The sharp constant for the weak (1, n/(n-s)) estimate of the Riesz potential for 0 < s < min(n,2) is explicitly computed and shown to be attained by an explicit radial function.
A dimension-free weak-type $(1,1)$ bound for the vector Riesz transform on $\mathbb{R}^n$
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abstract
We show that the best constant in the weak-type $(1,1)$ bound for the vector Riesz transform on $\mathbb{R}^n$ is at most $2$, independent of the dimension $n$. The proof relies on a new decomposition of the input data involving an obstacle problem for the fractional Laplacian and an associated Lewy-Stampacchia type estimate on an unbounded domain. This settles a problem posed by E. M. Stein at the 1986 International Congress of Mathematicians.
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A first-order method with O(L/T^3) convergence rate for convex quadratic minimization over the L1 ball is presented.
The vector Dunkl-Riesz transform satisfies a weak-type (1,1) bound with constant at most M_k+2 (and 2 for reflection-invariant functions), independent of the ambient dimension.
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Sharp constants for weak estimates of the Riesz Potentials
The sharp constant for the weak (1, n/(n-s)) estimate of the Riesz potential for 0 < s < min(n,2) is explicitly computed and shown to be attained by an explicit radial function.
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An $O(1/T^3)$ algorithm for minimizing convex quadratic functions over the $L_1$ ball
A first-order method with O(L/T^3) convergence rate for convex quadratic minimization over the L1 ball is presented.
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Dimension free weak-type endpoint estimates for the vectors of the Dunkl--Riesz transform
The vector Dunkl-Riesz transform satisfies a weak-type (1,1) bound with constant at most M_k+2 (and 2 for reflection-invariant functions), independent of the ambient dimension.
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