result of Whitney [22]; see also Theorem 2.1. Moreover, the function Mρ helps to normalize the con
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