The best constant in the usual $L^p$ norm inequality for the centered Hardy-Littlewood maximal function on ${ℝ}^1$ is obtained for the class of all “peak-shaped” functions. A function on the line is called peak-shaped if it is positive and convex except at one point. The techniques we use include variational methods.

We prove a sharp pointwise estimate of the nonincreasing rearrangement of the fractional maximal function of ⨍, $M_{\gamma }⨍$, by an expression involving the nonincreasing rearrangement of ⨍. This estimate is used to obtain necessary and sufficient conditions for the boundedness of $M_{\gamma }$ between classical Lorentz spaces.

We prove that Muckenhoupt's A1-weights satisfy a reverse Hölder inequality with an explicit and asymptotically sharp estimate for the exponent. As a by-product we get a new characterization of A1-weights.

We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space ${\dot{NL}}^{1,Q}$ by $L^{\infty }$ functions, generalizing a result of Bourgain–Brezis. We then use this to obtain a Gagliardo–Nirenberg inequality for $$ on the Heisenberg group ${ℍ}^n$.

Necessary conditions and sufficient conditions are derived in order that Bessel potential operator $J_{s,\lambda }$ is bounded from the weighted Lebesgue spaces $L_v^p=L^p({ℝ}^n,v\left(x\right)dx)$ into $L_u^q$ when $1<p\le q<\infty$.

We establish a variant sharp estimate for multilinear singular integral operators. As applications, we obtain the weighted norm inequalities on general weights and certain $Llo{g}^{+}L$ type estimates for these multilinear operators.

We prove a weighted vector-valued weak type (1,1) inequality for the Bochner-Riesz means of the critical order. In fact, we prove a slightly more general result.

In this paper we discuss a weighted version of Journé's covering lemma, a substitution for Whitney decomposition of an open set in R2 where squares are replaced by rectangles. We then apply this result to obtain a sharp version of the atomic decomposition of the weighted Hardy spaces Hu'p (R+2 x R+2) and a description of their duals when p is close to 1.

The Coifman-Fefferman inequality implies quite easily that a Calderón-Zygmund operator T acts boundedly in a Banach lattice X on ℝⁿ if the Hardy-Littlewood maximal operator M is bounded in both X and X'. We establish a converse result under the assumption that X has the Fatou property and X is p-convex and q-concave with some 1 < p, q < ∞: if a linear operator T is bounded in X and T is nondegenerate in a certain sense (for example, if T is a Riesz transform) then M is bounded in both X and...

A Carleson condition on the difference function for the coefficients of two elliptic-caloric operators is shown to give absolute continuity of one measure with respect to the other on the lateral boundary. The elliptic operators can have time dependent coefficients and only one of them is assumed to have a measure which is doubling. This theorem is an extension of a result of B. Dahlberg [4] on absolute continuity for elliptic measures to the case of the heat equation. The method of proof is an...

In the context of spaces of homogeneous type, we develop a method to deterministically construct dyadic grids, specifically adapted to a given combinatorial situation. This method is used to estimate vector-valued operators rearranging martingale difference sequences such as the Haar system.

Bellow and Calderón proved that the sequence of convolution powers ${\mu }_{n}f\left(x\right)={\sum }_{k\in \mathbb{Z}}{\mu }^n\left(k\right)f\left(T^{k}x\right)$ converges a.e, when $\mu$ is a strictly aperiodic probability measure on $\mathbb{Z}$ such that the expectation is zero, $E\left(\mu \right)=0$, and the second moment is finite, $m_2\left(\mu \right)\&lt;\infty$. In this paper we extend this result to cases where $m_2\left(\mu \right)=\infty$.