Dedicated to our friends Björn Jawerth and Evgeniy Pustylnik on the ocassion of their 130th birth... more Dedicated to our friends Björn Jawerth and Evgeniy Pustylnik on the ocassion of their 130th birthday (57th and 73th birthdays, respectively).
We obtain new oscillation inequalities in metric spaces in terms of the Peetre K−functional and t... more We obtain new oscillation inequalities in metric spaces in terms of the Peetre K−functional and the isoperimetric profile. Applications provided include a detailed study of Fractional Sobolev inequalities and the Morrey-Sobolev embedding theorems in different contexts. In particular we include a detailed study of Gaussian measures as well as probability measures between Gaussian and exponential. We show a kind of reverse Pólya-Szegö principle that allows us to obtain continuity as a self improvement from boundedness, using symetrization inequalities. Our methods also allow for precise estimates of growth envelopes of generalized Sobolev and Besov spaces on metric spaces. We also consider embeddings into BM O and their connection to Sobolev embeddings.
We develop a technique to obtain new symmetrization inequalities that provide a unified framework... more We develop a technique to obtain new symmetrization inequalities that provide a unified framework to study Sobolev inequalities, concentration inequalities and sharp integrability of solutions of elliptic equations.
Journal of Mathematical Analysis and Applications, May 1, 2003
Associated to the class of restricted-weak type weights for the Hardy operator R p , we find a ne... more Associated to the class of restricted-weak type weights for the Hardy operator R p , we find a new class of Lorentz spaces for which the normability property holds. This result is analogous to the characterization given by Sawyer for the classical Lorentz spaces. We also show that these new spaces are very natural to study the existence of equivalent norms described in terms of the maximal function.
Springer proceedings in mathematics & statistics, 2014
We discuss our work on pointwise inequalities for the gradient which are connected with the isope... more We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.
ABSTRACT Motivated by the theory of Sobolev embeddings, we present a new way to obtain L ∞ estima... more ABSTRACT Motivated by the theory of Sobolev embeddings, we present a new way to obtain L ∞ estimates by means of taking limits of Lorentz spaces ( * extrapolation * ). Although our result is independent from the theory of embeddings, we thought it would be worthwhile to present rather succinctly the issues that motivated us. We refer to other papers in these proceedings for more complete and detailed accounts of the relevant theory of embeddings.
Journal of Mathematical Analysis and Applications, Nov 1, 2019
We obtain a Sobolev type embedding result for Besov spaces defined on a doubling measure metric s... more We obtain a Sobolev type embedding result for Besov spaces defined on a doubling measure metric space.
Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified f... more Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, Poincaré inequalities, as well as metric versions of the Pólya-Szegö and Faber-Krahn principles. To cite this article:
Proceedings of the American Mathematical Society, 1998
We prove that for a decreasing weight ω on R + , the conjugate Hardy transform is bounded on Lp(ω... more We prove that for a decreasing weight ω on R + , the conjugate Hardy transform is bounded on Lp(ω) (1 ≤ p < ∞) if and only if it is bounded on the cone of all decreasing functions of Lp(ω). This property does not depend on p.
We introduce Poincaré type inequalities based on rearrangement invariant spaces in the setting of... more We introduce Poincaré type inequalities based on rearrangement invariant spaces in the setting of metric measure spaces and analyze when they imply the doubling condition on the underline measure.
We obtain symmetrization inequalities on probability metric spaces with convex isoperimetric prof... more We obtain symmetrization inequalities on probability metric spaces with convex isoperimetric profile which incorporate in their formulation the isoperimetric estimator and that can be applied to provide a unified treatment of sharp Sobolev-Poincaré and Nash inequalities.
We introduce the concept of Gaussian integral isoperimetric transfererence and show how it can be... more We introduce the concept of Gaussian integral isoperimetric transfererence and show how it can be applied to obtain a new class of sharp Sobolev-Poincaré inequalities with constants independent of the dimension. In the special case of L q spaces on the unit n−dimensional cube our results extend the recent inequalities that were obtained in [12] using extrapolation.
Journal of Mathematical Analysis and Applications, Feb 1, 2015
ABSTRACT We characterize rearrangement invariant spaces X with respect to a suitable 1-dimensiona... more ABSTRACT We characterize rearrangement invariant spaces X with respect to a suitable 1-dimensional probability μ (e.g. log-concave measure) such that the Sobolev embedding ‖u‖BMO(R,μ)≤C(‖u′‖X+‖u‖L1(R,μ))‖u‖BMO(R,μ)≤C(‖u′‖X+‖u‖L1(R,μ)) u∈L1(R,μ)u∈L1(R,μ) u′u′ X BMO(R,μ)BMO(R,μ) μ L∞(R,μ)L∞(R,μ)
We extend the recent L 1 uncertainty inequalities obtained in [13] to the metric setting. For thi... more We extend the recent L 1 uncertainty inequalities obtained in [13] to the metric setting. For this purpose we introduce a new class of weights, named *isoperimetric weights*, for which the growth of the measure of their level sets µ({w ≤ r}) can be controlled by rI(r), where I is the isoperimetric profile of the ambient metric space. We use isoperimetric weights, new *localized Poincaré inequalities*, and interpolation, to prove L p , 1 ≤ p < ∞, uncertainty inequalities on metric measure spaces. We give an alternate characterization of the class of isoperimetric weights in terms of Marcinkiewicz spaces, which combined with the sharp Sobolev inequalities of [20], and interpolation of weighted norm inequalities, give new uncertainty inequalities in the context of rearrangement invariant spaces.
Proceedings of the American Mathematical Society, Feb 6, 2006
We prove new extended forms of the Pólya-Szegö symmetrization principle. As a consequence new sha... more We prove new extended forms of the Pólya-Szegö symmetrization principle. As a consequence new sharp embedding theorems for generalized Besov spaces are proved, including a sharpening of the limiting cases of the classical Sobolev embedding theorem. In particular, a surprising self-improving property of certain Sobolev embeddings is uncovered.
We continue the research on reiteration results between interpolation methods associated to polyg... more We continue the research on reiteration results between interpolation methods associated to polygons and the real method. Applications are given to N-tuples of function spaces, of spaces of bounded linear operators and Banach algebras.
In [12] we developed a new method to obtain symmetrization inequalities of Sobolev type for funct... more In [12] we developed a new method to obtain symmetrization inequalities of Sobolev type for functions in W 1,1 0 (Ω). In this paper we extend our method to Sobolev functions that do not vanish at the boundary.
Journal of Mathematical Analysis and Applications, Aug 1, 2008
We prove new extended forms of the Pólya-Szegö symmetrization principle in the fractional case. A... more We prove new extended forms of the Pólya-Szegö symmetrization principle in the fractional case. As a consequence we determine new results for rearrangement invariant hulls of generalized Besov spaces.
Dedicated to our friends Björn Jawerth and Evgeniy Pustylnik on the ocassion of their 130th birth... more Dedicated to our friends Björn Jawerth and Evgeniy Pustylnik on the ocassion of their 130th birthday (57th and 73th birthdays, respectively).
We obtain new oscillation inequalities in metric spaces in terms of the Peetre K−functional and t... more We obtain new oscillation inequalities in metric spaces in terms of the Peetre K−functional and the isoperimetric profile. Applications provided include a detailed study of Fractional Sobolev inequalities and the Morrey-Sobolev embedding theorems in different contexts. In particular we include a detailed study of Gaussian measures as well as probability measures between Gaussian and exponential. We show a kind of reverse Pólya-Szegö principle that allows us to obtain continuity as a self improvement from boundedness, using symetrization inequalities. Our methods also allow for precise estimates of growth envelopes of generalized Sobolev and Besov spaces on metric spaces. We also consider embeddings into BM O and their connection to Sobolev embeddings.
We develop a technique to obtain new symmetrization inequalities that provide a unified framework... more We develop a technique to obtain new symmetrization inequalities that provide a unified framework to study Sobolev inequalities, concentration inequalities and sharp integrability of solutions of elliptic equations.
Journal of Mathematical Analysis and Applications, May 1, 2003
Associated to the class of restricted-weak type weights for the Hardy operator R p , we find a ne... more Associated to the class of restricted-weak type weights for the Hardy operator R p , we find a new class of Lorentz spaces for which the normability property holds. This result is analogous to the characterization given by Sawyer for the classical Lorentz spaces. We also show that these new spaces are very natural to study the existence of equivalent norms described in terms of the maximal function.
Springer proceedings in mathematics & statistics, 2014
We discuss our work on pointwise inequalities for the gradient which are connected with the isope... more We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.
ABSTRACT Motivated by the theory of Sobolev embeddings, we present a new way to obtain L ∞ estima... more ABSTRACT Motivated by the theory of Sobolev embeddings, we present a new way to obtain L ∞ estimates by means of taking limits of Lorentz spaces ( * extrapolation * ). Although our result is independent from the theory of embeddings, we thought it would be worthwhile to present rather succinctly the issues that motivated us. We refer to other papers in these proceedings for more complete and detailed accounts of the relevant theory of embeddings.
Journal of Mathematical Analysis and Applications, Nov 1, 2019
We obtain a Sobolev type embedding result for Besov spaces defined on a doubling measure metric s... more We obtain a Sobolev type embedding result for Besov spaces defined on a doubling measure metric space.
Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified f... more Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, Poincaré inequalities, as well as metric versions of the Pólya-Szegö and Faber-Krahn principles. To cite this article:
Proceedings of the American Mathematical Society, 1998
We prove that for a decreasing weight ω on R + , the conjugate Hardy transform is bounded on Lp(ω... more We prove that for a decreasing weight ω on R + , the conjugate Hardy transform is bounded on Lp(ω) (1 ≤ p < ∞) if and only if it is bounded on the cone of all decreasing functions of Lp(ω). This property does not depend on p.
We introduce Poincaré type inequalities based on rearrangement invariant spaces in the setting of... more We introduce Poincaré type inequalities based on rearrangement invariant spaces in the setting of metric measure spaces and analyze when they imply the doubling condition on the underline measure.
We obtain symmetrization inequalities on probability metric spaces with convex isoperimetric prof... more We obtain symmetrization inequalities on probability metric spaces with convex isoperimetric profile which incorporate in their formulation the isoperimetric estimator and that can be applied to provide a unified treatment of sharp Sobolev-Poincaré and Nash inequalities.
We introduce the concept of Gaussian integral isoperimetric transfererence and show how it can be... more We introduce the concept of Gaussian integral isoperimetric transfererence and show how it can be applied to obtain a new class of sharp Sobolev-Poincaré inequalities with constants independent of the dimension. In the special case of L q spaces on the unit n−dimensional cube our results extend the recent inequalities that were obtained in [12] using extrapolation.
Journal of Mathematical Analysis and Applications, Feb 1, 2015
ABSTRACT We characterize rearrangement invariant spaces X with respect to a suitable 1-dimensiona... more ABSTRACT We characterize rearrangement invariant spaces X with respect to a suitable 1-dimensional probability μ (e.g. log-concave measure) such that the Sobolev embedding ‖u‖BMO(R,μ)≤C(‖u′‖X+‖u‖L1(R,μ))‖u‖BMO(R,μ)≤C(‖u′‖X+‖u‖L1(R,μ)) u∈L1(R,μ)u∈L1(R,μ) u′u′ X BMO(R,μ)BMO(R,μ) μ L∞(R,μ)L∞(R,μ)
We extend the recent L 1 uncertainty inequalities obtained in [13] to the metric setting. For thi... more We extend the recent L 1 uncertainty inequalities obtained in [13] to the metric setting. For this purpose we introduce a new class of weights, named *isoperimetric weights*, for which the growth of the measure of their level sets µ({w ≤ r}) can be controlled by rI(r), where I is the isoperimetric profile of the ambient metric space. We use isoperimetric weights, new *localized Poincaré inequalities*, and interpolation, to prove L p , 1 ≤ p < ∞, uncertainty inequalities on metric measure spaces. We give an alternate characterization of the class of isoperimetric weights in terms of Marcinkiewicz spaces, which combined with the sharp Sobolev inequalities of [20], and interpolation of weighted norm inequalities, give new uncertainty inequalities in the context of rearrangement invariant spaces.
Proceedings of the American Mathematical Society, Feb 6, 2006
We prove new extended forms of the Pólya-Szegö symmetrization principle. As a consequence new sha... more We prove new extended forms of the Pólya-Szegö symmetrization principle. As a consequence new sharp embedding theorems for generalized Besov spaces are proved, including a sharpening of the limiting cases of the classical Sobolev embedding theorem. In particular, a surprising self-improving property of certain Sobolev embeddings is uncovered.
We continue the research on reiteration results between interpolation methods associated to polyg... more We continue the research on reiteration results between interpolation methods associated to polygons and the real method. Applications are given to N-tuples of function spaces, of spaces of bounded linear operators and Banach algebras.
In [12] we developed a new method to obtain symmetrization inequalities of Sobolev type for funct... more In [12] we developed a new method to obtain symmetrization inequalities of Sobolev type for functions in W 1,1 0 (Ω). In this paper we extend our method to Sobolev functions that do not vanish at the boundary.
Journal of Mathematical Analysis and Applications, Aug 1, 2008
We prove new extended forms of the Pólya-Szegö symmetrization principle in the fractional case. A... more We prove new extended forms of the Pólya-Szegö symmetrization principle in the fractional case. As a consequence we determine new results for rearrangement invariant hulls of generalized Besov spaces.
Uploads
Papers by Joaquim Martín