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Leonardo Colzani 《Annali di Matematica Pura ed Applicata》1984,137(1):207-215
Summary We prove direct and converse theorems of approximation for distributions in the Hardy spaces Hp, 0
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We give an alternative proof of a theorem of Stein and Weiss: The distribution function of the Hilbert transform of a characteristic function of a set E only depends on the Lebesgue measure |E| of such a set. We exploit a rational change of variable of the type used by George Boole in his paper “On the comparison of transcendents, with certain applications to the theory of definite integrals” together with the observation that if two functions f and g have the same Lp norm in a range of exponents p1<p<p2 then their distribution functions coincide. 相似文献
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Symmetrization and norm of the Hardy-Littlewood maximal operator on Lorentz and Marcinkiewicz spaces
Colzani Leonardo; Laeng Enrico; Morpurgo Carlo 《Journal London Mathematical Society》2008,77(2):349-362
We prove that when a function on the real line is symmetricallyrearranged, the distribution function of its uncentered Hardy–Littlewoodmaximal function increases pointwise, while it remains unchangedonly when the function is already symmetric. Equivalently, if is the maximal operator and the symmetrization, then f(x)f(x)for every x, and equality holds for all x if and only if, upto translations, f(x) = f(x) almost everywhere. Using theseresults, we then compute the exact norms of the maximal operatoracting on Lorentz and Marcinkiewicz spaces, and we determineextremal functions that realize these norms. 相似文献
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Leonardo Colzani Alice Cominardi Krzysztof Stempak 《Annali di Matematica Pura ed Applicata》2002,181(1):25-54
We study some boundedness properties of radial solutions to the Cauchy problem associated to the wave equation (∂
t
2-▵
x
)u(t,x)=0 and meanwhile we give a new proof of the solution formula.
Received: July 7, 1998?Published online: March 19, 2002 相似文献
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We study the maximal function Mf(x) = sup |f(x + y, t)| when Ω is a region in the (y,t) Ω upper half space R and f(x, t) is the harmonic extension to R+N+1 of a distribution in the Besov space Bαp,q(RN) or in the Triebel-Lizorkin space Fαp,q(RN). In particular, we prove that when Ω= {|y|N/ (N-αp) < t < 1} the operator M is bounded from F (RN) into Lp (RN). The admissible regions for the spaces B (RN) with p < q are more complicated. 相似文献
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Leonardo Colzani Leonede De Michele Delfina Roux 《Mediterranean Journal of Mathematics》2012,9(2):253-266
Although Fourier series or integrals of piecewise smooth functions may be slowly convergent, sometimes it is possible to accelerate their speed of convergence by adding and subtracting suitable combination of known functions. 相似文献
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We state a localization principle for expansions in eigenfunctions of a self-adjoint second order elliptic operator and we prove an equiconvergence result between eigenfunction expansions and trigonometric expansions. We then study the Gibbs phenomenon for eigenfunction expansions of piecewise smooth functions on two-dimensional manifolds. 相似文献
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