Weighted L p estimates for the area integral associated to self-adjoint operators

Abstract

Abstract. This article is concerned with some weighted norm inequalities for the so-called horizontal (i.e. involving time derivatives) area integrals associated to a non-negative selfadjoint operator satisfying a pointwise Gaussian estimate for its heat kernel, as well as the corresponding vertical (i.e. involving space derivatives) area integrals associated to a non-negative self-adjoint operator satisfying in addition a pointwise upper bounds for the gradient of the heat kernel. As applications, we obtain sharp estimates for the operator norm of the area integrals on L p

Lusin area integrals related to Jacobi expansions

“…Using [7, Theorem 1.1] we obtain, in particular, weighted weak type (1, 1) estimate for S α,β 1,0 = S α,β 1,0 with all A α,β 1 weights admitted. The assumptions imposed in [7] are indeed satisfied, since the Jacobi-heat kernel of the Jacobi-heat semigroup exp(−tJ α,β ) t>0 possesses the so-called Gaussian bound. The latter was recently established independently by Coulhon, Kerkyacharian, Petrushev in [6, Theorem 7.2] and by Nowak, Sjögren in [12, Theorem A].…”

“…Further, from now on we restrict our attention to |θ − θ ′ | ≤ t. Otherwise, an application of Lemma 3.6 shows that the left-hand side in question is controlled by a constant and the conclusion trivially follows. Using the estimate (7) with ξ = 2 we obtain…”

Local Hardy Spaces with Variable Exponents Associated with Non-negative Self-Adjoint Operators Satisfying Gaussian Estimates

“…Suppose that p − > 1. According to [45,Lemma 3.4] (see also [31,Lemma 5.1]) the area square integral S L defines a bounded operator from L 2 (R n , w) into itself, for every w ∈ A 2 (R n ). Here A 2 (R n ) denotes the Muckenhoupt class of weights and L 2 (R n , w) represents the weighted L 2 space.…”

Optimal orders of the best constants in the Littlewood-Paley inequalities

“…The case p > 2 is implicitly contained in [7], it can also be found in [29] if the aperture α is large enough, say α ≥ 3 √ d; for α < 3 √ d, one can adapt Wilson's argument. See [9,13,15] for related results.…”