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Extrapolation of Compactness on Banach Function Spaces
Full text linkWe prove an extrapolation of compactness theorem for operators on Banach function spaces satisfying certain convexity and concavity conditions. In particular, we show that the boundedness of an operator T in the weighted Lebesgue scale and the compactness of T in the unweighted Lebesgue scale yields compactness of T on a very general class of Banach function spaces. As our main new tool, we prove various characterizations of the boundedness of the Hardy-Littlewood maximal operator on such spaces and their associate spaces, using a novel sparse self-improvement technique. We apply our main results to prove compactness of the commutators of singular integral operators and pointwise multiplication by functions of vanishing mean oscillation on, for example, weighted variable Lebesgue spaces
Symplectic monodromy at radius zero and equimultiplicity of μ -constant families
No full textWe show that every family of isolated hypersurface singularities with constant Milnor number has constant multiplicity. To achieve this, we endow the A’Campo model of “radius zero” monodromy with a symplectic structure. This new approach allows us to generalize a spectral sequence of McLean converging to fixed point Floer homology of iterates of the monodromy to a more general setting that is well suited to study µ-constant families
Uniform resolvent estimates, smoothing effects and spectral stability for the Heisenberg sublaplacian
No full textWe establish global bounds for solutions to stationary and time-dependent Schrödinger equations associated with the sublaplacian on the Heisenberg group, as well as its pure fractional power s and conformally invariant fractional power s. The main ingredient is a new abstract uniform weighted resolvent estimate which is proved by using the method of weakly conjugate operators -- a variant of Mourre's commutator method -- and Hardy's type inequalities on the Heisenberg group. As applications, we show Kato-type smoothing effects for the time-dependent Schrödinger equation, and spectral stability of the sublaplacian perturbed by complex-valued decaying potentials satisfying an explicit subordination condition. In the local case s=1, we obtain uniform estimates without any symmetry or derivative loss, which improve previous results
Singular integrals along variable codimension one subspaces
No full textThis article deals with maximal operators on formed by taking arbitrary rotations of tensor products of a -dimensional H\"ormander--Mihlin multiplier with the identity in coordinates, in the particular \emph{codimension 1} case . These maximal operators are naturally connected to differentiation problems and maximally modulated singular integrals such as Sj\"olin's generalization of Carleson's maximal operator. Our main result, a weak-type -estimate on band-limited functions, leads to several corollaries. The first is a sharp estimate for the maximal operator restricted to a finite set of rotations in terms of the cardinality of the finite set. The second is a version of the Carleson--Sj\"olin theorem. In addition, we obtain that functions in the Besov space , , may be recovered from their averages along a measurable choice of codimension subspaces, a form of Zygmund's conjecture in general dimension .RYC2018-025477-I
IKERBASQU
Multiplication between elements in martingale Hardy spaces and their dual spaces
No full textIn this paper, we establish continuous bilinear decompositions that arise in the study of products between elements in martingale Hardy spaces Hp p0 ă p ⩽ 1q and functions in their dual spaces. Our decompositions are based on martingale paraproducts. As a consequence of our work, we also obtain analogous results for dyadic martingales on spaces of homogeneous type equipped with a doubling measure
On tiling spherical triangles into quadratic subpatches
Full text linkVarious interpolation and approximation methods arising in several practical applications in geometric modeling deal, at a particular step, with the problem of computing suitable rational patches (of low degree) on the unit sphere. Therefore, we are concerned with the construction of a system of spherical triangular patches with prescribed vertices that globally meet along common boundaries. In particular, we investigate various possibilities for tiling a given spherical triangular patch into quadratically parametrizable subpatches. We revisit the condition that the existence of a quadratic parameterization of a spherical triangle is equivalent
to the sum of the interior angles of the triangle being pi, and then circumvent this limitation by studying alternative scenarios and present constructions of spherical macro-elements of the lowest possible degree. Applications of our method include algorithms relying on the construction of (interpolation) surfaces from prescribed rational normal vector fields.RYC-2017-2264
Pre-asymptotic analysis of Lévy flights
Full text linkWe study the properties of Lévy flights with index 0 < α < 2 at elapsed times smaller than those required for reaching
the diffusive limit and we focus on the bulk of the walkers’ distribution rather than on its tails. On the basis of the
analogs of the Kramers–Moyal expansion and of the Pawula theorem, we show that, for any α ≤ 2/3, the bulk of
the walkers’ distribution occurs at wave-numbers greater than (2/α)
1/(2α) ≥ 1 and it remains non self-similar for a
time-scale longer than the Markovian time-lag of at least one order of magnitude. This result highlights the fact that
for Lévy flights the Markovianity time-lag is not the only time-scale of the process and indeed another and longer
time-scale controls the transition to the familiar power-law regime in the final diffusive limit. The magnitude of this
further time-scale is independent of the index α and may compromise the reliability of applications of Lévy flights to
real world cases related with recurrence and transience as optimal searching, animal foraging and site fidelity
A LATTICE APPROACH TO MATRIX WEIGHTS
Full text linkIn this paper we recontextualize the theory of matrix weights within the setting of Banach lattices. We define an intrinsic notion of directional Banach function spaces, generalizing matrix weighted Lebesgue spaces. Moreover, we prove an extrapola- tion theorem for these spaces based on the boundedness of the convex-set valued maximal operator. We also provide bounds and equivalences related to the convex body sparse operator. Finally, we introduce a weak-type analogue of directional Banach function spaces. In particular, we show that the weak-type boundedness of the convex-set val- ued maximal operator on matrix weighted Lebesgue spaces is equivalent to the matrix Muckenhoupt condition, with equivalent constants
WEIGHTED FRACTIONAL POINCARE ́ INEQUALITIES VIA ISOPERIMETRIC INEQUALITIES
Full text linkOur main result is a weighted fractional Poincar ́e–Sobolev inequality improving the celebrated estimate by Bourgain–Brezis–Mironescu. This also yields an improvement of the classical Meyers–Ziemer theorem in several ways. The proof is based on a fractional isoperimetric inequality and is new even in the non-weighted setting. We also extend the celebrated Poincar ́e– Sobolev estimate with Ap weights of Fabes–Kenig–Serapioni by means of a fractional type result in the spirit of Bourgain–Brezis–Mironescu. Examples are given to show that the corresponding Lp-versions of weighted Poincar ́e inequalities do not hold for p > 1