Analogues of Hausdorff-Young inequalities for the Dirac scattering transform (a.k.a. SU(1,1) nonlinear Fourier transform) were first established by Christ and Kiselev [1],[2]. Later Muscalu, Tao, and Thiele [5] raised a question if the constants can be chosen uniformly in $1\leq p\leq 2$. Here we give a positive answer to that question when the Euclidean real line is replaced by its Cantor group model.
We compute the exact Hausdorff and Packing measures of linear Cantor sets which might not be self similar or homogeneous . The calculation is based on the local behavior of the natural probability measure supported on the sets.
We study a generalization of Morán's sum sets, obtaining information about the $h$-Hausdorff and $h$-packing measures of these sets and certain of their subsets.
We establish formulas for bounds on the Haudorff measure of the intersection of certain Cantor sets with their translates. As a consequence we obtain a formula for the Hausdorff dimensions of these intersections.
We consider (not self-similar) Cantor sets defined by a sequence of piecewise linear functions. We prove that the dimension of the harmonic measure on such a set is strictly smaller than its Hausdorff dimension. Some Hausdorff measure estimates for these sets are also provided.
Let $K$ be the attractor of a linear iterated function system (IFS) $S_j(x)=ρ_jx+b_j,$ $j=1,\cdots,m$, on the real line satisfying the generalized finite type condition (whose invariant open set $\mathcal{O}$ is an interval) with an irreducible weighted incidence matrix. This condition was introduced by Lau \& Ngai recently as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of $K$ coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let …
Using well-known techniques, we establish Hardy-Littlewood-type and Hausdorff-Young-type inequalities for generalized Gegenbauer expansions and their unification.
For $λ\in(0,1/3]$ let $C_λ$ be the middle-$(1-2λ)$ Cantor set in $\mathbb R$. Given $t\in[-1,1]$, excluding the trivial case we show that \[ Λ(t):=\left\{λ\in(0,1/3]: C_λ\cap(C_λ+t)\ne\emptyset\right\} \] is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension. In particular, we calculate the local dimension of $Λ(t)$, which reveals a dimensional variation principle. Furthermore, for any $β\in[0,1]$ we show that the level set \[ Λ_β(t):=\left\{λ\inΛ(t): \dim_H(C_λ\cap(C_λ+t))=\dim_P(C_λ\cap(C_λ+t))=β\frac{\log 2}{-\log λ}\right\} \] has equal Hausdorff and packing d…
In this paper, we prove the boundedness of matrix Hausdorff operators and rough Hausdorff operators in the two weighted Herz-type Hardy spaces associated with both power weights and Muckenhoupt weights. By applying the fact that the standard infinite atomic decomposition norm on two weighted Herz-type Hardy spaces is equivalent to the finite atomic norm on some dense subspaces of them, we generalize some previous known results due to Chen et al. [7] and Ruan, Fan [34].
For any $α\in(0,d)$, we construct Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $α$ such that the associated natural measure $μ$ obeys the restriction estimate $\| \widehat{f dμ} \|_{p} \leq C_p \| f \|_{L^2(μ)}$ for all $p>2d/α$. This range is optimal except for the endpoint. This extends the earlier work of Chen-Seeger and Shmerkin-Suomala, where a similar result was obtained by different methods for $α=d/k$ with $k\in\mathbb{N}$. Our proof is based on the decoupling techniques of Bourgain-Demeter and a theorem of Bourgain on the existence of $Λ(p)$ sets.
Hausdorff-Young inequalities for $\mathrm{SU}(1,1)$ Fourier products indexed by binary cubes
Department of Mathematics, Faculty of Science, University of Split（斯普利特大学理学院数学系）；Faculty of Transport and Traffic Sciences, University of Zagreb（萨格勒布大学交通与运输科学学院）
In this paper we study the vector-valued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff--Young, Hardy--Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff--Young and Hardy--Littlewood inequalities and state that the Fourier transform is bounded from $L^p(\mathbb{R}^d,|\cdot|^{βp})$ into $L^q(\mathbb{R}^d,|\cdot|^{-γq})$ under certain condition on $p,q,β$ and $γ$. Vector-valued analogues are derived under geometric conditions on the underlying Banach space such as Fourier type and rela…
Let $N\ge 2$ and $ρ\in(0,1/N^2]$. The homogenous Cantor set $E$ is the self-similar set generated by the iterated function system \[ \left\{f_i(x)=ρx+\frac{i(1-ρ)}{N-1}: i=0,1,\ldots, N-1\right\}. \] Let $s=\dim_H E$ be the Hausdorff dimension of $E$, and let $μ=\mathcal H^s|_E$ be the $s$-dimensional Hausdorff measure restricted to $E$. In this paper we describe, for each $x\in E$, the pointwise lower $s$-density $Θ_*^s(μ,x)$ and upper $s$-density $Θ^{*s}(μ, x)$ of $μ$ at $x$. This extends some early results of Feng et al. (2000). Furthermore, we determine two critical values $a_c$ and $b_c$ …
We obtain several versions of the Hausdorff-Young and Hardy-Littlewood inequalities for the $(k,a)$-generalized Fourier transform recently investigated at length by Ben Saï d, Kobayashi, and Ø rsted. We also obtain a number of weighted inequalities - in particular Pitt's inequality - that have application to uncertainty principles. Specifically we obtain several analogs of the Heisenberg-Pauli-Weyl principle for $L^p$-functions, local Cowling-Price-type inequalities, Donoho-Stark-type inequalities and qualitative extensions. We finally use the Hausdorff-Young inequality as a means to obtain en…
