New estimates for the maximal singular integral

In this paper we pursue the study of the problem of controlling the maximal singular integral $T^{*}f$ by the singular integral $Tf$. Here $T$ is a smooth homogeneous Calder\'on-Zygmund singular integral of convolution type. We consider two forms of control, namely, in the L^2(\Rn) norm and via pointwise estimates of $T^{*}f$ by $M(Tf)$ or $M^2(Tf)$, where $M$ is the Hardy-Littlewood maximal operator and $M^2=M \circ M$ its iteration. It is known that the parity of the kernel plays an essential role in this question. In a previous article we considered the case of even kernels and here we deal with the odd case. Along the way, the question of estimating composition operators of the type $T^\star \circ T$ arises. It turns out that, again, there is a remarkable difference between even and odd kernels. For even kernels we obtain, quite unexpectedly, weak $(1,1)$ estimates, which are no longer true for odd kernels. For odd kernels we obtain sharp weaker inequalities involving a weak $L^1$ estimate for functions in $L LogL$.Comment: v2: 56 pages, with small changes made after acceptance by International Math. Research Notice

New estimates for the maximal singular integral

In this paper we pursue the study of the problem of controlling the maximal singular integral T∗ f by the singular integral T f. Here T is a smooth homogeneous Calder´on-Zygmund singular integral of convolution type. We consider two forms of control, namely, in the L2 (Rn) norm and via pointwise estimates of T∗ f by M(T f) or M2 (T f) , where M is the Hardy-Littlewood maximal operator and M2 = M ◦ M its iteration. It is known that the parity of the kernel plays an essential role in this question. In a previous article we considered the case of even kernels and here we deal with the odd case. Along the way, the question of estimating composition operators of the type e T ◦ T arises.. It turns out that, again, there is a remarkable difference between even and odd kernels. For even kernels we obtain, quite unexpectedly, weak (1, 1) estimates, which are no longer true for odd kernels. For odd kernels we obtain sharp weaker inequalities involving a weak L1 estimate for functions in L LogL.Generalitat de CatalunyaMinisterio de Educación y CienciaJunta de Andalucí

Beltrami equations with coefficient in the Sobolev space W1,p

We study the removable singularities for solutions to the Beltrami equation ∂f = µ ∂f, where µ is a bounded function, kµk∞ ≤ K−1 K+1 < 1, and such that µ ∈ W1,p for some p ≤ 2. Our results are based on an extended version of the well known Weyl's lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly supported Beltrami coefficient µ ∈ W1,p, 2K2 K2+1 < p ≤ 2, preserve compact sets of σ-finite length and vanishing analytic capacity, even though they need not be bilipschitz

Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings

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Calderón-Zygmund kernels and rectifiability in the plane

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Lipschitz approximation by harmonic functions and some applications to spectral synthesis

For 0 < s ≤ 1, we characterize those compact sets X with the property that each function harmonic in Ẋ and satisfying a little o Lipschitz condition of order s is the limit in the Lipschitz norm of orders of functions harmonic on neighbourhoods of X. As an application of the methods we give a spectral synthesis result in the space of locally integrable functions whose laplacian belongs to Bp(Rd), the containing Banach space of the Hardy space Hp(Rd)

Lipschitz approximation by harmonic functions and some applications to spectral synthesis

For 0 < s ≤ 1, we characterize those compact sets X with the property that each function harmonic in Ẋ and satisfying a little o Lipschitz condition of order s is the limit in the Lipschitz norm of orders of functions harmonic on neighbourhoods of X. As an application of the methods we give a spectral synthesis result in the space of locally integrable functions whose laplacian belongs to Bp(Rd), the containing Banach space of the Hardy space Hp(Rd)

Removable singularities for solutions of the fractional heat equation in time varying domains

In this paper we study removable singularities for solutions of the fractional heat equation in the spacial-time space. We introduce associated capacities and we study some of its metric and geometric properties

BMO Harmonic Approximation in the Plane and Spectral Synthesis for Hardy-Sobolev Spaces.

The spectral synthesis theorem for Sobolev spaces of Hedberg and Wolff [7] has been applied in combination with duality, to problems of Lq approximation by analytic and harmonic functions. In fact, such applications were one of the main motivations to consider spectral synthesis problems in the Sobolev space setting. In this paper we go the opposite way in the context of the BMO-H1 duality: we prove a BMO approximation theorem by harmonic functions and then we apply the ideas in its proof to produce a spectral synthesis result for variants of Sobolev spaces involving the Fefferman-Stein Hardy space H1

New estimates for the maximal singular integral

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