### Note on differentiation of integrals and the halo conjecture

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We extend some results of N. Yu. Antonov on convergence of Fourier series to more general settings. One special feature of our work is that we do not assume smoothness for the kernels in our hypotheses. This has interesting applications to convergence with respect to general orthonormal systems, like the Walsh-Fourier system, for which we prove a.e. convergence in the class L log L log log log L. Other applications are given in the theory of differentiation of integrals.

In recent work by Reguera and Thiele (2012) and by Reguera and Scurry (2013), two conjectures about joint weighted estimates for Calderón-Zygmund operators and the Hardy-Littlewood maximal function were refuted in the one-dimensional case. One of the key ingredients for these results is the construction of weights for which the value of the Hilbert transform is substantially bigger than that of the maximal function. In this work, we show that a similar construction is possible for classical Calderón-Zygmund...

Let Ω be homogeneous of degree 0 in R 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|, 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 finite...

In this paper we establish a formal connection between the average decay of the Fourier transform of functions with respect to a given measure and the of that measure. We also present a generalization of the classical restriction theorem of Stein and Tomas replacing the sphere with sets of prefixed Hausdorff dimension n - 1 + α, with 0 < α < 1.

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is ${u}_{t}=\nabla \xb7\left(u\nabla {(-\Delta )}^{-s}u\right),\phantom{\rule{1.0em}{0ex}}\phantom{\rule{4pt}{0ex}}0<s<1$. The problem is posed in $\{x\in {\mathbb{R}}^{n},t\in \mathbb{R}\}$ with nonnegative initial data $u(x,0)$ that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and ${C}^{\alpha}$ regularity of such weak solutions. Finally, we extend the existence...

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