RESEARCH PAPERS | Joan Verdera

@article{nokey,

title = { Boundary Regularity of Rotating Vortex Patches},

author = {Joan Mateu, Taoufik Hmidi Joan Verdera},

url = {http://mat.uab.cat/web/jvm/wp-content/uploads/sites/35/2024/01/Boundary_regularity_ARMA_2013.pdf},

doi = {https://doi.org/10.1007/s00205-013-0618-8},

year  = {2013},

date = {2013-01-01},

urldate = {2013-01-01},

journal = {Archive for Rational Mechanics and Analysis},

volume = {209},

number = {1},

pages = {171-208},

abstract = {A vortex patch is a special solution of the vorticity equation in the plane corresponding to an initial condition which is the characteristic function of a domain D0. In other words, at time 0 vorticity is 1 in D0 and 0 in the complement. Since vorticity is preserved along trajectories at time t vorticity is 1 on a domain Dt and 0 elsewhere. Numerical simulations show that Dt develops a very complicated structure. For instance the boundary ejects filaments and lenght increases much faster than exponentially. In spite of that, Kirchhoff had showed at the end of the nineteenth century that ellipses just rotate around the center of mass with a constant angular velocity determined by the lengths of the semi-axes. Other "rotating vortex patches" were discovered numerically by Deem and Zabusky (who called them "V-states") in the early eighties. Burbea succeeded in giving an analytic existence proof by bifurcation. In the pictures coming from the simulations one sees V-states bifurcating from the circle with m-fold symmetry, m=2, 3, ... which evolve till corners appear at the end of the bifurcation branch. These corners are at right angles and nobody knows the mechanism by which they are created. In this paper we show that the bifurcated V-state has smooth boundary provided it is close enough to the bifurcation circle in a Lipschitz sense. The proof uses classical regularity theorems for conformal mappings, estimates of singular integral operators on H\"{o}lder spaces and the special form of the boundary equation of V-states, which allows a subtle boot strap argument. Many interesting questions arise. For example, why corners are formed ? Can one find new V-states by new methods not involving bifurcation ? This is a modified version of the paper published in Arch. Ration. Mech. Anal. 209 (2013), no. 1, 171-208, which contains some simplified proofs (see at the end on the introduction a detailed account of the modifications). },

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Kari Astala, Albert Clop, Ignacio Uriarte-Tuero, Xavier Tolsa Joan Verdera

Quasiconformal distorsion of Riesz capacities and Hausdorff measures in the plane Journal Article

In: American Journal of Mathematics, vol. 135, no. 1, pp. 17-52, 2013.

Abstract | Links | BibTeX

2012

Roc Alabern, Joan Mateu; Joan Verdera

A new characterization of Sobolev spaces on R^n Journal Article

In: Mathematische Annalen, vol. 354, no. 2, pp. 589-626, 2012.

Abstract | Links | BibTeX

2011

Joan Verdera

The Maximal Singular Integral: estimates in terms of the Singular Integral Proceedings

XXXI Conference in Harmonic Analysis, 2011.

Abstract | Links | BibTeX

Laura Prat, Joan Mateu; Joan Verdera

Capacities associated with scalar signed Riesz kernels, and analytic capacity Journal Article

In: Indiana university mathematics journal, no. 4, pp. 1319–1361, 2011.

Abstract | Links | BibTeX

Joan Mateu, Joan Orobitg, Joan Verdera

Estimates for the maximal singular integral in terms of the singular integral: the case of even kernels Journal Article

In: Annals of Mathematics, vol. 174, iss. 3, pp. 1429-1483, 2011.

Abstract | Links | BibTeX

@article{nokey,

title = {Estimates for the maximal singular integral in terms of the singular integral: the case of even kernels },

author = {Joan Mateu, Joan Orobitg, Joan Verdera },

url = {http://mat.uab.cat/web/jvm/wp-content/uploads/sites/35/2024/01/MaximalSingular_2010.pdf},

doi = {https://doi.org/10.4007/annals.2011.174.3.2},

year  = {2011},

date = {2011-01-01},

urldate = {2011-01-01},

journal = {Annals of Mathematics},

volume = {174},

issue = {3},

pages = {1429-1483},

abstract = {Let T be a homogeneous smooth Calder\'{o}n-Zygmund operator in Euclidean space and let T^∗ stand for the maximal singular integral, i. e., the supremum of the truncations of T. In this paper we consider the problem of controlling T^∗(f) by T(f). We are mostly interested in estimating the L2 norm of T^∗(f) by the L^2 norm of T(f). The problem arose when trying to obtain existence of principal values from the L^2 boundedness of the singular integral in a rather general context in which the underlying measure in much more general than Lebesgue measure (see the paper "Convergence of singular integrals with general measures", by P.Mattila and J.Verdera). This in turn was motivated by a problem of David and Semmes, which asks whether rectifiability of the support of the underlying measure follows from L^2 boundedness for the Riesz kernels. In our problem the parity of the kernel plays an essential role and in this paper we deal only with even kernels. The odd case is, in some ways, easier and will be treated in a subsequent article. Our first result is that for even higher order Riesz transforms one has a pointwise estimate of T^∗(f) by a constant times M(T(f)), where M is the Hardy-Littlewood maximal function. This fails for odd kernels (indeed for the Hilbert Transform), as was shown by Mateu and Verdera (Math.Research Letters (2006)). Clearly the above pointwise estimate implies the L^2 estimate of T^∗(f) by T(f). Notice that our pointwise estimate improves substantially classical Cotlar's inequality, because the term M(f) is missing in the right hand side. We may then informally call it Super Cotlar's inequality. There are simple polynomial homogeneous Calder\'{o}n-Zygmund operators for which the L^2 control of T^∗(f) by T(f) is not possible. For example, for T=B+B^2, where B is the Beurling transform. Showing this requires some work involving multipliers. The question then arises of how one may characterize those T for which the above L^2 control is possible. The main result of the paper gives such a condition. The characterization is as follows: T must be of the form R composed with U, where R is a higher order Riesz transform associated with some harmonic polynomial P(x) and U is an invertible operator in the Calder\'{o}n-Zygmund algebra of all operators of the type λ I+S, where S is a smooth homogeneous Calder\'{o}n-Zygmund operator and λ a real number. Moreover the polynomial P(x) must divide, in the algebra of polynomials, all spherical harmonics appearing in the spherical harmonics expansion of the kernel of T. Thus the condition is purely algebraic and very simple to check in practice. One first shows that the L^2 control of T^∗(f) by T(f) implies the structure condition for T. Then that this condition implies Super Cotlar's inequality. Therefore Super Cotlar's inequality is equivalent to the L^2 control, a fact for which we do not have a direct proof. The proof of the main Theorem is somehow involved. In both the necessary and the sufficient conditions one has to first deal with polynomial operators (i.e., the kernel restricted to the unit sphere in given by a homogeneous polynomial). At this stage one works mainly with multipliers of operators and the new ideas required emerge rather naturally. In proving the necessary condition one resorts to a division process in which Hilbert's Nullstellensatz is used, although our polynomials are real. What saves us is that in our context the real polynomials involved have many more zeroes than one would expect a priori. Finally, to make the whole approach work smoothly one needs also to perform an intricate combinatorial work to compute explicitly certain constants which appear naturally along the proof. The sufficient condition follows from an elaborate argument part of which is performed in the Fourier Transform side. It is worth remarking that we use a lemma, which was already known to people working in regularity problems for the Euler equation, stating that the image of the characteristic function of a smoothly bounded domain under a smooth homogeneous even Calder\'{o}n -Zygmund operator is a bounded function. In a second step we must face the problem of obtaining estimates independent on the degree of the polynomial generating the kernel, because we need to use a compactness argument to pass to the general case. One of the main ingredients here is an old lemma of Calder\'{o}n and Zygmund concerning Bessel functions, which combines perfectly with our new ideas to complete the argument. },

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Let T be a homogeneous smooth Calderón-Zygmund operator in Euclidean space and let T^∗ stand for the maximal singular integral, i. e., the supremum of the truncations of T. In this paper we consider the problem of controlling T^∗(f) by T(f). We are mostly interested in estimating the L2 norm of T^∗(f) by the L^2 norm of T(f). The problem arose when trying to obtain existence of principal values from the L^2 boundedness of the singular integral in a rather general context in which the underlying measure in much more general than Lebesgue measure (see the paper "Convergence of singular integrals with general measures", by P.Mattila and J.Verdera). This in turn was motivated by a problem of David and Semmes, which asks whether rectifiability of the support of the underlying measure follows from L^2 boundedness for the Riesz kernels. In our problem the parity of the kernel plays an essential role and in this paper we deal only with even kernels. The odd case is, in some ways, easier and will be treated in a subsequent article. Our first result is that for even higher order Riesz transforms one has a pointwise estimate of T^∗(f) by a constant times M(T(f)), where M is the Hardy-Littlewood maximal function. This fails for odd kernels (indeed for the Hilbert Transform), as was shown by Mateu and Verdera (Math.Research Letters (2006)). Clearly the above pointwise estimate implies the L^2 estimate of T^∗(f) by T(f). Notice that our pointwise estimate improves substantially classical Cotlar's inequality, because the term M(f) is missing in the right hand side. We may then informally call it Super Cotlar's inequality. There are simple polynomial homogeneous Calderón-Zygmund operators for which the L^2 control of T^∗(f) by T(f) is not possible. For example, for T=B+B^2, where B is the Beurling transform. Showing this requires some work involving multipliers. The question then arises of how one may characterize those T for which the above L^2 control is possible. The main result of the paper gives such a condition. The characterization is as follows: T must be of the form R composed with U, where R is a higher order Riesz transform associated with some harmonic polynomial P(x) and U is an invertible operator in the Calderón-Zygmund algebra of all operators of the type λ I+S, where S is a smooth homogeneous Calderón-Zygmund operator and λ a real number. Moreover the polynomial P(x) must divide, in the algebra of polynomials, all spherical harmonics appearing in the spherical harmonics expansion of the kernel of T. Thus the condition is purely algebraic and very simple to check in practice. One first shows that the L^2 control of T^∗(f) by T(f) implies the structure condition for T. Then that this condition implies Super Cotlar's inequality. Therefore Super Cotlar's inequality is equivalent to the L^2 control, a fact for which we do not have a direct proof. The proof of the main Theorem is somehow involved. In both the necessary and the sufficient conditions one has to first deal with polynomial operators (i.e., the kernel restricted to the unit sphere in given by a homogeneous polynomial). At this stage one works mainly with multipliers of operators and the new ideas required emerge rather naturally. In proving the necessary condition one resorts to a division process in which Hilbert's Nullstellensatz is used, although our polynomials are real. What saves us is that in our context the real polynomials involved have many more zeroes than one would expect a priori. Finally, to make the whole approach work smoothly one needs also to perform an intricate combinatorial work to compute explicitly certain constants which appear naturally along the proof. The sufficient condition follows from an elaborate argument part of which is performed in the Fourier Transform side. It is worth remarking that we use a lemma, which was already known to people working in regularity problems for the Euler equation, stating that the image of the characteristic function of a smoothly bounded domain under a smooth homogeneous even Calderón -Zygmund operator is a bounded function. In a second step we must face the problem of obtaining estimates independent on the degree of the polynomial generating the kernel, because we need to use a compactness argument to pass to the general case. One of the main ingredients here is an old lemma of Calderón and Zygmund concerning Bessel functions, which combines perfectly with our new ideas to complete the argument.

Fulvio Ricci; Joan Verdera

Duality in spaces of finite linear combinations of atoms Journal Article

In: Transactions of the American Mathematical Society, vol. 363, no. 2011, pp. 1311-1323, 2011.

Abstract | Links | BibTeX

2010

Joan Mateu, Carlos Pérez, Joan Orobitg, Joan Verdera

New estimates for the maximal singular integral Journal Article

In: International Mathematics Research Notices, vol. 19, pp. 3658–3722, 2010.

Abstract | Links | BibTeX

2009

Pertti Mattila; Joan Verdera

Convergence of singular integrals with general measures Journal Article

In: Journal of the European Mathematical Society, vol. 11, no. 2, pp. 257-272, 2009.

Abstract | Links | BibTeX

J. Mateu, J. Orobitg; J. Verdera

Extra cancellation of even Calderón-Zygmund operators and Quasiconformal mappings Authors Journal Article

In: Journal de Mathématiques Pures et Appliqués, vol. 91, iss. 9, no. 4, pp. 402-431, 2009.

Abstract | Links | BibTeX

@article{nokey,

title = {Extra cancellation of even Calder\'{o}n-Zygmund operators and Quasiconformal mappings Authors},

author = {J. Mateu, J. Orobitg and J. Verdera},

url = {http://mat.uab.cat/web/jvm/wp-content/uploads/sites/35/2024/01/MOV_2009_JMPA.pdf},

year  = {2009},

date = {2009-01-01},

urldate = {2009-01-01},

journal = {Journal de Math\'{e}matiques Pures et Appliqu\'{e}s},

volume = {91},

number = {4},

issue = {9},

pages = {402-431},

abstract = {In this paper we discuss Lipschitz regularity for the Beltrami equation. An old result of Morrey states that there is a solution Φ of the Beltrami equation which is a homeomorphism of the plane. This homeomorphic solution is unique modulo simple normalizations. The standard proof shows immediately that Φ belongs locally to W(1,p), for some p larger than 2, and thus Φ satisfies a Holder condition of some positive order less than one. This extends to all quasiregular functions (i.e., solutions of the Beltarmi equation belonging locally to W(1,2)), via Stoilov's factorization theorem. This regularity result for the Beltrami equation may be understood as a precursor of the De Giorgi-Nash Theorem, since is not difficult to see that the real and imaginary parts of a quasiregular function satisfy a second order elliptic equation in divergence form with bounded mesurable coefficients. The question addressed in this paper in to find conditions on the Beltrami coefficient so that Φ is Lipschitz . In fact, under our hypothesis Φ turns out to be bilipischitz. The requirement on the Beltrami coefficient is that it must live in a domain with smooth boundary of class C1+ε and satisfy a Holder condition of order ε there. In fact, one may allow Beltrami coefficients supported on the closure of finitely many disjoint domains with boundary of class C1+ε, which satisfy a Holder condition of order ε in each domain. In particular, the boundaries of the domains may touch, as in the case of two tangent discs, or even have a piece in common of positive length. The result depends on the special cancellation properties of even Calder\'{o}n-Zygmund kernels, which yield mapping properties of the corresponding operators that fail in the odd case. We use the index theory for Fredholm operators for Banach spaces to study the invertibility of some operators. One also needs to show compactness of some natural operators on smoothtly bounded domains. },

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2007

Joan Verdera

Classical potential theory and analytic capacity Journal Article

In: World Scientific, vol. 2007, pp. 174-192, 2007.

Abstract | Links | BibTeX

2006

Joan Mateu Joan Verdera

Lp and weak L1 estimates for the maximal Riesz transform and the maximal Beurling transform Journal Article

In: Mathematical Research Letters, vol. 13, no. 5-6, 2006.

Links | BibTeX

Xavier Tolsa, Joan Verdera

May the Cauchy transform of a non-trivial finite measure vanish on the support of the measure? Journal Article

In: Annales Academiæ Scientiarum Fennicæ, vol. 31, pp. 479-494, 2006.

Abstract | Links | BibTeX

2005

Joan Mateu, Laura Prat; Joan Verdera

The capacities associated to signed Riesz kernels and Wolff potentials Journal Article

In: Journal für die reine und angewandte Mathematik, vol. 2005, no. 578, pp. 201-223, 2005.

Abstract | Links | BibTeX

@article{nokey,

title = {The capacities associated to signed Riesz kernels and Wolff potentials},

author = {Joan Mateu, Laura Prat and Joan Verdera},

url = {http://mat.uab.cat/web/jvm/wp-content/uploads/sites/35/2024/01/Wolffultim.pdf},

doi = {https://doi.org/10.1515/crll.2005.2005.578.201},

year  = {2005},

date = {2005-07-01},

urldate = {2005-07-01},

journal = {Journal f\"{u}r die reine und angewandte Mathematik},

volume = {2005},

number = {578},

pages = {201-223},

abstract = {We show that, for 0\<α\<1, the capacity γα associated with the signed vector valued Riesz kernel x∣∣x∣∣1+α in ℝn is comparable to the Riesz capacity C23(n−α),32 of non-linear potential theory. The result is surprising because the definition of γα involves a kernel with many cancellations, while the Riesz capacities are related to positive kernels. One would have expected a situation similar to that arising for analytic capacity. Recall that the analytic capacity of a segment is positive. Instead all existing possible Riesz capacities of homogeneity 1 or larger vanish on a segment or, more generally, on sets of σ finite length. The reason which explains the different behavior of γα,0\<α\<1, is that its homogeneity (or dimension) is α and thus non-integer for the range considered. In other words, there are no rectifiable sets in fractional dimensions. The proof uses special positivity properties of the Riesz kernels of the dimensions considered, which are very well known for the Cauchy kernel. This allows us to bring in Wolff's potentials and thus the capacities of non linear potential theory. The special relation of the indexes of the relevant non-linear capacity is intriguing. It is rather natural to conjecture that our main result holds for all non-integer indexes α betweeen 0 and n. For α\>1 this should be a difficult result, because one looses the positivity properties of the corresponding Riesz kernel alluded to before. In the proof we use Tolsa's scheme for the proof that analytic capacity and its positive counterpart are comparable. There is a major difficulty, which consists in showing that convolution with the kernel localizes uniform estimates. By this we mean that if T is a distribution such that xx1+α∗T is bounded in ℝ𝕟, then xx1+α∗φT is also bounded in ℝ𝕟 for each test function φ . For α=1 and n=2 this is a classical fact, easy to prove because the kernel is related to a first order differential operator, which is not the case for fractional α . },

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We show that, for 0<α<1, the capacity γα associated with the signed vector valued Riesz kernel x∣∣x∣∣1+α in ℝn is comparable to the Riesz capacity C23(n−α),32 of non-linear potential theory. The result is surprising because the definition of γα involves a kernel with many cancellations, while the Riesz capacities are related to positive kernels. One would have expected a situation similar to that arising for analytic capacity. Recall that the analytic capacity of a segment is positive. Instead all existing possible Riesz capacities of homogeneity 1 or larger vanish on a segment or, more generally, on sets of σ finite length. The reason which explains the different behavior of γα,0<α<1, is that its homogeneity (or dimension) is α and thus non-integer for the range considered. In other words, there are no rectifiable sets in fractional dimensions. The proof uses special positivity properties of the Riesz kernels of the dimensions considered, which are very well known for the Cauchy kernel. This allows us to bring in Wolff's potentials and thus the capacities of non linear potential theory. The special relation of the indexes of the relevant non-linear capacity is intriguing. It is rather natural to conjecture that our main result holds for all non-integer indexes α betweeen 0 and n. For α>1 this should be a difficult result, because one looses the positivity properties of the corresponding Riesz kernel alluded to before. In the proof we use Tolsa's scheme for the proof that analytic capacity and its positive counterpart are comparable. There is a major difficulty, which consists in showing that convolution with the kernel localizes uniform estimates. By this we mean that if T is a distribution such that xx1+α∗T is bounded in ℝ𝕟, then xx1+α∗φT is also bounded in ℝ𝕟 for each test function φ . For α=1 and n=2 this is a classical fact, easy to prove because the kernel is related to a first order differential operator, which is not the case for fractional α .

2004

Joan Verdera

Ensembles effaçables, ensembles invisibles et le problème du voyageur du commerce, ou comment l’analyse réelle aide l’analyse complexe Journal Article

In: Gazette des Mathématiciens, vol. 101, 2004.

Abstract | Links | BibTeX

Joan Verdera

Dalfsen slides Unpublished

2004.

Abstract | Links | BibTeX

2003

Joan Verdera

Conjunts evitables, conjunts invisibles i el viatjant de comerç o com l'anàlisi real ajuda l'anàlisi complexa Journal Article

In: Butlletí de la Societat Catalana de Matemàtiques, vol. 18, iss. 2, 2003.

Abstract | Links | BibTeX

Joan Mateu; Xavier Tolsa; Joan Verdera

On the semiadditivity of analytic capacity and planar Cantor sets Journal Article

In: Contemporary Mathematics, vol. 320, pp. 259-278, 2003.

Abstract | Links | BibTeX

Joan Verdera; Xavier Tolsa; Joan Mateu

The planar Cantor sets of zero analytic capacity and the local T(b) theorem Journal Article

In: Journal of the American Mathematical Society, vol. 16, pp. 19-28, 2003.

Abstract | Links | BibTeX

John Garnett, Joan Verdera

Analytic capacity, bilipschitz mappings and Cantor sets Journal Article

In: Mathematical Research Letters 10 (2003), 515--522, vol. 10, no. 4, pp. 515-522, 2003.

Abstract | Links | BibTeX

2002

Joan Verdera

The fall of the doubling condition in Calderón-Zygmund Theory Journal Article

In: Publicacions Matemàtiques, pp. 275-292, 2002.

Abstract | Links | BibTeX

2001

Joan Verdera

L^2 boundedness of the Cauchy Integral and Menger curvature Journal Article

In: Contemporary Mathematics, vol. 277, pp. 139-158, 2001.

Abstract | Links | BibTeX

2000

Joan Verdera

On the T(1)-Theorem for the Cauchy Integral Journal Article

In: Ark Mat, vol. 38, iss. 1, pp. 183-199, 2000.

Abstract | Links | BibTeX

1998

Joan Orobitg, Joan Verdera

Choquet Integrals, Hausdorff Content and the Hardy–Littlewood Maximal Operator Journal Article

In: Bulletin of the London Mathematical Society, vol. 30, iss. 2, pp. 145-150, 1998.

Links | BibTeX

1995

Mark Melnikov, Joan Verdera

A geometyric proof of the L2 Boundedness of the Cauchy Integral on Lipschitz graphs Journal Article

In: International Mathematics Research Notices, vol. 7, pp. 325-331, 1995.

Abstract | Links | BibTeX

1994

Joan Verdera

A new elementary proof of L2 estimates for the Cauchy Integral on Lipschitz graphs Unpublished

1994.

Abstract | Links | BibTeX

@unpublished{nokey,

title = {A new elementary proof of L2 estimates for the Cauchy Integral on Lipschitz graphs},

author = {Joan Verdera},

url = {http://mat.uab.cat/web/jvm/wp-content/uploads/sites/35/2024/01/paper1-1.pdf},

year  = {1994},

date = {1994-09-01},

urldate = {1994-09-01},

abstract = {Subtle simple positivity properties of the Cauchy kernel have been discovered recently by M. S. Melnikov. Given distinct 3 points in the plane, certain symmetrization of the Cauchy kernel involving only these 3 points turns out to be exactly the inverse of the square of the radius of the circle determined by the points. The following is a sketch of the author's short proof of the Coifman-McIntosh-Meyer Theorem on L2 boundedness of the Cauchy Integral on Lipschitz graphs.Take a Lipschitz graph in the plane and transform the Cauchy integral on the graph into an integral operator T on the line via the obvious parametrization of the graph. Given an interval I, the L2(I) norm of the Cauchy integral of the characteristic function of I may be expressed as the triple integral on I of the symmetrization of the Cauchy kernel plus an error term that may be estimated by a constant time the length of I . Using the geometric expression in terms of the radius and a Fourier transform computation one estimates the main term by constant times the length of I. The second step consists in showing that atoms are mapped into L1(R). By duality this implies that L∞(R) is mapped into BMO(R) and by interpolation that L2(R) is mapped into itself. In particular, one shows that the triple integral of the inverse of the square of the radius of the circle through 3 arbitrary points on an arc of a Lipschitz graph is finite. This may be understood as an L2 version of total curvature with respect to arc length measure. The same result holds for arcs of chord-arc curves. These are the first non-trivial examples of measures whose total curvature in the above sense is finite. },

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Joan Verdera

Removability, capacity and approximation Journal Article

In: Complex Potential Theory, Nato ASI series, vol 439. Springer, Dordrecht., vol. 439, pp. 419-473, 1994.

Abstract | Links | BibTeX