An RD-space $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type $𝒳$ having “dimension” $n$, there exists a $p_0\in (n/(n+1),1)$ such that for certain classes of distributions, the $L^p\left(𝒳\right)$ quasi-norms of their radial maximal functions and grand maximal functions are equivalent when $p\in (p_0,\infty ]$. This result yields a radial maximal function characterization for Hardy spaces on $𝒳$.

We derive weighted rearrangement estimates for a large class of area integrals. The main approach used earlier to study these questions is based on distribution function inequalities.

The aim of these lectures is to present a survey of some results on spaces of functions with dominating mixed smoothness. These results concern joint work with Winfried Sickel and Miroslav Krbec as well as the work which has been done by Jan Vybíral within his thesis. The first goal is to discuss the Fourier-analytical approach, equivalent characterizations with the help of derivatives and differences, local means, atomic and wavelet decompositions. Secondly, on this basis we study approximation...

We survey recent developments on the Kakeya problem.[Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].

We study the Hardy-Littlewood maximal operator defined via an unconditional norm, acting on block decreasing functions. We show that the uncentered maximal operator maps block decreasing functions of special bounded variation to functions with integrable distributional derivatives, thus improving their regularity. In the special case of the maximal operator defined by the ${\ell }_{\infty }$-norm, that is, by averaging over cubes, the result extends to block decreasing functions of bounded variation, not necessarily...

We characterize the class of weights, invariant under dilations, for which a modified fractional integral operator $I_{\alpha }$ maps weak weighted Orlicz$-\phi$ spaces into appropriate weighted versions of the spaces $BM{O}_{\psi }$, where $\psi \left(t\right)=t^{\alpha /n}{\phi }^{-1}(1/t)$. This generalizes known results about boundedness of $I_{\alpha }$ from weak $L^p$ into Lipschitz spaces for $p>n/\alpha$ and from weak $L^{n/\alpha }$ into $BMO$. It turns out that the class of weights corresponding to $I_{\alpha }$ acting on weak$-L_{\phi }$ for $\phi$ of lower type equal or greater than $n/\alpha$, is the same as the one solving the problem for weak...

We give a quantitative characterization of the pairs of weights $(w,v)$ for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak $(p,p)$ type inequality for $1\le p<\infty$. More precisely, given any measurable set $E_0$, the estimate $$w\left(\{x\in {ℝ}^n:M^{+,d}\left({𝒳}_{E_0}\right)\left(x\right)>t\}\right)\le \frac{C{\left[(w,v)\right]}_{A_p^{+,d}\left(ℛ\right)}^p}{t^p}v\left(E_0\right)$$ holds if and only if the pair $(w,v)$ belongs to $A_p^{+,d}\left(ℛ\right)$, that is, $$\frac{\left|E\right|}{\left|Q\right|}\le {\left[(w,v)\right]}_{A_p^{+,d}\left(ℛ\right)}{\left(\frac{v\left(E\right)}{w\left(Q\right)}\right)}^{1/p}$$ for every dyadic cube $Q$ and every measurable set $E\subset Q^{+}$. The proof follows some ideas appearing in S. Ombrosi (2005). We also obtain a similar quantitative characterization for the non-dyadic...

In connection with the $A_{\varphi }$ classes of weights (see [K-T] and [B-K]), we study, in the context of Orlicz spaces, the corresponding reverse-Hölder classes $R{H}_{\varphi }$. We prove that when ϕ is ${\Delta }_2$ and has lower index greater than one, the class $R{H}_{\varphi }$ coincides with some reverse-Hölder class $R{H}_q,q>1$. For more general ϕ we still get $R{H}_{\varphi }\subset A_{\infty }={\bigcup }_{q>1}R{H}_q$ although the intersection of all these $R{H}_{\varphi }$ gives a proper subset of ${\bigcap }_{q>1}R{H}_q$.

We prove $L^p$-bounds for the Riesz transforms associated to the Hodge-Laplacian equipped with absolute and relative boundary conditions in a Lipschitz subdomain of a (smooth) Riemannian manifold for $p$ in a certain interval depending on the Lipschitz character of the domain.

In this paper, we study the Marcinkiewicz integral operators MΩ,h on the product space Rn x Rm. We prove that MΩ,h is bounded on Lp(Rn x Rm) (1&lt; p &lt; ∞) provided that h is a bounded radial function and Ω is a function in certain block space Bq(0,0) (Sn−1 x Sm−1) for some q &gt; 1. We also establish the optimality of our condition in the sense that the space Bq(0,0) (Sn−1 x Sm−1) cannot be replaced by Bq(0,r) (Sn−1 x Sm−1) for any −1 &lt; r &lt; 0. Our results improve some...

Let Ω be homogeneous of degree 0 in Rn and integrable on the unit sphere. A rough maximal operator is obtained by inserting a factor Ω in the definition of the ordinary maximal function. Rough singular integral operators are given by principal value kernels Ω(y) / |y|n, provided that the mean value of Ω vanishes. In an earlier paper, the authors showed that a two-dimensional rough maximal operator is of weak type (1,1) when restricted to radial functions. This result is now extended to arbitrary...

2000 Mathematics Subject Classification: Primary 42B20; Secondary 42B15, 42B25In this paper, we establish the L^p boundedness of certain maximal oscillatory singular integral operators with rough kernels belonging to certain block spaces. Our L^p boundedness result improves previously known results.