The mollifier family generated by a unit-mass smooth bump

Depends on

• The spaces $C_c(\mathbb{R}^n)$ and $C_c^\infty(\mathbb{R}^n)$
• An $L^1$ approximate identity on $\mathbb{R}^n$

Used by

• Compactly supported smooth functions are dense in W^k,p(Rⁿ) Corollary
• Hilbert transform does not map L-infinity to L-infinity Counterexample
• Hilbert transform is not strong type (1,1) Counterexample
• Mollifying a zero extension leaks across the boundary Counterexample
• Pointwise convergent functions need not converge as distributions without local control Counterexample
• A radial mollifier family in Rn Definition
• Weighted L2 spaces and maximal dbar operators Definition
• Hilbert transform of an interval indicator Example
• Mollifying the absolute-value corner Example
• The weak Dirichlet Poisson problem on an interval Example
• A compact C¹ leaf has finitely generated fundamental group Lemma
• A function with nonnegative test pairings is nonnegative a.e. Lemma
• Complex translation, convolution, approximate identities, and mollification Lemma
• Cotlar's inequality for maximal truncations Lemma
• Finite sums of product tests are dense on product open sets Lemma
• Immersion extension on a disk: absolute and relative parametric forms Lemma
• Integration by parts for dual-exponent Sobolev functions Lemma
• Interior mollification commutes with weak derivatives Lemma
• Smooth compactly supported functions of an open set are dense in L² Lemma
• Smoothing continuous families of formal immersions Lemma
• Smoothing continuous families of genuine immersions Lemma
• The maximal distributional dbar operator is closed and densely defined Lemma
• Weak divergence-form equations are invariant under C² boundary charts Lemma
• A unit-mass smooth bump generates an L¹ approximate identity Proposition
• Convolution with a mollifier is smooth, and derivatives pass under the integral sign Theorem
• De Giorgi local boundedness with a scale-correct forcing term Theorem
• Local smooth approximation in integer-order Sobolev spaces Theorem
• Locally integrable functions embed in distributions Theorem
• Mollifier approximation in distributions Theorem
• Multidimensional Ito formula for Brownian-driven processes Theorem
• One-dimensional Ito formula Theorem
• Smale–Hirsch for open source manifolds Theorem
• Space-time harmonic functions yield Brownian local martingales up to exit lifetime Theorem
• The distributional Riesz functional of a subharmonic function is a positive Radon measure Theorem
• The measurable Riemann mapping theorem on the sphere Theorem
• The Mihlin–Hörmander Fourier multiplier theorem Theorem
• Vanishing viscosity selects the viscosity solution Theorem
• Zero weak gradient gives componentwise constants Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis (standard reference, not scraped)
• Walter Rudin, Real and Complex Analysis, 3rd ed. (standard reference, not scraped)