The pressure commutator #
The Fourier-defined double Riesz operator has the actual heat representation.
We first subtract the two time-integrable heat evolutions, insert the cutoff
difference, and only then use the absolute-integrability theorem to interchange
time and space. The resulting kernel has the proved radial L^(4/3) majorant.
Heat multipliers for the double Riesz transform #
The Fourier convention has a factor 2 * π in the character. Consequently the
heat semigroup has multiplier exp (-4 * π² * s * ‖ξ‖²). Integrating its second
spatial derivative over positive time recovers the double Riesz multiplier.
The Fourier multiplier of a second derivative of the heat semigroup.
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Positive-time integrability holds also at the zero frequency.
Integrating the heat second-derivative multiplier gives the double Riesz symbol.
The absolute time integral is the absolute value of the Riesz symbol.
A second derivative of the heat evolution, expressed in frequency space.
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- One or more equations did not get rendered due to their size.
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The positive-time representation converges absolutely at each spatial point.
The actual double Riesz test operator is the positive-time heat integral.
Fourier representation of the heat kernel #
The ordinary Gaussian Fourier formula is normalized to the three dimensional heat kernel used in the comparison proof. Two justified differentiations of the inverse Fourier integral identify the Hessian multiplier with its kernel.
Gaussian moments for Fourier differentiation #
The zeroth, first and second norm moments of a Gaussian are integrable. The only polynomial estimate used here absorbs the square of the norm into a Gaussian with half the decay rate.
A real Gaussian, regarded as a complex-valued function, is integrable.
The ordinary real Gaussian is integrable on three-dimensional space.
The second norm moment of a Gaussian is integrable.
The first norm moment follows from the zeroth and second moments.
Inverse Fourier transform of the heat semigroup's Gaussian multiplier.
The positive-time heat Hessian multiplier is integrable in frequency.
The inverse Fourier transform of the heat Hessian multiplier is its actual kernel.
Fubini after insertion of the cutoff difference #
The time kernel is multiplied by the cutoff difference before estimating its
absolute integral. The resulting radial majorant belongs to L^(4/3), so
Hölder with the L^4 test function proves integrability on space times time.
The complex pairing integrand, with cancellation already inserted.
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- NavierStokesR3.Comparison.cancelledComplexTimeIntegrand K φ r x ys = (K ys.2 (x - ys.1) * NavierStokesR3.Comparison.cutoffSquareDifference φ x ys.1) • r ys.1
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The cancelled integrand is absolutely integrable on the product space. The time-section hypothesis is needed only away from the origin; cancellation makes the diagonal section identically zero.
Cancellation justifies exchanging the heat-time and spatial integrals.
An inverse Fourier multiplier as a convolution #
For integrable A and ψ, the inverse Fourier transform of A * Fourier ψ
is the convolution of inverseFourier A with ψ. Absolute product
integrability also proves that the spatial convolution exists at every point;
no global integrability of inverseFourier A is required.
Convolution with an inverse Fourier transform is integrable at every fixed point.
Multiplication by an integrable Fourier multiplier becomes a spatial convolution.
Each positive-time Fourier multiplier is convolution with the actual heat Hessian.
The spatial heat-Hessian convolution is integrable at each positive time.
The pointwise Riesz commutator equals the absolutely convergent cancelled heat kernel. The cutoff difference is inserted before spatial/time Fubini.
The paired pressure commutator bound for the actual Fourier-defined Riesz operator.