Hilbert transform and singular integrals

On the line, for a Schwartz function define $Hf(x)$ by the limit of $\pi^{-1}\int_{|y|>\varepsilon}f(x-y)/y\,dy$ as $\varepsilon$ decreases to zero. Near the singularity, pair positive and negative y to obtain a numerator $f(x-y)-f(x+y)$ that vanishes linearly. Smoothness controls that difference, while rapid decay handles the far tail. The singular kernel itself is not locally absolutely integrable, so applying Young's L¹-kernel bound is not legitimate.

The principal value specifies symmetric exclusion around the singularity. It is not the same as assigning separate values to two divergent one-sided integrals and then subtracting them. Changing a truncation convention without compensating terms can alter the operator. For rough inputs, bounded extension and convergence of truncated formulas require additional theorems; the transparent Schwartz calculation is a safe initial domain. Cancellation is the structural reason the operator can act stably despite the pointwise singularity of its kernel.

The multiplier has magnitude one away from zero, so Plancherel extends H as an isometry of $L^2(\mathbb R)$. Its square is minus the identity there, since the single frequency zero has Lebesgue measure zero and does not affect an L² multiplier. On the period-two-pi circle, define the multiplier $-i\operatorname{sgn}(n)$ at integer n with zero value at n=0. The constant mode is now an actual basis vector rather than a negligible frequency point.

Consequently the periodic transform annihilates constants and its square equals minus the identity plus projection onto the mean. Positive-frequency cosines become sines, while positive-frequency sines become negative cosines. This phase rule preserves the energy of the mean-zero part, not the total energy of an arbitrary periodic input. Keeping the zero mode visible prevents a common mistake: applying the real-line square identity to a constant periodic function and claiming its transform can recover a constant that was removed.

Singular-integral theory gives bounded extensions on real-line Lᵖ for exponents strictly between one and infinity, together with a weak type (1,1) endpoint estimate. The transform is not generally bounded from L¹ to L¹ or from L∞ to L∞. These are not small technical omissions: simple interval inputs produce reciprocal tails and logarithmic singularities. Their local and distant behavior explain why a bounded input can generate an unbounded output and an integrable input can generate a nonintegrable absolute tail.

More general singular-integral operators require hypotheses on kernel size, regularity away from the diagonal, and a boundedness or cancellation structure that makes the initial operator meaningful. A reciprocal-looking formula alone does not imply the full Calderón–Zygmund conclusion. In proofs, decomposing an input into controlled and exceptional pieces ties these operators to maximal weak bounds. In applications, check the operator's actual domain and truncation, then identify whether the desired output estimate is strong, weak, or in a replacement endpoint space.