Let \Omega \Subset \mathbb{C}^{n} be a bounded strongly pseudoconvex domain. For any concave increasing weight \chi \colon \mathbb{R}^{-} \to \mathbb{R}^{-} such that \chi(0) = 0 , we introduce and study finite energy classes \mathcal{E}_{\chi}(\Omega) of plurisubharmonic functions, using the Orlicz space formalism. We investigate the range of the Monge–Ampère operator on these classes, and conjecture that this should lead to an integral characterization of the image of bounded plurisubharmonic functions, an open problem since the birth of pluripotential theory more than forty years ago.
In this article, we give a sufficient and necessary condition on the indices of Besov-type spaces on \mathbb{R} under which the characterization of these spaces in terms of the Haar system holds. This confirms that the sufficient condition obtained by H. Triebel is optimal. Moreover, a special case of our results also negatively answers a question, asked by H. Triebel, about the characterization of lifted local bounded mean oscillation spaces J^{s}{\mathrm{bmo}} on \mathbb{R} with s\in(-1,-1/2] in terms of the Haar system. As an application, we prove that characteristic functions of intervals are pointwise multipliers of Besov-type spaces with the aforementioned optimal indices.
We classify all positive solutions to -\Delta u=e^{-2u} in the half space \mathbb{R}^{N}_{+} , under Dirichlet boundary condition.
In this short note, we correct the lattice theoretic classification in [Rev. Mat. Iberoam. 38 (2022), no. 4, 1199–1218] of the first named author, including some missing cases and explaining how to recover them.
The L_{p} version (for p>1 ) of the dimensional Brunn-Minkowski inequality for the standard Gaussian measure \gamma_{n}(\cdot) on \mathbb{R}^{n} is shown. More precisely, we prove that for any 0 -symmetric convex sets with nonempty interior, any p>1 , and every \lambda \in (0,1) , \gamma_{n} ((1-\lambda)\cdot K+_{p} \lambda \cdot L )^{p/n}\geqslant (1-\lambda ) \gamma_{n}(K)^{p/n}+ \lambda \gamma_{n}(L)^{p/n}, with equality, for some \lambda \in (0,1) and p>1 , if and only if K=L . This result, recently established without the equality conditions by Hosle, Kolesnikov and Livshyts, by using a different and functional approach, turns out to be the L_{p} extension of a celebrated result for the Minkowski sum (that is, for p=1 ) by Eskenazis and Moschidis (2021) on a problem by Gardner and Zvavitch (2010). Moreover, an L_{p} Brunn–Minkowski type inequality is obtained for the classical Wills functional \mathcal{W}(\cdot) of convex bodies. These results are derived as a consequence of a more general approach, which provides us with other remarkable examples of functionals satisfying L_{p} Brunn–Minkowski type inequalities, such as different absolutely continuous measures with radially decreasing densities.
In this article, we explore the second integral homology, or Schur multiplier, of the special linear group SL(2.)Z(Z[1/n]for a positive integer n. We definitively calculate the group structure of H-2.SL2.Z(Z[1/n], Z/)when n is divisible by one of the primes 2, 3, 5, 7 or 13. For a general n > 1, we offer a partial description by placing the homology group within an exact sequence, and we investigate its rank. Finally, we propose a conjectural structure for H-2.SL2.Z(Z[1/n], Z/)when n is not divisible by any of those specific primes
We consider the balayage of a measure mu defined on a domain Omega onto its boundary partial derivative Omega. Assuming that Omega has a corner of opening pi alpha at a point z(0)is an element of partial derivative Omega for some 0 z(0) for some b>0, we obtain the precise rate of vanishing of the balayage of mu near z(0). The rate of vanishing is universal in the sense that it only depends on alpha and b. We also treat the case when the domain has multiple corners at the same point. Moreover, when 2b <= 1/alpha, we provide explicit constants for the upper and lower bounds.
We study isometric immersions f: M-n -> Hn+1 into hyperbolic space of dimension n + 1 of a complete Riemannian manifold of dimension n on which a compact connected group of intrinsic isometries acts with principal orbits of codimension one. We provide a characterization if either n >= 3 and M-n is compact, or n >= 5 and the connected components of the set where the sectional curvature is constant and equal to -1 are bounded.
We classify all positive solutions to -Delta u = e(-2u) in the half space R-+(N), under Dirichlet boundary condition.
A λ-convex body in a three-dimensional space form M^3(c) of constant curvature c is a compact convex set K whose boundary ∂ K has normal curvatures bounded below by a constant λ>0 (in a weak sense). Within this class, we prove a sharp reverse isoperimetric inequality: among all λ-convex bodies in M^3(c), with a fixed surface area, the body of minimal volume is the λ-convex lens, i.e., the domain bounded by two totally umbilical caps of curvature λ. Moreover, this minimizer is unique. This result confirms Borisenko's Conjecture in the three-dimensional model spaces of constant curvature for c≠ 0, and complements recent progress on the conjecture in the Euclidean case c=0. As a by-product, our method also yields an alternative proof of the corresponding reverse isoperimetric inequality in two-dimensional hyperbolic space.
For every p>(1+root 5)/2, we construct a uniformly discrete real sequence {lambda(n)}(n=1)(infinity) satisfying divided by lambda(n)divided by(infinity)n=1, a function g is an element of L-p(R), and continuous linear functionals {g(n)(& lowast;)}(n=1)(infinity) on L-p(R), such that every f is an element of L-p(R) admits a series expansion f(x)=& sum;(infinity)(n=1)g(n)(& lowast;)(f)g(x-lambda(n)) convergent in the L-p(R) norm. We moreover show that g can be chosen nonnegative.
The Dirac operator with MIT bag boundary condition in a bounded convex domain is shown to be always self-adjoint in the H^1-setting. This allows one to show that such operators appear as limit of Dirac operators with large positive mass outside the domain. Similar results were previously known for smooth domains only.
Weak solutions m:Omega subset of R-2 -> R-2 of the eikonal equation, |m|=1 a.e. and div m=0 arise naturally as sharp interface limits of bounded energy configurations in various physically motivated models, including the Aviles-Giga energy. The distributions mu(Phi) = div (Phi(m)) , defined for a class of smooth vector fields Phi called entropies, carry information about singularities and energy cost. If these entropy productions are Radon measures, a long-standing conjecture predicts that they must be concentrated on the 1-rectifiable jump set of m -as they do if m has bounded variation (BV) thanks to the chain rule. We establish this concentration property, for a large class of entropies, under the Besov regularity assumption m is an element of B-1/p (p, infinity) double left right arrow sup (2)(h is an element of R) \ {0} ||m(. + h)- m || Lp / |h| (1/P) < infinity for any 1 <= p < 3, thus going well beyond the BV setting (p = 1) and leaving only the borderline case p=3 open.
This paper investigates degenerate nonlocal free boundary problems arising in the context of superconductivity, extending the nonlocal counterpart to the work of Caffarelli and Salazar (2002) and Caffarelli, Salazar and Shahgholian (2004) in the local setting. In these models, no partial differential equation governs the moving sets where the gradient vanishes, meaning that test functions are only required to have a nonzero gradient. Our main results provide interior gradient Hölder regularity estimates for viscosity solutions.
For every $p > (1 + \sqrt{5})/2$ we construct a uniformly discrete real sequence $\{\lambda_n\}_{n=1}^\infty$ satisfying $|\lambda_n| = n + o(1)$, a function $g \in L^p(\mathbb{R})$, and continuous linear functionals $\{g^*_n\}_{n=1}^\infty$ on $L^p(\mathbb{R})$, such that every $f \in L^p(\mathbb{R})$ admits a series expansion \[ f(x) = \sum_{n=1}^{\infty} g_n^*(f) g(x-\lambda_n) \] convergent in the $L^p(\mathbb{R})$ norm. We moreover show that $g$ can be chosen nonnegative.
We classify, up to isomorphism, the group gradings on the non-exceptional classical simple Lie superalgebras, except for type A(1,1), over an algebraically closed field of characteristic zero. To this end, we study graded-simple and graded-superinvolution-simple associative superalgebras satisfying the descending chain condition on graded left superideals, which allows us to classify abelian group gradings on finite-dimensional simple and superinvolution-simple associative superalgebras over an algebraically closed field of characteristic different from 2.
We consider hyperbolic projections of orbits of holomorphic self-maps of the unit disc, onto curves landing on the unit circle with a given angle. We show that under certain, necessary, assumptions, the projections exhibit monotonicity properties akin to those present in continuous dynamics. Our techniques are purely hyperbolic-geometric in nature and provide the general framework for analysing projections of arbitrary sequences onto curves.
We introduce the concept of a nonassociative (i.e., not necessarily associative) inverse semialgebra over a field, the Lie version of which is inspired by the set of all partially defined derivations of a nonassociative algebra, whereas the associative case is based on such examples as the set of all partially defined linear maps of a vector space, the set of all sections of the structural sheaf of a scheme, the set of all regular functions defined on open subsets of an algebraic variety, and the set of all smooth real-valued functions defined on open subsets of a smooth manifold. Given a Lie algebra L , we define the notion of a partial action of L on a nonassociative algebra A as an appropriate premorphism and introduce a Lie inverse semialgebra E(L) , which is a Lie analogue of R. Exel’s inverse semigroup S(G) that governs the partial actions of a group G . We discuss how E(L) controls the premorphisms from L to A , obtaining results on its total control. We define the concept of a Lie F -inverse semialgebra and obtain Lie theoretic analogues of some classical results of the theory of inverse semigroups, namely, we show that the category of partial representations of L in meet semilattices is equivalent to the category \mathcal{F} of Lie F -inverse semialgebras with morphisms that preserve the greatest elements of \sigma -classes. In addition, we establish an adjunction between the category of Lie algebras and the category \mathcal{F} .
We consider Schrodinger operators on a bounded, smooth domain of dimension d >= 2 with Dirichlet boundary conditions and a properly scaled potential, which depends only on the distance to the boundary of the domain. Our aim is to analyse the convergence of these operators as the scaling parameter tends to zero. If the scaled potential is resonant, the limit in strong resolvent sense is a Robin Laplacian with boundary coefficient expressed in terms of the mean curvature of the boundary. A counterexample shows that norm resolvent convergence cannot hold in general in this setting. If the scaled potential is non-resonant and satisfies an explicit assumption on the smallness of the negative part, the limit in strong resolvent sense is the Dirichlet Laplacian. We conjecture that we can drop this additional assumption in the non-resonant case.
The L-p version (for p>1) of the dimensional Brunn-Minkowski inequality for the standard Gaussian measure gamma(n)(& sdot;) on R-n is shown. More precisely, we prove that for any 0-symmetric convex sets with nonempty interior, any p>1, and every lambda is an element of(0,1), gamma(n)((1-lambda)& sdot;K+(p)lambda & sdot;L)(p/n)>=(1-lambda)gamma(n)(K)(p/n)+lambda gamma(n)(L)(p/n), with equality, for some lambda is an element of(0,1) and p>1, if and only if K=L. This result, recently established without the equality conditions by Hosle, Kolesnikov and Livshyts, by using a different and functional approach, turns out to be the L-p extension of a celebrated result for the Minkowski sum (that is, for p=1) by Eskenazis and Moschidis (2021) on a problem by Gardner and Zvavitch (2010). Moreover, an L-p Brunn-Minkowski type inequality is obtained for the classical Wills functional W(& sdot;) of convex bodies. These results are derived as a consequence of a more general approach, which provides us with other remarkable examples of functionals satisfying L-p Brunn-Minkowski type inequalities, such as different absolutely continuous measures with radially decreasing densities.
