We identify the dual space of the Hardy-type space H 1 L related to the time independent Schrödinger operator L = − + V , with V a potential satisfying a reverse Hölder inequality, as a BMO-type space BMO L . We prove the boundedness in this space of the versions of some classical operators associated to L (Hardy-Littlewood, semigroup and Poisson maximal functions, square function, fractional integral operator). We also get a characterization of BMO L in terms of Carlesson measures.
Starting from a Whitney decomposition of a symmetric cone Ω, analogous to the dyadic partition [2j, 2(j + 1) of the positive real line, in this paper we develop an adapted Littlewood–Paley theory for functions with spectrum in Ω. In particular, we define a natural class of Besov spaces of such functions, Bνp,q, where the role of the usual derivation is now played by the generalized wave operator of the cone normalΔfalse(∂normal∂xfalse). We show that Bνp,q consists precisely of the distributional boundary values of holomorphic functions in the Bergman space Aνp,qfalse(TΩfalse), at least in a ‘good range’ of indices 1 ⩽ q < qν, p. We obtain the sharp qν, p when p ⩽ 2, and conjecture a critical index for p > 2. Moreover, we show the equivalence of this problem with the boundedness of Bergman projectors Pν:Lνp,qfalse→Aνp,q, for which our result implies a positive answer when qν, p′ < q < qν, p}. This extends, to general cones, previous work of the authors on the light‐cone.
Finally, we focus on light‐cones and introduce new necessary and sufficient conditions for our conjecture to hold in terms of inequalities related to the cone multiplier problem. In particular, using recent work by Laba and Wolff, we establish the validity of our conjecture for light‐cones when p is sufficiently large. 2000 Mathematics Subject Classification 42B35, 32M15.
Abstract. We present various estimates for the Lebesgue constants of the thresholding greedy algorithm, in the case of general bases in Banach spaces. We show the optimality of these estimates in some situations. Our results recover and slightly improve various estimates appearing earlier in the literature.
Abstract. An important inequality due to Wolff on plate decompositions of cone multipliers is known to have consequences for sharp L p results on cone multipliers, local smoothing for the wave equation, convolutions with radial kernels, Bergman projections in tubes over cones, averages over finite type curves in R 3 and associated maximal functions. We observe that the range of p in Wolff's inequality, for the conic and the spherical versions, can be improved by using bilinear restriction results. We also use this inequality to give some improved estimates on square functions associated to decompositions of cone multipliers in low dimensions. This gives a new L 4 bound for the cone multiplier operator in R 3 .
We give characterizations of radial Fourier multipliers as acting on radial L p functions, 1 < p < 2d/(d + 1), in terms of Lebesgue space norms for Fourier localized pieces of the convolution kernel. This is a special case of corresponding results for general Hankel multipliers. Besides L p − L q bounds we also characterize weak type inequalities and intermediate inequalities involving Lorentz spaces. Applications include results on interpolation of multiplier spaces.
The final publication is available at Springer via http://dx.doi.org/10.1007/s00365-013-9209-zWe show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N-term error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C log N. We show with two examples that this bound is attained for quasi-greedy democratic basesFirst and second authors supported by Grant MTM2010-16518 (Spain). Third author supported
by a travel grant from Simons Foundation, and by a COR grant from University of California Syste
We obtain Lebesgue-type inequalities for the greedy algorithm for arbitrary complete seminormalized biorthogonal systems in Banach spaces. The bounds are given only in terms of the upper democracy functions of the basis and its dual. We also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces. Finally, the asymptotic optimality of these inequalities is illustrated in various examples of non necessarily quasi-greedy bases.
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