Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDyadic analysis meets number theory
69
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We unite two themes in dyadic analysis and number theory by studying an analogue of the failure of the Hasse principle in harmonic analysis. Explicitly, we construct an explicit family of measures on the real line that are $p$-adic doubling for any finite set of primes, yet not doubling, and we apply these results to show analogous statements about the reverse Holder and Muckenhoupt $A_p$ classes of weights. The proofs involve a delicate interplay among several geometric and number theoretic properties.
rate research
Read More
Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G=Q_p, the field of p-adic rational numbers (as a group under addition), which has compact open subgroup H=Z_p, the ring of p-adic integers. Classical wavelet theories, which require a non-trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. A wavelet theory is developed on G using coset representatives of a quotient of the dual group of G. Wavelet bases are constructed by means of an iterative method giving rise to so-called wavelet sets in the dual group.
In this paper we show how to compute the $Lambda_{alpha}$ norm, $alphage 0$, using the dyadic grid. This result is a consequence of the description of the Hardy spaces $H^p(R^N)$ in terms of dyadic and special atoms.
We present in this paper some embeddings of various dyadic martingale Hardy-amalgam spaces $H^S_{p,q},,, H^s_{p,q},,,H^*_{p,q},,,mathcal{Q}_{p,q}$ and $mathcal{P}_{p,q}$ of the real line. In the same settings, we characterize the dual of $H^s_{p,q}$ for large $p$ and $q$. We also introduce a Garsia-type space $mathcal{G}_{p,q}$ and characterize its dual space.
Adjacent dyadic systems are pivotal in analysis and related fields to study continuous objects via collections of dyadic ones. In our prior work (joint with Jiang, Olson and Wei) we describe precise necessary and sufficient conditions for two dyadic systems on the real line to be adjacent. Here we extend this work to all dimensions, which turns out to have many surprising difficulties due to the fact that $d+1$, not $2^d$, grids is the optimal number in an adjacent dyadic system in $mathbb{R}^d$. As a byproduct, we show that a collection of $d+1$ dyadic systems in $mathbb{R}^d$ is adjacent if and only if the projection of any two of them onto any coordinate axis are adjacent on $mathbb{R}$. The underlying geometric structures that arise in this higher dimensional generalization are interesting objects themselves, ripe for future study; these lead us to a compact, geometric description of our main result. We describe these structures, along with what adjacent dyadic (and $n$-adic, for any $n$) systems look like, from a variety of contexts, relating them to previous work, as well as illustrating a specific example.
In the work of S. Petermichl, S. Treil and A. Volberg it was explicitly constructed that the Riesz transforms in any dimension $n geq 2$ can be obtained as an average of dyadic Haar shifts provided that an integral is nonzero. It was shown in the paper that when $n=2$, the integral is indeed nonzero (negative) but for $n geq 3$ the nonzero property remains unsolved. In this paper we show that the integral is nonzero (negative) for $n=3$. The novelty in our proof is the delicate decompositions of the integral for which we can either find their closed forms or prove an upper bound.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
