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Register a new userLower bounds for the truncated Hilbert transform
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Given two intervals $I, J subset mathbb{R}$, we ask whether it is possible to reconstruct a real-valued function $f in L^2(I)$ from knowing its Hilbert transform $Hf$ on $J$. When neither interval is fully contained in the other, this problem has a unique answer (the nullspace is trivial) but is severely ill-posed. We isolate the difficulty and show that by restricting $f$ to functions with controlled total variation, reconstruction becomes stable. In particular, for functions $f in H^1(I)$, we show that $$ |Hf|_{L^2(J)} geq c_1 exp{left(-c_2 frac{|f_x|_{L^2(I)}}{|f|_{L^2(I)}}right)} | f |_{L^2(I)} ,$$ for some constants $c_1, c_2 > 0$ depending only on $I, J$. This inequality is sharp, but we conjecture that $|f_x|_{L^2(I)}$ can be replaced by $|f_x|_{L^1(I)}$.
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In limited data computerized tomography, the 2D or 3D problem can be reduced to a family of 1D problems using the differentiated backprojection (DBP) method. Each 1D problem consists of recovering a compactly supported function $f in L^2(mathcal F)$, where $mathcal F$ is a finite interval, from its partial Hilbert transform data. When the Hilbert transform is measured on a finite interval $mathcal G$ that only overlaps but does not cover $mathcal F$ this inversion problem is known to be severely ill-posed [1]. In this paper, we study the reconstruction of $f$ restricted to the overlap region $mathcal F cap mathcal G$. We show that with this restriction and by assuming prior knowledge on the $L^2$ norm or on the variation of $f$, better stability with Holder continuity (typical for mildly ill-posed problems) can be obtained.
Consider the discrete cubic Hilbert transform defined on finitely supported functions $f$ on $mathbb{Z}$ by begin{eqnarray*} H_3f(n) = sum_{m ot = 0} frac{f(n- m^3)}{m}. end{eqnarray*} We prove that there exists $r <2$ and universal constant $C$ such that for all finitely supported $f,g$ on $mathbb{Z}$ there exists an $(r,r)$-sparse form ${Lambda}_{r,r}$ for which begin{eqnarray*} left| langle H_3f, g rangle right| leq C {Lambda}_{r,r} (f,g). end{eqnarray*} This is the first result of this type concerning discrete harmonic analytic operators. It immediately implies some weighted inequalities, which are also new in this setting.
Answering a key point left open in a recent work of Bongers, Guo, Li and Wick, we provide the following lower bound $$ |b|_{text{BMO}_{gamma}(mathbb{R}^2)}lesssim |[b,H_{gamma}]|_{L^p(mathbb{R}^2)to L^p(mathbb{R}^2)}, $$ where $H_{gamma}$ is the parabolic Hilbert transform.
Let $W$ denote a matrix $A_2$ weight. In this paper, we implement a scalar argument using the square function to deduce square-function type results for vector-valued functions in $L^2(mathbb{R},mathbb{C}^d)$. These results are then used to study the boundedness of the Hilbert transform and Haar multipliers on $L^2(mathbb{R},mathbb{C}^d)$. Our proof shortens the original argument by Treil and Volberg and improves the dependence on the $A_2$ characteristic. In particular, we prove that the Hilbert transform and Haar multipliers map $L^2(mathbb{R},W,mathbb{C}^d)$ to itself with dependence on on the $A_2$ characteristic at most $[W]_{A_2}^{frac{3}{2}} log [W]_{A_2}$.
Consider the discrete quadratic phase Hilbert Transform acting on $ell^{2}$ finitely supported functions $$ H^{alpha} f(n) : = sum_{m eq 0} frac{e^{2 pi ialpha m^2} f(n - m)}{m}. $$ We prove that, uniformly in $alpha in mathbb{T}$, there is a sparse bound for the bilinear form $langle H^{alpha} f , g rangle$. The sparse bound implies several mapping properties such as weighted inequalities in an intersection of Muckenhoupt and reverse Holder classes.
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