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Register a new userSingular integrals along lacunary directions in $mathbb{R}^n$
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A recent result by Parcet and Rogers is that finite order lacunarity characterizes the boundedness of the maximal averaging operator associated to an infinite set of directions in $mathbb{R}^n$. Their proof is based on geometric-combinatorial coverings of fat hyperplanes by two-dimensional wedges. Seminal results by Nagel-Stein-Wainger relied on geometric coverings of n-dimensional nature. In this article we find the sharp cardinality estimate for singular integrals along finite subsets of finite order lacunary sets in all dimensions. Previous results only covered the special case of the directional Hilbert transform in dimensions two and three. The proof is new in all dimensions and relies, among other ideas, on a precise covering of the n-dimensional Nagel-Stein-Wainger cone by two-dimensional Parcet-Rogers wedges.
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Here we present a method of constructing steerable wavelet frames in $L_2(mathbb{R}^d)$ that generalizes and unifies previous approaches, including Simoncellis pyramid and Riesz wavelets. The motivation for steerable wavelets is the need to more accurately account for the orientation of data. Such wavelets can be constructed by decomposing an isotropic mother wavelet into a finite collection of oriented mother wavelets. The key to this construction is that the angular decomposition is an isometry, whereby the new collection of wavelets maintains the frame bounds of the original one. The general method that we propose here is based on partitions of unity involving spherical harmonics. A fundamental aspect of this construction is that Fourier multipliers composed of spherical harmonics correspond to singular integrals in the spatial domain. Such transforms have been studied extensively in the field of harmonic analysis, and we take advantage of this wealth of knowledge to make the proposed construction practically feasible and computationally efficient.
We prove certain $L^p$ estimates ($1<p<infty$) for non-isotropic singular integrals along surfaces of revolution. As an application we obtain $L^p$ boundedness of the singular integrals under a sharp size condition on their kernels.
Let $D$ be a nonnegative integer and ${mathbf{Theta}}subset S^1$ be a lacunary set of directions of order $D$. We show that the $L^p$ norms, $1<p<infty$, of the maximal directional Hilbert transform in the plane $$ H_{{mathbf{Theta}}} f(x):= sup_{vin {mathbf{Theta}}} Big|mathrm{p.v.}int_{mathbb R }f(x+tv)frac{mathrm{d} t}{t}Big|, qquad x in {mathbb R}^2, $$ are comparable to $(log#{mathbf{Theta}})^frac{1}{2}$. For vector fields $mathsf{v}_D$ with range in a lacunary set of of order $D$ and generated using suitable combinations of truncations of Lipschitz functions, we prove that the truncated Hilbert transform along the vector field $mathsf{v}_D$, $$ H_{mathsf{v}_D,1} f(x):= mathrm{p.v.} int_{ |t| leq 1 } f(x+tmathsf{v}_D(x)) ,frac{mathrm{d} t}{t}, $$ is $L^p$-bounded for all $1<p<infty$. These results extend previous bounds of the first author with Demeter, and of Guo and Thiele.
Let $Omega$ be a function of homogeneous of degree zero and vanish on the unit sphere $mathbb {S}^n$. In this paper, we investigate the limiting weak-type behavior for singular integral operator $T_Omega$ associated with rough kernel $Omega$. We show that, if $Omegain Llog L(mathbb S^{n})$, then $lim_{lambdato0^+}lambda|{xinmathbb{R}^n:|T_Omega(f)(x)|>lambda}| = n^{-1}|Omega|_{L^1(mathbb {S}^n)}|f|_{L^1(mathbb{R}^n)},quad0le fin L^1(mathbb{R}^n).$ Moreover,$(n^{-1}|Omega|_{L^1(mathbb{S}^{n-1})}$ is a lower bound of weak-type norm of $T_Omega$ when $Omegain Llog L(mathbb{S}^{n-1})$. Corresponding results for rough bilinear singular integral operators defined in the form $T_{vecOmega}(f_1,f_2) = T_{Omega_1}(f_1)cdot T_{Omega_2}(f_2)$ have also been established.
In this paper, we prove an $L^2-L^2-L^2$ decay estimate for a trilinear oscillatory integral of convolution type in $mathbb{R}^d,$ which recovers the earlier result of Li (2013) when $d=1.$ We discuss the sharpness of our result in the $d=2$ case. Our main hypothesis has close connections to the property of simple nondegeneracy studied by Christ, Li, Tao and Thiele (2005).
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