Truncated Hilbert transform and principal value

Fix $1\le p<\mathrm{\infty }$ and a function $f\in L^p(\mathbb{R})$, with the $L^p$ conventions of Complex Lp classes and Euclidean test-function conventions. For $\epsilon >0$ and $x\in \mathbb{R}$ define the truncated Hilbert transform

$H_{\epsilon }f(x):=\frac{1}{\pi }{\int }_{\mathrm{\mid }x-y\mathrm{\mid }>\epsilon }\frac{f(y)}{x-y}dy=\frac{1}{\pi }{\int }_{\mathrm{\mid }t\mathrm{\mid }>\epsilon }\frac{f(x-t)}{t}dt\mathrm{.}$

Each truncation is an ordinary Lebesgue integral over the complement of an interval of length $2\epsilon$ around $x$, and it is absolutely convergent. For $p=1$ this follows from the pointwise bound $\mathrm{\mid }1\mathrm{/}(x-y)\mathrm{\mid }<{\epsilon }^{-1}$ on the domain of integration; for $1<p<\mathrm{\infty }$ it follows from Hölder's inequality applied to the two half-lines $x-y>\epsilon$ and $x-y<-\epsilon$, where $\mathrm{\mid }x-y{\mathrm{\mid }}^{-1}$ has finite $L^q$ norm, with $q$ conjugate to $p$ (Complex Holder, Minkowski, and the quotient norm). Changing $f$ on a null set changes no integral, so $H_{\epsilon }f(x)$ is a well-defined number attached to the class of $f$; and $H_{\epsilon }f$ is itself a measurable function of $x$.

The Hilbert transform in the principal-value sense is defined only where the truncations converge:

$H_{\mathrm{p}\mathrm{v}}f(x):={\lim }_{\epsilon \downarrow 0}{H}_{\epsilon }f(x),$

whenever this limit exists in $\mathbb{C}$. No almost-everywhere existence of this limit, and no bound of $H_{\mathrm{p}\mathrm{v}}f$ in any $L^p$ norm, is asserted by this definition. The definition also does not extend $H_{\epsilon }$ to $L^{\mathrm{\infty }}$: for the tail $\mathrm{\mid }x-y\mathrm{\mid }>\epsilon$ the bound $\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{/}\epsilon$ is finite but the integral over an unbounded domain is not controlled, so the truncation of a merely bounded $f$ need not converge absolutely at any $x$.

Three distinctions are recorded here for later use. First, $H_{\epsilon }f$ is an integral of a truncated singular kernel, while the pairing of a test function with the principal-value distribution of $1\mathrm{/}(\pi x)$ is a separate object; the two agree only under the convergence just defined. Second, the limit is taken symmetrically in $\epsilon$ about the singularity $y=x$, and unsymmetric truncations are a different object. Third, $H_{\mathrm{p}\mathrm{v}}f$ is a pointwise partial function, whereas the $L^2(\mathbb{R})$ extension constructed later on this page is a single bounded operator agreeing with $H_{\mathrm{p}\mathrm{v}}$ where the latter exists on a dense class.