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Discovery Projects - Grant ID: DP160100153

Title
Discovery Projects - Grant ID: DP160100153
Funding
ARC | Discovery Projects
Contract (GA) number
DP160100153
Start Date
2016/01/01
End Date
2020/12/31
Open Access mandate
no
Organizations
-
More information
http://purl.org/au-research/grants/arc/DP160100153

 

  • Sparse Endpoint Estimates for Bochner-Riesz Multipliers on the Plane

    Kesler, Robert; Lacey, Michael T. (2017)
    Projects: ARC | Discovery Projects - Grant ID: DP160100153 (DP160100153)
    For $ 0< \lambda < \frac{1}2$, let $ B_{\lambda }$ be the Bochner-Riesz multiplier of index $ \lambda $ on the plane. Associated to this multiplier is the critical index $1 < p_\lambda = \frac{4} {3+2 \lambda } < \frac{4}3$. We prove a sparse bound for $ B_{\lambda }$ with indices $ (p_\lambda , q)$, where $ p_\lambda ' < q < 4$. This is a further quantification of the endpoint weak $L^{p_\lambda}$ boundedness of $ B_{\lambda }$, due to Seeger. Indeed, the sparse bound immediately implies new...

    On logarithmic bounds of maximal sparse operators

    Karagulyan, Grigori A.; Lacey, Michael T. (2018)
    Projects: ARC | Discovery Projects - Grant ID: DP160100153 (DP160100153)
    Given sparse collections of measurable sets $\mathcal S_k$, $k=1,2,\ldots ,N$, in a general measure space $(X,\mathfrak M,\mu)$, let $ \Lambda_{\mathcal S_k}$ be the sparse operator, corresponding to $\mathcal S_k$. We show that the maximal sparse function $ \Lambda f = \max _{1\le k\le N} \Lambda_{\mathcal S_k} f $ satisfies \begin{align*} &\| \Lambda \| _{L^p(X) \mapsto L^{p,\infty}(X)} \lesssim \log N\cdot \|M_{\mathcal S}\|_{L^p(X) \mapsto L^{p,\infty}(X)},\,1\le p

    Sparse Bounds for Bochner-Riesz Multipliers

    The Bochner-Riesz multipliers $ B_{\delta }$ on $ \mathbb R ^{n}$ are shown to satisfy a range of sparse bounds, for all $0< \delta < \frac {n-1}2 $. The range of sparse bounds increases to the optimal range, as $ \delta $ increases to the critical value, $ \delta =\frac {n-1}2$, even assuming only partial information on the Bochner-Riesz conjecture in dimensions $ n \geq 3$. In dimension $n=2$, we prove a sharp range of sparse bounds. The method of proof is based upon a `single scale' analys...

    Sparse Bounds for Spherical Maximal Functions

    We consider the averages of a function $ f$ on $ \mathbb R ^{n}$ over spheres of radius $ 0< r< \infty $ given by $ A_{r} f (x) = \int_{\mathbb S ^{n-1}} f (x- r y) \; d \sigma (y)$, where $ \sigma $ is the normalized rotation invariant measure on $ \mathbb S ^{n-1}$. We prove a sharp range of sparse bounds for two maximal functions, the first the lacunary spherical maximal function, and the second the full maximal function. $$ M_{{lac}} f = \sup_{j\in \mathbb Z } A_{2^j} f , \qquad M_{{full}...
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