Sharp bounds for fractional type operators with $L^{\alpha,s}$-Hörmander conditions

Sharp Weighted Bounds for Fractional Integral Operators

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal...

Sharp lower bounds on the fractional matching number

A fractional matching of a graph G is a function f from E(G) to the interval [0, 1] such that ∑ e∈Γ(v) f(e) ≤ 1 for each v ∈ V (G), where Γ(v) is the set of edges incident to v. The fractional matching number of G, written α′∗(G), is the maximum of

Bounds for Singular Fractional Integrals and Related Fourier Integral Operators

Upper bounds for Stein-type operators

We present sharp bounds on the supremum norm of DjSh for j ≥ 2, where D is the differential operator and S the Stein operator for the standard normal distribution. The same method is used to give analogous bounds for the exponential, Poisson and geometric distributions, with D replaced by the forward difference operator in the discrete case. We also discuss applications of these bounds to the c...

BOUNDS FOR KAKEYA-TYPE MAXIMAL OPERATORS ASSOCIATED WITH k-PLANES

A (d, k) set is a subset of Rd containing a translate of every k-dimensional plane. Bourgain showed that for k ≥ kcr(d), where kcr(d) solves 2kcr−1 + kcr = d, every (d, k) set has positive Lebesgue measure. We give a short proof of this result which allows for an improved Lp estimate of the corresponding maximal operator, and which demonstrates that a lower value of kcr could be obtained if imp...