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48
Inequalities In Rearrangement Invariant Function Spaces
, 1995
"... Introduction. Set up by Hardy & Littlewood, a theory of rearrangements was popularized by the wellknown book [HLP]. Rearrangements of functions are frequently used in real and harmonic analysis, in investigations about singular integrals and function spaces  see, e.g., [He], [ON], [ONW], [SW ..."
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Cited by 43 (2 self)
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Introduction. Set up by Hardy & Littlewood, a theory of rearrangements was popularized by the wellknown book [HLP]. Rearrangements of functions are frequently used in real and harmonic analysis, in investigations about singular integrals and function spaces  see, e.g., [He], [ON], [ONW], [SW]. P'olya & Szego and their followers demonstrated a good many isoperimetric theorems and inequalities by means of rearrangements  see [PS], a source book on this matter. More recent investigations have shown 178 G. TALENTI that rearrangements of functions fit well also into the theory of elliptic secondorder partial differential equations  see, e.g., [Bae], [Ta3] and the bibliography therein. Several types of rearrangements are known  presentations are in [Ka] and [Bae]. Here we limit ourselves to rearrangements `a la Hardy & Littlewood. 1.2. Definitions and basic properties. Let G be a measurable subset of R<F
Quantile and probability curves without crossing
, 2007
"... The most common approach to estimating conditional quantile curves is to fit a curve, typically linear, pointwise for each quantile. Linear functional forms, coupled with pointwise fitting, are used for a number of reasons including parsimony of the resulting approximations and good computational ..."
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Cited by 35 (6 self)
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The most common approach to estimating conditional quantile curves is to fit a curve, typically linear, pointwise for each quantile. Linear functional forms, coupled with pointwise fitting, are used for a number of reasons including parsimony of the resulting approximations and good computational properties. The resulting fits, however, may not respect a logical monotonicity requirement – that the quantile curve be increasing as a function of probability. This paper studies the natural monotonization of these empirical curves induced by sampling from the estimated nonmonotone model, and then taking the resulting conditional quantile curves that by construction are monotone in the probability. This construction of monotone quantile curves may be seen as a bootstrap and also as a monotonic rearrangement of the original nonmonotone function. It is shown that the monotonized curves are closer to the true curves in finite samples, for any sample size. Under correct specification, the rearranged conditional quantile curves have the same asymptotic distribution as the original nonmonotone curves. Under misspecification, however, the asymptotics of the rearranged curves may partially differ from the asymptotics of the original nonmonotone curves. An analogous procedure is developed to monotonize the estimates of conditional distribution functions. The results are derived by establishing the compact (Hadamard) differentiability of the monotonized quantile and probability curves with respect to the original curves in discontinuous directions, tangentially to a set of continuous functions. In doing so, the compact differentiability of the rearrangementrelated operators is established.
Periodic Schrödinger operator with local defects and spectral pollution
 SIAM J. Numer. Anal. v
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Comparison theorems for exit times
 Geom. Funct. Anal
"... Abstract We study bounds on the exit time of Brownian motion from a set in terms of its size and shape, and the relation of such bounds with isoperimetric inequalities. The first result is an upper bound for the distribution function of the exit time from a subset of a sphere or hyperbolic space of ..."
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Cited by 15 (3 self)
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Abstract We study bounds on the exit time of Brownian motion from a set in terms of its size and shape, and the relation of such bounds with isoperimetric inequalities. The first result is an upper bound for the distribution function of the exit time from a subset of a sphere or hyperbolic space of constant curvature in terms of the exit time from a disc of the same volume. This amounts to a rearrangement inequality for the Dirichlet heat kernel. To connect this inequality with the classical isoperimetric inequality, we derive a formula for the perimeter of a set in terms of the heat flow over the boundary. An auxiliary result generalizes Riesz' rearrangement inequality to multiple integrals.
Two limiting cases of Sobolev imbeddings
 Houston J. Math
"... Communicated by H. Brezis. ABSTRACT. Generalizations of Trudinger's and Br•zisWainger's limiting imbedding theorems are proved and their connections are studied. 1. Introduction. Much attention has been paid to various generalizations of Trudinger's celebrated limiting imbedding theo ..."
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Cited by 14 (6 self)
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Communicated by H. Brezis. ABSTRACT. Generalizations of Trudinger's and Br•zisWainger's limiting imbedding theorems are proved and their connections are studied. 1. Introduction. Much attention has been paid to various generalizations of Trudinger's celebrated limiting imbedding theorem [22] to fractional order spaces H•N/p and other interesting refinements ([1], [12], [16], [20],...) and, recently, also to OrliczSobolev spaces close to H • in [9]. As is well known, a function
A Note on L(∞, q) Spaces and Sobolev Embeddings
 INDIANA UNIV. MATH. J
"... We prove a sharp version of the Sobolev embedding theorem using L(∞,n) spaces and we compare our result with embeddings due to Hansson, BrezisWainger and MalyPick. ..."
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Cited by 14 (6 self)
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We prove a sharp version of the Sobolev embedding theorem using L(∞,n) spaces and we compare our result with embeddings due to Hansson, BrezisWainger and MalyPick.
POINTWISE SYMMETRIZATION INEQUALITIES FOR SOBOLEV FUNCTIONS AND APPLICATIONS
, 2009
"... 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. ..."
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Cited by 10 (3 self)
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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.
SYMMETRIZATION OF LÉVY PROCESSES AND APPLICATIONS
, 907
"... Abstract. It is shown that many of the classical generalized isoperimetric inequalities for the Laplacian when viewed in terms of Brownian motion extend to a wide class of Lévy processes. The results are derived from the multiple integral inequalities of Brascamp, Lieb and Luttinger but the probabil ..."
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Cited by 6 (0 self)
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Abstract. It is shown that many of the classical generalized isoperimetric inequalities for the Laplacian when viewed in terms of Brownian motion extend to a wide class of Lévy processes. The results are derived from the multiple integral inequalities of Brascamp, Lieb and Luttinger but the probabilistic structure of the processes plays a crucial role in the proofs. 1.
Rearrangements of vector valued functions, with application to atmospheric and oceanic ows
"... This paper establishes the equivalence of four de nitions of two vector valued functions being rearrangements, and gives a characterisation of the set of rearrangements of a prescribed function. The theory of monotone rearrangement ofavector valued function is used to show the existence and uniquene ..."
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Cited by 5 (2 self)
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This paper establishes the equivalence of four de nitions of two vector valued functions being rearrangements, and gives a characterisation of the set of rearrangements of a prescribed function. The theory of monotone rearrangement ofavector valued function is used to show the existence and uniqueness of the minimiser of an energy functional arising from a model for atmospheric and oceanic ow. At each xed time solutions are shown to be equal to the gradient of a convex function, verifying the conjecture of Cullen, Norbury and Purser. Key words Rearrangement of functions, semigeostrophic, variational problems, generalised solution.
A NEW VARIATIONAL APPROACH TO THE STABILITY OF GRAVITATIONAL SYSTEMS
, 904
"... Abstract. We consider the three dimensional gravitational Vlasov Poisson system which describes the mechanical state of a stellar system subject to its own gravity. A wellknown conjecture in astrophysics is that the steady state solutions which are nonincreasing functions of their microscopic energ ..."
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Cited by 4 (2 self)
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Abstract. We consider the three dimensional gravitational Vlasov Poisson system which describes the mechanical state of a stellar system subject to its own gravity. A wellknown conjecture in astrophysics is that the steady state solutions which are nonincreasing functions of their microscopic energy are nonlinearly stable by the ow. This was proved at the linear level by Antonov in 1961. Since then, standard variational techniques based on concentration compactness methods as introduced by P.L. Lions in 1983 have led to the nonlinear stability of subclasses of stationary solutions of ground state type. In this paper, we propose a new variational approach based on the minimization of the Hamiltonian under equimeasurable constraints which are conserved by the nonlinear transport ow, and recognize any anisotropic steady state solution which is a decreasing function of its microscopic energy as a local minimizer. The outcome is the proof of its nonlinear stability of under radially symmetric perturbations.