Test functions as elements of Lp #
This file provides the generic bridge from compactly supported test functions on an open set to
Lp classes. A test function belongs to every Lᵖ space for any measure finite on compact sets.
The construction is used by the closed-graph presentations of weak Sobolev spaces.
The bridge is linear: TauCeti.testFunctionLp_add and TauCeti.testFunctionLp_smul record that
passing to the Lᵖ class commutes with the vector space structure of the test functions, and
TauCeti.testFunctionLpₗ bundles it as a linear map. For an inner product space,
TauCeti.gradientTestFunctionLp and TauCeti.gradientTestFunctionLpₗ provide the parallel
construction for the gradient. These facts make the image of C_c^∞(Ω) in an Lᵖ-based function
space a subspace. On the whole space, passing a test function or its gradient to Lᵖ
commutes with translation, as recorded by TauCeti.testFunctionLp_translateTestFunction and
TauCeti.gradientTestFunctionLp_translateTestFunction, and turns dilation of test functions into
the normalized Lᵖ dilation MeasureTheory.Measure.dilateLp, up to an explicit power of the
factor, as recorded by TauCeti.testFunctionLp_dilateTestFunction and
TauCeti.gradientTestFunctionLp_dilateTestFunction.
A distributional derivative bounded by the Lᵠ norm of test functions extends by Hahn–Banach
to a bounded functional on Lᵠ(Ω) for every q ≥ 1. At q = 2, the negative of its Riesz
representative is a weak derivative:
TauCeti.exists_norm_le_hasWeakLineDerivOn_of_abs_integral_lineDeriv_mul_le.
ContDiffOn.enorm_integral_lineDeriv_smul_le supplies the corresponding conjugate-exponent
test bound for a smooth map, using its actual Lᵖ derivative seminorm.
A test function on Omega lies in every Lᵠ(Omega): it is continuous with compact support.
A test function on Omega as an element of Lᵠ(Omega).
Instances For
The norm of a test function's Lᵖ class is its eLpNorm, converted to ℝ.
The extended norm of a test function's Lᵖ class is its eLpNorm for the ambient measure:
the function vanishes outside Omega, so the restriction may be dropped.
Passing from a test function on Omega to its class in Lᵠ(Omega), as a linear map.
Instances For
Passing from a test function to its Lᵠ class is injective for a measure that is positive on
nonempty open sets, such as an additive Haar measure.
A distributional directional test bound extends to a continuous functional on Lᵠ
with the same bound, for every q ≥ 1. No σ-finiteness assumption is needed.
An L²-bounded distributional derivative is an L² weak derivative. Let u be locally
integrable on Omega. If for some C ≥ 0 every test function phi on Omega satisfies
|∫ ∂_v phi * u| ≤ C ‖phi‖₂,
then u has a weak derivative in the direction v on Omega that lies in L²(Omega) and has
norm at most C. Only the restriction of mu to Omega needs to be locally finite.
Integration by parts and Hölder's inequality bound the action of a function's weak directional
derivative on a test function by its Lᵖ derivative seminorm. The function need only be C¹ on
the open test domain.
The gradient of a test function #
The gradient of a test function is smooth.
The gradient of a test function is continuous.
The gradient of a test function on Ω vanishes outside Ω.
The gradient of a test function on Ω is continuous with compact support, so it lies in every
Lᵠ(Ω).
The gradient of a test function on Ω, as an element of Lᵠ(Ω, E).
Instances For
The norm of a test function gradient's Lᵖ class is its eLpNorm, converted to ℝ.
The extended norm of a test function gradient's Lᵖ class is the eLpNorm of its Fréchet
derivative for the ambient measure: the gradient vanishes outside Omega, so the restriction
may be dropped, and ‖∇ phi x‖ = ‖fderiv ℝ phi x‖.
Passing from a test function on Omega to the class of its gradient in Lᵠ(Omega, E), as a
linear map.
Instances For
Translation on the whole space #
Passing a translated whole-space test function to Lᵖ translates its class. The sign
is reversed because translateTestFunction h evaluates at x - h, whereas translateLp h
evaluates at x + h.
Taking the gradient's Lᵖ class commutes with whole-space translation.
Dilation on the whole space #
Passing a dilated whole-space test function φ (r⁻¹ • ·) to Lᵖ gives r ^ (n / p) times
the normalized dilation of its class, where n is the dimension of E.
Passing the gradient of a dilated whole-space test function φ (r⁻¹ • ·) to Lᵖ gives
r ^ (n / p - 1) times the normalized dilation of the gradient's class, where n is the
dimension of E.