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Register a new userOn the maximal directional Hilbert transform in three dimensions
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We establish the sharp growth rate, in terms of cardinality, of the $L^p$ norms of the maximal Hilbert transform $H_Omega$ along finite subsets of a finite order lacunary set of directions $Omega subset mathbb R^3$, answering a question of Parcet and Rogers in dimension $n=3$. Our result is the first sharp estimate for maximal directional singular integrals in dimensions greater than 2. The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of two-dimensional angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting.
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We study the bilinear Hilbert transform and bilinear maximal functions associated to polynomial curves and obtain uniform $L^r$ estimates for $r>frac{d-1}{d}$ and this index is sharp up to the end point.
In the present paper, we study the geometric discrepancy with respect to families of rotated rectangles. The well-known extremal cases are the axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in all possible directions (polynomial discrepancy). We study several intermediate situations: lacunary sequences of directions, lacunary sets of finite order, and sets with small Minkowski dimension. In each of these cases, extensions of a lemma due to Davenport allow us to construct appropriate rotations of the integer lattice which yield small discrepancy.
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 $|f_x|_{L^1(I)}$.
We establish the sharp growth order, up to epsilon losses, of the $L^2$-norm of the maximal directional averaging operator along a finite subset $V$ of a polynomial variety of arbitrary dimension $m$, in terms of cardinality. This is an extension of the works by Cordoba, for one-dimensional manifolds, Katz for the circle in two dimensions, and Demeter for the 2-sphere. For the case of directions on the two-dimensional sphere we improve by a factor of $sqrt{log N}$ on the best known bound, due to Demeter, and we obtain a sharp estimate for our model operator. Our results imply new $L^2$-estimates for Kakeya-type maximal functions with tubes pointing along polynomial directions. Our proof technique is novel and in particular incorporates an iterated scheme of polynomial partitioning on varieties adapted to directional operators, in the vein of Guth, Guth-Katz, and Zahl.
Let $ v$ be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform operatorname H_{v, epsilon}f(x) := text{p.v.}int_{-epsilon}^ epsilon f(x-yv(x)) frac{dy}y where $ epsilon $ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.thinspace M.thinspace Stein, that if $ v$ is Lipschitz, there is a positive $ epsilon $ for which the transform above is bounded on $ L ^{2}$. Our principal result gives a sufficient condition in terms of the boundedness of a maximal function associated to $ v$. This sufficient condition is that this new maximal function be bounded on some $ L ^{p}$, for some $ 1<p<2$. We show that the maximal function is bounded from $ L ^{2}$ to weak $ L ^{2}$ for all Lipschitz maximal function. The relationship between our results and other known sufficient conditions is explored.
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