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In this paper we study properties of a variant of the $1/2$-caloric capacity, called $1/2$-symmetric caloric capacity. The latter is associated simultaneously with the $1/2$-fractional heat equation and its conjugate. We establish its semi-additivity in $\mathbb{R}^2$ and, moreover, we compute explicitly the $1/2$-symmetric caloric capacity of rectangles, which illustrates its anisotropic behaviour.

In this paper, we address for the 2D Euler equations the existence of rigid time periodic solutions close to stationary radial vortices of type $f_0(|x|){\bf 1}_{\mathbb{D}}(x)$, with $\mathbb{D}$ the unit disc and $f_0$ being a strictly monotonic profile with constant sign. We distinguish two scenarios according to the sign of the profile: defocusing and focusing. In the first regime, we have scarcity of the bifurcating curves associated with lower symmetry. However in the focusing case we get a countable family of bifurcating solutions associated with large symmetry. The approach developed in this work is new and flexible, and the explicit expression of the radial profile is no longer required as in [41] with the quadratic shape. The alternative for that is a refined study of the associated spectral problem based on Sturm-Liouville differential equation with a variable potential that changes the sign depending on the shape of the profile and the location of the time period. Deep hidden structure on positive definiteness of some intermediate integral operators are also discovered and used in a crucial way. Notice that a special study will be performed for the linear problem associated with the first mode founded on Pr\"{u}fer transformation and Kneser's Theorem on the non-oscillation phenomenon., Comment: 50 pages

In this paper we consider a general class of anisotropic energies in three dimensions and give a complete characterisation of their minimisers. We show that, depending on the Fourier transform of the interaction potential, the minimiser is either the normalised characteristic function of an ellipsoid or a measure supported on a two-dimensional ellipse. In particular, it is always an ellipsoid if the transform is strictly positive, while when the Fourier transform is degenerate both cases can occur. Finally, we show an explicit example where loss of dimensionality of the minimiser does occur.

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

In this paper we characterise the minimiser for a class of nonlocal perturbations of the Coulomb energy. We show that the minimiser is the normalised characteristic function of an ellipsoid, under the assumption that the perturbation kernel has the same homogeneity as the Coulomb potential, is even, smooth off the origin and sufficiently small. This result can be seen as the stability of ellipsoids as energy minimisers, since the minimiser of the Coulomb energy is the normalised characteristic function of a ball.

In this paper we study removable singularities for regular $(1,1/2)$-Lipschitz solutions of the heat equation in time varying domains. We introduce an associated Lipschitz caloric capacity and we study its metric and geometric properties and the connection with the $L^2$ boundedness of the singular integral whose kernel is given by the gradient of the fundamental solution of the heat equation., Comment: 37 pages

This paper aims to study time periodic solutions for 3D inviscid quasi-geostrophic model. We show the existence of non trivial rotating patches by suitable perturbation of stationary solutions given by generic revolution shapes around the vertical axis. The construction of those special solutions are done through bifurcation theory. In general, the spectral problem is very delicate and strongly depends on the shape of the initial stationary solutions. More specifically, the spectral study can be related to an eigenvalue problem of a self-adjoint compact operator. We are able to implement the bifurcation only from the largest eigenvalues of the operator, which are simple. Additional difficulties generated by the singularities of the poles are solved through the use of suitable function spaces with Dirichlet boundary condition type and refined potential theory with anisotropic kernels., Comment: 105 pages

We construct an example of a purely unrectifiable measure $\mu$ in $\mathbb{R}^d$ for which the singular integrals associated to the kernels $\displaystyle{K(x)=\frac{P_{2k+1}(x)}{|x|^{2k+d}}}$, with $k\geq 1$ and $P_{2k+1}$ a homogeneous harmonic polynomial of degree $2k+1$, are bounded in $L^2(\mu)$. This contrasts starkly with the results concerning the Riesz kernel $\displaystyle{\frac{x}{|x|^{d}}}$ in $\mathbb{R}^d$.

We study the behaviour of singular integral operators $T_{k_t}$ of convolution type on $\mathbb{C}$ associated with the parametric kernels $$ k_t(z):=\frac{(\Re z)^{3}}{|z|^{4}}+t\cdot \frac{\Re z}{|z|^{2}}, \quad t\in \mathbb{R},\qquad k_\infty(z):=\frac{\Re z}{|z|^{2}}\equiv \Re \frac{1}{z},\quad z\in \mathbb{C}\setminus\{0\}. $$ It is shown that for any positive locally finite Borel measure with linear growth the corresponding $L^2$-norm of $T_{k_0}$ controls the $L^2$-norm of $T_{k_\infty}$ and thus of the Cauchy transform. As a corollary, we prove that the $L^2(\mathcal{H}^1\lfloor E)$-boundedness of $T_{k_t}$ with a fixed $t\in (-t_0,0)$, where $t_0>0$ is an absolute constant, implies that $E$ is rectifiable. This is so in spite of the fact that the usual curvature method fails to be applicable in this case. Moreover, as a corollary of our techniques, we provide an alternative and simpler proof of the bi-Lipschitz invariance of the $L^2$-boundedness of the Cauchy transform, which is the key ingredient for the bi-Lipschitz invariance of analytic capacity., Comment: In this version, we corrected several inaccuracies and added more corollaries. 1 figure

We prove well-posedness of linear scalar conservation laws using only assumptions on the growth and the modulus of continuity of the velocity field, but not on its divergence. As an application, we obtain uniqueness of solutions in the atomic Hardy space, H1, for the scalar conservation law induced by a class of vector fields whose divergence is an unbounded BMO function.

The well-known curvature method initiated in works of Melnikov and Verdera is now commonly used to relate the $L^2(\mu)$-boundedness of certain singular integral operators to the geometric properties of the support of measure $\mu$, e.g. rectifiability. It can be applied however only if Menger curvature-like permutations, directly associated with the kernel of the operator, are non-negative. We give an example of an operator in the plane whose corresponding permutations change sign but the $L^2(\mu)$-boundedness of the operator still implies that the support of $\mu$ is rectifiable. To the best of our knowledge, it is the first example of this type. We also obtain several related results with Ahlfors-David regularity conditions., Comment: 1 figure

In this paper, we study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations and the $(\hbox{SQG})_\alpha$ equations with $\alpha\in (0,1).$ From the numerical experiments implemented for Euler equations in \cite{DZ, humbert, S-Z} it is conjectured the existence of a curve of steady vortex pairs passing through the point vortex pairs. There are some analytical proofs based on variational principle \cite{keady, Tur}, however they do not give enough information about the pairs such as the uniqueness or the topological structure of each single vortex. We intend in this paper to give direct proofs confirming the numerical experiments and extend these results for the $(\hbox{SQG})_\alpha$ equation when $\alpha\in (0,1)$. The proofs rely on the contour dynamics equations combined with a desingularization of the point vortex pairs and the application of the implicit function theorem., Comment: 39 pages, we unified some sections

In this note, we study the well-posedness of the Cauchy problem for the transport equation in the BMO space and certain Triebel-Lizorkin spaces., Comment: 14 pages

In this paper we study the existence of doubly-connected rotating patches for Euler equations when the classical non-degeneracy conditions are not satisfied. We prove the bifurcation of the V-states with two-fold symmetry, however for higher $m-$fold symmetry with $m\geq3$ the bifurcation does not occur. This answers to a problem left open in \cite{H-F-M-V}. Note that, contrary to the known results for simply-connected and doubly-connected cases where the bifurcation is pitchfork, we show that the degenerate bifurcation is actually transcritical. These results are in agreement with the numerical observations recently discussed in \cite{H-F-M-V}. The proofs stem from the local structure of the quadratic form associated to the reduced bifurcation equation., Comment: 39 pages

In this paper, we prove the existence of $m$-fold rotating patches for the Euler equations in the disc, for both simply-connected and doubly-connected cases. Compared to the planar case, the rigid boundary introduces rich dynamics for the lowest symmetries $m=1$ and $m=2$. We also discuss some numerical experiments highlighting the interaction between the boundary of the patch and the rigid one., Comment: 56 pages

In this paper we prove the existence of countable branches of rotating patches bifurcating from the ellipses at some implicit angular velocities., Comment: 21 pages

In this paper, we study flows associated to Sobolev vector fields with subexponentially integrable divergence. Our approach is based on the transport equation following DiPerna-Lions [DPL89]. A key ingredient is to use a quantitative estimate of solutions to the Cauchy problem of transport equation to obtain the regularity of density functions., Comment: 24 pages, revised

We face the well-posedness of linear transport Cauchy problems $$\begin{cases}\dfrac{\partial u}{\partial t} + b\cdot\nabla u + c\,u = f&(0,T)\times{\mathbb R}^n\\u(0,\cdot)=u_0\in L^\infty&{\mathbb R}^n\end{cases}$$ under borderline integrability assumptions on the divergence of the velocity field $b$. For $W^{1,1}_{loc}$ vector fields $b$ satisfying $\frac{|b(x,t)|}{1+|x|}\in L^1(0,T; L^1)+L^1(0,T; L^\infty)$ and $$\operatorname{div} b\in L^1(0,T;L^\infty) + L^1\left(0,T; \operatorname{Exp}\left(\frac{L}{\log L}\right)\right),$$ we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every $\gamma>1$, we construct an example of a bounded autonomous velocity field $b$ with $$\operatorname{div} b\in \operatorname{Exp}\left(\frac{L}{\log^\gamma L}\right) ,$$ for which the associate Cauchy problem for the transport equation admits infinitely many solutions. Stability questions and further extensions to the $BV$ setting are also addressed.

We prove existence of doubly connected V-states for the planar Euler equations which are not annuli. The proof proceeds by bifurcation from annuli at simple "eigenvalues". The bifurcated $V$-states we obtain enjoy a $m$-fold symmetry for some $m\ge 3.$ The existence of doubly connected $V$-states of strict $2$-fold symmetry remains open., Comment: 40 pages

It is known that the improved Cotlar's inequality $B^{*}f(z) \le C M(Bf)(z)$, $z\in\mathbb C$, holds for the Beurling transform $B$, the maximal Beurling transform $B^{*}f(z)=$ $\displaystyle\sup_{\varepsilon >0}\left|\int_{|w|>\varepsilon}f(z-w) \frac{1}{w^2} \,dw\right|$, $z\in\mathbb C$, and the Hardy--Littlewood maximal operator $M$. In this note we consider the maximal Beurling transform associated with squares, namely, $B^{*}_Sf(z)=\displaystyle\sup_{\varepsilon >0}\left|\int_{w\notin Q(0,\varepsilon)}f(z-w) \frac{1}{w^2} \,dw \right|$, $z\in\mathbb C$, $Q(0,\varepsilon)$ being the square with sides parallel to the coordinate axis of side length $\varepsilon$. We prove that $B_{S}^{*}f(z) \le C M^2(Bf)(z)$, $z\in\mathbb C$, where $M^2=M \circ M$ is the iteration of the Hardy--Littlewood maximal operator, and $M^2$ cannot be replaced by $M$., Comment: 3 figures

In this paper we consider rotating doubly connected vortex patches for the Euler equations in the plane. When the inner interface is an ellipse we show that the exterior interface must be a confocal ellipse. We then discuss some relations, first found by Flierl and Polvani, between the parameters of the ellipses, the velocity of rotation and the magnitude of the vorticity in the domain enclosed by the inner ellipse., Comment: 30 pages

This paper continues the study, initiated in the works {MOV} and {MOPV}, 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 operator of convolution type. We consider two forms of control, namely, in the weighted $L^p(\omega)$ 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. The novelty with respect to the aforementioned works, lies in the fact that here $p$ is different from 2 and the $L^p$ space is weighted.

We show that the boundary of a rotating vortex patch (or V-state, in the terminology of Deem and Zabusky) is of class C^infinity provided the patch is close enough to the bifurcation circle in the Lipschitz norm. The rotating patch is convex if it is close enough to the bifurcation circle in the C^2 norm. Our proof is based on Burbea's approach to V-states. Thus conformal mapping plays a relevant role as well as estimating, on H\"older spaces, certain non-convolution singular integral operators of Calder\'on-Zygmund type., Comment: Various proofs have been shortened. One added reference

Our goal in this work is to present some function spaces on the complex plane $\C$, $X(\C)$, for which the quasiregular solutions of the Beltrami equation, $\bar\partial f (z) = \mu(z) \partial f (z)$, have first derivatives locally in $X(\C)$, provided that the Beltrami coefficient $\mu$ belongs to $X(\C)$.

Analytic capacity is associated with the Cauchy kernel $1/z$ and the $L^\infty$-norm. For $n\in\mathbb{N}$, one has likewise capacities related to the kernels $K_i(x)=x_i^{2n-1}/|x|^{2n}$, $1\le i\le 2$, $x=(x_1,x_2)\in\mathbb{R}^2$. The main result of this paper states that the capacities associated with the vectorial kernel $(K_1, K_2)$ are comparable to analytic capacity., Comment: To appear in Potential Analysis. arXiv admin note: text overlap with 1004.2170

Let $E \subset \C$ be a Borel set with finite length, that is, $0<\mathcal{H}^1 (E)<\infty$. By a theorem of David and L\'eger, the $L^2 (\mathcal{H}^1 \lfloor E)$-boundedness of the singular integral associated to the Cauchy kernel (or even to one of its coordinate parts $x / |z|^2,y / |z|^2,z=(x,y) \in \C$) implies that $E$ is rectifiable. We extend this result to any kernel of the form $x^{2n-1} /|z|^{2n}, z=(x,y) \in \C,n \in \mathbb{N}$. We thus provide the first non-trivial examples of operators not directly related with the Cauchy transform whose $L^2$-boundedness implies rectifiability.

In this paper we present a new characterization of Sobolev spaces on Euclidian spaces ($\mathbb{R}^n$). Our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of Euclidean space. An interesting feature of our condition is that depends only on the metric of $\mathbb{R}^n$ and the Lebesgue measure, so that one can define Sobolev spaces of any order of smoothness on any metric measure space., Comment: Two references added. Some statements have been improved and proofs made clearer. Typos corrected

The real and imaginari parts of the Cauchy kernel in the plane are scalar Riesz kernels of homogeneity -1. One can associate with each of them a natural notion of capacity related to bounded potentials. The main result of the paper asserts that these capacities are comparable to classical analytic capacity, thus stressing the real variables nature of analytic capacity. Higher dimensional versions of this result are also considered., Comment: 37 pages, ams.org

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 Notices

We discuss a special class of Beltrami coefficients whose associated quasiconformal mapping is bilipschitz. These are of the form the characteristic function of a planar bounded domain with smooth boundary of class C 1+epsilon times a density of class Lip epsilon on the domain. The crucial fact in the argument is the special extracancellation property of even Calderon-Zygmund kernels, namely that they have zero integral on half the unit ball. This property is expressed in a particularly suggestive way and is shown to have far-reaching consequences. The main result may also be viewed as a Lipschitz regularity result for the Beltrami equation, and so for certain planar second order elliptic equations in divergence form., Comment: 44 pages, 7 figures, new references, new results

The purpose of this paper is to describe the smooth homogeneous Calderon-Zygmund operators for which the maximal singular integral T*f may be controlled by the singular integral Tf. We consider two types of control. The first is the L2 estimate of T*f by Tf, namely the estimate of the L2 norm of T*f by a constant times the L2 norm of Tf. The second is the pointwise estimate of T*f(x) by a constant times M(Tf)(x), where M denotes the Hardy-Littlewood maximal operator. Notice that this is an improved variant of Cotlar's inequality, because the term Mf(x) is missing on the right hand side. Our main result states that, for even operators, both are equivalent to a purely algebraic condition formulated in terms of the expansion of the kernel in spherical harmonics. The condition holds by higher order Riesz transforms, which then satisfy an improved version of Cotlar's inequality, Comment: Minor misprints and English inaccuracies corrected, references updated. To appear in Annals on Math

We study the removable singularities for solutions to the Beltrami equation $\bar\partial f=\mu \partial f$, assuming that the coefficient $\mu$ lies on some Sobolev space $W^{1,p}$, $p\leq 2$. 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 $\mu\in W^{1,2}$ preserve compact sets of $\sigma$-finite length and vanishing analytic capacity, even though they need not be bilipschitz., Comment: 1 figure

The classical Painlev\'e theorem tells that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general $K$-quasiregular mappings in planar domains the corresponding critical dimension is $\frac{2}{K+1}$. We show that when $K>1$, unexpectedly one has improved removability. More precisely, we prove that sets $E$ of $\sigma$-finite Hausdorff $\frac{2}{K+1}$-measure are removable for bounded $K$-quasiregular mappings. On the other hand, $\dim(E) = \frac{2}{K+1}$ is not enough to guarantee this property. We also study absolute continuity properties of pull-backs of Hausdorff measures under $K$-quasiconformal mappings, in particular at the relevant dimensions 1 and $\frac{2}{K+1}$. For general Hausdorff measures ${\cal H}^t$, $0 < t < 2$, we reduce the absolute continuity properties to an open question on conformal mappings., Comment: 31 pages, 1 figure

We estimate in Lp the maximal Riesz transform in terms of the Riesz transform itself for p greater than 1. In the limiting case p=1 the weak L1 inequality is shown to fail. Surprisingly, the weak L1 inequality for the maximal Beurling transform in terms of the Beurling transform does hold., Comment: 12 pages

There is a natural capacity associated to any vector valued Riesz kernel of a given homogeneity. If we are in the plane and the kernel is the Cauchy kernel, then this capacity is analytic capacity. Our main result states that if the homogeneity of the kernel is negative and larger than minus one, then the capacity is comparable to one of the well studied capacities of non-linear potential theory., Comment: 26 pages, to appear in J. Reine Angew. Math